This notebook delves into the challenges and methodologies associated with Hypothesis Testing for Sample Covariance Matrices in ultra-high dimensional scenarios.

Initially, we provide an illustrative walkthrough based on the research by X. Ding and Z. Wang, titled "Global and local CLTs for linear spectral statistics of general sample covariance matrices when the dimension is much larger than the sample size with applications". The paper can be accessed on arXiv: 2308.08646.

Subsequent sections explore:

• One-sample hypothesis testing using real-world data.
• Two-sample hypothesis testing for covariance matrices.

These explorations are precursors to our forthcoming research in this domain.

Prerequisite

How to use

Within the section for most figures, the codes are divided into three blocks.

• Data generating
• Saved data
• Plot

What you need to do depends on how you want to use it

• If you just want to see some existing results, no need to do anything except reading.
• If you just want to see some existing data and results, go to the output of the block "Plot" directly, or run "Saved data" then "Plot".
• If you want to check, replicate or change parameters, modify the subsection "Data generating" accordingly, then run the block "Plot".

Additional note if you want to run this notebook locally

This notebook use the Google jax to accelerate the sampling and calculation of large random matrices, for jax we refer to https://github.com/google/jax.

The jax framework has been readily to use in the colab. To use the notebook in a local server/environment, you may need to install jax first. To install jax, use the command

## cpu version
!pip install --upgrade pip
!pip install --upgrade "jax[cpu]"

# ##or
# ## gpu version
# !pip install --upgrade pip

# # CUDA 12 installation
# # Note: wheels only available on linux.
# !pip install --upgrade "jax[cuda12_pip]" -f https://storage.googleapis.com/jax-releases/jax_cuda_releases.html

# # CUDA 11 installation
# # Note: wheels only available on linux.
# !pip install --upgrade "jax[cuda11_pip]" -f https://storage.googleapis.com/jax-releases/jax_cuda_releases.html

This file was created with Google Colab, so for a ready-to-use environment you can also view the document with the link https://colab.research.google.com/drive/1xTWEHXadpq0Rb8Y0ADE-7lzpAtBDH-KK?usp=sharing where you can run or modify without setting up environment locally.

Necessary Functions

This block contain most of the global functions needed. If you want to replicate the results, please run this block before any other blocks.

import numpy as np
import matplotlib.pyplot as plt
import jax.numpy as jnp
from jax import grad, jit, random
from jax.random import normal, choice, split, PRNGKey
import scipy.integrate as integrate
from google.colab import files
from scipy.stats import norm



def bold_legend(legend):
    """Utility function to set legend text to bold."""
    for text in legend.get_texts():
        text.set_fontweight('bold')

def count_outside_interval(arr, a, b):
    outside_interval = np.logical_or(arr < a, arr > b)
    count = np.sum(outside_interval)
    return count

def count_inside_interval(arr, a, b):
    inside_interval = np.logical_and(arr >= a, arr <= b)
    count = np.sum(inside_interval)
    return count


def extract_elements(arr, a, b):
    return arr[(arr >= a) & (arr <= b)]

###########################
####Here the example functions are mainly written for $\Sigma=\operatorname{Tr}\{a,\ldots,a,b,\ldots,b\}$ where they are a repeated (s*m) times and b repeated (1-s)*m times in the diagonal.
#########################
# Calculate the Stieltjes transform
def stieltjes(z,phi,r,s,a,b):
    p=[a*b*z, (a+b)*r*z+a*b*(1-phi),phi*z+(a+b)*r-s*a*phi*r-(1-s)*b*phi*r,phi]
    ro=np.roots(p)
    if (np.imag(ro[0])>0):
      st=ro[0];
    elif (np.imag(ro[1])>0):
      st=ro[1];
    elif (np.imag(ro[2])>0):
      st=ro[2];
    else:
      print("error")
    return st


## Calculate the approximate density function while taking the  imaginary part to close to zero
def lsd(x,phi,r,s,a,b):
    return np.imag(stieltjes(x+0.001j,phi,r,s,a,b))/np.pi

def generate_eigenvalues(key, Sigmad, m, n):
    X = random.normal(key, shape=(m, n))
    X = X * jnp.sqrt(Sigmad)[:, None]  # Scale the rows of X by the square root of the diagonal elements of Sigma
    H = jnp.matmul(X.T, X) / jnp.sqrt(n * m)
    E0 = jnp.linalg.eigvalsh(H)
    return E0
def plot_eigenvalues_histogram(phi=0.9, R=20, n=800, a=1, b=15, s=1/2):
    ratio = phi
    r = np.sqrt(phi)
    E_start = -1
    E_end = (s*a+(1-s)*b)*r + 2*np.sqrt(s*a**2+(1-s)*b**2) + 25
    E_values = np.arange(E_start, E_end, 0.01)
    m_imag_values = [lsd(E, phi, r, s, a, b) if E >= 0.1 else 0 for E in E_values]

    m = int(n * phi)
    F1 = jnp.array([])
    Sigmad = jnp.concatenate([jnp.repeat(a, int(s * m)), jnp.repeat(b, int((1 - s) * m))])

    key = random.PRNGKey(0)

    for i in range(R):
        key, subkey = random.split(key)
        E0 = generate_eigenvalues(subkey, Sigmad, m, n)
        F1 = jnp.concatenate([F1, E0]) if F1.shape[0] > 0 else E0

    # Clear unneeded data
    del Sigmad
    del E0

    F1_filtered = F1[F1 > 0.01]
    FF1 = np.array(F1_filtered)
    # Plotting
    if phi<1:
        plt.xlim(-1, (s*a+(1-s)*b)*(1/np.sqrt(phi))+40)
    else:
        plt.xlim( (s*a+(1-s)*b)*(np.sqrt(phi))-30, (s*a+(1-s)*b)*(np.sqrt(phi))+50)
    # plt.ylim(0, 0.35)

    if phi > 1:
        # Use histogram directly for phi > 1
        plt.hist(FF1, bins=200, density=True, alpha=0.9, label='histogram')
    else:
        # Use bar plot for phi <= 1
        scaling_factor = phi
        hist_values, bin_edges = np.histogram(FF1, bins=200, density=True)
        scaled_hist_values = hist_values * scaling_factor
        bin_widths = bin_edges[1:] - bin_edges[:-1]
        plt.bar(bin_edges[:-1], scaled_hist_values, width=bin_widths, align='edge', alpha=0.9, label='histogram')

    plt.plot(E_values, m_imag_values, color='red', label="theoretical")

    # Legend and formatting
    legend = plt.legend(loc="upper right", fontsize='xx-large')
    for text in legend.get_texts():
        text.set_fontweight('bold')
    ax = plt.gca()
    for label in ax.get_xticklabels():
        label.set_fontweight('bold')
        label.set_fontsize(12)
    for label in ax.get_yticklabels():
        label.set_fontweight('bold')
        label.set_fontsize(12)
    plt.tight_layout()
    plt.show()




def ecdf(data):
    """Compute ECDF for a one-dimensional array of measurements."""
    n = len(data)
    x = np.sort(data)
    y = np.arange(1, n+1) / n
    return x, y


## Compare the ECDFs of two data sets, specifically when $kappa_4 = or \neq 0$
## dot plot version
# def plot_ecdf2(data1, data2, label1="", label2="", filename=None):
#     # Normalize data using the mean and variance of data1
#     mean_data1 = np.mean(data1)
#     std_data1 = np.std(data1)

#     data1_norm = (data1 - mean_data1) / std_data1
#     data2_norm = (data2 - mean_data1) / std_data1

#     # Compute ECDF for normalized data
#     x_data1_norm, y_data1_norm = ecdf(data1_norm)
#     x_data2_norm, y_data2_norm = ecdf(data2_norm)

#     # Plot ECDFs
#     plt.figure(figsize=(10, 7))
#     plt.plot(x_data1_norm, y_data1_norm, marker='.', linestyle='none', color='blue', label=label1)
#     plt.plot(x_data2_norm, y_data2_norm, marker='.', linestyle='none', color='red', label=label2)

#     # Formatting
#     plt.xticks(fontweight='bold', fontsize=16)
#     plt.yticks(fontweight='bold', fontsize=16)
#     plt.legend(fontsize=25)
#     plt.tight_layout()

#     # Save the plot if filename is provided
#     if filename:
#         plt.savefig(filename)
#         files.download(filename)

#     # Display the plot
#     plt.show()

## line plot version
def plot_ecdf2(data1, data2, label1="", label2="", filename=None):
    # Normalize data using the mean and variance of data1
    mean_data1 = np.mean(data1)
    std_data1 = np.std(data1)

    data1_norm = (data1 - mean_data1) / std_data1
    data2_norm = (data2 - mean_data1) / std_data1

    # Compute ECDF for normalized data
    x_data1_norm, y_data1_norm = ecdf(data1_norm)
    x_data2_norm, y_data2_norm = ecdf(data2_norm)

    # Plot ECDFs
    plt.figure(figsize=(8, 5))
    plt.plot(x_data1_norm, y_data1_norm, linestyle='-', color='blue', label=label1)
    plt.plot(x_data2_norm, y_data2_norm, linestyle='-', color='red', label=label2)

    # Formatting
    plt.xticks(fontweight='bold', fontsize=16)
    plt.yticks(fontweight='bold', fontsize=16)
    plt.legend(fontsize=25)
    plt.tight_layout()

    # Save the plot if filename is provided
    if filename:
        plt.savefig(filename)
        files.download(filename)

    # Display the plot
    plt.show()




## Compare the ECDFs of multiple normalized data sets with ECDF of standard normal
def plot_ecdf_and_standard_normal_cdf(*data_sets, labels=None):

    # Ensure that the number of datasets matches the number of labels
    if len(data_sets) != len(labels):
        raise ValueError("Number of datasets must match the number of labels.")

    plt.rcParams['text.usetex'] = False
    normal_x = np.arange(-4, 4.001, 0.001)
    standard_normal_cdf = norm.cdf(normal_x)

    for idx, data in enumerate(data_sets):
        sorted_data = np.sort(data)
        cumulative_probabilities = np.arange(1, len(sorted_data) + 1) / len(sorted_data)
        label = labels[idx] if labels else f"Data {idx+1}"
        plt.plot(sorted_data, cumulative_probabilities, linestyle='-', linewidth=0.8, label=label)

    plt.plot(normal_x, standard_normal_cdf, linestyle='-', linewidth=0.8, label=r'$\mathbf{N(0,1)}$')
    plt.ylabel('Cumulative Probability', fontweight='bold', fontsize=18)
    legend = plt.legend(loc="upper left", fontsize=16)
    plt.xlim(-4, 4)
    bold_legend(legend)
    plt.xticks(fontsize=16)
    plt.yticks(fontsize=16)
    ax = plt.gca()
    for label in ax.get_xticklabels():
        label.set_fontweight('bold')
    for label in ax.get_yticklabels():
        label.set_fontweight('bold')
    plt.show()

##############################################################
############Functions for doing hypothesis testing for $\Sigma=Id$

import numpy as np
from scipy.stats import chi2
from numpy.linalg import svd
from statsmodels.stats.correlation_tools import cov_nearest









from scipy.stats import norm
import numpy as np
def sample_covariance_matrix(data):
    n, p = data.shape  ###nneed fix but not essential
    # mean = np.mean(data, axis=0)
    # centered_data = data - mean
    centered_data=data
    cov_matrix = np.dot(centered_data,centered_data.T) / n
    return cov_matrix

def CLRT(data, Sigma="identity", c=0, K4=0, alternative='two-sided'):
    # p, n = data.shape #n> p
    n, p = data.shape #p>n
    S = sample_covariance_matrix(data)
    n,p=p,n
    log_det_S = np.log(np.linalg.det(S))
    # print(log_det_S)
    # log_det_I = np.log(np.linalg.det(np.eye(p)))
    # print(log_det_I)
    # Calculate likelihood ratio test statistic
    CLRT_stat = (-log_det_S+p*np.log(np.trace(S)/p)+(p-n)*np.log(1-p/n)-p+1/2*np.log(1-p/n))/np.sqrt(-2*np.log(1-p/n)-2*p/n)
    z_score=CLRT_stat
    if alternative == 'two-sided':
        p_value = 2 * (1 - norm.cdf(abs(z_score)))
    elif alternative == 'greater':
        p_value = 1 - norm.cdf(z_score)
    elif alternative == 'less':
        p_value = norm.cdf(z_score)
    else:
        raise ValueError("alternative must be 'two-sided', 'greater' or 'less'")

    return p_value


def CLRTpln(data, Sigma="identity", c=0, K4=0, alternative='two-sided'):
    # p, n = data.shape #n> p
    n, p = data.shape #p>n
    S = sample_covariance_matrix(data)
    log_det_S = np.log(np.linalg.det(S))
    # print(log_det_S)
    # log_det_I = np.log(np.linalg.det(np.eye(p)))
    # print(log_det_I)
    # Calculate likelihood ratio test statistic
    CLRT_stat = (-log_det_S+p*np.log(np.trace(S)/p)+(p-n)*np.log(1-p/n)-p+1/2*np.log(1-p/n))/np.sqrt(-2*np.log(1-p/n)-2*p/n)
    z_score=CLRT_stat
    if alternative == 'two-sided':
        p_value = 2 * (1 - norm.cdf(abs(z_score)))
    elif alternative == 'greater':
        p_value = 1 - norm.cdf(z_score)
    elif alternative == 'less':
        p_value = norm.cdf(z_score)
    else:
        raise ValueError("alternative must be 'two-sided', 'greater' or 'less'")

    return p_value



def DW2023Global(X, Sigma="identity", c=0, K4=0, alternative='two-sided'):
    # m=X.shape[0]
    # n=X.shape[1]
    m=X.shape[1]
    n=X.shape[0]
    phi=m/n
    r=np.sqrt(phi)
    centerc=r+1/r+c
    H = np.matmul(X, X.T) / np.sqrt(n * m)-centerc*np.identity(n)
    TestST=np.trace(H)
    z_score=(TestST-n*(c-1/r))/np.sqrt(2+K4)
    if alternative == 'two-sided':
        p_value = 2 * (1 - norm.cdf(abs(z_score)))
    elif alternative == 'greater':
        p_value = 1 - norm.cdf(z_score)
    elif alternative == 'less':
        p_value = norm.cdf(z_score)
    else:
        raise ValueError("alternative must be 'two-sided', 'greater' or 'less'")
    return p_value




### Global Test based on x^2
def DW2023Global2(X, Sigma="identity", c=0, K4=0, alternative='two-sided'):
    # m=X.shape[0]
    # n=X.shape[1]
    m=X.shape[1]
    n=X.shape[0]
    phi=m/n
    r=np.sqrt(phi)
    centerc=r+1/r+c
    H = np.matmul(X, X.T) / np.sqrt(n * m)-centerc*np.identity(n)
    TestST=np.trace(np.matmul(H, H))
    # print(TestST)
    z_score=(TestST-(n*(1+c**2-2*c/r+1/phi)+1))/np.sqrt(4+8*c**2) ### not yet modified by kappa4
    # print(z_score)
    if alternative == 'two-sided':
        p_value = 2 * (1 - norm.cdf(abs(z_score)))
    elif alternative == 'greater':
        p_value = 1 - norm.cdf(z_score)
    elif alternative == 'less':
        p_value = norm.cdf(z_score)
    else:
        raise ValueError("alternative must be 'two-sided', 'greater' or 'less'")
    return p_value


### Global Test based on x^2 excluding spikes or near 0 eigenvalues
def DW2023Global2modified(X, Sigma="identity", c=0, K4=0, alternative='two-sided'):
    m = X.shape[1]
    n = X.shape[0]
    phi = m/n
    r = np.sqrt(phi)
    centerc = r + 1/r + c
    H = np.matmul(X, X.T) / np.sqrt(n * m) - centerc * np.identity(n)

    # Compute eigenvalues of H
    eigvals = np.linalg.eigvalsh(H)

    # Filter eigenvalues below 10^-3 and above 100
    filtered_eigvals = eigvals[(eigvals < 1e-3) | (eigvals > 100)]

    # Subtract the sum of squares of filtered eigenvalues from TestST
    TestST = np.trace(np.matmul(H, H)) - np.sum(filtered_eigvals**2)

    # Record the number of such eigenvalues as ns
    ns = len(filtered_eigvals)

    z_score = (TestST - (ns * (1 + c**2 - 2*c/r + 1/phi) + 1)) / np.sqrt(4 + 8*c**2)  # Modified by kappa4

    if alternative == 'two-sided':
        p_value = 2 * (1 - norm.cdf(abs(z_score)))
    elif alternative == 'greater':
        p_value = 1 - norm.cdf(z_score)
    elif alternative == 'less':
        p_value = norm.cdf(z_score)
    else:
        raise ValueError("alternative must be 'two-sided', 'greater' or 'less'")

    return p_value

# def DW2023Global2modified(X, Sigma="identity", c=0, K4=0, alternative='two-sided'):
#     # m=X.shape[0]
#     # n=X.shape[1]
#     m=X.shape[1]
#     n=X.shape[0]
#     phi=m/n
#     r=np.sqrt(phi)
#     centerc=r+1/r+c
#     H = np.matmul(X, X.T) / np.sqrt(n * m)-centerc*np.identity(n)

#     eigvals = np.linalg.eigvalsh(H)  # Compute eigenvalues of H, in ascending order
#     max_eigval = eigvals[-1]  # Extract the largest eigenvalue

#     TestST = np.trace(np.matmul(H, H)) - max_eigval**2
#     print(TestST)
#     z_score=(TestST-(n*(1+c**2-2*c/r+1/phi)+1))/np.sqrt(4+8*c**2) ### not yet modified by kappa4
#     print(z_score)
#     if alternative == 'two-sided':
#         p_value = 2 * (1 - norm.cdf(abs(z_score)))
#     elif alternative == 'greater':
#         p_value = 1 - norm.cdf(z_score)
#     elif alternative == 'less':
#         p_value = norm.cdf(z_score)
#     else:
#         raise ValueError("alternative must be 'two-sided', 'greater' or 'less'")
#     return p_value




def DW2023Local(X, Sigma="identity", c=0, K4=0, alternative='two-sided'):
    # m=X.shape[0]
    # n=X.shape[1]
    m=X.shape[1]
    n=X.shape[0]
    phi=m/n
    r=np.sqrt(phi)
    leftb=r+1/r-2+c
    H = np.matmul(X, X.T) / np.sqrt(n * m)
    E0, _ = np.linalg.eigh(H)
    eta=1/(n**(1/4))
    # print(E0)
    Et = np.array(extract_elements(E0, leftb-eta, leftb+eta))
    # print(Eb)
    # print(E0[-k:])
    Eb = (Et - leftb*np.ones(Et.shape))/eta
    # print(Eb)
    TestST=np.sum(Eb)
    z_score=(TestST-Mean_x)/np.sqrt(Variance_x)
    if alternative == 'two-sided':
        p_value = 2 * (1 - norm.cdf(abs(z_score)))
    elif alternative == 'greater':
        p_value = 1 - norm.cdf(z_score)
    elif alternative == 'less':
        p_value = norm.cdf(z_score)
    else:
        raise ValueError("alternative must be 'two-sided', 'greater' or 'less'")

    return p_value


def DW2023Local_two_sample(X, Y, alternative='two-sided'):
    m=X.shape[1]
    n=X.shape[0]
    phi=m/n
    cov1=X@np.transpose(X)/ np.sqrt(n * m)
    E01,_=np.linalg.eigh(cov1)
    E1=np.array(E01)
    E1.sort()
    E=E1[-3]
    # E=np.median(X)
    eta=2*np.std(X)
    HX = np.matmul(X, X.T) / np.sqrt(n * m)
    HY = np.matmul(Y, Y.T) / np.sqrt(n * m)
    EX, _ = np.linalg.eigh(HX)
    EY, _ = np.linalg.eigh(HY)
    EXt = np.array(extract_elements(EX, E-eta, E+eta))
    EYt = np.array(extract_elements(EY, E-eta, E+eta))
    # print(EXt)
    # print(EYt)
    EXb = (EXt - E*np.ones(EXt.shape))/eta
    EYb = (EYt - E*np.ones(EYt.shape))/eta
    TestSTX=np.sum(EXb)
    TestSTY=np.sum(EYb)
    # print(TestSTX-TestSTY)
    z_score=(TestSTX-TestSTY)/np.sqrt(4/np.pi/np.pi)
    print(z_score)
    # z_score=(TestSTX-TestSTY)/np.sqrt(8/3/np.pi/np.pi)
    if alternative == 'two-sided':
        p = 2 * (1 - norm.cdf(abs(z_score)))
    elif alternative == 'greater':
        p = 1 - norm.cdf(z_score)
    elif alternative == 'less':
        p = norm.cdf(z_score)
    else:
        raise ValueError("alternative must be 'two-sided', 'greater' or 'less'")

    return True if -1.96<p<1.96 else False



def Nagao1973(x, significant_level=0.05, Sigma = "identity"):
    if isinstance(x, np.ndarray):
        return Nagao1973_matrix(x, significant_level, Sigma)
    elif isinstance(x, np.cov):
        return Nagao1973_covariance(x, significant_level, Sigma)

def Nagao1973_covariance(x, significant_level=0.05, Sigma = "identity"):
    p = x.shape[1]
    # n = x.shape[0] + 1
    n = x.shape[0]
    S = x

    if Sigma == "identity":
        S_ = x
    else:
        svCov = svd(x)
        sv = svd(Sigma)
        x_ = svCov[0] @ np.diag(np.sqrt(svCov[1])) @ np.linalg.inv(sv[0] @ np.diag(np.sqrt(sv[1])))
        S_ = x_.T @ x_


    statistic = Nagao1973_(n, p, S_)
    parameter = p * (p + 1) / 2
    p_value = 1 - chi2.cdf(statistic, parameter)
    # null_value=(p_value > significant_level)
    # estimate = S
    # estimate = None if estimate.shape[0] > 5 else estimate

    # obj = {'statistic': statistic,
    #        'parameter': parameter,
    #        'p.value': p_value,
    #        'estimate': estimate,
    #        'null.value': null_value,
    #        'alternative': "two.sided",
    #        'method': "Nagao 1973 Test of Covariance Matrix Structure"}

    return p_value

def Nagao1973_matrix(x,significant_level=0.05, Sigma = "identity"):

    p = x.shape[1]
    n = x.shape[0]

    S = np.cov(x, rowvar=False)

    if Sigma == "identity":
        S_ = S
    else:
        sv = svd(Sigma)
        svDf = svd(S)
        x_ = svDf[0] @ np.diag(np.sqrt(sv[1])) @ np.linalg.inv(sv[0] @ np.diag(np.sqrt(sv[1])))
        S_ = x_.T @ x_

    statistic = Nagao1973_(n, p, S_)
    parameter = p * (p + 1) / 2
    p_value = 1 - chi2.cdf(statistic, parameter)
    # null_value=(p_value > significant_level)

    # estimate = S
    # estimate = None if estimate.shape[0] > 5 else estimate

    # obj = {'statistic': statistic,
    #        'parameter': parameter,
    #        'p.value': p_value,
    #        'estimate': estimate,
    #        'null.value': null_value,
    #        'alternative': "two.sided",
    #        'method': "Nagao 1973 Test of Covariance Matrix Structure"}

    return p_value

def Nagao1973_(n, p, S_):
    mid = S_ - np.eye(p)
    return n * p * (np.trace(mid @ mid) / p) / 2




import numpy as np
from scipy.stats import norm
from numpy.linalg import svd
from statsmodels.stats.correlation_tools import cov_nearest


def Srivastava2005(x, significant_level=0.05, Sigma = "identity"):
    if isinstance(x, np.ndarray):
        return Srivastava2005_matrix(x, significant_level, Sigma = "identity")
    elif isinstance(x, np.cov):
        return Srivastava2005_covariance(x, significant_level, Sigma = "identity")

def Srivastava2005_covariance(x, significant_level=0.05, Sigma = "identity"):
    p = x.shape[1]
    n = x.shape[0] + 1
    S = x

    if Sigma == "identity":
        S_ = x
    else:
        svCov = svd(x)
        sv = svd(Sigma)
        x_ = svCov[0] @ np.diag(np.sqrt(svCov[1])) @ np.linalg.inv(sv[0] @ np.diag(np.sqrt(sv[1])))
        S_ = x_.T @ x_

    statistic = Srivastava2005Stat(S_)  # replace Srivastava2005Stat with the correct function
    parameter = np.array([0, 1])

    p_value = 1 - norm.cdf(np.abs(statistic))
    # null_value =  (p_value > significant_level)

    # estimate = S
    # estimate = None if estimate.shape[0] > 5 else estimate

    # obj = {'statistic': statistic,
    #        'parameter': parameter,
    #        'p.value': p_value,
    #        'estimate': estimate,
    #        'null.value': null_value,
    #        'alternative': "two.sided",
    #        'method': "Srivastava 2005 Test of Covariance Matrix Structure"}

    return p_value

def Srivastava2005_matrix(x, significant_level=0.05, Sigma = "identity"):
    p = x.shape[1]
    n = x.shape[0]
    S = np.cov(x, rowvar=False)

    if Sigma == "identity":
        S_ = S
    else:
        sv = svd(Sigma)
        svDf = svd(S)
        x_ = svDf[0] @ np.diag(np.sqrt(sv[1])) @ np.linalg.inv(sv[0] @ np.diag(np.sqrt(sv[1])))
        S_ = x_.T @ x_

    statistic = Srivastava2005Stat(S_)  # replace Srivastava2005Stat with the correct function
    parameter = np.array([0, 1])
    p_value = 1 - norm.cdf(np.abs(statistic))
    # null_value = (p_value > significant_level)
    # estimate = S
    # estimate = None if estimate.shape[0] > 5 else estimate

    # obj = {'statistic': statistic,
    #        'parameter': parameter,
    #        'p.value': p_value,
    #        'estimate': estimate,
    #        'null.value': null_value,
    #        'alternative': "two.sided",
    #        'method': "Srivastava 2005 Test of Covariance Matrix Structure"}

    return p_value



def Srivastava2005Stat(x):
    ncol = x.shape[1]
    nrow = x.shape[0]

    return nrow * ((np.power(nrow, 2.0) / (ncol * (nrow - 1.0) * (nrow + 2.0))) *
                   (np.trace(np.dot(x, x)) - np.power(np.trace(x), 2.0) / nrow) -
                   2.0 * np.trace(x) / ncol + 1.0) / 2.0





from scipy.stats import norm
from scipy.linalg import svd
import numpy as np

def Srivastava2011(x, significant_level=0.05, Sigma = "identity"):
    if isinstance(x, np.ndarray):
        return Srivastava2011_matrix(x, significant_level, Sigma = "identity")
    elif isinstance(x, np.cov):
        return Srivastava2011_covariance(x, significant_level, Sigma = "identity")

def Srivastava2011_covariance(x, significant_level=0.05, Sigma = "identity"):
    p = x.shape[1]
    n = x.shape[0]
    S = x

    if Sigma == "identity":
        S_ = x
    else:
        svCov_u, svCov_d, _ = svd(x)
        sv_u, sv_d, _ = svd(Sigma)
        x_ = svCov_u @ np.diag(np.sqrt(svCov_d)) @ np.linalg.inv(sv_u @ np.diag(np.sqrt(sv_d)))
        S_ = x_.T @ x_

    statistic = Srivastava2011_(n - 1, p, S_)
    parameter = np.array([0, 1])

    p_value = 1 - norm.cdf(abs(statistic))
    # null_value = (p_value>significant_level)

    # estimate = S
    # if estimate.shape[0] > 5:
    #     estimate = None

    # result = {"statistic": statistic,
    #           "parameter": parameter,
    #           "p.value": p_value,
    #           "estimate": estimate,
    #           "null.value": null_value,
    #           "alternative": "two.sided",
    #           "method": "Srivastava et al. 2011 Test of Covariance Matrix Structure"}

    return p_value


def Srivastava2011_matrix(x, significant_level=0.05, Sigma = "identity"):
    p = x.shape[1]
    n = x.shape[0]
    S = np.cov(x, rowvar=False)

    if Sigma == "identity":
        S_ = S
    else:
        sv_u, sv_d, _ = svd(Sigma)
        svDf_u, svDf_d, _ = svd(S)
        x_ = svDf_u @ np.diag(np.sqrt(sv_d)) @ np.linalg.inv(sv_u @ np.diag(np.sqrt(sv_d)))
        S_ = x_.T @ x_

    statistic = Srivastava2011_(n - 1, p, S_)
    parameter = np.array([0, 1])

    p_value = 1 - norm.cdf(abs(statistic))
    # null_value = (p_value>significant_level)
    # estimate = S
    # if estimate.shape[0] > 5:
    #     estimate = None

    # result = {"statistic": statistic,
    #           "parameter": parameter,
    #           "p.value": p_value,
    #           "estimate": estimate,
    #           "null.value": null_value,
    #           "alternative": "two.sided",
    #           "method": "Srivastava et al. 2011 Test of Covariance Matrix Structure"}

    return p_value


def Srivastava2011_(n, p, S_):
    S_squared = S_ @ S_
    tr_S_squared = np.trace(S_squared)
    tr_S = np.trace(S_)
    return n * (((n ** 2) / ((n - 1) * (n + 2))) * (tr_S_squared - (tr_S ** 2) / n) / p -
                2 * (tr_S / p) + 1) / 2



from scipy.stats import norm
from numpy.linalg import svd, solve
import numpy as np

def Chen2010(x, significant_level=0.05, Sigma = "identity"):
    if type(x) is np.ndarray:
        return Chen2010_matrix(x, significant_level, Sigma = "identity")
    else:
        return Chen2010_covariance(x, significant_level, Sigma = "identity")

def Chen2010_covariance(x, significant_level=0.05, Sigma = "identity"):
    p = x.shape[1]
    n = x.shape[0] + 1
    S = x

    if Sigma == "identity":
        svCov_u, svCov_d, _ = svd(x)
        x_ = svCov_u @ np.diag(np.sqrt(svCov_d))
    else:
        svCov_u, svCov_d, _ = svd(x)
        sv_u, sv_d, _ = svd(Sigma)
        x_ = svCov_u @ np.diag(np.sqrt(svCov_d)) @ solve(sv_u @ np.diag(np.sqrt(sv_d)), np.eye(p))

    statistic = Chen2010Stat(x_)
    parameter = [0, 1]


    p_value = 1 - norm.cdf(abs(statistic))
    # null_value = (p_value>significant_level)
    # estimate = None if S.shape[0] > 5 else S

    # result = {"statistic": statistic,
    #           "parameter": parameter,
    #           "p.value": p_value,
    #           "estimate": estimate,
    #           "null.value": null_value,
    #           "alternative": "two.sided",
    #           "method": "Chen et al. 2010 Test of Covariance Matrix Structure"}

    return p_value

def Chen2010_matrix(x, significant_level=0.05, Sigma = "identity"):
    p = x.shape[1]
    n = x.shape[0]
    S = np.cov(x.T)

    if Sigma == "identity":
        x_ = x
        # print(1)
    else:
        # print(2)
        sv_u, sv_d, _ = svd(Sigma)
        x_ = x @ solve(sv_u @ np.diag(np.sqrt(sv_d)), np.eye(p))

    statistic = Chen2010Stat(x_)
    parameter = [0, 1]


    p_value = 1 - norm.cdf(abs(statistic))
    # null_value = (p_value>significant_level)
    # estimate = None if S.shape[0] > 5 else S

    # result = {"statistic": statistic,
    #           "parameter": parameter,
    #           "p.value": p_value,
    #           "estimate": estimate,
    #           "null.value": null_value,
    #           "alternative": "two.sided",
    #           "method": "Chen et al. 2010 Test of Covariance Matrix Structure"}

    return p_value


# import scipy.special as sp
# def Chen2010Stat(x):
#     nrow, ncol = x.shape
#     quadraNum = 0
#     bilinearoffNum = 0
#     bilinearcubeNum = 0
#     bilinearsquareNum = 0
#     bilinearquadNum = 0

#     for i in range(nrow):
#         quadraNum += (x[i, :].reshape(-1, 1).T @ x[i, :].reshape(-1, 1)).item()
#         for j in range(i + 1, nrow):
#             bilinearoffNum += (x[i, :].reshape(-1, 1).T @ x[j, :].reshape(-1, 1)).item()
#             bilinearsquareNum += (x[i, :].reshape(-1, 1).T @ x[j, :].reshape(-1, 1) @ x[i, :].reshape(-1, 1).T @ x[j, :].reshape(-1, 1)).item()
#             for k in range(j + 1, nrow):
#                 bilinearcubeNum += (x[i, :].reshape(-1, 1).T @ x[j, :].reshape(-1, 1) @ x[j, :].reshape(-1, 1).T @ x[k, :].reshape(-1, 1)).item()
#                 for l in range(k + 1, nrow):
#                     bilinearquadNum += (x[i, :].reshape(-1, 1).T @ x[j, :].reshape(-1, 1) @ x[k, :].reshape(-1, 1).T @ x[l, :].reshape(-1, 1)).item()

#     bilinearquad = bilinearquadNum * 24.0 / (sp.gamma(nrow + 1.0) / sp.gamma(nrow - 3.0))
#     bilinearsquare = bilinearsquareNum * 2.0 / (sp.gamma(nrow + 1.0) / sp.gamma(nrow - 1.0))
#     bilinearcube = bilinearcubeNum * 6.0 / (sp.gamma(nrow + 1.0) / sp.gamma(nrow - 2.0))
#     bilinearoff = bilinearoffNum * 2.0 / (sp.gamma(nrow + 1.0) / sp.gamma(nrow - 1.0))
#     quadra = quadraNum / nrow

#     return nrow * (bilinearsquare / ncol - 2.0 * bilinearcube / ncol + bilinearquad / ncol - 2.0 * quadra / ncol + 2.0 * bilinearoff / ncol + 1.0) / 2.0



# import numpy as np
# from scipy.special import gamma

# def Chen2010Stat(x):
#     nrow, ncol = x.shape
#     quadraNum = 0
#     bilinearoffNum = 0
#     bilinearcubeNum = 0
#     bilinearsquareNum = 0
#     bilinearquadNum = 0

#     for i in range(nrow):
#         quadraNum += np.sum(np.dot(x[i, :], x[i, :]))
#         for j in range(i + 1, nrow):
#             bilinearoffNum += np.sum(np.dot(x[i, :], x[j, :]))
#             bilinearsquareNum += np.sum(np.dot(x[i, :], x[j, :]) ** 2)
#             for k in range(j + 1, nrow):
#                 bilinearcubeNum += np.sum(np.dot(x[i, :], x[j, :]) * np.dot(x[j, :], x[k, :]))
#                 for l in range(k + 1, nrow):
#                     bilinearquadNum += np.sum(np.dot(x[i, :], x[j, :]) * np.dot(x[k, :], x[l, :]))

#     bilinearquad = bilinearquadNum * 24.0 / (gamma(nrow + 1.0) / gamma(nrow - 3.0))
#     bilinearsquare = bilinearsquareNum * 2.0 / (gamma(nrow + 1.0) / gamma(nrow - 1.0))
#     bilinearcube = bilinearcubeNum * 6.0 / (gamma(nrow + 1.0) / gamma(nrow - 2.0))
#     bilinearoff = bilinearoffNum * 2.0 / (gamma(nrow + 1.0) / gamma(nrow - 1.0))
#     quadra = quadraNum / nrow

#     return nrow * (bilinearsquare / ncol - 2.0 * bilinearcube / ncol + bilinearquad / ncol - 2.0 * quadra / ncol + 2.0 * bilinearoff / ncol + 1.0) / 2.0

import numpy as np
from scipy.special import gamma

# def Chen2010Stat(x):
#     nrow, ncol = x.shape
#     quadraNum = 0
#     bilinearoffNum = 0
#     bilinearcubeNum = 0
#     bilinearsquareNum = 0
#     bilinearquadNum = 0

#     for i in range(nrow):
#         quadraNum += np.sum(np.dot(x[i, :], x[i, :]))
#         for j in range(i + 1, nrow):
#             bilinearoffNum += np.sum(np.dot(x[i, :], x[j, :]))
#             bilinearsquareNum += np.sum(np.dot(x[i, :], x[j, :]) ** 2)
#             for k in range(j + 1, nrow):
#                 bilinearcubeNum += np.sum(np.dot(x[i, :], x[j, :]) * np.dot(x[j, :], x[k, :]))
#                 for l in range(k + 1, nrow):
#                     bilinearquadNum += np.sum(np.dot(x[i, :], x[j, :]) * np.dot(x[k, :], x[l, :]))

#     bilinearquad = bilinearquadNum * 24.0 / (gamma(nrow + 1.0) / gamma(nrow - 3.0))
#     bilinearsquare = bilinearsquareNum * 2.0 / (gamma(nrow + 1.0) / gamma(nrow - 1.0))
#     bilinearcube = bilinearcubeNum * 6.0 / (gamma(nrow + 1.0) / gamma(nrow - 2.0))
#     bilinearoff = bilinearoffNum * 2.0 / (gamma(nrow + 1.0) / gamma(nrow - 1.0))
#     quadra = quadraNum / nrow

#     return nrow * (bilinearsquare / ncol - 2.0 * bilinearcube / ncol + bilinearquad / ncol - 2.0 * quadra / ncol + 2.0 * bilinearoff / ncol + 1.0) / 2.0

## vect trial
# def Chen2010Stat(x):
#     nrow, ncol = x.shape
#     quadraNum = 0
#     bilinearoffNum = 0
#     bilinearcubeNum = 0
#     bilinearsquareNum = 0
#     bilinearquadNum = 0
#     wholerow=np.zeros(p)
#     Y1=0
#     for i in range(nrow):
#         wholerow+=np.sum(x[i,:])
#         Y1+=np.sum(np.dot(x[i, :], x[i, :]))

#     for i in range(nrow):
#         quadraNum += np.sum(np.dot(x[i, :], x[i, :]))
#         for j in range(i + 1, nrow):
#             bilinearoffNum += np.sum(np.dot(x[i, :], x[j, :]))
#             bilinearsquareNum += np.sum(np.dot(x[i, :], x[j, :]) ** 2)
#             for k in range(j + 1, nrow):
#                 bilinearcubeNum += np.sum(np.dot(x[i, :], x[j, :]) * np.dot(x[j, :], x[k, :]))
#                 for l in range(k + 1, nrow):
#                     bilinearquadNum += np.sum(np.dot(x[i, :], x[j, :]) * np.dot(x[k, :], x[l, :]))

#     bilinearquad = bilinearquadNum * 24.0 / (gamma(nrow + 1.0) / gamma(nrow - 3.0))
#     bilinearsquare = bilinearsquareNum * 2.0 / (gamma(nrow + 1.0) / gamma(nrow - 1.0))
#     bilinearcube = bilinearcubeNum * 6.0 / (gamma(nrow + 1.0) / gamma(nrow - 2.0))
#     bilinearoff = bilinearoffNum * 2.0 / (gamma(nrow + 1.0) / gamma(nrow - 1.0))
#     quadra = quadraNum / nrow

#     return nrow * (bilinearsquare / ncol - 2.0 * bilinearcube / ncol + bilinearquad / ncol - 2.0 * quadra / ncol + 2.0 * bilinearoff / ncol + 1.0) / 2.0



# import numpy as np
# from scipy.special import gamma

# def Chen2010Stat(x):
#     nrow, ncol = x.shape

#     quadraNum = np.sum(x * x)
#     bilinearoffNum = np.sum(np.dot(x, x.T)) - quadraNum

#     bilinearcubeNum = 0
#     bilinearsquareNum = 0
#     bilinearquadNum = 0

#     for i in range(nrow):
#         for j in range(i + 1, nrow):
#             bilinearsquareNum += np.sum(np.dot(x[i, :], x[j, :]) ** 2)
#             for k in range(j + 1, nrow):
#                 bilinearcubeNum += np.sum(np.dot(x[i, :], x[j, :]) * np.dot(x[j, :], x[k, :]))
#                 for l in range(k + 1, nrow):
#                     bilinearquadNum += np.sum(np.dot(x[i, :], x[j, :]) * np.dot(x[k, :], x[l, :]))

#     bilinearquad = bilinearquadNum * 24.0 / (gamma(nrow + 1.0) / gamma(nrow - 3.0))
#     bilinearsquare = bilinearsquareNum * 2.0 / (gamma(nrow + 1.0) / gamma(nrow - 1.0))
#     bilinearcube = bilinearcubeNum * 6.0 / (gamma(nrow + 1.0) / gamma(nrow - 2.0))
#     bilinearoff = bilinearoffNum * 2.0 / (gamma(nrow + 1.0) / gamma(nrow - 1.0))
#     quadra = quadraNum / nrow

#     return nrow * (bilinearsquare / ncol - 2.0 * bilinearcube / ncol + bilinearquad / ncol - 2.0 * quadra / ncol + 2.0 * bilinearoff / ncol + 1.0) / 2.0

from numba import njit
import numpy as np
from scipy.special import gamma

@njit
def Chen2010Stat_numba_helper(nrow, ncol, x):
    bilinearcubeNum = 0
    bilinearsquareNum = 0
    bilinearquadNum = 0

    for i in range(nrow):
        for j in range(i + 1, nrow):
            dot_product_ij = np.dot(x[i, :], x[j, :])
            bilinearsquareNum += dot_product_ij ** 2
            for k in range(j + 1, nrow):
                dot_product_jk = np.dot(x[j, :], x[k, :])
                bilinearcubeNum += dot_product_ij * dot_product_jk
                for l in range(k + 1, nrow):
                    dot_product_kl = np.dot(x[k, :], x[l, :])
                    bilinearquadNum += dot_product_ij * dot_product_kl

    return bilinearsquareNum, bilinearcubeNum, bilinearquadNum


def Chen2010Stat(x):
    nrow, ncol = x.shape
    quadraNum = np.sum(x * x)
    bilinearoffNum = np.sum(np.dot(x, x.T)) - quadraNum
    bilinearsquareNum, bilinearcubeNum, bilinearquadNum = Chen2010Stat_numba_helper(nrow, ncol, x)

    bilinearquad = bilinearquadNum * 24.0 / (gamma(nrow + 1.0) / gamma(nrow - 3.0))
    bilinearsquare = bilinearsquareNum * 2.0 / (gamma(nrow + 1.0) / gamma(nrow - 1.0))
    bilinearcube = bilinearcubeNum * 6.0 / (gamma(nrow + 1.0) / gamma(nrow - 2.0))
    bilinearoff = bilinearoffNum * 2.0 / (gamma(nrow + 1.0) / gamma(nrow - 1.0))
    quadra = quadraNum / nrow

    return nrow * (bilinearsquare / ncol - 2.0 * bilinearcube / ncol + bilinearquad / ncol - 2.0 * quadra / ncol + 2.0 * bilinearoff / ncol + 1.0) / 2.0



from scipy.stats import chi2
from numpy.linalg import svd, solve
import numpy as np

def LedoitWolf2002(x, significant_level=0.05, Sigma = "identity"):
    if type(x) is np.ndarray:
        return LedoitWolf2002_matrix(x, significant_level, Sigma = "identity")
    else:
        return LedoitWolf2002_covariance(x, significant_level, Sigma = "identity")

def LedoitWolf2002_covariance(x, significant_level=0.05, Sigma = "identity"):
    p = x.shape[1]
    n = x.shape[0] + 1
    S = x

    if Sigma == "identity":
        S_ = x
    else:
        svCov_u, svCov_d, _ = svd(x)
        sv_u, sv_d, _ = svd(Sigma)
        x_ = svCov_u @ np.diag(np.sqrt(svCov_d)) @ solve(sv_u @ np.diag(np.sqrt(sv_d)), np.eye(p))
        S_ = x_.T @ x_

    statistic = LedoitWolf2002_stat(n, p, S_)

    parameter = p * (p + 1) / 2



    p_value = 1 - chi2.cdf(statistic, parameter)
    # null_value = (p_value>significant_level)
    # result = {"statistic": statistic,
    #           "parameter": parameter,
    #           "p.value": p_value,
    #           "estimate": S,
    #           "null.value": null_value,
    #           "alternative": "two.sided",
    #           "method": "Ledoit and Wolf 2002 Test of Covariance Matrix Structure"}

    return p_value

def LedoitWolf2002_matrix(x, significant_level=0.05, Sigma = "identity"):
    p = x.shape[1]
    n = x.shape[0]
    S = np.cov(x.T)

    if Sigma == "identity":
        S_ = S
    else:
        sv_u, sv_d, _ = svd(Sigma)
        svDf_u, svDf_d, _ = svd(S)
        x_ = svDf_u @ np.diag(np.sqrt(sv_d)) @ solve(sv_u @ np.diag(np.sqrt(sv_d)), np.eye(p))
        S_ = x_.T @ x_

    statistic = LedoitWolf2002_stat(n, p, S_)

    parameter = p * (p + 1) / 2


    p_value = 1 - chi2.cdf(statistic, parameter)
    # null_value = (p_value>significant_level)
    # result = {"statistic": statistic,
    #           "parameter": parameter,
    #           "p.value": p_value,
    #           "estimate": S,
    #           "null.value": null_value,
    #           "alternative": "two.sided",
    #           "method": "Ledoit and Wolf 2002 Test of Covariance Matrix Structure"}

    return p_value

def LedoitWolf2002_stat(n, p, S_):
    mid = S_ - np.eye(p)
    statistic = n * p * (np.trace(mid @ mid) / p - (p / n) * ((np.trace(S_) / p) ** 2) + (p / n)) / 2
    return statistic





from scipy.stats import norm
from numpy.linalg import svd, solve
import numpy as np

def Fisher2012(x, significant_level=0.05, Sigma = "identity"):
    if type(x) is np.ndarray:
        return Fisher2012_matrix(x, significant_level, Sigma = "identity")
    else:
        return Fisher2012_covariance(x, significant_level, Sigma = "identity")

def Fisher2012_covariance(x, significant_level, Sigma = "identity"):
    p = x.shape[1]
    n = x.shape[0] + 1
    S = x

    if Sigma == "identity":
        S_ = x
    else:
        svCov_u, svCov_d, _ = svd(x)
        sv_u, sv_d, _ = svd(Sigma)
        x_ = svCov_u @ np.diag(np.sqrt(svCov_d)) @ solve(sv_u @ np.diag(np.sqrt(sv_d)), np.eye(p))
        S_ = x_.T @ x_

    statistic = Fisher2012_stat(n - 1, p, S_)

    parameter = [0, 1]



    p_value = 1 - norm.cdf(abs(statistic))
    # null_value = (p_value>significant_level)
    # result = {"statistic": statistic,
    #           "parameter": parameter,
    #           "p.value": p_value,
    #           "estimate": S if S.shape[0] <= 5 else None,
    #           "null.value": null_value,
    #           "alternative": "two.sided",
    #           "method": "Fisher 2012 Test of Covariance Matrix Structure"}

    return p_value

def Fisher2012_matrix(x, significant_level=0.05, Sigma = "identity"):
    p = x.shape[1]
    n = x.shape[0]
    S = np.cov(x.T)

    if Sigma == "identity":
        S_ = S
    else:
        sv_u, sv_d, _ = svd(Sigma)
        svDf_u, svDf_d, _ = svd(S)
        x_ = svDf_u @ np.diag(np.sqrt(sv_d)) @ solve(sv_u @ np.diag(np.sqrt(sv_d)), np.eye(p))
        S_ = x_.T @ x_

    statistic = Fisher2012_stat(n - 1, p, S_)

    parameter = [0, 1]



    p_value = 1 - norm.cdf(abs(statistic))
    # null_value = (p_value>significant_level)
    # result = {"statistic": statistic,
    #           "parameter": parameter,
    #           "p.value": p_value,
    #           "estimate": S if S.shape[0] <= 5 else None,
    #           "null.value": null_value,
    #           "alternative": "two.sided",
    #           "method": "Fisher 2012 Test of Covariance Matrix Structure"}

    return p_value

def Fisher2012_stat(n, p, S_):
    c = p / n
    ahat2 = (n ** 2 / ((n - 1) * (n + 2) * p)) * (np.trace(S_ @ S_) - (np.trace(S_) ** 2) / n)
    gamma = (n ** 5 * (n ** 2 + n + 2)) / ((n + 1) * (n + 2) * (n + 4) * (n + 6) * (n - 1) * (n - 2) * (n - 3))
    ahat4 = (gamma / p) * (np.trace(S_ @ S_ @ S_ @ S_) -(4 / n) * np.trace(S_ @ S_ @ S_) *np.trace(S_) -((2 * n ** 2 + 3 * n - 6) / (n * (n ** 2 + n + 2))) *
            (np.trace(S_ @ S_) ** 2) +((2 * (5 * n + 6)) / (n * (n ** 2 + n + 2))) *np.trace(S_ @ S_) * (np.trace(S_) ** 2) -((5 * n + 6) / ((n ** 2) * (n ** 2 + n + 2))) *(np.trace(S_) ** 4))
    return (n / np.sqrt(8 * (c ** 2 + 12 * c + 8))) * (ahat4 - 2 * ahat2 + 1)




import numpy as np
from scipy.stats import norm
from numpy.linalg import svd

def Ahmad2015(x, significant_level=0.05, Sigma = "identity"):
    if isinstance(x, np.ndarray):
        if len(x.shape) == 1 or x.shape[0] == 1 or x.shape[1] == 1:
            raise ValueError("Input must be a matrix or a 2D array.")
        elif x.shape[0] != x.shape[1]:  # this is a matrix
            return Ahmad2015_matrix(x, significant_level, Sigma = "identity")
        else:  # this is a covariance matrix
            return Ahmad2015_covariance(x, significant_level, Sigma = "identity")
    else:
        raise TypeError("Input must be a matrix or a 2D array.")

def Ahmad2015_covariance(x, significant_level=0.05, Sigma = "identity"):
    p = x.shape[1]
    n = x.shape[0] + 1
    S = x

    if Sigma == "identity":
        svCov = svd(x)
        x_ = svCov[0] @ np.diag(np.sqrt(svCov[1]))
    else:
        svCov = svd(x)
        sv = svd(Sigma)
        x_ = svCov[0] @ np.diag(np.sqrt(svCov[1])) @ np.linalg.inv(sv[0] @ np.diag(np.sqrt(sv[1])))

    # statistic = Ahmad2015Stat(x_)
    statistic = Ahmad2015Stat(x_)  # You need to provide the Ahmad2015Stat function

    parameter = [0, 4 * (2 / (p / n + 1))]



    p_value = 1 - norm.cdf(abs(statistic), 0, 4 * (2 / (p / n + 1)))
    # null_value = (p_value>significant_level)
    # estimate = S if S.shape[0] <= 5 else None

    # result= {
    #     "statistic": statistic,
    #     "parameter": parameter,
    #     "p_value": p_value,
    #     "estimate": estimate,
    #     "null_value": null_value,
    #     "alternative": "two.sided",
    #     "method": "Ahmad and Rosen 2015 Test of Covariance Matrix Structure"
    # }
    return p_value

def Ahmad2015_matrix(x, significant_level=0.05, Sigma = "identity"):
    p = x.shape[1]
    n = x.shape[0]
    S = np.cov(x, rowvar=False)

    if Sigma == "identity":
        x_ = x
    else:
        sv = svd(Sigma)
        x_ = x @ np.linalg.inv(sv[0] @ np.diag(np.sqrt(sv[1])))

    # statistic = Ahmad2015Stat(x_)
    statistic = Ahmad2015Stat(x_)  # You need to provide the Ahmad2015Stat function

    parameter = [0, 4 * (2 / (p / n + 1))]


    p_value = 1 - norm.cdf(abs(statistic), 0, 4 * (2 / (p / n + 1)))
    # null_value = (p_value>significant_level)
    # estimate = S if S.shape[0] <= 5 else None

    # result={
    #     "statistic": statistic,
    #     "parameter": parameter,
    #     "p_value": p_value,
    #     "estimate": estimate,
    #     "null_value": null_value,
    #     "alternative": "two.sided",
    #     "method": "Ahmad and Rosen 2015 Test of Covariance Matrix Structure"
    # }
    return p_value



def Ahmad2015Stat(x):
    ncol = x.shape[1]
    nrow = x.shape[0]
    c3num = 0
    c1num = 0

    for i in range(nrow):
        c1num += np.sum(x[i, :] ** 2)
        for j in range(i + 1, nrow):
            c3num += np.sum(x[i, :] * x[j, :]) ** 2

    c3 = c3num * 2.0 / (nrow * (nrow - 1.0))

    c1 = c1num / nrow

    return nrow * (c3 / ncol - 2.0 * c1 / ncol + 1.0)

One-Sample Hypothesis Testing: Simulated Data

Figure 1:

This part shows the comparison of

• the histogram of nonzero eigenvalues of $Q$ (or $\mathcal{Q}$)

• the theoretical density curve (generated from solving the equation for Stieltjes transform $m(z)$ where $z=E+i\eta$ then taking the imaginary part $\eta$ ↓ $0^+$)

after excluding the zero eigenvalues.

The histogram of nonzero eigenvalues is straightforward.

To calculate the theoretical density curve we adopt the following scheme.

1. Stieltjes Transform: The Stieltjes transform of a measure $\mu$ at a point $z$ in the complex plane is given by: $$ S_\rho(z)=\int \frac{1}{x-z} d \rho(x) $$ where $z$ is a complex number with a positive imaginary part.
2. With the Lemma 2.3 of our paper, the Stieltjes Transform can be obtained via solving the following equation $$ \frac{1}{m}=-z+\int \frac{\phi}{\phi^{1 / 2} x^{-1}+m} \pi(\mathrm{d} x) $$ and talking the branch which maps $\mathbb{C}_+$ to $\mathbb{C}_+$.
3. Obtaining the Density Curve: The density of the limiting spectral distribution (LSD) can be obtained from the imaginary part of the Stieltjes transform as: $$ \rho(x)=\frac{1}{\pi} \lim _{y \rightarrow 0^{+}} \operatorname{Im}\left(S_\rho(x+i y)\right) $$ where $y$ is a small positive number.

The default parameter is $n=800$, $\phi=0.6$ (Fig. 1(a)) vs $\phi=100$ (Fig. 1(b)). You may modify the phi parameter in the blocks below to see results for other phi's.

Figure 1(a)

Figure 1(a) shows the density curve of $\phi=0.6$, $n=800$.

## $\Sigma=\operatorname{Tr}\{a,\ldots,a,b,\ldots,b\}$ where they are a repeated (s*m) times and b repeated (1-s)*m times in the diagonal.

s = 1/2
a = 1
b = 15

## m=n*phi, R=number of repeating
phi=0.6
R=20
n=800

plot_eigenvalues_histogram(phi, R, n, a, b, s)

Figure 1(b)

Figure 1(b) shows the histogram and the density curve of lsd for $\phi=100$, $n=800$.

## $\Sigma=\operatorname{Tr}\{a,\ldots,a,b,\ldots,b\}$ where they are a repeated (s*m) times and b repeated (1-s)*m times in the diagonal.

s = 1/2
a = 1
b = 15

## m=n*phi, R=number of repeating
phi=100
R=20
n=800

plot_eigenvalues_histogram(phi, R, n, a, b, s)

Figure 2:

In Fig. 2, we show the ECDF for

• the Global test statistics $\mathcal{T}^g_1$
• the Local test statistics $\mathcal{T}^l_1$

when $\Sigma=\operatorname{Id}$ and the entries of the matrix $X$ have $\kappa_4 =0$ vs $\kappa_4\ne 0$ respectively.

It becomes visually clear that for Global test statistics, we can see the distributional difference when the entries of $X$ have $\kappa=0$ (Gaussian) or $\kappa\ne 0$ (2PN). The "2PN" here indicates the discrete random variable which takes value $\sqrt{2}$ with probability $1/3$ and value $-1/\sqrt{2}$ with probability $2/3$.

While for the Local test statistics, it's visually almost non-distinguishable that which one is from $X$ with Gaussian entries while which one is from $X$ with 2PN entries.

Figure 2(a) Global

Data generating

Generate the data for the Global statistics

%%time
import jax
import jax.numpy as jnp
from jax import grad, jit
from jax.random import normal, choice, split, PRNGKey
import numpy as np
import matplotlib.pyplot as plt

def generate_matrix_normal(key, m, n):
    return normal(key, shape=(m, n)) / jnp.sqrt(jnp.sqrt(n * m))


def generate_matrix_choice(key, m, n, values, probabilities):
    return choice(key, a=values, shape=(m, n), p=probabilities) / jnp.sqrt(jnp.sqrt(n * m))

def compute_global_test_statistic(X, center):
    H = jnp.matmul(jnp.transpose(X), X)
    E0 = jnp.linalg.eigh(H)[0]
    Eb = E0 - center
    return jnp.sum(Eb)

def simulate_global_test_statistic(phi, n, R, c, method="normal", values=None, probabilities=None):
    m = int(n * phi)
    center = (jnp.sqrt(phi) + 1/jnp.sqrt(phi) - c) * jnp.ones(n)
    T = jnp.zeros(R)
    key = PRNGKey(0)

    for k in range(R):
        key, subkey = split(key)
        if method == "normal":
            X = generate_matrix_normal(subkey, m, n)
        elif method == "choice":
            X = generate_matrix_choice(subkey, m, n, values, probabilities)
        else:
            raise ValueError("Invalid matrix generation method.")
        T = T.at[k].set(compute_global_test_statistic(X, center))

    return np.array(T)
values = jnp.array([np.sqrt(2), -1/np.sqrt(2)])
probabilities = jnp.array([1/3, 2/3])
c=3
phi=100
n=200
m = int(n*phi)
R = 2000

TestST1 = simulate_global_test_statistic(phi, n, R, c, "normal")
TestST2 = simulate_global_test_statistic(phi, n, R, c, "choice", values, probabilities)


# plot_ecdf2(TestST1, TestST2, r'$\mathbf{\kappa_4\ =\ 0}$', r'$\mathbf{\kappa_4\ \neq\ 0}$')

CPU times: user 24.7 s, sys: 2.16 s, total: 26.9 s
Wall time: 23.8 s

Saved data

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579.9866 , 580.327 , 579.8214 , 580.23926 , 579.27856 , 578.8534 , 580.2618 , 579.03424 , 579.85913 , 580.25433 , 580.0444 , 578.98303 , 580.0833 , 580.15466 , 580.37134 , 580.27905 , 579.7601 , 578.31635 , 580.4238 , 579.36865 , 579.02655 , 579.4999 , 580.916 , 580.2874 , 579.8036 , 579.7068 , 579.754 , 579.7796 , 580.27075 , 580.2521 , 580.0706 , 579.7158 , 580.45764 , 580.44324 , 579.6002 , 579.9048 , 581.5736 , 580.8018 , 579.4031 , 579.7106 , 579.775 , 580.1007 , 579.73694 , 580.22345 , 579.97595 , 579.2726 , 579.49304 , 580.8049 , 579.27045 , 579.1908 , 580.38477 , 580.1537 , 580.07056 , 579.8779 , 581.1715 , 580.7793 , 580.0309 , 579.40094 , 580.1883 , 580.2438 , 580.3324 , 579.5298 , 579.83435 , 579.916 , 579.0687 , 578.9441 , 580.6271 , 579.4316 , 580.3601 , 579.9895 , 580.6315 , 580.29333 , 579.1196 , 581.2459 , 579.8155 , 579.11646 , 580.23926 , 579.676 , 580.02844 , 579.5334 , 578.8496 , 579.1976 , 579.78186 , 580.86414 , 580.55054 , 579.68665 , 580.97577 , 581.4356 , 580.12683 , 580.3534 , 581.4716 , 581.0133 , 579.7667 , 579.70386 , 581.1079 , 580.86487 , 580.31805 , 579.7571 , 578.7722 , 580.1322 , 579.7699 , 579.7743 , 578.72205 , 579.85986 , 579.93274 , 579.33185 , 578.8962 , 579.9701 , 578.7521 , 580.7606 , 578.66956 , 580.3849 , 579.96924 , 578.8279 , 579.2937 , 580.6841 , 580.0061 , 578.9666 , 580.34955 , 580.40076 , 579.97003 , 579.56055 , 579.1819 , 579.51337 , 579.72644 , 579.58673 , 578.56244 , 579.85547 , 580.0001 , 580.37897 , 579.2696 , 580.1104 , 580.3969 , 579.403 , 579.7916 , 580.6293 , 579.8667 , 579.46606 , 580.17633 , 580.18896 , 579.9918 , 581.2556 , 580.15906 , 579.6663 , 580.31195 , 579.3311 , 579.8507 , 579.67834 , 580.4358 , 578.9193 , 580.9331 , 580.52954 , 578.96954 , 581.2872 , 580.17633 , 579.4856 , 578.8346 , 580.4103 , 581.24133 , 581.3456 , 579.08203 , 579.87775 , 579.6835 , 580.18756 , 578.63806 , 580.8362 , 579.7143 , 579.9469 , 579.83136 , 580.53345 , 579.7849 , 579.8974 , 580.01587 , 578.9089 , 580.44336 , 579.8554 , 580.1644 , 580.0564 , 579.31836 , 580.5574 , 579.90405 , 579.6454 , 580.2483 , 578.8054 , 578.97107 , 578.4258 , 580.2319 , 578.4807 , 579.7968 , 579.9115 , 579.27716 , 580.42163 , 579.7982 , 578.8338 , 580.5289 , 578.81665 , 580.7334 , 579.43994 , 580.40204 , 581.30426 , 579.98584 , 581.4303 , 578.8142 , 579.33563 , 578.7528 , 580.5963 , 579.9409 , 579.58984 , 578.14294 , 580.45306 , 579.5995 , 580.83936 , 580.7793 , 579.2568 , 580.558 , 579.18713 , 579.58545 , 579.9873 , 581.1709 , 579.5125 , 578.55853 , 580.3429 , 580.24304 , 579.5576 , 579.70386 , 579.07166 , 579.0175 , 579.94305 , 580.63464 , 579.6971 , 580.2295 , 580.1328 , 580.15686 , 580.0872 , 580.8273 , 579.45483 , 580.0383 , 580.7208 , 580.2019 , 580.7081 , 579.42114 , 579.3432 , 579.51636 , 578.77905 , 580.39307 , 579.0027 , 580.15826 , 579.7046 , 579.56573 , 580.5041 , 579.2914 , 580.71857 , 580.4688 , 578.4551 , 579.98425 , 579.8801 , 579.79236 , 579.5575 , 579.29504 , 581.33356 , 579.1429 , 580.39246 , 579.85455 , 580.6811 , 580.6345 , 580.4449 , 581.3215 , 580.65027 , 580.8318 , 578.0546 , 579.66187 , 581.1227 , 578.83826 , 580.09753 , 580.42664 , 579.1173 , 580.45605 , 579.7939 , 579.73376 , 579.52765 , 580.6977 , 580.0331 , 580.32715 , 580.26483 , 579.3424 , 579.5989 , 579.4601 , 579.844 , 580.0226 , 579.61615 , 579.65283 , 579.93195 , 579.60626 , 579.9342 , 580.0249 , 580.08636 , 581.1034 , 579.5073 , 580.15466 , 579.39105 , 580.9885 , 579.72473 , 579.5149 , 579.0785 , 580.12537 , 581.0696 , 579.4039 , 579.57544 , 579.4113 , 580.0189 , 579.8598 , 581.3103 , 580.44855 , 580.2993 , 580.603 , 579.7046 , 578.9838 , 580.8446 , 580.2732 , 579.8523 , 580.072 , 580.0609 , 580.25134 , 579.09863 , 580.9715 , 581.03656 , 580.4877 , 580.71564 , 580.2964 , 579.8913 , 579.6325 , 581.41455 , 579.622 , 579.6843 , 580.0571 , 579.63873 , 579.58453 , 579.1338 , 579.0918 , 579.511 , 580.8461 , 580.8874 , 579.181 , 578.8751 , 580.3016 , 579.4098 , 579.79675 , 580.0128 , 580.4028 , 580.5512 , 580.3286 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Plot

Now we plot the two ECDFs of the Global test statistics $\mathcal{T}_1^g$ for $\kappa_4=0$ and $\kappa_4\neq 0$ respectively.

Here we used a line plot instead of the dot plot we showed in our preprint. If you want to see the dot plot version, uncomment the dot version plot_ecdf2 function in the section Necessary Functions.

plot_ecdf2(TestST1, TestST2, r'$\mathbf{\kappa_4\ =\ 0}$', r'$\mathbf{\kappa_4\ \neq\ 0}$')

Figure 2(b) Local

Discussion: it suffices to consider only sum of functions of eigenvalues within an interval when it comes to Local Statistics.

In our theorem, we introduce the the factor $\mathcal{K}(x)$ which is a mollifier to ensure that the test functions $g(x)=h(x)\mathcal{K}(x)$ will satisfy the assumption that $g\in\mathcal{C}^2_c(\mathbb{R})$. $$\mathcal{K}(x) \equiv \mathcal{K}_{a, b}(x):= \begin{cases}0 & |x| \geq a+b \\ 1 & |x| \leq b \\ \exp \left(\frac{1}{a^2}-\frac{1}{a^2-(x+b)^2}\right) & -(b+a)<x<-b \\ \exp \left(\frac{1}{a^2}-\frac{1}{a^2-(x-b)^2}\right) & b<x<b+a\end{cases}.$$

For simplicity of implementation later, we simplify the discussion of local statistics to omit $\mathcal{K}(x)$. That is, we will directly use $h(x)$ in the implementation of local statistics later. In order to show that there is little harm in doing this. We can do two simple realizations of $T_1^l$ with $h(x)$ and $h(x)\mathcal{K}(x)$ respectively. Then show that the distributions of the realized statistics are extremely close when the tuning parameters $a,b$ in $\mathcal{K}$ are properly chosen.

Below we show that there is little difference using test function $$\frac{x-E}{\eta}1_{x-E\in[-\eta,\eta]}$$ vs $$\frac{x-E}{\eta}\mathcal{K}_{0.05,1}\left(\frac{x-E}{\eta}\right)$$

import numpy as np

def K(x, a, b):
    if x >= a+b:
        return 0
    elif x<-b-a:
        return 0
    elif abs(x) <= b:
        return 1
    elif -(b + a) < x < -b:
        return np.exp(1 / a**2 - 1 / (a**2 - (x + b)**2))
    elif b < x < b + a:
        return np.exp(1 / a**2 - 1 / (a**2 - (x - b)**2))
    else:
        return 0

def Testfunc1(x, E, eta):
    a = 0.05 * eta
    b = eta
    return (x-E)/eta*K(x - E, a, b)


# Vectorize the Testfunc
vTestfunc1 = np.vectorize(Testfunc1)

%%time
import jax
import jax.numpy as jnp
from jax import grad, jit, random
from jax.random import normal, choice, split, PRNGKey
import numpy as np
import matplotlib.pyplot as plt

# Constants
t = 0.000
n = 800
R0 = 1000
c = 1
phi = 100
m = int(n * phi)
r = jnp.sqrt(phi)

### you may choose between left edge leftb, right edge rightb, middle point midb and any other points in the spectra
leftb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi) - 2)
midb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi))
rightb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi) + 2)


eta1 = 1/(n**(1/4))

## Parameter for 2PN
values = jnp.array([np.sqrt(2), -1/np.sqrt(2)])
probabilities = jnp.array([1/3, 2/3])



def generate_matrix_normal(key, m, n):
    return normal(key, shape=(m, n)) / jnp.sqrt(jnp.sqrt(n * m))


def generate_matrix_choice(key, m, n, values, probabilities):
    return choice(key, a=values, shape=(m, n), p=probabilities) / jnp.sqrt(jnp.sqrt(n * m))

def compute_local_test_statistic(E0, midb, eta1):
    Et = jnp.array(extract_elements(E0, midb-eta1, midb+eta1))
    Eb = (Et - midb * jnp.ones(Et.shape)) / eta1
    return jnp.sum(Eb)

def simulate_local_test_statistics_normal(R0):
    T0 = jnp.zeros(R0)
    key = PRNGKey(0)
    for r in range(R0):
        key, subkey = split(key)
        X = generate_matrix_normal(subkey, m, n)
        H = jnp.matmul(X.T, X)
        E00, _ = jnp.linalg.eigh(H)
        E0 = np.array(E00)
        t0 = compute_local_test_statistic(E0, midb, eta1)
        T0 = T0.at[r].set(t0)
    return np.array(T0)

def simulate_local_test_statistics_choice(R0):
    T0 = jnp.zeros(R0)
    key = PRNGKey(0)
    for r in range(R0):
        key, subkey = split(key)
        X=generate_matrix_choice(key, m, n, values, probabilities)
        H = jnp.matmul(X.T, X)
        E00, _ = jnp.linalg.eigh(H)
        E0 = np.array(E00)
        t0 = compute_local_test_statistic(E0, midb, eta1)
        T0 = T0.at[r].set(t0)
    return np.array(T0)



# Simulate test statistics for normal and choice methods
TestST1 = simulate_local_test_statistics_normal(R0)
Mean_x1 = jnp.mean(TestST1)
Variance_x1 = jnp.var(TestST1)
print(Mean_x1)
print(Variance_x1)
Normalize_TestST_1 = (TestST1 - Mean_x1) / np.sqrt(Variance_x1)

TestST2 = simulate_local_test_statistics_choice(R0)
Mean_x2 = jnp.mean(TestST2)
Variance_x2 = jnp.var(TestST2)
print(Mean_x2)
print(Variance_x2)
Normalize_TestST_2 = (TestST2 - Mean_x1) / np.sqrt(Variance_x1)

-0.57790804
1.154052
-0.58190227
1.1411626
CPU times: user 2min 29s, sys: 2.89 s, total: 2min 32s
Wall time: 2min 29s

%%time
import jax
import jax.numpy as jnp
from jax import grad, jit, random
from jax.random import normal, choice, split, PRNGKey
import numpy as np
import matplotlib.pyplot as plt

# Constants
t = 0.000
n = 800
R0 = 1000
c = 1
phi = 100
m = int(n * phi)
r = jnp.sqrt(phi)

### you may choose between left edge leftb, right edge rightb, middle point midb and any other points in the spectra
leftb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi) - 2)
midb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi))
rightb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi) + 2)


eta1 = 1/(n**(1/4))

## Parameter for 2PN
values = jnp.array([np.sqrt(2), -1/np.sqrt(2)])
probabilities = jnp.array([1/3, 2/3])



def generate_matrix_normal(key, m, n):
    return normal(key, shape=(m, n)) / jnp.sqrt(jnp.sqrt(n * m))


def generate_matrix_choice(key, m, n, values, probabilities):
    return choice(key, a=values, shape=(m, n), p=probabilities) / jnp.sqrt(jnp.sqrt(n * m))

def compute_local_test_statistic_K(E0, midb, eta1):
    Eb =vTestfunc1(E0,midb,eta1)
    return jnp.sum(Eb)

def simulate_local_test_statistics_normal(R0):
    T0 = jnp.zeros(R0)
    key = PRNGKey(0)
    for r in range(R0):
        key, subkey = split(key)
        X = generate_matrix_normal(subkey, m, n)
        H = jnp.matmul(X.T, X)
        E00, _ = jnp.linalg.eigh(H)
        E0 = np.array(E00)
        t0 = compute_local_test_statistic_K(E0, midb, eta1)
        T0 = T0.at[r].set(t0)
    return np.array(T0)

def simulate_local_test_statistics_choice(R0):
    T0 = jnp.zeros(R0)
    key = PRNGKey(0)
    for r in range(R0):
        key, subkey = split(key)
        X=generate_matrix_choice(key, m, n, values, probabilities)
        H = jnp.matmul(X.T, X)
        E00, _ = jnp.linalg.eigh(H)
        E0 = np.array(E00)
        t0 = compute_local_test_statistic_K(E0, midb, eta1)
        T0 = T0.at[r].set(t0)
    return np.array(T0)



# Simulate test statistics for normal and choice methods
TestST1 = simulate_local_test_statistics_normal(R0)
Mean_x1 = jnp.mean(TestST1)
Variance_x1 = jnp.var(TestST1)
print(Mean_x1)
print(Variance_x1)
Normalize_TestST_1 = (TestST1 - Mean_x1) / np.sqrt(Variance_x1)

TestST2 = simulate_local_test_statistics_choice(R0)
Mean_x2 = jnp.mean(TestST2)
Variance_x2 = jnp.var(TestST2)
print(Mean_x2)
print(Variance_x2)
Normalize_TestST_2 = (TestST2 - Mean_x1) / np.sqrt(Variance_x1)

-0.57521427
1.1631669
-0.58357704
1.1412829
CPU times: user 2min 27s, sys: 1.87 s, total: 2min 29s
Wall time: 2min 26s

We can see from the output that the difference is rather minor. Therefore, we will use out simplified realizations in the sections after.

Data generating

Generate the data for the local statistics with gaussian and 2PN entries.

%%time
import jax
import jax.numpy as jnp
from jax import grad, jit, random
from jax.random import normal, choice, split, PRNGKey
import numpy as np
import matplotlib.pyplot as plt

# Constants
t = 0.000
n = 800
R0 = 1000
c = 1
phi = 100
m = int(n * phi)
r = jnp.sqrt(phi)

### you may choose between left edge leftb, right edge rightb, middle point midb and any other points in the spectra
leftb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi) - 2)
midb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi))
rightb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi) + 2)


eta1 = 1/(n**(1/4))

## Parameter for 2PN
values = jnp.array([np.sqrt(2), -1/np.sqrt(2)])
probabilities = jnp.array([1/3, 2/3])



def generate_matrix_normal(key, m, n):
    return normal(key, shape=(m, n)) / jnp.sqrt(jnp.sqrt(n * m))


def generate_matrix_choice(key, m, n, values, probabilities):
    return choice(key, a=values, shape=(m, n), p=probabilities) / jnp.sqrt(jnp.sqrt(n * m))

def compute_local_test_statistic(E0, midb, eta1):
    Et = jnp.array(extract_elements(E0, midb-eta1, midb+eta1))
    Eb = (Et - midb * jnp.ones(Et.shape)) / eta1
    return jnp.sum(Eb)

def simulate_local_test_statistics_normal(R0):
    T0 = jnp.zeros(R0)
    key = PRNGKey(0)
    for r in range(R0):
        key, subkey = split(key)
        X = generate_matrix_normal(subkey, m, n)
        H = jnp.matmul(X.T, X)
        E00, _ = jnp.linalg.eigh(H)
        E0 = np.array(E00)
        t0 = compute_local_test_statistic(E0, midb, eta1)
        T0 = T0.at[r].set(t0)
    return np.array(T0)

def simulate_local_test_statistics_choice(R0):
    T0 = jnp.zeros(R0)
    key = PRNGKey(0)
    for r in range(R0):
        key, subkey = split(key)
        X=generate_matrix_choice(key, m, n, values, probabilities)
        H = jnp.matmul(X.T, X)
        E00, _ = jnp.linalg.eigh(H)
        E0 = np.array(E00)
        t0 = compute_local_test_statistic(E0, midb, eta1)
        T0 = T0.at[r].set(t0)
    return np.array(T0)



# Simulate test statistics for normal and choice methods
TestST1 = simulate_local_test_statistics_normal(R0)
Mean_x1 = jnp.mean(TestST1)
Variance_x1 = jnp.var(TestST1)
print(Mean_x1)
print(Variance_x1)
Normalize_TestST_1 = (TestST1 - Mean_x1) / np.sqrt(Variance_x1)

TestST2 = simulate_local_test_statistics_choice(R0)
Mean_x2 = jnp.mean(TestST2)
Variance_x2 = jnp.var(TestST2)
print(Mean_x2)
print(Variance_x2)
Normalize_TestST_2 = (TestST2 - Mean_x1) / np.sqrt(Variance_x1)

-0.57790804
1.154052
-0.58190227
1.1411626
CPU times: user 2min 32s, sys: 3.34 s, total: 2min 35s
Wall time: 2min 32s

Saved data

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Plot

Similarly, to show the closeness to the ECDF of N(0,1) visually, we normalize both statistics (with mean and variance of the case $\kappa_4=0$) so that it serves the purpose of comparison while retain a relatively standard x-coordinates.

plot_ecdf2(TestST1, TestST2, r'$\mathbf{\kappa_4\ =\ 0}$', r'$\mathbf{\kappa_4\ \neq\ 0}$')

Figure 3: ECDF different $\phi$, Global

In this section, we display the comparison of ECDFs of normalized test statistics $\mathcal{T}^g_1$ (and $\mathcal{T}^l_1$, respectively) versus the CDF of standard normal distribution. Here the normalizations are done with our theoretical value for the Global statistics and with calibration for the Local statistics).

For quicker execution

• I changed the number of repeat $R$ from 10000 (the version in paper) to 2000 so that saving via print will be easier and less time will be needed (less than 1 min with a T4 gpu machine each).

• However, the plot could be less smooth than the ones in the paper, to replicate, change the value of $R$ to 10000; to get more smooth curves, change $R$ and $n$ to larger quantities.

$T_1^g$

In practice, $x-\sqrt{\phi}-1/\sqrt{\phi}+c$ is used where $c=3$.

Data generating

%%time
import jax
import jax.numpy as jnp
from jax import grad, jit
from jax.random import normal
import numpy as np

R = 2000
n = 200
c = 3

def simulate_global_test_statistics1(phi, n, R, c):
    m = int(n * phi)
    center = (jnp.sqrt(phi) + 1/jnp.sqrt(phi) - c) * jnp.ones(n)

    @jit
    def objective1(X):
        H = jnp.matmul(jnp.transpose(X), X)
        E0 = jnp.linalg.eigh(H)[0]
        Eb = E0 - center
        return jnp.sum(Eb)

    T = jnp.zeros(R)
    key = jax.random.PRNGKey(0)

    for k in range(R):
        key, subkey = jax.random.split(key)
        X = normal(subkey, shape=(m, n)) / jnp.sqrt(jnp.sqrt(n * m))
        T = T.at[k].set(objective1(X))

    return np.array(T)



TestST1 = simulate_global_test_statistics1(50,  n, R, 3)
TestST2 = simulate_global_test_statistics1(100, n, R, 3)
TestST3 = simulate_global_test_statistics1(150, n, R, 3)
TestST4 = simulate_global_test_statistics1(200, n, R, 3)

CPU times: user 32.4 s, sys: 1.75 s, total: 34.2 s
Wall time: 31.3 s

Saved data

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TestST2=[581.8905 , 577.13184 , 582.865 , 579.22266 , 581.2446 , 579.2334 , 578.31226 , 581.1064 , 579.64465 , 582.0055 , 578.58167 , 578.6935 , 582.4458 , 578.7346 , 580.98206 , 578.9697 , 579.5271 , 579.5486 , 580.0008 , 582.287 , 580.4851 , 577.51764 , 581.4924 , 581.0149 , 580.6128 , 580.0384 , 579.024 , 579.5055 , 580.9938 , 580.35754 , 581.33435 , 580.0991 , 581.3005 , 579.0812 , 579.06055 , 579.3107 , 580.10016 , 578.7818 , 581.82166 , 579.0411 , 577.3502 , 581.6246 , 579.0169 , 578.18335 , 579.3432 , 581.1167 , 580.1179 , 578.0752 , 578.25574 , 581.2135 , 579.3932 , 581.8179 , 578.5685 , 578.463 , 579.62744 , 581.6946 , 580.8406 , 580.6653 , 580.27344 , 578.94763 , 580.8428 , 580.3493 , 581.214 , 577.2184 , 580.10657 , 579.65015 , 579.41754 , 579.3157 , 580.1062 , 581.9437 , 580.5542 , 582.27747 , 580.6162 , 581.3917 , 580.86597 , 581.2231 , 579.6072 , 578.95325 , 578.5316 , 579.33997 , 581.1282 , 579.64825 , 578.9053 , 577.18494 , 579.82153 , 579.8206 , 580.614 , 581.4748 , 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Plot

R=2000
c=3
n=200
phi_values = [50, 100, 150, 200]
test_statistics = [TestST1, TestST2, TestST3, TestST4]

def normalize_test_statistics1(test_statistic, n, c, r):
    return (test_statistic - n * (c - 1/r) * np.ones(np.size(test_statistic))) / np.sqrt(2)


normalized_test_statistics = [normalize_test_statistics1(ts, n, c, np.sqrt(phi)) for ts, phi in zip(test_statistics, phi_values)]
labels = [f"$\mathbf{{\phi={phi}}}$" for phi in phi_values]

plot_ecdf_and_standard_normal_cdf(*normalized_test_statistics, labels=labels)

$T_2^g$

In practice, $(x-\sqrt{\phi}-1/\sqrt{\phi}+c)^2$ is used where, $c=3$

Data generating

Here we generate the test statistics with 2000 repetitions.

%%time
import jax
import jax.numpy as jnp
from jax import grad, jit
from jax.random import normal
import numpy as np

R = 2000
n = 200

def simulate_test_statistics2(phi, n, R, c):
    m = int(n * phi)
    center = (jnp.sqrt(phi) + 1/jnp.sqrt(phi) - c) * jnp.ones(n)
    pn = 2 * jnp.ones(n)

    @jit
    def objective(X):
        H = jnp.matmul(jnp.transpose(X), X)
        E0 = jnp.linalg.eigh(H)[0]
        Eb = E0 - center
        loE = jnp.power(Eb, pn)
        return jnp.sum(loE)

    T = jnp.zeros(R)
    key = jax.random.PRNGKey(0)

    for k in range(R):
        key, subkey = jax.random.split(key)
        X = normal(subkey, shape=(m, n)) / jnp.sqrt(jnp.sqrt(n * m))
        T = T.at[k].set(objective(X))

    return np.array(T)

TestST1 = simulate_test_statistics2(50,  n, R, 3)
TestST2 = simulate_test_statistics2(100, n, R, 3)
TestST3 = simulate_test_statistics2(150, n, R, 3)
TestST4 = simulate_test_statistics2(200, n, R, 3)

CPU times: user 33 s, sys: 1.79 s, total: 34.8 s
Wall time: 31.8 s

Saved data

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Plot

n = 200
c = 3
R = 2000

def normalize_test_statistics2(test_statistics, n, c, r):
    return (test_statistics - (n * (1 + c**2 - 2 * c / r + 1 /r/r) + 1) * np.ones(np.size(test_statistics))) / np.sqrt(4 + 8 * c**2)

phi_values = [50, 100, 150, 200]
test_statistics = [TestST1, TestST2, TestST3, TestST4]
normalized_test_statistics = [normalize_test_statistics2(ts, n, c, np.sqrt(phi)) for ts, phi in zip(test_statistics, phi_values)]
labels = [f"$\mathbf{{\phi={phi}}}$" for phi in phi_values]

plot_ecdf_and_standard_normal_cdf(*normalized_test_statistics, labels=labels)

$T_3^g$

In practice, $-\log(x-\sqrt{\phi}-1/\sqrt{\phi}+t+1/t) +(x-\sqrt{\phi}-1/\sqrt{\phi}+t+1/t)$ is used, where $t=3$

Data generating

%%time
import jax
import jax.numpy as jnp
from jax import grad, jit
from jax.random import normal
import numpy as np

R = 2000
n = 200
c = 3.3333333

def simulate_test_statistics3(phi, n, R, c):
    m = int(n * phi)
    center = (jnp.sqrt(phi) + 1/jnp.sqrt(phi) - c) * jnp.ones(n)

    @jit
    def objective(X):
        H = jnp.matmul(jnp.transpose(X), X)
        E0 = jnp.linalg.eigh(H)[0]
        Eb = E0 - center
        return -jnp.sum(jnp.log(Eb)) + jnp.sum(Eb)

    T = jnp.zeros(R)
    key = jax.random.PRNGKey(0)

    for k in range(R):
        key, subkey = jax.random.split(key)
        X = normal(subkey, shape=(m, n)) / jnp.sqrt(jnp.sqrt(n * m))
        T = T.at[k].set(objective(X))

    return np.array(T)

TestST1 = simulate_test_statistics3(50,  n, R, c)
TestST2 = simulate_test_statistics3(100, n, R, c)
TestST3 = simulate_test_statistics3(150, n, R, c)
TestST4 = simulate_test_statistics3(200, n, R, c)

CPU times: user 33.1 s, sys: 1.8 s, total: 34.9 s
Wall time: 31.9 s

Saved data

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Plot

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm
import scipy.integrate as integrate

# Functions for Stieltjes transform and its derivatives
def stieltjesp(z, phi, r):
    gammap = r + 1/r + 2
    gammam = r + 1/r - 2
    s = (r - 1/r - z + np.sqrt((z - gammam) * (gammap - z)) * 1j) / (2 * z/r)
    return s

def lsd(x, phi, r):
    return np.imag(stieltjesp(x + 0.00001j, phi, r)) / np.pi

def testf(x, phi, r, c):
    return np.log(x - r - 1/r + c)

def compute_nlsd(phi, r, c):
    lower_limit = r - 2.3
    upper_limit = r + 2.3
    result, _ = integrate.quad(lambda x: testf(x, phi, r, c) * lsd(x, phi, r), lower_limit, upper_limit)
    return result

def normalize_test_statistics3(TestST, nlsd, n, r, c):
    t1 = (c + np.sqrt(c**2 - 4)) / 2
    return (TestST - (-nlsd + n * (c - 1/r) - 0.5 * np.log(1 - 1/t1/t1)) * np.ones(len(TestST))) / np.sqrt(2 * (1 - 2/t1) + 2 * np.log(t1) - 2 * np.log(t1 - 1/t1))

# Parameters
phi_values = [50, 100, 150, 200]
r_values = [np.sqrt(phi) for phi in phi_values]
c = 10/3
n = 200  # Assuming n is 200 as it was not provided in the code


## Note that for $\log$ test functions, we don't have explicit formulas, therefore we also need to calculate $n\int\log \mathrm{d}\varrho$ numerically.
# Compute nlsd for each phi and r
nlsd_values = [n * compute_nlsd(phi, r, c) for phi, r in zip(phi_values, r_values)]
# for nlsd in nlsd_values:
#     print(nlsd)

# Assuming you have the data sets TestST1, TestST2, TestST3, and TestST4
normalized_data_sets = [normalize_test_statistics3(TestST, nlsd, n, r, c)
    for TestST, nlsd, r in zip([TestST1, TestST2, TestST3, TestST4], nlsd_values, r_values)
]

labels = [f"$\mathbf{{\phi={phi}}}$" for phi in phi_values]
plot_ecdf_and_standard_normal_cdf(*normalized_data_sets, labels=labels)

$T_4^g$

In practice, we use $n^2\frac{T_2^g}{(T_1^g)^2}$ where $T_1^g,T_2^g$ are the realizations in the above subsections.

Data generating

%%time
import jax
import jax.numpy as jnp
from jax import grad, jit
from jax.random import normal
import numpy as np

R = 2000
n = 200
c = 3
pn=2*jnp.ones(n)

def simulate_test_statistics4(phi, n, R, c):
    m = int(n * phi)
    center = (jnp.sqrt(phi) + 1/jnp.sqrt(phi) - c) * jnp.ones(n)

    @jit
    def objective(X):
        H = jnp.matmul(jnp.transpose(X), X)
        E0 = jnp.linalg.eigh(H)[0]
        Eb = E0 - center
        loE=jnp.power(Eb,pn)
        t = n*n*jnp.sum(loE)/jnp.power(jnp.sum(Eb),2)
        return t


    T = jnp.zeros(R)
    key = jax.random.PRNGKey(0)

    for k in range(R):
        key, subkey = jax.random.split(key)
        X = normal(subkey, shape=(m, n)) / jnp.sqrt(jnp.sqrt(n * m))
        T = T.at[k].set(objective(X))

    return np.array(T)

TestST1 = simulate_test_statistics4(50,  n, R, c)
TestST2 = simulate_test_statistics4(100, n, R, c)
TestST3 = simulate_test_statistics4(150, n, R, c)
TestST4 = simulate_test_statistics4(200, n, R, c)

CPU times: user 33.1 s, sys: 1.8 s, total: 34.9 s
Wall time: 32.1 s

Saved data

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223.2985 , 223.4681 , 223.71182 , 223.92883 , 223.21225 , 223.24608 , 223.77579 , 223.75687 , 223.29607 , 223.6511 , 223.32675 , 223.19318 , 223.40648 , 223.85796 , 223.22935 , 223.64964 , 223.92151 , 223.23936 , 223.45334 , 223.68774 , 223.77788 , 223.49756 , 223.51561 , 223.64769 , 222.89583 , 223.24634 , 223.32893 , 223.68571 , 223.68028 , 223.56061 , 223.2174 , 223.6075 , 223.53831 , 223.51701 , 222.87375 , 223.65578 , 223.34407 , 223.62704 , 223.55023 , 223.23357 , 222.83913 , 223.45055 , 223.45448 , 223.2655 , 223.49217 , 223.38132 , 223.66534 , 223.21379 , 223.61722 , 223.32623 , 223.07498 , 223.90146 , 223.08173 , 223.47794 , 223.01842 , 223.18033 , 223.42015 , 223.6286 , 223.3766 , 223.3767 , 223.31725 , 223.4009 , 223.02982 , 223.16583 , 223.82182 , 223.40941 , 223.99397 , 223.35497 , 223.74161 , 223.22638 , 223.76587 , 223.4144 , 223.43819 , 223.10196 , 223.0637 , 223.46812 , 222.88896 , 223.78085 , 223.06873 , 222.87074 , 223.5554 , 223.2738 , 222.7615 , 223.62143 , 223.42897 , 223.3527 , 223.98569 , 223.44739 , 223.09886 , 223.35054 , 223.76874 , 223.45111 , 223.53064 , 223.70477 , 223.64609 , 223.20207 , 223.83665 , 223.66336 , 222.701 , 223.44037 , 224.03445 , 223.87305 , 223.5135 , 223.30342 , 223.60657 , 223.77748 , 222.9121 , 223.75417 , 223.63397 , 223.15874 , 223.30844 , 223.91351 , 223.69322 , 223.27448 , 223.21785 , 223.43741 , 223.34859 , 223.27919 , 223.11552 , 223.46034 , 223.69963 , 223.56377 , 223.02155 , 223.49338 , 223.60509 , 223.57394 , 223.47476 , 223.21222 , 223.59195 , 223.59837 , 223.18713 , 224.38506 , 223.8252 , 223.5005 , 223.762 , 223.46573 , 223.2143 , 223.55203 , 223.35045 , 223.57147 , 223.57181 , 223.20653 , 223.18259 , 223.08209 , 223.06465 , 223.3842 , 222.98431 , 223.47806 , 223.53596 , 223.4826 , 223.40292 , 223.7696 , 223.6563 , 223.33649 , 223.49973 , 223.66515 , 223.41536 , 223.32574 , 223.53308 , 223.27109 , 223.88948 , 223.41235 , 223.28773 , 223.352 , 223.73602 , 223.94196 , 223.61801 , 223.75296 , 223.63936 , 223.65349 , 223.3688 , 223.92258 , 223.46529 , 223.32076 , 223.61043 , 223.35466 , 223.4344 , 223.50197 , 223.25705 , 223.5795 , 223.1511 , 223.534 , 222.92738 , 223.51456 , 223.12177 , 223.64426 , 223.32767 , 223.56084 , 223.81487 , 223.4986 , 223.16443 , 223.39386 , 223.31113 , 223.8015 , 223.23126 , 223.63965 , 222.73285 , 223.25262 , 223.18452 , 223.65103 , 223.4252 , 223.13907 , 223.46173 , 223.38582 , 223.5022 , 223.37888 , 223.23328 , 223.23888 , 223.54016 , 223.7384 , 223.1863 , 223.7245 , 223.18541 , 223.46696 , 223.65787 , 223.63544 , 223.5643 , 223.5507 , 223.42116 , 223.3312 , 223.48505 , 223.77332 , 223.44173 , 223.89722 , 223.29372 , 223.67018 , 223.88866 , 223.61508 , 223.84071 , 223.67831 , 223.61626 , 223.57489 , 223.22168 , 223.85342 , 223.99884 , 223.34335 , 223.98347 , 223.42897 , 223.23814 , 223.05066 , 223.738 , 222.98895 , 223.68735 , 223.14906 , 223.1406 , 223.64868 , 223.22012 , 223.64157 , 223.43118 , 223.32971 , 223.24178 , 223.61594 , 223.825 , 223.13503 , 223.08192 , 223.32346 , 223.35112 , 222.74535 , 223.23387 , 223.36722 , 223.07416 , 223.40839 , 223.67728 , 223.31316 , 223.15977 , 223.11795 , 223.18849 , 223.11351 , 224.12282 , 222.9305 , 223.51143 , 223.11084 , 223.59396 , 223.24268 , 223.36879 , 223.6951 , 223.7016 , 223.50171 , 223.3773 , 222.9516 , 223.60956 , 223.2828 , 223.27995 , 223.50691 , 223.09618 , 223.54132 , 223.15198 , 223.34341 , 223.50548 , 223.54985 , 223.55026 , 223.78177 , 223.49112 , 223.54607 , 223.53114 , 223.06987 , 223.37434 , 223.38205 , 223.63872 , 223.496 , 223.37364 , 223.36783 , 223.79727 , 223.73882 , 223.17883 , 223.46803 , 222.84082 , 223.22131 , 223.7128 , 223.02972 , 223.52563 , 223.19331 , 223.75713 , 223.35721 , 223.76 , 223.27773 , 223.50885 , 223.27312 , 222.72604 , 223.07663 , 223.70802 , 223.71298 , 223.04227 , 223.74309 , 223.33269 , 223.2801 , 223.27202 , 223.34149 , 223.15613 , 223.06198 , 223.30124 , 223.41307 , 223.22379 , 224.01282 , 223.1886 , 223.53561 , 223.06335 , 223.33772 , 223.43626 , 223.2968 , 223.48718 , 223.0495 , 223.9255 , 223.52705 , 223.82362 , 224.08798 , 223.65054 , 223.31346 , 223.65187 , 223.15746 , 223.63628 , 223.25487 , 223.0373 , 223.62022 , 223.49536 , 223.44278 , 223.87143 , 223.23729 , 223.22128 , 222.98067 , 223.1445 , 223.63802 , 223.42601 , 223.71558 , 223.18181 , 223.41127 , 223.28273 , 223.08195 , 223.65974 , 223.69276 , 223.85318 , 223.04224 , 223.6281 , 223.57465 , 223.42691 , 223.4662 , 223.70918 , 223.36496 , 223.85185 , 223.34868 , 222.97293 , 223.12317 , 223.46884 , 223.49522 , 223.51097 , 222.9521 , 223.85971 , 223.78123 , 223.1033 , 223.40237 , 224.02762 , 223.66441 , 223.10461 , 223.6346 , 223.43353 , 223.88498 , 223.40816 , 223.28242 , 222.90833 , 223.49963 , 222.99113 , 223.20287 , 223.01338 , 223.36949 , 223.57033 , 223.78932 , 223.51863 , 223.13046 , 223.89134 , 223.33873 , 223.35791 , 223.3434 , 223.13957 , 223.57921 , 223.29082 , 223.3857 , 223.7748 , 223.0685 , 223.52061 , 223.65034 , 223.42175 , 223.57103 , 223.27885 , 223.64417 , 223.81953 , 223.33937 , 223.64655 , 223.59598 , 223.05182 , 223.24612 , 223.72739 , 223.30305 , 223.46439 , 223.44327 , 223.58421 , 223.71782 , 223.1252 , 223.11784 , 223.29996 , 223.2364 , 223.60751 , 223.54442 , 223.42564 , 223.51695 , 223.55614 , 223.257 , 223.49577 , 223.50987 , 223.53024 , 223.23302 , 223.75151 , 223.85074 , 223.16927 , 223.18677 , 223.38533 , 223.37099 , 223.37378 , 223.5321 , 223.63454 , 223.5803 , 223.32593 , 223.11284 , 223.31264 , 223.60779 , 223.16412 , 224.04926 , 223.71779 , 223.74707 , 223.16736 , 223.27411 , 223.903 , 223.57639 , 223.41484 , 223.79497 , 223.12119 , 223.38489 , 223.34624 , 223.59633 , 223.35423 , 223.41373 , 223.28783 , 223.67558 , 223.08101 , 223.18619 , 223.4632 , 223.4547 , 223.97446 , 223.42758 , 223.03838 , 223.52608 , 223.51497 , 223.21082 , 222.96695 , 223.60114 , 223.6965 , 223.45776 , 223.61238 , 223.09026 , 223.8389 , 223.57646 , 223.79634 , 223.10608 , 223.65678 , 222.96011 , 223.64784 , 223.54372 , 223.25931 , 223.37186 , 223.56548 , 223.17784 , 223.4518 , 223.15848 , 223.66061 , 223.54964 , 223.7 , 223.18086 , 223.32294 , 223.32568 , 223.57002 , 223.53387 , 223.5455 , 223.75264 , 223.31886 , 223.37946 , 223.2545 , 223.39607 , 223.11176 , 223.3795 , 223.30615 , 223.61513 , 223.41013 , 223.59315 , 223.2768 , 223.28609 , 223.22725 , 223.29938 , 223.37778 , 223.45807 , 223.64742 , 223.58133 , 223.92566 , 223.58221 , 222.96521 , 223.66937 , 223.92166 , 223.58696 , 223.9424 , 223.41238 , 223.52261 , 223.40977 , 223.42328 , 223.48051 , 222.96591 , 223.7525 , 223.6611 , 223.55983 , 223.78517 , 223.75766 , 223.29117 , 223.72777 , 223.40213 , 223.44489 , 223.23985 , 223.4613 , 223.07837 , 223.82034 , 223.59068 , 223.53041 , 223.69916 , 223.4305 , 223.59349 , 223.76219 , 223.57616 , 223.47289 , 224.03885 , 223.81363 , 223.7747 , 223.0863 , 223.32364 , 223.63528 , 223.73946 , 223.61308 , 223.37888 , 223.72148 , 223.23912 , 223.23064 , 222.7099 , 223.2259 , 224.09848 , 223.44609 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223.25728 , 223.30916 , 223.39847 , 223.5736 , 223.90439 , 223.33014 , 223.33012 , 223.02379 , 223.44131 , 223.07133 , 223.68752 , 222.94395 , 223.5269 , 223.65495 , 223.47989 , 223.53967 , 223.28346 , 223.63591 , 224.07697 , 223.594 , 223.22694 , 223.37436 , 223.12843 , 223.29071 , 223.25206 , 223.05136 , 223.91992 , 223.1616 , 223.2663 , 223.66919 , 223.65158 , 223.32588 , 223.96954 , 223.80067 , 223.48264 , 223.6067 , 223.47804 , 223.1232 , 223.5648 , 223.76178 , 223.432 , 224.04033 , 223.44626 , 223.4293 , 223.15614 , 223.5282 , 223.40005 , 223.61931 , 223.88501 , 223.75249 , 223.08665 , 223.82225 , 223.32271 , 223.94756 , 223.27617 , 223.64893 , 223.20561 , 223.34859 , 223.98273 , 223.37747 , 223.3377 , 223.289 , 223.4348 , 223.31616 , 223.44228 , 223.31396 , 223.50214 , 223.49979 , 223.47421 , 223.38083 , 223.5437 , 223.44069 , 223.18582 , 223.5217 , 223.42873 , 223.21597 , 223.24654 , 223.53748 , 223.44334 , 223.51318 , 223.50003 , 223.13797 , 223.37787 , 223.645 , 223.66275 , 223.29987 , 223.56836 , 222.99379 , 223.57515 , 223.58832 , 223.5116 , 223.36694 , 223.546 , 223.08243 , 223.58582 , 223.34698 , 223.15247 , 223.33379 , 223.2437 , 223.31363 , 223.66333 , 224.07726 , 223.58257 , 223.31837 , 223.56296 , 223.40222 , 223.5327 , 223.49945 , 223.26811 , 223.13487 , 223.05276 , 223.48953 , 223.31284 , 223.04355 , 223.34727 , 223.74706 , 223.53368 , 223.37047 , 223.0563 , 223.39412 , 223.16014 , 223.85934 , 223.44252 , 223.73524 , 224.1261 , 223.2832 , 223.31497 , 223.4263 , 223.28867 , 223.54071 , 223.34955 , 223.18144 , 223.24466 , 223.322 , 223.17644 , 223.69804 , 223.30473 , 223.38824 , 223.78772 , 223.10136 , 223.2068 , 223.42555 , 223.78676 , 223.48206 , 223.52054 , 223.07164 , 223.47667 , 223.56729 , 223.69615 , 223.4807 , 223.11728 , 223.93053 , 223.84117 , 223.46326 , 223.67824 , 223.57605 , 223.3182 , 223.65543 , 223.35107 , 223.8799 , 223.56676 , 223.52249 , 223.32881 , 223.31151 , 223.11159 , 223.53348 , 223.52504 , 223.58723 , 223.20187 , 223.44888 , 223.32039 , 223.45554 , 222.98329 , 223.60625 , 223.46982 , 223.36198 , 223.51877 , 223.46959 , 223.47238 , 223.50056 , 223.14171 , 223.76817 , 223.01596 , 223.4202 , 223.08995 , 223.46619 , 223.33327 , 223.95624 , 223.78732 , 223.29025 , 223.52954 , 223.27809 , 223.1728 , 223.35162 , 223.32407 , 223.09854 , 223.00464 , 223.1015 , 223.01602 , 223.90144 , 223.41089 , 223.2455 , 223.17729 , 223.11891 , 223.51868 , 223.19305 , 223.81764 , 223.46321 , 223.47165 , 223.9151 , 223.29431 , 223.35313 , 223.56526 , 223.62946 , 223.65869 , 223.49463 , 223.47952 , 223.26111 , 223.04778 , 223.29512 , 223.78513 , 223.55305 , 223.57104 , 223.72893 , 223.22028 , 223.82388 , 223.19827 , 223.02557 , 223.13297 , 223.24722 , 223.2185 , 223.78928 , 223.36154 , 223.69542 , 223.37128 , 223.52931 , 223.74347 , 223.42516 , 223.56665 , 223.19359 , 222.90475 , 223.55806 , 223.76842 , 223.6739 , 223.61748 , 223.40364 , 223.41525 , 223.10365 , 223.15622 , 223.74174 , 223.23618 , 223.34396 , 222.84099 , 223.35991 , 224.06119 , 223.79173 , 223.88477 , 223.55424 , 223.2446 , 222.81386 , 223.57024 , 223.99513 , 223.43153 , 223.1641 , 223.81964 , 223.0981 , 223.87769 , 223.38359 , 223.39043 , 223.1295 , 223.46927 , 223.094 , 223.22873 , 223.35521 , 223.04623 , 223.82301 , 223.2718 , 223.5761 , 223.20912 , 223.27255 , 223.21475 , 222.95976 , 223.39484 , 223.3098 , 223.38669 , 223.68712 , 223.14905 , 223.79356 , 223.4413 , 223.41345 , 223.46967 , 223.34723 , 223.56258 , 223.48997 , 223.80669 , 223.1723 , 223.28674 , 223.61179 , 223.99881 , 223.28313 , 223.13383 , 223.19078 , 223.208 , 222.89064 , 223.46414 , 223.26685 , 223.15451 , 223.23886 , 223.66576 , 223.48602 , 223.58366 , 223.19623 , 223.70073 , 223.75836 , 223.3502 , 223.86113 , 223.28583 , 223.3788 , 223.64952 , 223.97472 , 223.31752 , 223.30812 , 223.56818 , 223.15329 , 222.95943 , 223.40031 , 224.18524 , 223.05006 , 223.27911 , 223.39223 , 223.24686 , 223.49095 , 224.1674 , 223.23065 , 223.28696 , 223.68597 , 223.51073 , 223.68553 , 223.71263 , 223.45918 , 223.12277 , 223.40178 , 223.15213 , 222.96594 , 222.80846 , 223.7778 , 223.32484 , 223.44476 , 224.07074 , 222.98997 , 223.34467 , 223.3111 , 223.74452 , 223.5378 , 223.53905 , 223.34483 , 223.4899 , 223.47328 , 223.45155 , 223.38501 , 223.51079 , 223.07436 , 223.55313 , 223.35793 , 223.29602 , 223.54794 , 223.9102 , 223.44844 , 223.2556 , 223.537 , 223.37027 , 223.7462 , 223.16498 , 223.99448 , 223.50513 , 223.18394 , 223.39978 , 223.76959 , 223.63919 , 223.28728 , 223.07506 , 223.40382 , 223.08163 , 223.40689 , 223.48782 , 223.32808 , 223.5034 , 223.50656 , 222.84917 , 223.16882 , 223.57446 , 223.68396 , 222.98953 , 223.60828 , 222.85893 , 223.3061 , 222.97949 , 223.42232 , 223.19218 , 223.6725 , 223.09666 , 223.56387 , 223.09134 , 223.23557 , 223.19156 , 223.05742 , 223.3902 , 223.39314 , 222.98355 , 223.03015 , 223.11798 , 222.79411 , 223.37791 , 223.21661 , 223.465 , 223.66118 , 223.20695 , 223.44435 , 223.43388 , 223.45456 , 223.40887 , 223.83316 , 223.65347 , 223.13934 , 223.39928 , 223.63722 , 223.47272 , 223.91663 , 223.59949 , 223.43419 , 222.93344 , 223.12401 , 223.14908 , 223.9061 , 223.79985 , 223.57857 , 223.6299 , 223.72316 , 223.30371 , 223.53397 , 223.31673 , 223.5075 , 223.4948 , 223.39725 , 223.10463 , 223.83649 , 223.27968 , 223.31511 , 223.61996 , 223.49368 , 223.72093 , 222.71927 , 223.26344 , 223.70892 , 223.2429 , 223.65585 , 223.21321 , 223.39218 , 223.69128 , 223.56393 , 223.84677 , 223.51872 , 222.81242 , 223.4307 , 222.93195 , 223.18823 , 223.61157 , 223.35121 , 223.49521 , 223.44026 , 223.178 , 223.8178 , 223.2065 , 223.4588 , 223.82126 , 222.91595 , 223.50195 , 223.2362 , 223.32175 , 223.46036 , 223.44864 , 223.61336 , 223.626 , 222.9814 , 223.7648 , 223.22998 , 223.28795 , 223.47813 , 223.54445 , 223.17828 , 223.14714 , 223.3673 , 223.38268 , 223.77603 , 224.01834 , 223.30772 , 223.56699 , 223.14891 , 223.98694 , 223.31978 , 223.30731 , 223.15756 , 223.45787 , 223.4943 , 223.70839 , 223.26073 , 223.64995 , 222.97316 , 223.22403 , 223.61127 , 223.96204 , 223.29262 , 223.89986 , 223.52496 , 223.14568 , 223.1308 , 223.36595 , 223.35034 , 223.63033 , 223.2663 , 222.97752 , 223.24783 , 223.43098 , 223.03917 , 223.29315 , 223.76537 , 223.2923 , 223.54137 , 223.4052 , 223.19247 , 223.88728 , 223.17947 , 223.38504 , 223.70137 , 223.20137 , 223.5328 , 223.4985 , 223.31633 , 223.24855 , 223.46107 , 223.59488 , 223.66212 , 223.12842 , 223.10217 , 223.14496 , 223.22435 , 223.25682 , 223.7292 , 223.24023 , 223.01633 , 223.16588 , 223.6319 , 223.7791 , 223.06973 , 223.77223 , 223.98451 , 223.1409 , 223.34912 , 223.44893 , 223.36983 , 223.43817 , 223.40488 , 223.37369 , 222.891 , 223.9089 , 223.05133 , 223.36128 , 223.68605 , 222.83026 , 223.35641 , 223.66023 , 223.44666 , 223.20448 , 223.13245 , 223.40123 , 223.16957 , 223.72379 , 223.51431 , 223.3719 , 223.38489 , 223.47133 , 223.20761 , 223.81587 , 223.35321 , 223.44759 , 223.69803 , 223.65393 , 223.55658 , 223.39424 , 223.42818 , 223.27895 , 223.703 , 223.38823 , 223.57602 , 223.27351 , 223.74342 , 223.57454 , 223.46062 , 223.68848 , 223.25119 , 223.33887 , 223.91164 , 223.67088 , 223.58882 , 223.18977 , 222.87134 , 223.20483 , 223.22229 , 223.24496 , 223.16373 , 223.1976 , 223.90572 , 223.91228 , 223.61322 , 223.57253 , 223.73099 , 223.41356 , 223.7682 , 223.07404 , 223.42122 , 223.57079 , 223.66489 , 223.53662 , 223.59802 , 223.40564 , 223.69748 , 223.70036 , 223.44116 , 223.98769 , 223.13164 , 222.85107 , 223.92767 , 223.35825 , 223.66702 , 223.75641 , 223.0488 , 223.04025 , 223.3549 , 223.57092 , 223.44768 , 223.40622 , 223.259 , 223.24939 , 223.64435 , 223.67575 , 223.57039 , 223.15915 , 223.27652 , 223.28467 , 223.55304 , 223.38509]

Plot

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm


# Parameters
R = 2000
c = 3
phi_values = [50, 100, 150, 200]

# Normalize the test statistics for each phi value
normalized_data_sets = []
for phi in phi_values:
    center = n * (1 + 1 / (c - 1 / np.sqrt(phi))**2) + 1 / (c - 1 / np.sqrt(phi))**2
    variance = (4 + 4 * 2 * (-1 / np.sqrt(phi) + 1 / (c - 1 / np.sqrt(phi)) + 1 / n / (c - np.sqrt(phi)))**2) / (c - 1 / np.sqrt(phi))**4
    normalized_data = (eval(f"TestST{phi//50}") - center * np.ones(R)) / np.sqrt(variance)
    normalized_data_sets.append(normalized_data)

# Plot the ECDFs
labels = [f"$\mathbf{{\phi={phi}}}$" for phi in phi_values]
plot_ecdf_and_standard_normal_cdf(*normalized_data_sets, labels=labels)

Figure 4: ECDF different $\eta$, Local

In this section, we display the comparison of ECDFs of normalized test statistics $\mathcal{T}^g_1$ (and $\mathcal{T}^l_1$, respectively) versus the CDF of standard normal distribution. Here the normalizations are done with our theoretical value for the Global statistics and with calibration for the Local statistics).

$T_1^l$ (bulk)

Data generating

%%time
import jax
import jax.numpy as jnp
from jax import grad, jit, random
from jax.random import normal
import numpy as np
import matplotlib.pyplot as plt

n = 800
R0 = 2000
c = 1
phi = 100
m = int(n * phi)
midb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi))
leftb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi)-2)
rightb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi)+2)



def simulate_local_test_statistics1(eta, E, m, n, R0):
    """
    Compute the test statistics for a given value of eta.
    """
    T0 = jnp.zeros(R0)
    key = jax.random.PRNGKey(0)
    for r in range(R0):
        key, subkey = jax.random.split(key)
        X = random.normal(key, shape=(m, n))
        H = jnp.matmul(X.T, X)
        E00, _ = jnp.linalg.eigh(H)
        E0 = np.array(E00) / np.sqrt(n * m)
        Et = jnp.array(extract_elements(E0, E-eta, E+eta))
        Eb = (Et - E * np.ones(Et.shape)) / eta
        t0 = jnp.sum(Eb)
        T0 = T0.at[r].set(t0)

    TestST = np.array(T0)
    Mean_x = jnp.mean(TestST)
    Variance_x = jnp.var(TestST)
    Normalize_TestST = (TestST - Mean_x) / np.sqrt(Variance_x)
    return Normalize_TestST

etas = [1/(n**(1/4)), 1/(n**(1/3)), 1/(n**(1/2)), 1/(n**(3/5))]

Normalize_TestST_1 = simulate_local_test_statistics1(etas[0], midb, m, n, R0)
Normalize_TestST_2 = simulate_local_test_statistics1(etas[1], midb, m, n, R0)
Normalize_TestST_3 = simulate_local_test_statistics1(etas[2], midb, m, n, R0)
Normalize_TestST_4 = simulate_local_test_statistics1(etas[3], midb, m, n, R0)

CPU times: user 7min 49s, sys: 9.81 s, total: 7min 59s
Wall time: 7min 46s

Saved data

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Plot

n = 800
R0 = 2000
c = 1
phi = 100
labels = ['$\mathbf{\eta_0=n^{-1/4}}$', '$\mathbf{\eta_0=n^{-1/3}}$', '$\mathbf{\eta_0=n^{-1/2}}$', '$\mathbf{\eta_0=n^{-3/5}}$']
plot_ecdf_and_standard_normal_cdf(Normalize_TestST_1, Normalize_TestST_2, Normalize_TestST_3, Normalize_TestST_4, labels=labels)

$T_1^l$ (edge)

Here $E_0=$ left edge $\gamma_-$ and the main part of the test function is in $[E_0-2*\eta,E_0+2*\eta]$

n or the local interval need to be large enough to avoid the situation when there are no eigenvalues in the local interval empirically. So, adjustment is made when the imaginary part is too small.

Data generating

import jax
import jax.numpy as jnp
from jax import grad, jit, random
from jax.random import normal
import numpy as np
import matplotlib.pyplot as plt

n = 800
R0 = 2000
c = 1
phi = 100
m = int(n * phi)
midb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi))
leftb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi)-2)
rightb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi)+2)



def simulate_local_test_statistics1(eta, E, m, n, R0):
    """
    Compute the test statistics for a given value of eta.
    """
    T0 = jnp.zeros(R0)
    key = jax.random.PRNGKey(0)
    for r in range(R0):
        key, subkey = jax.random.split(key)
        X = random.normal(key, shape=(m, n))
        H = jnp.matmul(X.T, X)
        E00, _ = jnp.linalg.eigh(H)
        E0 = np.array(E00) / np.sqrt(n * m)
        Et = jnp.array(extract_elements(E0, E-eta, E+eta))
        Eb = (Et - E * np.ones(Et.shape)) / eta
        t0 = jnp.sum(Eb)
        T0 = T0.at[r].set(t0)

    TestST = np.array(T0)
    Mean_x = jnp.mean(TestST)
    Variance_x = jnp.var(TestST)
    Normalize_TestST = (TestST - Mean_x) / np.sqrt(Variance_x)
    return Normalize_TestST

etas = [1/(n**(1/4)), 1/(n**(1/3)), 1/(n**(1/2)), 1/(n**(3/5))]

Normalize_TestST_1 = simulate_local_test_statistics1(etas[0], leftb, m, n, R0)
Normalize_TestST_2 = simulate_local_test_statistics1(etas[1], leftb, m, n, R0)
Normalize_TestST_3 = simulate_local_test_statistics1(etas[2], leftb, m, n, R0)
Normalize_TestST_4 = simulate_local_test_statistics1(etas[3], leftb, m, n, R0)

Saved data

Normalize_TestST_1=[0.32757175 , 1.6343967 , -0.5739341 , 0.60261613 , -0.79334795 , 2.3761516 , -0.14965487 , -0.1558663 , 1.841156 , 0.6115311 , 2.0494478 , 0.18570155 , 0.100029565 , -0.72848934 , -0.6146979 , -1.7974417 , -0.6397626 , -0.31970564 , 0.6323801 , -0.062129628 , -0.7002782 , 1.0323788 , -0.038327612 , -0.7320991 , -0.06544664 , -0.022063324 , 0.350829 , -1.1729176 , -1.3071151 , 0.85747844 , 0.20051226 , -0.92809534 , -2.3598695 , 1.6390705 , 1.0375643 , -1.5703679 , -1.8885436 , 0.53678256 , 0.2654625 , -0.14803836 , -0.16045868 , 1.4334567 , -1.6358757 , -0.4566805 , -0.18291913 , 1.8234738 , 1.2570927 , -0.25099546 , -0.39340532 , -0.6358524 , 1.4815419 , -0.20196839 , 0.39941058 , 0.14796454 , 1.38292 , 1.092146 , 0.74581265 , -0.53427255 , -1.0816145 , 0.41357213 , 1.6062033 , 0.3996575 , 0.7096565 , -0.9268582 , -1.2812358 , 1.4814885 , 0.27976406 , 0.82424474 , 0.21017817 , -0.87518615 , -2.0676444 , 0.3545304 , 2.4243436 , -0.9331409 , 0.110517725 , 0.25220463 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Plot

n = 800
R0 = 2000
c = 1
phi = 100
labels = ['$\mathbf{\eta_0=n^{-1/4}}$', '$\mathbf{\eta_0=n^{-1/3}}$', '$\mathbf{\eta_0=n^{-1/2}}$', '$\mathbf{\eta_0=n^{-3/5}}$']
plot_ecdf_and_standard_normal_cdf(Normalize_TestST_1, Normalize_TestST_2, Normalize_TestST_3, Normalize_TestST_4, labels=labels)

$T_2^l$

One can see from the above 2 subsections that when $n$ is large enough, there is little visual difference between bulk E and edge E. But when n is not large enough, bulk eigenvalues typically have less eigen gaps. Therefore, in what follows, we use bulk E unless stated otherwise.

Data generating

%%time
import jax
import jax.numpy as jnp
from jax import grad, jit, random
from jax.random import normal
import numpy as np
import matplotlib.pyplot as plt

n = 800
R0 = 2000
c = 1
phi = 100
m = int(n * phi)
midb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi))
leftb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi)-2)
rightb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi)+2)



def simulate_local_test_statistics2(eta, E, m, n, R0):
    """
    Compute the test statistics for a given value of eta.
    """
    T0 = jnp.zeros(R0)
    key = jax.random.PRNGKey(0)
    for r in range(R0):
        key, subkey = jax.random.split(key)
        X = random.normal(key, shape=(m, n))
        H = jnp.matmul(X.T, X)
        E00, _ = jnp.linalg.eigh(H)
        E0 = np.array(E00) / np.sqrt(n * m)
        Et = jnp.array(extract_elements(E0, E-eta, E+eta))
        Eb = (Et - E * np.ones(Et.shape)) / eta
        t0 = jnp.sum(np.square(Eb))
        T0 = T0.at[r].set(t0)
    TestST = np.array(T0)
    Mean_x = jnp.mean(TestST)
    Variance_x = jnp.var(TestST)
    Normalize_TestST = (TestST - Mean_x) / np.sqrt(Variance_x)
    return Normalize_TestST



etas = [1/(n**(1/4)), 1/(n**(1/3)), 1/(n**(1/2)), 1/(n**(3/5))]
normalized_test_stats = []

Normalize_TestST_1 = simulate_local_test_statistics2(etas[0], leftb, m, n, R0)
Normalize_TestST_2 = simulate_local_test_statistics2(etas[1], leftb, m, n, R0)
Normalize_TestST_3 = simulate_local_test_statistics2(etas[2], leftb, m, n, R0)
Normalize_TestST_4 = simulate_local_test_statistics2(etas[3], leftb, m, n, R0)

Saved data

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Plot

n = 800
R0 = 2000
c = 1
phi = 100
labels = ['$\mathbf{\eta_0=n^{-1/4}}$', '$\mathbf{\eta_0=n^{-1/3}}$', '$\mathbf{\eta_0=n^{-1/2}}$', '$\mathbf{\eta_0=n^{-3/5}}$']
plot_ecdf_and_standard_normal_cdf(Normalize_TestST_1, Normalize_TestST_2, Normalize_TestST_3, Normalize_TestST_4, labels=labels)

$T_3^l$

Data generating

import jax
import jax.numpy as jnp
from jax import grad, jit, random
from jax.random import normal
import numpy as np
import matplotlib.pyplot as plt

n = 800
R0 = 2000
c = 1
phi = 100
m = int(n * phi)
midb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi))
leftb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi)-2)
rightb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi)+2)



def simulate_local_test_statistics3(eta, E, m, n, R0):
    """
    Compute the test statistics for a given value of eta.
    """
    T0 = jnp.zeros(R0)
    key = jax.random.PRNGKey(0)
    for r in range(R0):
        key, subkey = jax.random.split(key)
        X = random.normal(key, shape=(m, n))
        H = jnp.matmul(X.T, X)
        E00, _ = jnp.linalg.eigh(H)
        E0 = np.array(E00) / np.sqrt(n * m)
        Et = jnp.array(extract_elements(E0, E-2*eta, E+3*eta))
        Eb = (Et - E * np.ones(Et.shape)) / eta
        t0=-jnp.sum(np.log(Eb+3*np.ones(np.size(Eb))))+jnp.sum(Eb)
        T0 = T0.at[r].set(t0)
    TestST = np.array(T0)
    Mean_x = jnp.mean(TestST)
    Variance_x = jnp.var(TestST)
    Normalize_TestST = (TestST - Mean_x) / np.sqrt(Variance_x)
    return Normalize_TestST


etas = [1/(n**(1/4)), 1/(n**(1/3)), 1/(n**(1/2)), 1/(n**(3/5))]
normalized_test_stats = []

Normalize_TestST_1 = simulate_local_test_statistics3(etas[0], leftb, m, n, R0)
Normalize_TestST_2 = simulate_local_test_statistics3(etas[1], leftb, m, n, R0)
Normalize_TestST_3 = simulate_local_test_statistics3(etas[2], leftb, m, n, R0)
Normalize_TestST_4 = simulate_local_test_statistics3(etas[3], leftb, m, n, R0)

Saved data

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Plot

n = 800
R0 = 2000
c = 1
phi = 100
labels = ['$\mathbf{\eta_0=n^{-1/4}}$', '$\mathbf{\eta_0=n^{-1/3}}$', '$\mathbf{\eta_0=n^{-1/2}}$', '$\mathbf{\eta_0=n^{-3/5}}$']
plot_ecdf_and_standard_normal_cdf(Normalize_TestST_1, Normalize_TestST_2, Normalize_TestST_3, Normalize_TestST_4, labels=labels)

$T_4^l$

Data generating

import jax
import jax.numpy as jnp
from jax import grad, jit, random
from jax.random import normal
import numpy as np
import matplotlib.pyplot as plt

n = 800
R0 = 2000
c = 1
phi = 100
m = int(n * phi)
midb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi))
leftb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi)-2)
rightb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi)+2)



def simulate_local_test_statistics4(eta, E, m, n, R0):
    """
    Compute the test statistics for a given value of eta.
    """
    T0 = jnp.zeros(R0)
    key = jax.random.PRNGKey(0)
    for r in range(R0):
        key, subkey = jax.random.split(key)
        X = random.normal(key, shape=(m, n))
        H = jnp.matmul(X.T, X)
        E00, _ = jnp.linalg.eigh(H)
        E0 = np.array(E00) / np.sqrt(n * m)
        Et = jnp.array(extract_elements(E0, E-5*eta, E+7*eta))
        Eb = (Et - E * np.ones(Et.shape)) / eta
        loE=jnp.power(Eb,2*jnp.ones(np.size(Eb)))
        t0 = n*n*jnp.sum(loE)/jnp.power(jnp.sum(Eb),2)
        T0 = T0.at[r].set(t0)
    TestST = np.array(T0)
    Mean_x = jnp.mean(TestST)
    Variance_x = jnp.var(TestST)
    Normalize_TestST = (TestST - Mean_x) / np.sqrt(Variance_x)
    return Normalize_TestST



etas = [1/(n**(1/4)), 1/(n**(1/3)), 1/(n**(1/2)), 1/(n**(3/5))]


Normalize_TestST_1 = simulate_local_test_statistics4(etas[0], leftb, m, n, R0)
Normalize_TestST_2 = simulate_local_test_statistics4(etas[1], leftb, m, n, R0)
Normalize_TestST_3 = simulate_local_test_statistics4(etas[2], leftb, m, n, R0)
Normalize_TestST_4 = simulate_local_test_statistics4(etas[3], leftb, m, n, R0)

Saved data

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-0.9319251 , 0.005282067 , -0.42953634 , 0.5028244 , 0.12471847 , -0.43589148 , 0.28126347 , 0.18725835 , 0.12947184 , -1.0834132 , -1.2194017 , 0.6079508 , 0.35627666 , -0.2244629 , 0.2284105 , -1.9442012 , -0.19839817 , -0.9076893 , -0.13652308 , -0.54430753 , 0.21839954 , -0.67066103 , 0.36337492 , 0.00022224824 , 0.5276085 , -1.3058143 , -1.6579778 , -1.6447986 , -1.1643684 , 0.55048835 , 1.1723762 , -0.31072947 , 0.45638922 , -0.051769156 , -0.4439962 , -1.8709484 , -0.106335506 , -0.16405544 , -1.2623074 , 2.6740782 , 0.25232127 , -1.3484526 , 0.816824 , 1.4215919 , 0.7582885 , -0.2382981 , -0.17392248 , -0.2478254 , 0.96383584 , 0.83351517 , 2.0971413 , -0.18320794 , 0.3575475 , -0.14307989 , 0.30397588 , -0.08464721 , -0.87144125 , -0.32018533 , 2.891994 , -0.30608675 , -0.35487562 , -1.3231937 , -1.2521456 , 0.076371156 , -0.48365527 , 1.6235793 , 1.0207852 , -0.53855646 , -1.4058592 , -0.13983527 , 0.37149334 , 0.68622327 , -0.07298848 , 0.72218543 , 0.051477395 , -0.29169154 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0.15031716 , -1.8872637 , 1.8797767 , 0.7783486 , 0.23343116 , 0.13263325 , -0.042554177 , -0.63462156 , -0.97300184 , 0.05734984 , 0.60827583 , -0.36126795 , 0.45413345 , -0.68137884 , -0.5124516 , 2.6862588 , -0.0055023576 , -0.11074914 , -0.7804184 , -0.9628068 , 0.35868812 , -0.3631125 , -0.7713052 , -0.58273196 , -0.17503175 , 2.9720633 , 1.5471866 , -0.6759235 , -0.6598472 , -0.5950653 , -0.48553607 , 0.80918044 , 0.9217154 , 0.36015671 , -0.9695771 , 8.2013235 , -0.5029752 , -0.93587756 , 0.9225564 , 0.52746755 , 0.62554264 , 2.0038998 , 0.1011112 , -1.0501602 , 2.645272 , -1.0488394 , 0.10395833 , -0.2877488 , -0.35055697 , -0.7166106 , -0.6785239 , 0.06686735 , -0.8243197 , -0.21429528 , 2.4013112 , -0.1969893 , -0.87430114 , 0.9130389 , 1.0215763 , -0.11252125 , -0.83762234 , 2.48345 , -0.34987456 , -1.0403519 , -0.99969906 , -0.56308603 , 1.7761405 , -0.28795052 , -0.15564328 , 1.0187302 , -0.8904577 , -0.5897372 , 0.97848463 , -0.1293054 , 0.26992196 , -1.2563713 , 1.0250452 , -1.5261034 , 0.2690467 , 1.0675298 , -0.38521984 , 1.0223262 , -0.8799386 , 0.100655936 , -0.10350013 , 0.05118955 , 1.3831478 , 1.3600966 , -0.9100449 , 2.5938945 , -0.10884681 , 2.5210853 , 1.5549026 , 1.216723 , -0.810742 , 0.42226973 , -1.0441585 , 0.5345139 , 0.2131723 , 3.2323542 , 0.4219946 , -1.1118512 , -0.039438784 , -1.7339437 , 1.2888802 , 0.12396655 , -1.3229107 , -0.78014815 , 0.39893168 , 1.0234747 , 0.40885356 , 0.81782067 , 0.8351688 , -1.3670901 , 0.8878005 , -0.057650413 , -1.0986837 , -0.80250615 , -1.1928945 , -0.1735005 , 0.14147325 , -0.5128716 , 0.1979478 , -0.19176401 , -0.5267685 , 0.8883605 , 0.8968363 , 2.589201 , 0.5647328 , -1.165929 , -0.8411842 , 1.0197288 , 0.91107094 , -0.048944548 , 2.0791726 , 0.3143354 , 0.31127483 , -0.031758003 , -0.562666 , 0.028127622 , -0.48087275 , 1.0092841 , -0.18694602 , -0.12061325 , 0.58369344 , 0.1527991 , -0.16462916 , -1.4837782 , 1.5148343 , 0.5789606 , -0.25448796 , 2.8219576 , -0.3390725 , 0.21357763 , -0.052610174 , 0.22294241 , 1.1596522 , 0.954706 , 0.2708658 , -1.1898447 , -1.140015 , 0.25893977 , -0.8822443 , 0.17529905 , -0.44013965 , -0.35707852 , -1.1362976 , 0.62259275 , -1.0685881 , 0.48227185 , -0.35931864 , 0.048336547 , -1.4320816 , 1.3850061 , 0.37065232 , 1.7819933 , -1.6073669 , 0.75894547 , 1.0886835 , -0.69170314 , -1.5131855 , 1.3220892 , 0.56899667 , 0.4063638 , -0.56540936 , -0.016795898 , 1.0319535 , 0.36830255 , -1.5734627 , 1.4761983 , 1.0653162 , -1.5867213 , -0.83490735 , 1.5792011 , -1.5752681 , -1.8764606 , -0.41499427 , -0.4447393 , 0.1307505 , -0.4621628 , -0.21768874 , -0.5003944 , 0.7688536 , -0.8424805 , 0.8405713 , -0.09208322 , -1.356761 , -0.29458466 , -0.98035854 , 0.504902 , 0.80998427 , -0.94218963 , -0.8384966 , 0.42702994 , 0.08357709 , -0.17942874 , 1.097596 , -2.3771281 , -0.4919842 , -0.80227506 , 1.4914199 , 0.71204424 , -1.2522221 , -1.5698187 , -0.17803064 , -0.16810486 , -0.18902948 , 2.0514698 , -1.2015533 , 1.3623769 , -0.17606664 , -1.0348465 , -0.7747535 , -0.25123844 , 0.6702841 , -0.11103013 , 0.3862048 , 0.61774147 , 1.9388555 , -2.0499876 , -0.68809724 , -0.17855738 , 1.506052 , -1.1955419 , -1.2174094 , -1.0709487 , -0.85135865 , -0.32943064 , -1.0381509 , 3.2015114 , -0.25026622 , -0.03979908 , -0.5790115 , -0.7137527 , -2.0277655 , -0.9007986 , 0.015352752 , 0.94477046 , 0.13807099 , -0.3200414 , -1.2688084 , -0.2261136 , 0.42901257 , -0.62154025 , -0.48357204 , -0.601533 , -0.66293424 , -1.4960508 , -0.65151244 , 1.3020008 , -0.39442995 , 1.0437013 , -1.0364914 , 0.35568923 , 0.4343984 , -0.6627815 , -0.104396954 , 0.23129973 , 1.1456124 , 0.3592922 , 0.9262132 , -0.73945713 , -0.73289055 , -0.7609144 , -0.008022477 , -0.13379541 , -1.0547392 , 1.3843932 , -2.4132576 , -0.30177495 , 0.43838027 , -0.1930887 , -0.41546422 , -0.719772 , 0.039479904 , 1.4382526 , -0.4721845 , -0.79888946 , 0.64993614 , 0.28864467 , 0.5714531 , -1.2346145 , 0.036111914 , 1.7256294 , -1.4915717 , -1.1606783 , -1.4436619 , 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-2.033781 , 0.47075704 , -0.68870425 , 0.6200109 ]

Plot

n = 800
R0 = 2000
c = 1
phi = 100
labels = ['$\mathbf{\eta_0=n^{-1/4}}$', '$\mathbf{\eta_0=n^{-1/3}}$', '$\mathbf{\eta_0=n^{-1/2}}$', '$\mathbf{\eta_0=n^{-3/5}}$']
plot_ecdf_and_standard_normal_cdf(Normalize_TestST_1, Normalize_TestST_2, Normalize_TestST_3, Normalize_TestST_4, labels=labels)

Figure 5: Empirical power for $T_1^g$-$T_4^g$, $T_1^l$-$T_4^l$

In this section, we compare the power of our global and local statistics

It's too time consuming to run all these tests all the way to 0.4. Also, the domain of the log function will affect how large $ɛ$ can be so the below tests only run from $ɛ=0.01$ to $ɛ=0.2$, then 20 1's are attached behind the results.

Figure 5(a) global

Data generating

import jax
import jax.numpy as jnp
from jax import grad, jit,random
from jax.random import normal
import numpy as np
import matplotlib.pyplot as plt





n = 200
R = 1000
c=2
T = jnp.zeros(R)
# key = jax.random.PRNGKey(0)

# for k in range(R):
#     key, subkey = jax.random.split(key)
#     X = normal(subkey, shape=(m, n))
#     T = T.at[k].set(objective(X))


TestST=np.array(T)

def count_outside_interval(arr, a, b):
    outside_interval = np.logical_or(arr < a, arr > b)
    count = np.sum(outside_interval)
    return count



####################################################

K4=0

# Normalize_TestST=(TestST-(n*(1+1/(c-1/np.sqrt(phi))**2)+1/(c-1/np.sqrt(phi))**2)*np.ones(R))/np.sqrt((4+4*2*(-1/np.sqrt(phi)+1/(c-1/np.sqrt(phi))+1/n/(c-np.sqrt(phi)))**2)/((c-1/np.sqrt(phi))**4))
# plot_ecdf_and_standard_normal_cdf(Normalize_TestST)


pn=2*jnp.ones(n)


def power_phi(n,phi,R,c,t):
    m=int(n*phi)
    r=jnp.sqrt(phi)
    Sigmad = jnp.repeat(jnp.array([1, 1+t]), m//2)
    centerc = (jnp.sqrt(phi)+1/jnp.sqrt(phi)-c)*jnp.ones(n)
    # objective_grad = jit(grad(objective))
    T = jnp.zeros(R)
    key = jax.random.PRNGKey(0)
    for k in range(R):
        key, subkey = jax.random.split(key)
        X = random.normal(subkey, shape=(m, n))
        X = X * jnp.sqrt(Sigmad)[:, None]  # Scale the rows of X by the square root of the diagonal elements of Sigma
        H = jnp.matmul(X.T, X) / jnp.sqrt(n * m)
        E0 = jnp.linalg.eigvalsh(H)
        Eb = E0 - centerc
        # pn=2*jnp.ones(n)
        # loE=jnp.power(Eb,pn)
        # t = n*n*jnp.sum(loE)/jnp.power(jnp.sum(Eb),2)
        t=jnp.sum(Eb)
        T = T.at[k].set(t)
    TestST=np.array(T)
    Normalize_TestST=(TestST-n*(c-1/r)*np.ones(R))/np.sqrt(2+K4)
    # plot_ecdf_and_standard_normal_cdf(Normalize_TestST)
    print(jnp.mean(Normalize_TestST))
    print(jnp.var(Normalize_TestST))
    True_Pos=count_outside_interval(Normalize_TestST,-1.96,1.96)
    Power=True_Pos/R
    return(Power)
P1=[]
temp=0
phi=100
repeat=20
for s in range(repeat):
    P1.append(power_phi(n,phi,R,c,0.01*s))
print(P1)
x_values = [0.01*s for s in range(repeat)]
plt.plot(x_values, P1)
plt.show()

################################################################


c=1
T = jnp.zeros(R)

pn=2*jnp.ones(n)


def power_phi(n,phi,R,c,t):
    m=int(n*phi)
    r=jnp.sqrt(phi)
    Sigmad = jnp.repeat(jnp.array([1, 1+t]), m//2)
    centerc = (jnp.sqrt(phi)+1/jnp.sqrt(phi)-c)*jnp.ones(n)
    # objective_grad = jit(grad(objective))
    T = jnp.zeros(R)
    key = jax.random.PRNGKey(0)
    for k in range(R):
        key, subkey = jax.random.split(key)
        X = random.normal(subkey, shape=(m, n))
        X = X * jnp.sqrt(Sigmad)[:, None]  # Scale the rows of X by the square root of the diagonal elements of Sigma
        H = jnp.matmul(X.T, X) / jnp.sqrt(n * m)
        E0 = jnp.linalg.eigvalsh(H)
        Eb = E0 - centerc
        pn=2*jnp.ones(n)
        loE=jnp.power(Eb,pn)
        # t = n*n*jnp.sum(loE)/jnp.power(jnp.sum(Eb),2)
        t=jnp.sum(loE)
        T = T.at[k].set(t)
    TestST=np.array(T)
    Normalize_TestST=(TestST-(n*(1+c**2-2*c/r+1/phi)+1+K4)*np.ones(R))/np.sqrt(4+4*K4*c**2+8*c**2)
    # plot_ecdf_and_standard_normal_cdf(Normalize_TestST)
    print(jnp.mean(Normalize_TestST))
    print(jnp.var(Normalize_TestST))
    True_Pos=count_outside_interval(Normalize_TestST,-1.96,1.96)
    Power=True_Pos/R
    return(Power)
P2=[]
temp=0
phi=100

for s in range(repeat):
    P2.append(power_phi(n,phi,R,c,0.01*s))
print(P2)
x_values = [0.01*s for s in range(repeat)]
plt.plot(x_values, P2)
plt.show()



############################################################


t=0.000

n = 200
R = 1000
c=6
T = jnp.zeros(R)
# key = jax.random.PRNGKey(0)

# for k in range(R):
#     key, subkey = jax.random.split(key)
#     X = normal(subkey, shape=(m, n))
#     T = T.at[k].set(objective(X))


TestST=np.array(T)




import numpy as np
#import jax.numpy as jnp
def stieltjesp(z,phi,r):
    # phi=100
    # r=np.sqrt(phi)
    gammap=r+1/r+2
    gammam=r+1/r-2
    s=(r-1/r-z +np.sqrt((z-gammam)*(gammap-z))*1j)/(2*z/r)
    return s
def stieltjesm(z,phi,r):
    return np.conjugate(stieltjesp(np.conjugate(z),phi,r))
def deriv_stp(z,phi,r):
    m=stieltjesp(z,phi,r)
    mp=(-r*m-np.power(m,2))/(2*z*m+r*z+1-phi)
    return mp
def deriv_stm(z,phi,r):
    m=stieltjesm(z,phi,r)
    mp=(-r*m-np.power(m,2))/(2*z*m+r*z+1-phi)
    return mp

def lsd(x,phi,r):
    return np.imag(stieltjesp(x+0.00001j,phi,r))/np.pi

import scipy.integrate as integrate


def testf(x,phi,r,c):
    return -np.log(x-r-1/r+c)





pn=2*jnp.ones(n)
T1 = jnp.zeros(R)
phi=100
m=int(n*phi)
key = jax.random.PRNGKey(0)
Sigmad = jnp.repeat(jnp.array([1]), m)
centerc = (jnp.sqrt(phi)+1/jnp.sqrt(phi)-c)*jnp.ones(n)
center = (jnp.sqrt(phi)+1/jnp.sqrt(phi))*jnp.ones(n)
for k in range(R):
      key, subkey = jax.random.split(key)
      X = random.normal(subkey, shape=(m, n))
      X = X * jnp.sqrt(Sigmad)[:, None]  # Scale the rows of X by the square root of the diagonal elements of Sigma
      H = jnp.matmul(X.T, X) / jnp.sqrt(n * m)
      E0 = jnp.linalg.eigvalsh(H)
      Eb= E0- centerc
      loE=jnp.log(Eb)
      # tt1=-jnp.sum(loE)
      tt1=-jnp.sum(loE)+jnp.sum(Eb)
      T1 = T1.at[k].set(tt1)




def power_phi(n,phi,R,c,t):
    m=int(n*phi)
    r=jnp.sqrt(phi)
    Sigmad = jnp.repeat(jnp.array([1,1+t]), m//2)
    center = (jnp.sqrt(phi)+1/jnp.sqrt(phi))*jnp.ones(n)
    centerc = (jnp.sqrt(phi)+1/jnp.sqrt(phi)-c)*jnp.ones(n)
    # objective_grad = jit(grad(objective))
    # T1 = jnp.zeros(R)
    T2 = jnp.zeros(R)
    key = jax.random.PRNGKey(0)
    for k in range(R):
        key, subkey = jax.random.split(key)
        X = random.normal(subkey, shape=(m, n))
        X = X * jnp.sqrt(Sigmad)[:, None]  # Scale the rows of X by the square root of the diagonal elements of Sigma
        H = jnp.matmul(X.T, X) / jnp.sqrt(n * m)
        E0 = jnp.linalg.eigvalsh(H)
        Ec= E0- center
        Eb = E0 - centerc
        # pn=2*jnp.ones(n)
        loE=jnp.log(Eb)
        # t = n*n*jnp.sum(loE)/jnp.power(jnp.sum(Eb),2)
        # tt1=-jnp.sum(loE)
        tt2=-jnp.sum(loE)+jnp.sum(Eb)
        #
        # T1 = T1.at[k].set(tt1)
        T2 = T2.at[k].set(tt2)
    t1=(c+np.sqrt(c**2-4))/2
    K4=0
    # result = integrate.quad(lambda x: testf(x,phi,r,c)*lsd(x,phi,r), r-2.2,r+2.2)
    # nlsd=n*result[0]

    # print(nlsd+0.5*np.log(1-1/(t1**2))-K4/2/t1/t1)
    TestST=np.array(T2)
    print(jnp.mean(TestST))
    # Normalize_TestST=(TestST-(nlsd+0.5*np.log(1-1/(t1**2))-n/r/c-K4/2/t1/t1)*np.ones(R))/np.sqrt(2/c/c-4/t1/c+2*np.log(t1)-2*np.log(t1-1/t1)+K4/t1/t1)
    Normalize_TestST=(TestST-jnp.mean(T1)*np.ones(R))/np.sqrt((2+K4)*(1-1/t1)+2*np.log(t1)-2*np.log(t1-1/t1)+K4/t1/t1)

    # plot_ecdf_and_standard_normal_cdf(Normalize_TestST)
    print(jnp.mean(Normalize_TestST))
    print(jnp.var(Normalize_TestST))
    True_Pos=count_outside_interval(Normalize_TestST,-1.96,1.96)
    Power=True_Pos/R
    return(Power)

P3=[]
temp=0
phi=100

for s in range(repeat):
    P3.append(power_phi(n,phi,R,c,0.01*s))
print(P3)
x_values = [0.01*s for s in range(repeat)]
plt.plot(x_values, P3)
plt.show()




############################################################


n = 200
R = 1000
c = 1

pn=2*jnp.ones(n)

@jit
def objective(X,Sigma,centerc):
    H = jnp.matmul(jnp.matmul(jnp.transpose(X), Sigma), X)
    E0 = jnp.linalg.eigh(H)[0]
    Eb = E0 - centerc
    pn=2*jnp.ones(n)
    loE=jnp.power(Eb,pn)
    t = n*n*jnp.sum(loE)/jnp.power(jnp.sum(Eb),2)
    return t


def power_phi(n,phi,R,c,t):
    m=int(n*phi)
    # print(m)
    # print(n)
    Sigmad = jnp.repeat(jnp.array([1, 1+t]), m//2)
    Sigma = jnp.diag(Sigmad)
    # print(Sigma)
    centerc = (jnp.sqrt(phi)+1/jnp.sqrt(phi)-c)*jnp.ones(n)
    objective_grad = jit(grad(objective))
    T = jnp.zeros(R)
    key = jax.random.PRNGKey(0)
    for k in range(R):
        key, subkey = jax.random.split(key)
        X = normal(subkey, shape=(m, n))/jnp.sqrt(jnp.sqrt(n*m))
        T = T.at[k].set(objective(X,Sigma,centerc))
    TestST=np.array(T)
    Normalize_TestST=(TestST-(n*(1+1/(c-1/np.sqrt(phi))**2)+1/(c-1/np.sqrt(phi))**2)*np.ones(R))/np.sqrt((4+4*2*(-1/np.sqrt(phi)+1/(c-1/np.sqrt(phi))+1/n/(c-np.sqrt(phi)))**2)/((c-1/np.sqrt(phi))**4))
    # plot_ecdf_and_standard_normal_cdf(Normalize_TestST)
    True_Pos=count_outside_interval(Normalize_TestST,-1.96,1.96)
    Power=True_Pos/R
    return(Power)


P4=[]
temp=0
phi=100

for s in range(repeat):
    P4.append(power_phi(n,phi,R,c,0.01*s))
print(P4)
x_values = [0.01*s for s in range(repeat)]
plt.plot(x_values, P4)
plt.show()

Saved data

P1=[0.052, 0.9415, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
P2=[0.0485, 0.8425, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
P3=[0.0504, 0.2661, 0.7171, 0.9463, 0.9931, 0.9994, 1.0, 1.0, 1.0, 1.0,  1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
P4=[0.053, 0.7945, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]

Plot

from google.colab import files
import matplotlib.pyplot as plt

# P1=[0.052, 0.9415, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
# P2=[0.0485, 0.8425, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
# P3=[0.0504, 0.2661, 0.7171, 0.9463, 0.9931, 0.9994, 1.0, 1.0, 1.0, 1.0,  1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
# P4=[0.053, 0.7945, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]

#### If u want to plot directly with the newly ran output,
#### remember to add 20 1's behind each vector P1, P2, P3 , P4 to make dimension the same
#### or use repeat=20
repeat=40
#repeat=20
x_values = [0.01*s for s in range(repeat)]
plt.plot(x_values, P1, label=r'$\mathbf{\mathcal{T}_1^{\mathrm{g}}}$')
plt.plot(x_values, P2, label=r'$\mathbf{\mathcal{T}_2^{\mathrm{g}}}$')
plt.plot(x_values, P3, label=r'$\mathbf{\mathcal{T}_3^{\mathrm{g}}}$')
plt.plot(x_values, P4, label=r'$\mathbf{\mathcal{T}_4^{\mathrm{g}}}$')
# plt.title(r"Power vs. $\varepsilon$")
# Adjusting xlabel, ylabel, title with bold font and larger fontsize
plt.xlabel(r"value of $\varepsilon$", fontweight='bold', fontsize=14)
plt.ylabel("Power", fontweight='bold', fontsize=14)
# plt.title(r"Power vs. $\varepsilon$", fontweight='bold', fontsize=16)  # Uncomment if you want a title

# Making legend text bold and larger
legend = plt.legend(fontsize='xx-large')
for text in legend.get_texts():
    text.set_fontweight('bold')

# Setting bold x-tick and y-tick labels with larger font
ax = plt.gca()
for label in ax.get_xticklabels():
    label.set_fontweight('bold')
    label.set_fontsize(15)
for label in ax.get_yticklabels():
    label.set_fontweight('bold')
    label.set_fontsize(15)

plt.tight_layout()
plt.savefig("global.eps")
# Download the figure
files.download("global.eps")

WARNING:matplotlib.backends.backend_ps:The PostScript backend does not support transparency; partially transparent artists will be rendered opaque.

Figure 5(b) local

Data generating

import jax
import jax.numpy as jnp
from jax import grad, jit,random
from jax.random import normal
import numpy as np
import matplotlib.pyplot as plt

# for t=0: 1: 0.1
t=0.000


n = 200
R = 200
c=1
T = jnp.zeros(R)
# key = jax.random.PRNGKey(0)

# for k in range(R):
#     key, subkey = jax.random.split(key)
#     X = normal(subkey, shape=(m, n))
#     T = T.at[k].set(objective(X))


TestST=np.array(T)

def count_outside_interval(arr, a, b):
    outside_interval = np.logical_or(arr < a, arr > b)
    count = np.sum(outside_interval)
    return count


def extract_elements(arr, a, b):
    return arr[(arr >= a) & (arr <= b)]



K4=0

# Normalize_TestST=(TestST-(n*(1+1/(c-1/np.sqrt(phi))**2)+1/(c-1/np.sqrt(phi))**2)*np.ones(R))/np.sqrt((4+4*2*(-1/np.sqrt(phi)+1/(c-1/np.sqrt(phi))+1/n/(c-np.sqrt(phi)))**2)/((c-1/np.sqrt(phi))**4))
# plot_ecdf_and_standard_normal_cdf(Normalize_TestST)


pn=2*jnp.ones(n)

eta=1/(n**(1/4))

k=int(n*eta)
R0=2000


#calculate the null distribution
phi=100
m=int(n*phi)
r=jnp.sqrt(phi)

leftb = (jnp.sqrt(phi)+1/jnp.sqrt(phi)-2)
    # objective_grad = jit(grad(objective))
T0 = jnp.zeros(R0)
key = jax.random.PRNGKey(0)
for r in range(R0):
    key, subkey = jax.random.split(key)
    X = random.normal(key, shape=(m, n))
    H = jnp.matmul(X.T, X) / jnp.sqrt(n * m)
    E00, _ = jnp.linalg.eigh(H)
    E0=np.array(E00)
    Et = jnp.array(extract_elements(E0, 8.1, 8.25))
    # print(E0[-k:])
    Eb = (Et - leftb*jnp.ones(Et.shape))/eta
    t0=jnp.sum(Eb)
    T0 = T0.at[r].set(t0)
TestST0=np.array(T0)
Mean_x=jnp.mean(TestST0)
Variance_x=jnp.var(TestST0)
print(Mean_x)
print(Variance_x)




def power_phi(n,phi,R,c,t):
    m=int(n*phi)
    r=jnp.sqrt(phi)
    Sigmad = jnp.repeat(jnp.array([1, 1+t]), m//2)
    eta=1/(n**(1/4))
    k=int(n*eta)

    leftb = (jnp.sqrt(phi)+1/jnp.sqrt(phi)-2)
    # objective_grad = jit(grad(objective))
    T = jnp.zeros(R)
    key = jax.random.PRNGKey(0)
    for r in range(R):
        key, subkey = jax.random.split(key)
        X = random.normal(subkey, shape=(m, n))
        X = X * jnp.sqrt(Sigmad)[:, None]  # Scale the rows of X by the square root of the diagonal elements of Sigma
        H = jnp.matmul(X.T, X) / jnp.sqrt(n * m)
        E00, _ = jnp.linalg.eigh(H)
        E0=np.array(E00)
        Et = jnp.array(extract_elements(E0, 8.1, 8.25))
        # print(E0[-k:])
        Eb = (Et - leftb*jnp.ones(Et.shape))/eta
        t1=jnp.sum(Eb)
        T = T.at[r].set(t1)
        TestST=np.array(T)
    # print(TestST)
    Normalize_TestST=(TestST-Mean_x*jnp.ones(R))/np.sqrt(Variance_x)
    # plot_ecdf_and_standard_normal_cdf(Normalize_TestST)
    print(jnp.mean(Normalize_TestST))
    print(jnp.var(Normalize_TestST))
    True_Pos=count_outside_interval(Normalize_TestST,-1.96,1.96)
    Power=True_Pos/R
    return(Power)

L1=[]
temp=0
phi=100

for s in range(repeat):
    L1.append(power_phi(n,phi,R,c,0.01*s))
print(L1)
x_values = [0.01*s for s in range(repeat)]
plt.plot(x_values, L1)
plt.show()

#########################################################################


# for t=0: 1: 0.1
t=0.000


n = 200
R = 2000
c=1
T = jnp.zeros(R)
# key = jax.random.PRNGKey(0)

# for k in range(R):
#     key, subkey = jax.random.split(key)
#     X = normal(subkey, shape=(m, n))
#     T = T.at[k].set(objective(X))


TestST=np.array(T)

def count_outside_interval(arr, a, b):
    outside_interval = np.logical_or(arr < a, arr > b)
    count = np.sum(outside_interval)
    return count


def extract_elements(arr, a, b):
    return arr[(arr >= a) & (arr <= b)]



K4=0

# Normalize_TestST=(TestST-(n*(1+1/(c-1/np.sqrt(phi))**2)+1/(c-1/np.sqrt(phi))**2)*np.ones(R))/np.sqrt((4+4*2*(-1/np.sqrt(phi)+1/(c-1/np.sqrt(phi))+1/n/(c-np.sqrt(phi)))**2)/((c-1/np.sqrt(phi))**4))
# plot_ecdf_and_standard_normal_cdf(Normalize_TestST)


pn=2*jnp.ones(n)

eta=1/(n**(1/4))

k=int(n*eta)
R0=2000


#calculate the null distribution
phi=100
m=int(n*phi)
r=jnp.sqrt(phi)

leftb = (jnp.sqrt(phi)+1/jnp.sqrt(phi)-2)
    # objective_grad = jit(grad(objective))
T0 = jnp.zeros(R0)
key = jax.random.PRNGKey(0)
for r in range(R0):
    key, subkey = jax.random.split(key)
    X = random.normal(key, shape=(m, n))
    H = jnp.matmul(X.T, X) / jnp.sqrt(n * m)
    E00, _ = jnp.linalg.eigh(H)
    E0=np.array(E00)
    Et = jnp.array(extract_elements(E0, 8.1, 8.25))
    # print(E0[-k:])
    Eb = (Et - leftb*jnp.ones(Et.shape))/eta
    t0=jnp.sum(jnp.square(Eb))
    T0 = T0.at[r].set(t0)
TestST0=np.array(T0)
Mean_x=jnp.mean(TestST0)
Variance_x=jnp.var(TestST0)
print(Mean_x)
print(Variance_x)




def power_phi(n,phi,R,c,t):
    m=int(n*phi)
    r=jnp.sqrt(phi)
    Sigmad = jnp.repeat(jnp.array([1, 1+t]), m//2)
    eta=1/(n**(1/4))
    k=int(n*eta)

    leftb = (jnp.sqrt(phi)+1/jnp.sqrt(phi)-2)
    # objective_grad = jit(grad(objective))
    T = jnp.zeros(R)
    key = jax.random.PRNGKey(0)
    for r in range(R):
        key, subkey = jax.random.split(key)
        X = random.normal(subkey, shape=(m, n))
        X = X * jnp.sqrt(Sigmad)[:, None]  # Scale the rows of X by the square root of the diagonal elements of Sigma
        H = jnp.matmul(X.T, X) / jnp.sqrt(n * m)
        E00, _ = jnp.linalg.eigh(H)
        E0=np.array(E00)
        Et = jnp.array(extract_elements(E0, 8.1, 8.25))
        # print(E0[-k:])
        Eb = (Et - leftb*jnp.ones(Et.shape))/eta
        t1=jnp.sum(jnp.square(Eb))
        T = T.at[r].set(t1)
        TestST=np.array(T)
    print(TestST)
    Normalize_TestST=(TestST-Mean_x*jnp.ones(R))/np.sqrt(Variance_x)
    # plot_ecdf_and_standard_normal_cdf(Normalize_TestST)
    print(jnp.mean(Normalize_TestST))
    print(jnp.var(Normalize_TestST))
    True_Pos=count_outside_interval(Normalize_TestST,-1.96,1.96)
    Power=True_Pos/R
    if t==0.9:
      print(Normalize_TestST)
    return(Power)

L2=[]
temp=0
phi=100

for s in range(repeat):
    L2.append(power_phi(n,phi,R,c,0.01*s))
print(L2)
x_values = [0.01*s for s in range(repeat)]
plt.plot(x_values, L2)
plt.show()


################################################################


t=0.000


n = 200
R = 2000
c=1
T = jnp.zeros(R)
# key = jax.random.PRNGKey(0)

# for k in range(R):
#     key, subkey = jax.random.split(key)
#     X = normal(subkey, shape=(m, n))
#     T = T.at[k].set(objective(X))


TestST=np.array(T)

def count_outside_interval(arr, a, b):
    outside_interval = np.logical_or(arr < a, arr > b)
    count = np.sum(outside_interval)
    return count


def extract_elements(arr, a, b):
    return arr[(arr >= a) & (arr <= b)]



K4=0

# Normalize_TestST=(TestST-(n*(1+1/(c-1/np.sqrt(phi))**2)+1/(c-1/np.sqrt(phi))**2)*np.ones(R))/np.sqrt((4+4*2*(-1/np.sqrt(phi)+1/(c-1/np.sqrt(phi))+1/n/(c-np.sqrt(phi)))**2)/((c-1/np.sqrt(phi))**4))
# plot_ecdf_and_standard_normal_cdf(Normalize_TestST)


# Normalize_TestST=(TestST-(n*(1+1/(c-1/np.sqrt(phi))**2)+1/(c-1/np.sqrt(phi))**2)*np.ones(R))/np.sqrt((4+4*2*(-1/np.sqrt(phi)+1/(c-1/np.sqrt(phi))+1/n/(c-np.sqrt(phi)))**2)/((c-1/np.sqrt(phi))**4))
# plot_ecdf_and_standard_normal_cdf(Normalize_TestST)


pn=2*jnp.ones(n)

eta=1/(n**(1/4))

k=int(n*eta)
R0=200


#calculate the null distribution
phi=100
m=int(n*phi)
r=jnp.sqrt(phi)

leftb = (jnp.sqrt(phi)+1/jnp.sqrt(phi)-2)
    # objective_grad = jit(grad(objective))
T0 = jnp.zeros(R0)
key = jax.random.PRNGKey(0)
for r in range(R0):
    key, subkey = jax.random.split(key)
    X = random.normal(key, shape=(m, n))
    H = jnp.matmul(X.T, X) / jnp.sqrt(n * m)
    E00, _ = jnp.linalg.eigh(H)
    E0=np.array(E00)
    Et = jnp.array(extract_elements(E0, 8.1, 8.25))
    # print(E0[-k:])
        # print(E0[-k:])
    Eb = (Et - leftb*jnp.ones(Et.shape))/eta
    t0=jnp.sum(np.log(Eb+2*np.ones(np.size(Eb))))+jnp.sum(Eb)
    T0 = T0.at[r].set(t0)

Mean_x=jnp.mean(T0)
Variance_x=jnp.var(T0)
print(Mean_x)
print(Variance_x)





def power_phi(n,phi,R,c,t):
    m=int(n*phi)
    r=jnp.sqrt(phi)
    Sigmad = jnp.repeat(jnp.array([1, 1+t]), m//2)
    eta=1/(n**(1/4))
    k=int(n*eta)

    leftb = (jnp.sqrt(phi)+1/jnp.sqrt(phi)-2)
    # objective_grad = jit(grad(objective))
    T = jnp.zeros(R)
    key = jax.random.PRNGKey(0)
    for r in range(R):
        key, subkey = jax.random.split(key)
        X = random.normal(subkey, shape=(m, n))
        X = X * jnp.sqrt(Sigmad)[:, None]  # Scale the rows of X by the square root of the diagonal elements of Sigma
        H = jnp.matmul(X.T, X) / jnp.sqrt(n * m)
        E00, _ = jnp.linalg.eigh(H)
        E0=np.array(E00)
        Et = jnp.array(extract_elements(E0, 8.1, 8.25))
        # print(E0[-k:])
        Eb = (Et - leftb*jnp.ones(Et.shape))/eta
        t0=jnp.sum(np.log(Eb+2*np.ones(np.size(Eb))))+jnp.sum(Eb)
        T = T.at[r].set(t0)
        TestST=np.array(T)
    print(TestST)
    Normalize_TestST=(TestST-Mean_x*jnp.ones(R))/np.sqrt(Variance_x)
    # plot_ecdf_and_standard_normal_cdf(Normalize_TestST)
    print(jnp.mean(Normalize_TestST))
    print(jnp.var(Normalize_TestST))
    True_Pos=count_outside_interval(Normalize_TestST,-1.96,1.96)
    Power=True_Pos/R
    if t==0.9:
      print(Normalize_TestST)
    return(Power)

L3=[]
temp=0
phi=100

for s in range(repeat):
    L3.append(power_phi(n,phi,R,c,0.01*s))
print(L3)
x_values = [0.01*s for s in range(repeat)]
plt.plot(x_values, L3)
plt.show()




##############################################################



# for t=0: 1: 0.1
t=0.000


n = 200
R = 200
c=1
T = jnp.zeros(R)
# key = jax.random.PRNGKey(0)

# for k in range(R):
#     key, subkey = jax.random.split(key)
#     X = normal(subkey, shape=(m, n))
#     T = T.at[k].set(objective(X))


TestST=np.array(T)

def count_outside_interval(arr, a, b):
    outside_interval = np.logical_or(arr < a, arr > b)
    count = np.sum(outside_interval)
    return count


def extract_elements(arr, a, b):
    return arr[(arr >= a) & (arr <= b)]



K4=0

# Normalize_TestST=(TestST-(n*(1+1/(c-1/np.sqrt(phi))**2)+1/(c-1/np.sqrt(phi))**2)*np.ones(R))/np.sqrt((4+4*2*(-1/np.sqrt(phi)+1/(c-1/np.sqrt(phi))+1/n/(c-np.sqrt(phi)))**2)/((c-1/np.sqrt(phi))**4))
# plot_ecdf_and_standard_normal_cdf(Normalize_TestST)


pn=2*jnp.ones(n)

eta=1/(n**(1/4))

k=int(n*eta)
R0=2000


#calculate the null distribution
phi=100
m=int(n*phi)
r=jnp.sqrt(phi)



leftb = (jnp.sqrt(phi)+1/jnp.sqrt(phi)-2)
    # objective_grad = jit(grad(objective))
T0 = jnp.zeros(R0)
key = jax.random.PRNGKey(0)
for r in range(R0):
    key, subkey = jax.random.split(key)
    X = random.normal(key, shape=(m, n))
    H = jnp.matmul(X.T, X) / jnp.sqrt(n * m)
    E00, _ = jnp.linalg.eigh(H)
    E0=np.array(E00)
    Et = jnp.array(extract_elements(E0, 8.1, 8.25))
    # print(E0[-k:])
    Eb = (Et - leftb*jnp.ones(Et.shape))/eta
    loE=jnp.power(Eb,2*jnp.ones(np.size(Eb)))
    t0 = n*n*jnp.sum(loE)/jnp.power(jnp.sum(Eb),2)
    T0 = T0.at[r].set(t0)
TestST0=np.array(T0)
Mean_x=jnp.mean(TestST0)
Variance_x=jnp.var(TestST0)
print(Mean_x)
print(Variance_x)




def power_phi(n,phi,R,c,t):
    m=int(n*phi)
    r=jnp.sqrt(phi)
    Sigmad = jnp.repeat(jnp.array([1, 1+t]), m//2)
    eta=1/(n**(1/4))
    k=int(n*eta)

    leftb = (jnp.sqrt(phi)+1/jnp.sqrt(phi)-2)
    # objective_grad = jit(grad(objective))
    T = jnp.zeros(R)
    key = jax.random.PRNGKey(0)
    for r in range(R):
        key, subkey = jax.random.split(key)
        X = random.normal(subkey, shape=(m, n))
        X = X * jnp.sqrt(Sigmad)[:, None]  # Scale the rows of X by the square root of the diagonal elements of Sigma
        H = jnp.matmul(X.T, X) / jnp.sqrt(n * m)
        E00, _ = jnp.linalg.eigh(H)
        E0=np.array(E00)
        Et = jnp.array(extract_elements(E0, 8.1, 8.25))
    # print(E0[-k:])
        Eb = (Et - leftb*jnp.ones(Et.shape))/eta
        loE=jnp.power(Eb,2*jnp.ones(np.size(Eb)))
        t1 = n*n*jnp.sum(loE)/jnp.power(jnp.sum(Eb),2)
        T = T.at[r].set(t1)
        TestST=np.array(T)
    # print(TestST)
    Normalize_TestST=(TestST-Mean_x*jnp.ones(R))/np.sqrt(Variance_x)
    # plot_ecdf_and_standard_normal_cdf(Normalize_TestST)
    print(jnp.mean(Normalize_TestST))
    print(jnp.var(Normalize_TestST))
    True_Pos=count_outside_interval(Normalize_TestST,-1.96,1.96)
    Power=True_Pos/R
    return(Power)

L4=[]
temp=0
phi=100

for s in range(repeat):
    L4.append(power_phi(n,phi,R,c,0.01*s))
print(L4)
x_values = [0.01*s for s in range(repeat)]
plt.plot(x_values, L4)
plt.show()

Saved data

L1=[0.06, 0.125, 0.285, 0.705, 0.935, 0.99, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
L2=[0.053, 0.0685, 0.1935, 0.598, 0.916, 0.991, 0.9995, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
L3=[0.0445, 0.1825, 0.7355, 0.982, 0.9995, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
L4=[0.03, 0.365, 0.89, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0,  1.0, 1.0, 1.0,1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0,1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]

Plot

# L1=[0.06, 0.125, 0.285, 0.705, 0.935, 0.99, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
# L2=[0.053, 0.0685, 0.1935, 0.598, 0.916, 0.991, 0.9995, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
# L3=[0.0445, 0.1825, 0.7355, 0.982, 0.9995, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
# L4=[0.03, 0.365, 0.89, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0,  1.0, 1.0, 1.0,1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0,1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]

from google.colab import files

#### Similar to the global case, if u want to plot directly with the newly obtained output,
#### remember to add 20 1's behind each vector L1, L2, L3 , L4 to make dimension the same
#### or use repeat=20

repeat=40
x_values = [0.01*s for s in range(repeat)]

plt.plot(x_values, L1, label=r'$\mathbf{\mathcal{T}_1^{\mathrm{l}}}$')
plt.plot(x_values, L2, label=r'$\mathbf{\mathcal{T}_2^{\mathrm{l}}}$')
plt.plot(x_values, L3, label=r'$\mathbf{\mathcal{T}_3^{\mathrm{l}}}$')
plt.plot(x_values, L4, label=r'$\mathbf{\mathcal{T}_4^{\mathrm{l}}}$')
# plt.title(r"Power vs. $\varepsilon$")
# Adjusting xlabel, ylabel, title with bold font and larger fontsize
plt.xlabel(r"value of $\varepsilon$", fontweight='bold', fontsize=14)
plt.ylabel("Power", fontweight='bold', fontsize=14)
# plt.title(r"Power vs. $\varepsilon$", fontweight='bold', fontsize=16)  # Uncomment if you want a title

# Making legend text bold and larger
legend = plt.legend(fontsize='xx-large')
for text in legend.get_texts():
    text.set_fontweight('bold')

# Setting bold x-tick and y-tick labels with larger font
ax = plt.gca()
for label in ax.get_xticklabels():
    label.set_fontweight('bold')
    label.set_fontsize(15)
for label in ax.get_yticklabels():
    label.set_fontweight('bold')
    label.set_fontsize(15)

plt.tight_layout()


plt.savefig("local.eps")

# Download the figure
files.download("local.eps")

WARNING:matplotlib.backends.backend_ps:The PostScript backend does not support transparency; partially transparent artists will be rendered opaque.

Figure 6: ROC

In this section, we display the ROC curve of our proposed tests compared with some of the existing literature.

Be cautious, this part of the simulation is extremely time-consuming due to the fact that around 10 methods are replicated. If you want to replicate the results here, use smaller $n$, $p$ or be prepared for the longer hours (days).

But if you only want to try our method, comment out the other lines and it will not take long.

Figure 6(a) $\varepsilon=0.03$

Data generating

import numpy as np
import matplotlib.pyplot as plt
from sklearn.metrics import roc_curve, auc

def perform_test(TestFunc, Sigmad, m=2000, n=40, num_matrices=1000):
    p_values0 = []
    p_values1 = []

    for _ in range(num_matrices):
        data0 = np.random.normal(0, 1, (m, n))
        p0 = TestFunc(data0.T)
        if p0 is not None:
            p_values0.append(p0)
        else:
            p_values0.append(0)
        del data0

        data0_prime = np.random.normal(0, 1, (m, n))
        data1 = data0_prime * np.sqrt(Sigmad)[:, None]
        p1 = TestFunc(data1.T)
        if p1 is not None:
            p_values1.append(p1)
        else:
            p_values1.append(0)
        del data1, data0_prime

    return np.array(p_values0), np.array(p_values1)
from sklearn.metrics import roc_curve, auc

def generate_roc_curve(p_values0, p_values1, label, lw=2, linestyle='-', alpha=0.8):
    # calculate TPRs and FPRs for each threshold
    thresholds = np.linspace(0, 1, 100)
    tprs = []
    fprs = []

    for threshold in thresholds:
        tpr = np.sum(p_values1 < threshold) / num_matrices
        fpr = np.sum(p_values0 < threshold) / num_matrices

        tprs.append(tpr)
        fprs.append(fpr)

    # calculate AUC
    roc_auc = auc(fprs, tprs)

    # # plot ROC curve
    # # Use string formatting to control the presentation of AUC in the legend
    # legend_text = "{:<16} ({:0.2f})".format(label, roc_auc)

    # # plot ROC curve
    # plt.plot(fprs, tprs, linestyle=linestyle, alpha=alpha,lw=lw, label=legend_text)

    # plt.plot(fprs, tprs, linestyle=linestyle, alpha=alpha,lw=lw, label=f'({roc_auc:.2f}){label} ')

    plt.plot(fprs, tprs, linestyle=linestyle, alpha=alpha,lw=lw, label=f'{label}  ({roc_auc:.2f}) ')
    # label_padded = label.ljust(15)
    # plt.plot(fprs, tprs, linestyle=linestyle, alpha=alpha,lw=lw, label=f'{label_padded} (AUC: {roc_auc:.2f})')


import matplotlib.pyplot as plt
# Initialize the plot
plt.figure()

# Set parameters
# m = 20000
# n = 200
m = 10000
n = 100
num_matrices = 100
t = 0.03
Sigmad = np.repeat(np.array([1, 1+t]), m//2)

# Perform Test1 and plot its ROC curve
p_values0_test1, p_values1_test1 = perform_test(Nagao1973, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test1, p_values1_test1, 'Nagao73',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test1)
# print(p_values1_test1)
# Perform Test2 and plot its ROC curve
p_values0_test2, p_values1_test2 = perform_test(Srivastava2005, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test2, p_values1_test2, 'Srivastava05',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test2)
# print(p_values1_test2)
p_values0_test3, p_values1_test3 = perform_test(Srivastava2011, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test3, p_values1_test3, 'Srivastava11',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test3)
# print(p_values1_test3)
p_values0_test4, p_values1_test4 = perform_test(Chen2010, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test4, p_values1_test4, 'Chen10',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test4)
# print(p_values1_test4)

p_values0_test5, p_values1_test5 = perform_test(LedoitWolf2002, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test5, p_values1_test5, 'Wolf02',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test5)
# print(p_values1_test5)
p_values0_test6, p_values1_test6 = perform_test(Fisher2012, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test6, p_values1_test6, 'Fisher12',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test6)
# print(p_values1_test6)
p_values0_test7, p_values1_test7 = perform_test(Ahmad2015, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test7, p_values1_test7, 'Ahmad15',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test7)
# print(p_values1_test7)
p_values0_test8, p_values1_test8 = perform_test(DW2023Global, Sigmad, m, n, num_matrices)
p_values0_test10, p_values1_test10 = p_values0_test8, p_values1_test8
generate_roc_curve(p_values0_test8, p_values1_test8, 'Qiu21',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test10)
# print(p_values1_test10)
generate_roc_curve(p_values0_test8, p_values1_test8, 'PM1',lw=2, linestyle='--', alpha=1)
# print(p_values0_test8)
# print(p_values1_test8)
p_values0_test9, p_values1_test9 = perform_test(DW2023Local, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test9, p_values1_test9, 'PM2',lw=2, linestyle='--', alpha=1)
# print(p_values0_test9)
# print(p_values1_test9)

p_values0_test11, p_values1_test11 = perform_test(CLRT, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test11, p_values1_test11, 'CLRT',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test11)
# print(p_values1_test11)















# plt.plot([0, 1], [0, 1], color='navy', lw=2, linestyle='dotted')
# plt.xlim([0.0, 1.0])
# plt.ylim([0.0, 1.05])
# # ... [Your other code]

# # Setting bold x-label and y-label
# plt.xlabel('False Positive Rate', fontweight='bold')
# plt.ylabel('True Positive Rate', fontweight='bold')
# plt.title('Receiver Operating Characteristic', fontweight='bold')

# # Setting the legend and then making its text bold
# legend = plt.legend(loc="lower right", fontsize='small')
# for text in legend.get_texts():
#     text.set_fontweight('bold')

# # Setting bold x-tick and y-tick labels
# ax = plt.gca()  # Get current axis
# for label in ax.get_xticklabels():
#     label.set_fontweight('bold')

# for label in ax.get_yticklabels():
#     label.set_fontweight('bold')

# plt.show()

Saved data

p_values0_test1=[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
p_values1_test1=[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
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p_values1_test9=[0.07039294204031754, 0.1164076389642581, 0.8786374876329686, 0.009544007264397303, 0.044999062088537656, 0.06228753285634436, 0.028521120436967307, 0.045052548671586035, 0.011685896471893331, 0.01373056633607228, 0.05115724740559813, 0.0074782847404955, 0.033607387847615966, 0.012657904556399702, 0.1783080668890873, 0.0014879135470073912, 0.20150598783127371, 0.3827375311308687, 0.5726141854758495, 0.027841214735923048, 0.013455978349000919, 0.5694323863665991, 0.011962287461888987, 0.1848055918430349, 0.1549374893843718, 0.023148138154342357, 0.08140701709626508, 0.05312033757678947, 0.7105521462254472, 0.8582901410641017, 0.06139092356913678, 1.256423388107386e-05, 0.5017799763903261, 0.17103096402718765, 0.04880282887002352, 0.029943465542798364, 0.1943908781260113, 0.2095529446859956, 0.3517946189671659, 0.07600445912117526, 0.020172879815893108, 0.01694865130675227, 0.04097546559853393, 0.10056948416292455, 0.02572233302057314, 0.24644008568605846, 0.04492305848333622, 0.05440500049765018, 0.04104689377348314, 0.7069644085600917, 0.0025284713984163876, 0.07586417952853175, 0.021046000633547113, 0.036440285247616444, 0.0013015223176793267, 0.0027874496779196622, 0.4492260197507958, 0.08431234127347698, 0.3633575933319073, 0.48605075541413933, 0.43756767836331045, 0.0017894226621462117, 0.009752095772410652, 0.8751518106468659, 0.041665723712704716, 0.9802892415504765, 0.16185622768403007, 0.5290936062378921, 0.03653425521502296, 0.5161748139679576, 0.4889570357928812, 0.0014203726055153076, 0.6733917754725534, 0.7224774999768151, 0.1548999263079216, 0.00654921909099393, 0.3874031354297416, 0.4760519593145838, 0.004651874754067364, 0.350833217942891, 0.35378219208282724, 0.2761352299652571, 0.026161281313607176, 0.161368726981316, 0.0417797621241871, 0.07798925547123847, 0.837497849994586, 0.0546331320347182, 0.04629416582883672, 0.02392672052100564, 0.37072433235246827, 0.033199269257943476, 0.32482814268420324, 0.23922156314428378, 0.15944713045691294, 0.9799659975493, 0.037995856092295055, 0.03981486219669517, 0.2031674904636409, 0.3461023336437392]
p_values0_test10=[0.9774664234370511, 0.19358334582183812, 0.48917930433742685, 0.7484328524808119, 0.7198448763809364, 0.7105619645880754, 0.4088078557894921, 0.34808634234440605, 0.5859629961077106, 0.9558657197450151, 0.7334071212922071, 0.30156881526447554, 0.6918549734775221, 0.5350919375179501, 0.6954532033729453, 0.7962529922119905, 0.25966243246147425, 0.20295255963192904, 0.6167449270212582, 0.959557346829125, 0.006850505370554982, 0.036101928912106285, 0.5787341415395568, 0.17484883255359351, 0.46377041936664454, 0.9947189368542635, 0.6788493666612949, 0.1556824354212225, 0.45095218898762424, 0.10889833690150641, 0.2779094056157141, 0.9718480634030691, 0.3799254856940568, 0.7176288061115992, 0.7702693686156792, 0.9643958022642289, 0.5164982946986822, 0.6713967753606125, 0.8672034387829224, 0.8326890833514544, 0.8334800251221979, 0.09227091363363815, 0.10883522177213378, 0.7718176033467867, 0.7724897308493475, 0.7034999994708078, 0.17068754982596301, 0.817894027736517, 0.5327043020264077, 0.46530145416355273, 0.9640607936221619, 0.12600199714289606, 0.4159471040582503, 0.1913192009240139, 0.1715502369386761, 0.04054745740082444, 0.9904713288501468, 0.9379387433869901, 0.5682932300858314, 0.746756183193271, 0.7467963399912607, 0.4968377385915166, 0.3680973888028234, 0.548285732629473, 0.2631260588339197, 0.06568800315199907, 0.44556417970243767, 0.17279801701638364, 0.10033879647356048, 0.6143664492089298, 0.1357558084582493, 0.41261119530842505, 0.5803488272441779, 0.7463945757992294, 0.9869482840389345, 0.8040993023401823, 0.6335599752320298, 0.6886721089060104, 0.7456727001458634, 0.5973113507749512, 0.7743958168644163, 0.6471598322960799, 0.4019365483649311, 0.531649871383189, 0.9484202029730684, 0.9795489970277478, 0.9329204662659738, 0.8540487409934685, 0.13219437098245113, 0.8079439001955138, 0.35093986936433996, 0.2671148210142609, 0.42019901179513397, 0.0026232402133588018, 0.7476493832543871, 0.3285199169098916, 0.2879998548637872, 0.6671838838845048, 0.8279178435736907, 0.4439588290381522]
p_values1_test10=[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 8.881784197001252e-16, 0.0, 0.0, 0.0, 1.5019097077129118e-12, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
p_values0_test11=[0.1887616675799313, 0.9296974959386481, 0.3295736189173657, 0.43460864737953697, 0.5085032820771471, 0.7680384507754916, 0.031124306144526548, 0.48554660520105886, 0.9779473184622516, 0.991167052751158, 0.01177299761012729, 0.5901585541619996, 0.8466863288740261, 0.4065597098078526, 0.9542591175441038, 0.23734533581034456, 0.8497131397206645, 0.6038157309522725, 0.14519681945526375, 0.38945849551678524, 0.22762421688590662, 0.1892392645114498, 0.7441282722234197, 0.01433648622847472, 0.7728579253071688, 0.8383333008401799, 0.7557514005528754, 0.23962545443864913, 0.9412933042929739, 0.8867022717119271, 0.5614138188995195, 0.39478234722731687, 0.5512746629819376, 0.041154769360477905, 0.9058217657114143, 0.8204541207381499, 0.049782289083723175, 0.9080261172705864, 0.26672934786742, 0.82915990766621, 0.5504110027422537, 0.840285672119929, 0.22610667084034874, 0.04181191249094751, 0.69188381904705, 0.09623240308159908, 0.37596296787292305, 0.7675173580837997, 0.00013222435477255878, 0.7141645861858148, 0.08769848644420764, 0.961098387278567, 0.14773627892632923, 0.6787033250206804, 0.14137860808156932, 0.8765479334556543, 0.8846277929202588, 0.5111137199873985, 0.13807768605808612, 0.9589220222823582, 0.4488913852181904, 0.6938829184045994, 0.5677957815728405, 0.8401623433638292, 0.584899311508714, 0.8568405882583925, 0.3643061104854317, 0.11037081000670979, 0.8155504175991735, 0.22904599637933698, 0.37967264401648393, 0.4002968213179945, 0.9202305641069413, 0.3740842536717519, 0.683106011345572, 0.5675649058236045, 0.993372886219785, 0.15309155237614114, 0.8050586384014276, 0.11441791701801796, 0.8488044968212916, 0.40487613425360713, 0.6664364793295876, 0.34431662541246455, 0.14084206241989272, 0.9772980371356879, 0.8819236125688299, 0.38170470894351594, 0.29813650274322434, 0.5191860547870957, 0.45205167320955497, 0.9012763736507987, 0.10685963463046666, 0.5790638704222757, 0.4381606611583977, 0.9885926712806683, 0.17891799966666433, 0.6295528970752207, 0.4413961199141472, 0.34543121286988265]
p_values1_test11=[0.5015141259906777, 0.7368909681570839, 0.9849452522317315, 0.11107886907953013, 0.09503049575653444, 0.5260016135532517, 0.8869129675226148, 0.6006911795680094, 0.27653172119997804, 0.8270763144374389, 0.3952760161482707, 0.9859027193734979, 0.934467867128002, 0.36632226429000103, 0.7848157711365613, 0.45493551352785033, 0.9835359837551085, 0.8799459388541548, 0.39641844016205763, 0.442153404368514, 0.26483416439023166, 0.3278657942286314, 0.8676226351389074, 0.3761625753325246, 0.8658602712911185, 0.7204653558740923, 0.24637879687042474, 0.26229370162593035, 0.9825537720838784, 0.316499620493389, 0.24878023980148, 0.6191760360643221, 0.4262110337075531, 0.033197141405008956, 0.9759536712039727, 0.5904112200584577, 0.7033465439112461, 0.7277772066558204, 0.7036077326622601, 0.6165899864758195, 0.21603496695635438, 0.317747264508653, 0.8901024769443857, 0.13816565277228543, 0.007330432154567079, 0.05570750751349207, 0.6328643395569911, 0.6996964055228527, 0.15026947092084808, 0.6925552971272619, 0.576058210642246, 0.7996715075141754, 0.5143120890628459, 0.36181399327458497, 0.08799790299093058, 0.7927245851016222, 0.24942299011568103, 0.6499580443436233, 0.551189747266047, 0.6981258968453847, 0.6503874828644713, 0.2677645506661337, 0.7095818565294674, 0.3432599619718548, 0.6905723546612987, 0.1709015759043928, 0.7925238202983302, 0.7052515516502162, 0.1536547777889845, 0.1824158981599715, 0.946487652490001, 0.9202383856765923, 0.6848772589253769, 0.7190193631064203, 0.2707926706923742, 0.6509275156564001, 0.8197494279134747, 0.338018108930525, 0.5833061184305368, 0.7786715159544855, 0.020741715682890627, 0.6263444461223961, 0.9405210132617736, 0.7337784045954432, 0.887697222310907, 0.8657270197610007, 0.01414645804575021, 0.27073980426077693, 0.8725327873999045, 0.44524317837638616, 0.09496763140470788, 0.6593358636462963, 0.497823044338916, 0.20059627926742518, 0.6427979475578454, 0.25899215546372156, 0.3100872215998649, 0.6408420992393871, 0.1998623210048902, 0.8064130151700284]

Plot

import numpy as np
import matplotlib.pyplot as plt
from sklearn.metrics import roc_curve, auc
from google.colab import files
import matplotlib.gridspec as gridspec
num_matrices=100

def generate_roc_curve(p_values0, p_values1, label, lw=2, linestyle='-', alpha=0.8):
    # calculate TPRs and FPRs for each threshold
    thresholds = np.linspace(0, 1, 100)
    tprs = []
    fprs = []

    for threshold in thresholds:
        tpr = np.sum(p_values1 < threshold) / num_matrices
        fpr = np.sum(p_values0 < threshold) / num_matrices

        tprs.append(tpr)
        fprs.append(fpr)

    # calculate AUC
    roc_auc = auc(fprs, tprs)

    # # plot ROC curve
    # # Use string formatting to control the presentation of AUC in the legend
    # legend_text = "{:<16} ({:0.2f})".format(label, roc_auc)

    # # plot ROC curve
    # plt.plot(fprs, tprs, linestyle=linestyle, alpha=alpha,lw=lw, label=legend_text)

    # plt.plot(fprs, tprs, linestyle=linestyle, alpha=alpha,lw=lw, label=f'({roc_auc:.2f}){label} ')

    plt.plot(fprs, tprs, linestyle=linestyle, alpha=alpha,lw=lw, label=f'{label}  ({roc_auc:.2f}) ')
    # label_padded = label.ljust(15)
    # plt.plot(fprs, tprs, linestyle=linestyle, alpha=alpha,lw=lw, label=f'{label_padded} (AUC: {roc_auc:.2f})')

def bold_legend(legend):
    """Utility function to set legend text to bold."""
    for text in legend.get_texts():
        text.set_fontweight('bold')


def set_ticks_bold(ax):
    """Utility function to set tick labels to bold."""
    for label in ax.get_xticklabels():
        label.set_fontweight('bold')
    for label in ax.get_yticklabels():
        label.set_fontweight('bold')

# Create a GridSpec layout
gs = gridspec.GridSpec(1, 2, width_ratios=[1.5, 0.2])

fig = plt.figure(figsize=(8.5, 6))

# First subplot
ax0 = fig.add_subplot(gs[0])

# Perform Test1 and plot its ROC curve
# p_values0_test1, p_values1_test1 = perform_test(Nagao1973, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test1, p_values1_test1, 'Nagao73',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test1)
# print(p_values1_test1)
# Perform Test2 and plot its ROC curve
# p_values0_test2, p_values1_test2 = perform_test(Srivastava2005, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test2, p_values1_test2, 'Srivastava05',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test2)
# print(p_values1_test2)
# p_values0_test3, p_values1_test3 = perform_test(Srivastava2011, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test3, p_values1_test3, 'Srivastava11',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test3)
# print(p_values1_test3)
# p_values0_test4, p_values1_test4 = perform_test(Chen2010, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test4, p_values1_test4, 'Chen10',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test4)
# print(p_values1_test4)

# p_values0_test5, p_values1_test5 = perform_test(LedoitWolf2002, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test5, p_values1_test5, 'Wolf02',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test5)
# print(p_values1_test5)
# p_values0_test6, p_values1_test6 = perform_test(Fisher2012, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test6, p_values1_test6, 'Fisher12',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test6)
# print(p_values1_test6)
# p_values0_test7, p_values1_test7 = perform_test(Ahmad2015, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test7, p_values1_test7, 'Ahmad15',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test7)
# print(p_values1_test7)
# p_values0_test8, p_values1_test8 = perform_test(DW2023Global, Sigmad, m, n, num_matrices)
# p_values0_test10, p_values1_test10 = p_values0_test8, p_values1_test8
generate_roc_curve(p_values0_test8, p_values1_test8, 'Qiu21',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test10)
# print(p_values1_test10)

# p_values0_test11, p_values1_test11 = perform_test(CLRT, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test11, p_values1_test11, 'CLRT',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test11)
# print(p_values1_test11)

generate_roc_curve(p_values0_test8, p_values1_test8, 'PM1',lw=2, linestyle='--', alpha=1)
# print(p_values0_test8)
# print(p_values1_test8)
# p_values0_test9, p_values1_test9 = perform_test(DW2023Local, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test9, p_values1_test9, 'PM2',lw=2, linestyle='--', alpha=1)
# print(p_values0_test9)
# print(p_values1_test9)



set_ticks_bold(ax0)
ax0.plot([-0.1, 1], [0, 1], color='navy', lw=2, linestyle='dotted')
ax0.set_xlim([0.0, 1.0])
ax0.set_ylim([0.0, 1.05])
ax0.set_xlabel('False Positive Rate', fontweight='bold', fontsize=14)
ax0.set_ylabel('True Positive Rate', fontweight='bold', fontsize=14)

ax0.set_title('Receiver Operating Characteristic', fontweight='bold', fontsize=16)
ax0.tick_params(axis='both', which='major', labelsize=18,  width=1, length=5)



# Second subplot for the legend
ax1 = fig.add_subplot(gs[1])
ax1.axis('off')
handles, labels = ax0.get_legend_handles_labels()
legend = ax1.legend(handles, labels, loc='center',handlelength=0.5)
bold_legend(legend)


plt.tight_layout()
plt.savefig("n100p10000t0dot03.eps",dpi=300, bbox_inches='tight')
files.download("n100p10000t0dot03.eps")
plt.show()

WARNING:matplotlib.backends.backend_ps:The PostScript backend does not support transparency; partially transparent artists will be rendered opaque.

Figure 6(b) $\varepsilon=0.3$

Data generating

import numpy as np
import matplotlib.pyplot as plt
from sklearn.metrics import roc_curve, auc

def perform_test(TestFunc, Sigmad, m=2000, n=40, num_matrices=1000):
    p_values0 = []
    p_values1 = []

    for _ in range(num_matrices):
        data0 = np.random.normal(0, 1, (m, n))
        p0 = TestFunc(data0.T)
        if p0 is not None:
            p_values0.append(p0)
        else:
            p_values0.append(0)
        del data0

        data0_prime = np.random.normal(0, 1, (m, n))
        data1 = data0_prime * np.sqrt(Sigmad)[:, None]
        p1 = TestFunc(data1.T)
        if p1 is not None:
            p_values1.append(p1)
        else:
            p_values1.append(0)
        del data1, data0_prime

    return np.array(p_values0), np.array(p_values1)
from sklearn.metrics import roc_curve, auc

def generate_roc_curve(p_values0, p_values1, label, lw=2, linestyle='-', alpha=0.8):
    # calculate TPRs and FPRs for each threshold
    thresholds = np.linspace(0, 1, 100)
    tprs = []
    fprs = []

    for threshold in thresholds:
        tpr = np.sum(p_values1 < threshold) / num_matrices
        fpr = np.sum(p_values0 < threshold) / num_matrices

        tprs.append(tpr)
        fprs.append(fpr)

    # calculate AUC
    roc_auc = auc(fprs, tprs)

    # # plot ROC curve
    # # Use string formatting to control the presentation of AUC in the legend
    # legend_text = "{:<16} ({:0.2f})".format(label, roc_auc)

    # # plot ROC curve
    # plt.plot(fprs, tprs, linestyle=linestyle, alpha=alpha,lw=lw, label=legend_text)

    # plt.plot(fprs, tprs, linestyle=linestyle, alpha=alpha,lw=lw, label=f'({roc_auc:.2f}){label} ')

    plt.plot(fprs, tprs, linestyle=linestyle, alpha=alpha,lw=lw, label=f'{label}  ({roc_auc:.2f}) ')
    # label_padded = label.ljust(15)
    # plt.plot(fprs, tprs, linestyle=linestyle, alpha=alpha,lw=lw, label=f'{label_padded} (AUC: {roc_auc:.2f})')
import matplotlib.pyplot as plt
# Initialize the plot
plt.figure()

# Set parameters
# m = 20000
# n = 200
m = 10000
n = 100
num_matrices = 100
t = 0.03
Sigmad = np.repeat(np.array([1, 1+t]), m//2)

# Perform Test1 and plot its ROC curve
p_values0_test1, p_values1_test1 = perform_test(Nagao1973, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test1, p_values1_test1, 'Nagao73',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test1)
# print(p_values1_test1)
# Perform Test2 and plot its ROC curve
p_values0_test2, p_values1_test2 = perform_test(Srivastava2005, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test2, p_values1_test2, 'Srivastava05',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test2)
# print(p_values1_test2)
p_values0_test3, p_values1_test3 = perform_test(Srivastava2011, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test3, p_values1_test3, 'Srivastava11',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test3)
# print(p_values1_test3)
p_values0_test4, p_values1_test4 = perform_test(Chen2010, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test4, p_values1_test4, 'Chen10',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test4)
# print(p_values1_test4)

p_values0_test5, p_values1_test5 = perform_test(LedoitWolf2002, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test5, p_values1_test5, 'Wolf02',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test5)
# print(p_values1_test5)
p_values0_test6, p_values1_test6 = perform_test(Fisher2012, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test6, p_values1_test6, 'Fisher12',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test6)
# print(p_values1_test6)
p_values0_test7, p_values1_test7 = perform_test(Ahmad2015, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test7, p_values1_test7, 'Ahmad15',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test7)
# print(p_values1_test7)
p_values0_test8, p_values1_test8 = perform_test(DW2023Global, Sigmad, m, n, num_matrices)
p_values0_test10, p_values1_test10 = p_values0_test8, p_values1_test8
generate_roc_curve(p_values0_test8, p_values1_test8, 'Qiu21',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test10)
# print(p_values1_test10)
generate_roc_curve(p_values0_test8, p_values1_test8, 'PM1',lw=2, linestyle='--', alpha=1)
# print(p_values0_test8)
# print(p_values1_test8)
p_values0_test9, p_values1_test9 = perform_test(DW2023Local, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test9, p_values1_test9, 'PM2',lw=2, linestyle='--', alpha=1)
# print(p_values0_test9)
# print(p_values1_test9)

p_values0_test11, p_values1_test11 = perform_test(CLRT, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test11, p_values1_test11, 'CLRT',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test11)
# print(p_values1_test11)

Saved data

p_values0_test1=[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
p_values1_test1=[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
p_values0_test2=[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
p_values1_test2=[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
p_values0_test3=[0.23692729768650422, 0.24315018704945968, 0.155507468111022, 0.3496193209078652, 0.1512861826386831, 0.14558273799959187, 0.1057522881876991, 0.07664152544918801, 0.37697882972514796, 0.1774121338681749, 0.004362561481954685, 0.14533329525508099, 0.4260128415461333, 0.3523860668903849, 0.20967916093181027, 0.03598015568155055, 0.4628632033988209, 0.303925509118104, 0.08593855259248262, 0.3209145929312167, 0.06632116826505485, 0.3638830859484825, 0.37738731990337415, 0.18782633680955163, 0.34126318822508295, 0.4011212977351566, 0.2446815614946185, 0.34912766713846377, 0.04898604981881194, 0.05425731164795733, 0.005718671832747324, 0.34514801902757464, 0.22157482090697767, 0.15782383570357372, 0.23465712333822886, 0.2548578803670264, 0.09386832099373943, 0.40946687299412265, 0.2935981589308343, 0.36044760780491436, 0.42598291959290413, 0.17625233416633912, 0.33074962557172116, 0.31053748996904484, 0.20085438385167764, 0.09573598062443611, 0.02870464633180525, 0.11174046604581189, 0.48017388530479, 0.18396876411190977, 0.4052226605025604, 0.08697061390270966, 0.10950728339188232, 0.2764560185822773, 0.23523668147052057, 0.2393329524153598, 0.05185324555037396, 0.22664279585024805, 0.3507780742078702, 0.4052268667020519, 0.4602500539196527, 0.03334219790763304, 0.017711665000346044, 0.11340238329162644, 0.14048325050768828, 0.46548960950075435, 0.46176661406755193, 0.1296470654368711, 0.13975221864530196, 0.15664017416837572, 0.4805734667550463, 0.4772547386749496, 0.25580914136366095, 0.0646196095699838, 0.1490737110409649, 0.35308191064745387, 0.20717516983879514, 0.44258274026813127, 0.015192844534477445, 0.3662426411705114, 0.04394188069158744, 0.14753494324326277, 0.18795889484788286, 0.4554094527421515, 0.06041801586575435, 0.30415005412565355, 0.17167142292590531, 0.4407104581981791, 0.4458255588960468, 0.3001698784634077, 0.4442464908664663, 0.09069142380628326, 0.35005034802396473, 0.4563234812555742, 0.0049929383085935974, 0.44603638773343435, 0.07672452430911558, 0.20192854705223606, 0.36114756623424193, 0.40802976263452195]
p_values1_test3=[0.03520337388899397, 0.19073180871091266, 0.04684526992357552, 0.08153913130155344, 7.468330622817376e-07, 0.035976295391845325, 0.0008130917517700231, 0.0002476970553603186, 0.13666436338370969, 0.15180047317713175, 3.0410807127956296e-05, 0.00015273156899808793, 0.49605507510982105, 2.967931088670994e-07, 0.00140961521952776, 0.29326402613817404, 6.247471595610676e-07, 0.020334918726483964, 0.25511946578730293, 7.74369481870707e-08, 0.23093841107877, 0.16568985219880905, 0.2446869160731, 0.03231411075276103, 0.46424569807758487, 0.0328054573538582, 0.03989589402261928, 9.821172197033157e-05, 0.054034771867072506, 1.8784364601232006e-05, 0.013559349042003399, 0.001961115606146868, 0.004441159908318837, 0.0007987793200519011, 0.09394628724553777, 0.012310013838091827, 0.11798213797224377, 0.008948096576063769, 0.04606543055681622, 0.00024281507488066456, 0.0002680723276573538, 0.0006482200408844951, 0.4106172425722179, 0.0003543960666739787, 0.01516356064508495, 0.004894592408234755, 0.02759481143778042, 0.004495170256898939, 0.215459940335764, 0.0033431189829533636, 9.464410411830215e-06, 2.095206823704565e-05, 0.4448503647406866, 0.00014235894286573902, 0.0011866144045018512, 0.0012980090838523584, 0.03960690019255286, 0.0023105261661638954, 0.044989487259893046, 0.25307467876359346, 0.23499115959572192, 2.6321141372021017e-06, 7.302458016611446e-06, 0.0002461714563466444, 3.907807945013886e-07, 0.2573807451831205, 0.006948340438198475, 8.77762242479374e-06, 0.07117161264283312, 0.0003336353643145218, 0.02694323602455806, 0.37150309858310615, 0.045739395068019495, 0.0006400130118023872, 0.014759409463538598, 0.08866365574201218, 1.8991311168115033e-07, 0.06601550745709839, 0.00047829806851917844, 0.010441702668528063, 0.14927636793661936, 0.4713945939923744, 0.0006378431546062435, 0.015481122094316402, 0.00393287088772365, 0.14555466614481127, 0.03867679618487807, 0.13401720662532468, 0.08325971443417002, 0.20683440947252785, 0.00010176209770074163, 0.007291981860597541, 0.08143638505757256, 0.04830146164714022, 0.24757979895310134, 0.11792025272212459, 0.3788170582649901, 0.0006249280779390132, 0.004739847644780748, 0.0006937845302824597]
p_values0_test4=[0.17548502705921998, 0.31251642191381757, 0.335703905534956, 0.45727041477529895, 0.25782024152480654, 0.42009941478276047, 0.19949776748280124, 0.40664204638404766, 0.40708693283171893, 0.18995514588919005, 0.08378293451645402, 0.10337287612911006, 0.12039516722445764, 0.2594491792190967, 0.1784927373780747, 0.11602421337751323, 0.19381344737166173, 0.1791629030558033, 0.10303202421879631, 0.23386377732524388, 0.3128773006992598, 0.007456518962193903, 0.06665837461973245, 0.14618573186289208, 0.28883987323008675, 0.296795952387219, 0.05200686498115725, 0.25701160977197657, 0.2771969618978788, 0.4878387914572492, 0.03023663886447736, 0.12268666195037081, 0.4689496522353829, 0.04221409277572996, 0.30618608408423964, 0.41546755943784697, 0.26826427339765624, 0.02871472582796286, 0.0360802316436436, 0.3001862706516637, 0.18208030318496682, 0.29703968329568053, 0.14017822741694674, 0.12900670541947057, 0.19981969637636054, 0.046570220688109565, 0.15326135449489686, 0.4969441758583615, 0.4756472052176699, 0.4208987023383388, 0.2860584996020732, 0.15299579106970684, 0.3109672991930529, 0.03354915517120882, 0.421628203006409, 0.46452425554073773, 0.16395750566389578, 0.1473781328982008, 0.48250074371897733, 0.24647368295698324, 0.47012817687778763, 0.1314399425770011, 0.10949558328435804, 0.024003079750737566, 0.36439398290321057, 0.3004213911958259, 0.05927847807311204, 0.4438934053147072, 0.011321107968801258, 0.44691199306312723, 0.09277441877032944, 0.42696003988566533, 0.005352814960582708, 0.20028255790751626, 0.1721669385146114, 0.09262035058504503, 0.02604664349201269, 0.06722087601075089, 0.4176350857189508, 0.2856909135621917, 0.3351830533664426, 0.38553798081336055, 0.4802816889314233, 0.014912006817176748, 0.22414152683513178, 0.13571555998652185, 0.23205929287726923, 0.2697197725413738, 0.11402652983359274, 0.30118236753697336, 0.19615283315733112, 0.232837099749593, 0.044790844124551255, 0.47729829580777094, 0.3637279924856024, 0.458692281219644, 0.05830794534944905, 0.12474071372395379, 0.1922864697466269, 0.2810786609219934]
p_values1_test4=[0.06960297151830785, 0.029165449776972996, 1.0531136438340916e-05, 0.10701329794984793, 0.09301284813303701, 0.1460043286299174, 0.006226073406840005, 0.17641859885326605, 0.2673140014788046, 0.22535617981714506, 0.06133948970972036, 0.03255938481933329, 0.0005297693972364792, 0.02372223648066729, 0.009033283605784592, 0.005235474546630314, 0.0011610636950388775, 0.01418590747859838, 0.05529688289668033, 0.43708378004430826, 0.03540953104275246, 0.05153240573897078, 0.4402421702859127, 0.061081069595914306, 0.16862047436008964, 0.007577962155872697, 0.000781323721394056, 1.2583549521272275e-06, 0.0001268719381783967, 0.0005595835916664127, 0.013032274804726462, 0.03244346711099233, 0.0018463693820941485, 0.07732733392685254, 0.11273634932987653, 3.7999583561543204e-05, 0.09659721339618299, 0.0009926502451583774, 0.3970157576733283, 0.0005289528830847434, 0.014909312398172725, 0.0009512898392419755, 0.041988854184400015, 0.38636163417179037, 0.02679959624405137, 6.215078311222477e-06, 0.016455855107433415, 0.2839081403456831, 0.014105703518341506, 0.0005351697792324028, 0.001198456380363977, 0.0017857404614683992, 0.3905277304778826, 0.0031324764524240933, 0.05586123571306756, 0.00012033472180639215, 0.009290033017412713, 0.09575587498165872, 0.0014016834502924569, 0.0012628129865294158, 0.0981088664980947, 0.013049379477997336, 0.36902349629594156, 0.0029727231525600306, 0.36452225779791836, 1.1013708912654963e-06, 0.0002969416562867977, 0.030250513734253137, 0.0018118272685386794, 0.012137809349920148, 0.0007371484464322453, 0.022953926330355756, 0.1050346546562454, 0.029776952701388804, 0.20606656083335873, 0.017030261625100573, 3.033779584216134e-06, 0.0005937495866428222, 0.1615166663665617, 0.007877080069069842, 0.00024122642890245238, 0.022486058128789788, 4.707899731204446e-05, 3.9851450028827173e-08, 1.795992208331043e-05, 0.02299816574118796, 0.0034499906265239577, 0.0014524439396444988, 0.006369795708277448, 0.11664416758814644, 0.0015124550793474967, 0.011693890090266668, 0.24602143324454495, 0.0049225557367523765, 0.14826641774305926, 0.42807276146056616, 0.00817129108978043, 7.672031483663133e-06, 0.00019809638992562828, 0.449981797296516]
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p_values0_test10=[0.6671434906750298, 0.29880768855635, 0.6674952063726991, 0.8818844463756548, 0.08780357444116449, 0.4041771308141695, 0.6258231696599865, 0.9214077540818242, 0.45890037223814995, 0.9858593334871564, 0.7755495907102665, 0.2815625503869048, 0.4627416276168532, 0.1311630582507064, 0.3824995743987216, 0.6624164146723441, 0.4783241083067815, 0.1890982942626711, 0.278639679998248, 0.14760704043336847, 0.7413786320567546, 0.3900872474170445, 0.15360651074887866, 0.34160885345003833, 0.6848522867872546, 0.9504690019287358, 0.8760732810028569, 0.08159144774945148, 0.4716417632059322, 0.532849796014192, 0.14021451575415478, 0.6265198225165038, 0.771004876803477, 0.14701875485152494, 0.025510901210538117, 0.23709690345331058, 0.3178098593347496, 0.7225293971653866, 0.6484965178891668, 0.05295617499470451, 0.8488780601781671, 0.927663294727407, 0.23736508550536994, 0.792568051169042, 0.7396645085534692, 0.6617974569298615, 0.1547549155065393, 0.8428061935988755, 0.0865815272416901, 0.7389569896677759, 0.28781079750053795, 0.47719276837852354, 0.8188516996616544, 0.06860934166223887, 0.9016999579282228, 0.5418196849883161, 0.41061292774950764, 0.2986909627534855, 0.04553717501358845, 0.7257850255779039, 0.017006915540242584, 0.9380369199939833, 0.2801039551825628, 0.040840705706497005, 0.056072028330951484, 0.5399777947770079, 0.5482002527631427, 0.5417428743946178, 0.6881564613332594, 0.9198952388521802, 0.11843163599519535, 0.0816994614847355, 0.6666990264501214, 0.19607210971808176, 0.2857652629344014, 0.2833172768966019, 0.7514383329404997, 0.059417132102534964, 0.07332069347451142, 0.10410933174669323, 0.04741783304649694, 0.9691542439662071, 0.9341037318715728, 0.6738624970737517, 0.1359491664225283, 0.9234995233445586, 0.8090513926788427, 0.9115462525350857, 0.22007347012242806, 0.682845571551342, 0.4357841156564346, 0.016049086678028956, 0.3705275905089771, 0.04652469783543589, 0.6105138416420732, 0.4742950636170944, 0.7141754096796737, 0.018399428333405465, 0.8115485212420539, 0.8680984100810323]
p_values1_test10=[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
p_values0_test11=[0.17706855336626903, 0.272261796465755, 0.19020920235224925, 0.7732251503041341, 0.5970291402366639, 0.4476183943275709, 0.41220396547455707, 0.7469570882194099, 0.061431190637930344, 0.03565714760929661, 0.4587584766376196, 0.9796002703880919, 0.768811079372554, 0.9913202461539106, 0.10345672843237796, 0.4731353146764534, 0.38765376557494746, 0.6078474287220224, 0.7534536631001918, 0.9858919380267601, 0.6365139370425434, 0.9538977202570336, 0.8139409141005196, 0.4173128523349141, 0.8442555288321554, 0.9125625677105762, 0.17580300905440116, 0.29861821399414845, 0.12006027322351298, 0.7333474346611331, 0.3389379989139125, 0.5310457051737221, 0.6124286563135741, 0.5528450735014636, 0.02826432454339356, 0.9609682698088537, 0.9081564906754096, 0.5807453037128436, 0.022126138249320748, 0.7530126147262126, 0.9850432862010634, 0.8087208923502776, 0.7733056495552038, 0.5696006117413757, 0.09035288641102168, 0.7516566121818729, 0.9397668585778889, 0.05188189717204161, 0.8353470974451982, 0.6411331324388576, 0.49155485590942183, 0.8639863218663826, 0.8092847499216069, 0.2682506538843019, 0.4983033055983621, 0.31523877567101133, 0.8225998737808142, 0.16093933861788012, 0.09525314694434006, 0.09279379102693142, 0.16756441761898966, 0.6192878517883205, 0.04660217791133414, 0.17668573195734183, 0.22397949960784613, 0.4997480624945805, 0.8359717688889188, 0.7720661581151402, 0.6812699378371565, 0.4963675018327076, 0.532163660360788, 0.9100600470824629, 0.12261056607573173, 0.35901197808307717, 0.8667921643514709, 0.16047124839079396, 0.5170019892266371, 0.8531419836178065, 0.7765311539705528, 0.6346820543552278, 0.7250379681259356, 0.4382084405936786, 0.15707159607981835, 0.3384949416378058, 0.7862600445901289, 0.646474030563239, 0.5191080521789755, 0.3027025009649571, 0.9295498597004099, 0.9630218228903282, 0.7662685190437073, 0.4844732873090378, 0.5920193450933742, 0.991256548362764, 0.12220471382427656, 0.7818213518139638, 0.2763579354766412, 0.2658185819391108, 0.1666769498226479, 0.13914893522105753]
p_values1_test11=[0.9942154932912088, 0.18411115287084856, 0.183710272909126, 0.2988282599379919, 0.018357778811993297, 0.010237429411041665, 0.3981678065155534, 0.7813680388960624, 0.22620518760120945, 0.10544026916814375, 0.046844349502929816, 0.19512624607053097, 0.8753017117538826, 0.42341417335226295, 0.3377482220571375, 0.11164162330685468, 0.6536312165753739, 0.46466806752710044, 0.6427816877125561, 0.5335015228371205, 0.11937303341046457, 0.25869000313880597, 0.7773045039325837, 0.06134861477246867, 0.017762145089615045, 0.03592225385415704, 0.910220226067046, 0.7717317883605364, 0.4642301611130901, 0.5467054620144454, 0.7265840775085843, 0.21196078598315937, 0.056403196965733926, 0.30873363860657066, 0.4628104764923373, 0.004022971245150275, 0.3879534608683197, 0.9656598139038621, 0.3484749592672609, 0.19673784395965233, 0.9304117006154977, 0.1892578920847936, 0.9804182376862363, 0.8568828994646531, 0.9438001574666393, 0.38576958267272077, 0.03741253047677673, 0.007337690225904803, 0.8780386900425017, 0.1556704182475075, 0.061776323081465856, 0.9852702263666826, 0.35295850342884005, 0.32422847987406067, 0.6105770967135924, 0.21953752643059543, 0.18377993875767862, 0.3219123235558006, 0.3348541965213323, 0.07733787753010635, 0.40477787919760955, 0.03479206761303044, 0.8211122555729722, 0.439701107797007, 0.8545887832625099, 0.3316392052411512, 0.3608644961338623, 0.20648891676894987, 0.45316625315343795, 0.04511738901041196, 0.7964707707049716, 0.5371993769723147, 0.09686271312139705, 0.018511946605605845, 0.30312034088995077, 0.8682680533245009, 0.2990664064552153, 0.42704032979627016, 0.06108483956140098, 0.19877945756642146, 0.1732646926129653, 0.6709670324665642, 0.43692759952164106, 0.42438214217306536, 0.048070478690638074, 0.9383249293666667, 0.7221025742282932, 0.38240758860896173, 0.05906529855549825, 0.04925842260632707, 0.49154423659377655, 0.06782801916427705, 0.7350889994383984, 0.08802198115455662, 0.020724340743764058, 0.5249344549832644, 0.2656125153293334, 0.04274217639132538, 0.3921682753800191, 0.2584364509093615]

Plot

import numpy as np
import matplotlib.pyplot as plt
from sklearn.metrics import roc_curve, auc
from google.colab import files
import matplotlib.gridspec as gridspec
num_matrices=100

def generate_roc_curve(p_values0, p_values1, label, lw=2, linestyle='-', alpha=0.8):
    # calculate TPRs and FPRs for each threshold
    thresholds = np.linspace(0, 1, 100)
    tprs = []
    fprs = []

    for threshold in thresholds:
        tpr = np.sum(p_values1 < threshold) / num_matrices
        fpr = np.sum(p_values0 < threshold) / num_matrices

        tprs.append(tpr)
        fprs.append(fpr)

    # calculate AUC
    roc_auc = auc(fprs, tprs)

    # # plot ROC curve
    # # Use string formatting to control the presentation of AUC in the legend
    # legend_text = "{:<16} ({:0.2f})".format(label, roc_auc)

    # # plot ROC curve
    # plt.plot(fprs, tprs, linestyle=linestyle, alpha=alpha,lw=lw, label=legend_text)

    # plt.plot(fprs, tprs, linestyle=linestyle, alpha=alpha,lw=lw, label=f'({roc_auc:.2f}){label} ')

    plt.plot(fprs, tprs, linestyle=linestyle, alpha=alpha,lw=lw, label=f'{label}  ({roc_auc:.2f}) ')
    # label_padded = label.ljust(15)
    # plt.plot(fprs, tprs, linestyle=linestyle, alpha=alpha,lw=lw, label=f'{label_padded} (AUC: {roc_auc:.2f})')

def bold_legend(legend):
    """Utility function to set legend text to bold."""
    for text in legend.get_texts():
        text.set_fontweight('bold')


def set_ticks_bold(ax):
    """Utility function to set tick labels to bold."""
    for label in ax.get_xticklabels():
        label.set_fontweight('bold')
    for label in ax.get_yticklabels():
        label.set_fontweight('bold')

# Create a GridSpec layout
# Create a GridSpec layout
gs = gridspec.GridSpec(1, 2, width_ratios=[1.5, 0.2])

fig = plt.figure(figsize=(8.5, 6))

# First subplot
ax0 = fig.add_subplot(gs[0])

# Perform Test1 and plot its ROC curve
# p_values0_test1, p_values1_test1 = perform_test(Nagao1973, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test1, p_values1_test1, 'Nagao73',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test1)
# print(p_values1_test1)
# Perform Test2 and plot its ROC curve
# p_values0_test2, p_values1_test2 = perform_test(Srivastava2005, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test2, p_values1_test2, 'Srivastava05',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test2)
# print(p_values1_test2)
# p_values0_test3, p_values1_test3 = perform_test(Srivastava2011, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test3, p_values1_test3, 'Srivastava11',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test3)
# print(p_values1_test3)
# p_values0_test4, p_values1_test4 = perform_test(Chen2010, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test4, p_values1_test4, 'Chen10',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test4)
# print(p_values1_test4)

# p_values0_test5, p_values1_test5 = perform_test(LedoitWolf2002, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test5, p_values1_test5, 'Wolf02',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test5)
# print(p_values1_test5)
# p_values0_test6, p_values1_test6 = perform_test(Fisher2012, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test6, p_values1_test6, 'Fisher12',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test6)
# print(p_values1_test6)
# p_values0_test7, p_values1_test7 = perform_test(Ahmad2015, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test7, p_values1_test7, 'Ahmad15',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test7)
# print(p_values1_test7)
# p_values0_test8, p_values1_test8 = perform_test(DW2023Global, Sigmad, m, n, num_matrices)
# p_values0_test10, p_values1_test10 = p_values0_test8, p_values1_test8
generate_roc_curve(p_values0_test8, p_values1_test8, 'Qiu21',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test10)
# print(p_values1_test10)

# p_values0_test11, p_values1_test11 = perform_test(CLRT, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test11, p_values1_test11, 'CLRT',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test11)
# print(p_values1_test11)

generate_roc_curve(p_values0_test8, p_values1_test8, 'PM1',lw=2, linestyle='--', alpha=1)
# print(p_values0_test8)
# print(p_values1_test8)
# p_values0_test9, p_values1_test9 = perform_test(DW2023Local, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test9, p_values1_test9, 'PM2',lw=2, linestyle='--', alpha=1)
# print(p_values0_test9)
# print(p_values1_test9)




set_ticks_bold(ax0)
ax0.plot([-0.1, 1], [0, 1], color='navy', lw=2, linestyle='dotted')
ax0.set_xlim([0.0, 1.0])
ax0.set_ylim([0.0, 1.05])
ax0.set_xlabel('False Positive Rate', fontweight='bold', fontsize=14)
ax0.set_ylabel('True Positive Rate', fontweight='bold', fontsize=14)

ax0.set_title('Receiver Operating Characteristic', fontweight='bold', fontsize=16)
ax0.tick_params(axis='both', which='major', labelsize=18,  width=1, length=5)



# Second subplot for the legend
ax1 = fig.add_subplot(gs[1])
ax1.axis('off')
handles, labels = ax0.get_legend_handles_labels()
legend = ax1.legend(handles, labels, loc='center',handlelength=0.5)
bold_legend(legend)


plt.tight_layout()
plt.savefig("n100p10000t0dot3.eps",dpi=300, bbox_inches='tight')
files.download("n100p10000t0dot3.eps")
plt.show()

WARNING:matplotlib.backends.backend_ps:The PostScript backend does not support transparency; partially transparent artists will be rendered opaque.

Figure 7: ROC

In this section, we display the ROC curve of our proposed tests compared with some of the existing literature when a wrongly estimated 4th cumulant is provided to the tests.

ROC Curve with wrongly estimated $\kappa_4$, $\varepsilon=0.08$

Data generating

import jax
import jax.numpy as jnp
from jax import grad, jit,random
from jax.random import normal
import numpy as np
import matplotlib.pyplot as plt


k4=4
prob=0.5+0.5*np.sqrt((2+k4)/(6+k4))
# for t=0: 1: 0.1
t=0.000
values = np.array([np.sqrt((1-prob)/(prob)), -np.sqrt((prob)/(1-prob))  ])
probabilities =np.array( [prob,1-prob])


n = 100
R = 1000
c=0
T = jnp.zeros(R)


TestST=np.array(T)

def count_outside_interval(arr, a, b):
    outside_interval = np.logical_or(arr < a, arr > b)
    count = np.sum(outside_interval)
    return count


def extract_elements(arr, a, b):
    return arr[(arr >= a) & (arr <= b)]





pn=2*jnp.ones(n)

eta=1/(n**(1/4))

# k=int(n*eta)
R0=10000


#calculate the null distribution
phi=100
m=int(n*phi)
r=jnp.sqrt(phi)

leftb = jnp.sqrt(phi)+1/jnp.sqrt(phi)-2
    # objective_grad = jit(grad(objective))
T0 = jnp.zeros(R0)
key = jax.random.PRNGKey(4)
for r in range(R0):
    key, subkey = jax.random.split(key)
    X = random.choice(key,values, shape=(m, n),p=probabilities)
    H = jnp.matmul(X.T, X) / jnp.sqrt(n * m)
    E00, _ = jnp.linalg.eigh(H)
    E0=np.array(E00)
    Et = jnp.array(extract_elements(E0, leftb-eta, leftb+eta))
    # print(E0[-k:])
    Eb = (Et - leftb*jnp.ones(Et.shape))/eta
    t0=jnp.sum(Eb)
    T0 = T0.at[r].set(t0)
TestST0=np.array(T0)
Mean_x=jnp.mean(TestST0)
Variance_x=jnp.var(TestST0)
print(Mean_x)
print(Variance_x)

import numpy as np
import matplotlib.pyplot as plt
from sklearn.metrics import roc_curve, auc

def perform_test(TestFunc, Sigmad, m=2000, n=40, num_matrices=1000):
    p_values0 = []
    p_values1 = []

    for _ in range(num_matrices):
        data0 = np.random.choice(values, size=(m, n), p=probabilities)
        p0 = TestFunc(data0.T)
        if p0 is not None:
            p_values0.append(p0)
        else:
            p_values0.append(0)
        # del data0

        # data0_prime = np.random.choice(values, size=(m, n), p=probabilities)
        # data1 = data0_prime * np.sqrt(Sigmad)[:, None]
        data1 = data0 * np.sqrt(Sigmad)[:, None]
        p1 = TestFunc(data1.T)
        if p1 is not None:
            p_values1.append(p1)
        else:
            p_values1.append(0)
        # del data1, data0_prime
        del data1, data0

    return np.array(p_values0), np.array(p_values1)



def generate_roc_curve(p_values0, p_values1, label, lw=2, linestyle='-', alpha=0.8):
    # calculate TPRs and FPRs for each threshold
    thresholds = np.linspace(0, 1, 100)
    tprs = []
    fprs = []

    for threshold in thresholds:
        tpr = np.sum(p_values1 < threshold) / num_matrices
        fpr = np.sum(p_values0 < threshold) / num_matrices

        tprs.append(tpr)
        fprs.append(fpr)

    # calculate AUC
    roc_auc = auc(fprs, tprs)

    # plot ROC curve
    plt.plot(fprs, tprs, linestyle=linestyle, alpha=alpha,lw=lw, label=f'{label} (AUC = {roc_auc:.2f})')



# Initialize the plot
plt.figure()

# Set parameters
# m = 20000
# n = 200
m = 10000
n = 100
num_matrices = 100
t = 0.08
Sigmad = np.repeat(np.array([1, 1+t]), m//2)

# Perform Test and plot its ROC curve
p_values0_test1, p_values1_test1 = perform_test(Nagao1973, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test1, p_values1_test1, 'Nagao73')

p_values0_test2, p_values1_test2 = perform_test(Srivastava2005, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test2, p_values1_test2, 'Srivastava05')

p_values0_test3, p_values1_test3 = perform_test(Srivastava2011, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test3, p_values1_test3, 'Srivastava11')

p_values0_test4, p_values1_test4 = perform_test(Chen2010, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test4, p_values1_test4, 'Chen10')

p_values0_test5, p_values1_test5 = perform_test(LedoitWolf2002, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test5, p_values1_test5, 'Wolf02')

p_values0_test6, p_values1_test6 = perform_test(Fisher2012, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test6, p_values1_test6, 'Fisher12')

p_values0_test7, p_values1_test7 = perform_test(Ahmad2015, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test7, p_values1_test7, 'Ahmad15')

p_values0_test11, p_values1_test11 = perform_test(CLRT, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test11, p_values1_test11, 'CLRT')

p_values0_test8, p_values1_test8 = perform_test(DW2023Global, Sigmad, m, n, num_matrices)
p_values0_test10, p_values1_test10 = p_values0_test8, p_values1_test8
# generate_roc_curve(p_values0_test8, p_values1_test8, 'Qiu21')
generate_roc_curve(p_values0_test8, p_values1_test8, 'PM1',linestyle='--',alpha=1) ## since test stat will be basically the same for Identity

p_values0_test9, p_values1_test9 = perform_test(DW2023Local, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test9, p_values1_test9, 'PM2',linestyle='--',alpha=1)

Saved data

#p_values0_test1=[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 2.1826984664130578e-13, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.3631011880899102e-05, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
p_values1_test1=[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
p_values0_test2=[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
p_values1_test2=[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
p_values0_test3=[0.14444037439075996, 0.007331859748987091, 7.231103663563054e-05, 0.1711474599070888, 0.308505004200748, 0.043386952350843, 0.19646902901995134, 0.26836584132297703, 0.013973354487776923, 0.39203170160307843, 0.11670077581720706, 0.13207224785823746, 0.10453335145239917, 0.018561290606681657, 0.009908934834340188, 8.07072549946497e-06, 8.640000992643504e-05, 0.03353432969451098, 1.6245553436933236e-06, 0.00027650916178945106, 0.012018629080801091, 0.012638613236756524, 0.022535739437377966, 0.008953854067071942, 0.020005190387807237, 0.2759682239986081, 1.772478538175637e-05, 0.40152963333443814, 6.184675971365738e-05, 0.0060553348196831225, 0.005021538947650783, 0.15780410581029336, 0.009553368328714784, 0.04633759365429524, 0.085180214752136, 0.18324731393469385, 0.0007868053278433651, 0.006392329827196508, 2.427965250273445e-05, 0.00409620761181162, 0.0479671207806146, 0.1605644288526451, 0.46448088897810536, 0.4124541895739545, 0.09643229811411191, 0.07727232011449359, 0.013012348482968394, 0.3268380694097959, 0.14401283639263518, 0.021337286669555144, 1.0312432058401377e-05, 0.3922607304945338, 0.028311853940801046, 0.007081010961484835, 0.044513056520347205, 0.004759875326874163, 0.035204467572570564, 4.0166463697755006e-05, 0.001698093000141454, 0.028770662011301962, 0.03863281827860909, 0.29258904952121023, 0.018029180285584978, 0.07033550356385809, 0.01967246953722024, 0.020560318895908924, 0.0178264082707561, 0.13756624078061352, 0.44940575482729117, 0.18198731718135208, 0.13095185525428876, 0.025733919881463896, 0.09448554525398978, 0.09271657734237737, 0.0604255036293575, 0.0018543434703043316, 0.004734605802687475, 0.0002076497874722305, 0.3431795273415025, 0.13355937264516515, 0.023023807438889432, 0.11185353804069731, 0.2482120012856761, 0.049887436076558433, 0.07738724856116286, 0.202988062633516, 0.0007569474638203788, 0.05491047751000011, 0.2654569940097039, 0.034247842356012526, 5.146930540100847e-05, 0.07186726405956279, 0.013577249611519071, 0.015140594505205662, 0.0011736112335489768, 0.1076226307228687, 0.24747287620744607, 0.3261749859790425, 0.0011194508720507867, 0.12666054102701152]
p_values1_test3=[0.0993393368222768, 0.0027570161173889884, 1.303917177108982e-05, 0.11401662182966388, 0.24451793262532862, 0.018492094239092816, 0.13370097539739256, 0.19746661439368673, 0.004602076888761242, 0.43940345086264665, 0.07761270618106053, 0.0916451556405743, 0.0629302684499563, 0.006960211286221374, 0.003830822522100097, 5.519677983567561e-07, 1.5414527031354375e-05, 0.013591962012162284, 1.6533250146721912e-07, 6.113587674050347e-05, 0.004688029047929865, 0.006506667022703527, 0.009103267397348791, 0.0025904314911103787, 0.007014635172447825, 0.2021760244474229, 1.1778558899333547e-06, 0.42937834500598493, 8.39412926723071e-06, 0.0015578930445846462, 0.0013485070380416797, 0.10750037139256541, 0.002596366078844503, 0.028153532890247912, 0.04858499844928155, 0.13777008697860738, 0.00013671943703985612, 0.0018857377016793242, 4.153813924157035e-06, 0.001485028939952615, 0.02063956952019652, 0.09769978541813296, 0.47856113937260514, 0.34021013919728305, 0.057556988921864005, 0.04414769497509641, 0.0038937069579869643, 0.2545848027465949, 0.1575953944577242, 0.008239799770767497, 1.3538113543543062e-06, 0.3432371613041467, 0.01593321643099077, 0.0026385979138763282, 0.025652002940127705, 0.0012085214517247866, 0.01887983163510487, 6.846123131842852e-06, 0.0005331362811200613, 0.013890415125514699, 0.016815169655289397, 0.2017269847012335, 0.008414146794966326, 0.04143401879711894, 0.0129711261295401, 0.009688965943210115, 0.00750171698934754, 0.09208381986748071, 0.3870252158586335, 0.12077520686894183, 0.1107204070378558, 0.009667068484886632, 0.051765408545881186, 0.04770421185503215, 0.027847510899257744, 0.0004916453279428801, 0.001497728393152431, 1.6722997947749008e-05, 0.27110003339140354, 0.08163434298509664, 0.009662710772647687, 0.06708025155334685, 0.17673508611661637, 0.0241232105992224, 0.03959977505535828, 0.15189484618784355, 0.00014208175317032, 0.02410370352858915, 0.18719861326087894, 0.019041153442164283, 1.0678245052964108e-05, 0.051551030584598756, 0.006634878928875154, 0.0056772565170886136, 0.0003623650586145333, 0.06631949004279625, 0.17189109938838332, 0.4098362312478073, 0.00024733854628500307, 0.06643425008444304]
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p_values0_test10=[0.8422526427519785, 0.10418318572711849, 0.23533604880386427, 0.7646237134821516, 0.0030089366602077394, 0.2553585192052277, 0.29213017016612675, 0.01953658415947257, 0.007727165535075908, 0.2031780808498458, 0.003654104727765173, 5.995863318797312e-05, 0.8723514798047776, 0.1045698328173541, 0.9243580142427217, 0.1287563851460809, 0.0667969537347024, 0.5926945885450052, 0.16694295160748185, 0.5813821247438984, 0.294646495515126, 0.9985376433549165, 0.09337368711991334, 0.13967078229988683, 0.031641914051960374, 0.045390220799202075, 0.5889125626229343, 0.0001337446766096484, 0.6678821893000966, 0.11137575105965647, 0.20253559022400136, 0.05273250219248116, 0.5299723993131851, 0.3415005964664499, 0.4981882414507053, 0.08283867532662259, 0.8408357635169832, 0.7369134329009823, 0.40305211824932696, 0.6441388763556024, 0.04033368632022749, 0.23318189963686642, 0.9854278715617368, 0.13107367286484783, 0.4618639735100474, 0.011097935403904913, 0.11638465029297906, 0.031498554956905345, 0.988352190042701, 0.6520145653736547, 0.09736459414844534, 0.9679549063066819, 0.37098995629576326, 0.03207874731558702, 1.1288135032705426e-05, 0.07012090521104475, 0.02460839195363751, 0.10456983281735344, 0.06053789051055669, 0.07656458404531974, 0.00048153067692968676, 0.08381437703209382, 0.010982175835906105, 0.2191725584701123, 0.49818824145071017, 0.43240898062503197, 0.003918302619960734, 0.5751317557368967, 0.03372614735583035, 0.058560882902950695, 0.20644319598786232, 0.2701335167798202, 0.008608173987934231, 0.2797655195305855, 0.10812899761761918, 0.3022824106412312, 0.7869934330154751, 0.038075995471362534, 0.6130409067815261, 9.700971213622012e-05, 0.47530625976717156, 0.35652508615385825, 0.757658211000044, 0.2064431959878621, 0.2130718405825287, 0.8322794492628716, 0.6507093778147865, 0.8251533101441253, 0.9504973125798108, 0.6015525127881824, 0.0022683226865287676, 0.004447938935986384, 0.7286679458425811, 0.00012984738500554371, 0.519264336398332, 0.03418747638832831, 2.6145590266146712e-05, 0.0025603317962321626, 0.16031809305823863, 0.39387317635676045]
p_values1_test10=[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
p_values0_test11=[0.15499111710743718, 0.06404358457775228, 0.03530435232941165, 0.14402805454398004, 2.9020397750434412e-05, 0.2217210042780362, 0.02514532067846864, 0.011454437485514113, 0.21662780003797244, 0.03887201367161963, 0.127139823598579, 0.9284010504768747, 0.014736196752053399, 0.8297374626985443, 0.008381068909510292, 6.544921145223981e-05, 0.6001662117323596, 0.4041922541356948, 0.27185604484275805, 0.013801333088710344, 0.08369260085827168, 0.018366089434033395, 0.00037672774771801976, 0.013225474462768894, 0.3790384016042978, 0.0041219335792728184, 6.971970303304431e-07, 0.00013943492188284168, 0.0007913343746648316, 0.004915775251616905, 0.038053589295575785, 0.01644043104945281, 0.18535865460753875, 0.023374772128968768, 0.014053960856090342, 0.010190631382954507, 0.11891093510707984, 0.0030244753619415032, 0.0015352941145030385, 0.0071700428787038195, 0.02154741050935005, 0.20801397461399862, 0.09080762971445999, 0.7382672661488883, 0.0258200043902459, 0.005876720583163131, 0.05506764973692513, 0.20906115734946606, 0.39990430300950597, 0.007102873775415519, 0.0032345941878126894, 0.010075772377853287, 0.7510278712785186, 0.37105132452929634, 9.236722930006103e-05, 0.0014619582728225833, 0.13880849566823628, 0.0010621402934849478, 0.0003320976483509597, 0.1579816148487332, 0.008720126125259853, 0.2443422579603749, 0.11234976353524306, 0.14148726652925636, 0.04816680964648179, 0.02156811385491464, 0.21528021069504222, 0.5948795223934051, 0.1331218730412851, 0.07095173635351171, 0.22231320546769995, 0.004999522322852501, 0.30066440588655174, 0.04933898624337707, 0.026846732677247465, 0.03496577275940105, 0.002930747017210633, 0.02649612232749088, 0.018071098921272677, 0.0046529607724667965, 0.03922645042055706, 0.0011236096822753172, 0.019391566486395728, 7.295316592825785e-06, 0.0007288560182634463, 0.31837252108492464, 0.16079769732165916, 0.0012463307533860934, 0.3123239103556157, 0.845403025254196, 0.39876160137111016, 0.4636023696481315, 0.24401886064545786, 0.0027340328497806876, 0.10337413007252061, 0.027224472567085378, 0.6743486066734559, 0.7898984447011914, 0.0002842903340702918, 0.49697876262441665]
p_values1_test11=[0.1455185908193466, 0.05308771631608855, 0.042107012202098115, 0.10840071750308233, 1.2589729295076069e-05, 0.22456717995400188, 0.017652153680836502, 0.006840478198847588, 0.18829244066414086, 0.025069302563623053, 0.11251589480905366, 0.896399399489199, 0.012190444622867114, 0.9703897577382348, 0.006415167481009787, 7.294054534168204e-05, 0.5234331803009393, 0.3470007633785177, 0.22704669258080323, 0.010421991487338245, 0.07035309161255054, 0.0168979975310124, 0.00019932308009473765, 0.010818660042251071, 0.3319559925513029, 0.0027745560005469905, 3.8484733400068194e-07, 0.00010340548787346471, 0.0005178149398359366, 0.003404185036137264, 0.03343198093196964, 0.017175297708968085, 0.1663167461425603, 0.018849748986028425, 0.008669255044420954, 0.008476400296945963, 0.09219695734781119, 0.0030649331411307656, 0.0010822716586238457, 0.00442381794559199, 0.021540598095874053, 0.18163232410260854, 0.06279215584988984, 0.6767244731922137, 0.023037216765312563, 0.005616899277643084, 0.047398584995630655, 0.20474779012405087, 0.348452284067311, 0.0051462615220243535, 0.0028643100699139445, 0.006039495748146884, 0.6837714590983055, 0.3628542862618711, 7.970473609342221e-05, 0.001158029872700972, 0.11570092007036226, 0.0007491513148678575, 0.00023541178044284017, 0.1547570354235699, 0.006074569705734056, 0.2462195813151904, 0.10326180452596145, 0.11263481755228288, 0.04695434804570864, 0.019373695070680208, 0.17623220229682923, 0.4618749023779998, 0.12134482381678358, 0.05795072075031893, 0.19549891520034457, 0.00429600451751333, 0.2907685227414376, 0.04620844985700501, 0.02461749330133589, 0.030404159070227088, 0.002138584158056789, 0.01977144613450399, 0.017404943923492455, 0.005221527134760429, 0.022968829034013316, 0.0010104490082087914, 0.014026951134878285, 9.392313679024156e-06, 0.00043951342190262643, 0.30174136174281063, 0.16350637236513865, 0.001199830281000125, 0.27567463578666107, 0.7435511525898344, 0.3783050359462765, 0.4118426081084485, 0.1950464283612936, 0.0017462746231802218, 0.08438470568772294, 0.020910504536891805, 0.62658881065579, 0.9004332255613645, 0.00014407653752379623, 0.416989529691441]

Plot

import numpy as np
import matplotlib.pyplot as plt
from sklearn.metrics import roc_curve, auc
from google.colab import files
import matplotlib.gridspec as gridspec
num_matrices=100

def generate_roc_curve(p_values0, p_values1, label, lw=2, linestyle='-', alpha=0.8):
    # calculate TPRs and FPRs for each threshold
    thresholds = np.linspace(0, 1, 100)
    tprs = []
    fprs = []

    for threshold in thresholds:
        tpr = np.sum(p_values1 < threshold) / num_matrices
        fpr = np.sum(p_values0 < threshold) / num_matrices

        tprs.append(tpr)
        fprs.append(fpr)

    # calculate AUC
    roc_auc = auc(fprs, tprs)

    # # plot ROC curve
    # # Use string formatting to control the presentation of AUC in the legend
    # legend_text = "{:<16} ({:0.2f})".format(label, roc_auc)

    # # plot ROC curve
    # plt.plot(fprs, tprs, linestyle=linestyle, alpha=alpha,lw=lw, label=legend_text)

    # plt.plot(fprs, tprs, linestyle=linestyle, alpha=alpha,lw=lw, label=f'({roc_auc:.2f}){label} ')

    plt.plot(fprs, tprs, linestyle=linestyle, alpha=alpha,lw=lw, label=f'{label}  ({roc_auc:.2f}) ')
    # label_padded = label.ljust(15)
    # plt.plot(fprs, tprs, linestyle=linestyle, alpha=alpha,lw=lw, label=f'{label_padded} (AUC: {roc_auc:.2f})')

def bold_legend(legend):
    """Utility function to set legend text to bold."""
    for text in legend.get_texts():
        text.set_fontweight('bold')


def set_ticks_bold(ax):
    """Utility function to set tick labels to bold."""
    for label in ax.get_xticklabels():
        label.set_fontweight('bold')
    for label in ax.get_yticklabels():
        label.set_fontweight('bold')

# Create a GridSpec layout
gs = gridspec.GridSpec(1, 2, width_ratios=[1.5, 0.2])

fig = plt.figure(figsize=(8.5, 6))

# First subplot
ax0 = fig.add_subplot(gs[0])

# Perform Test1 and plot its ROC curve
p_values0_test1, p_values1_test1 = perform_test(Nagao1973, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test1, p_values1_test1, 'Nagao73',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test1)
# print(p_values1_test1)
# Perform Test2 and plot its ROC curve
p_values0_test2, p_values1_test2 = perform_test(Srivastava2005, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test2, p_values1_test2, 'Srivastava05',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test2)
# print(p_values1_test2)
p_values0_test3, p_values1_test3 = perform_test(Srivastava2011, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test3, p_values1_test3, 'Srivastava11',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test3)
# print(p_values1_test3)
p_values0_test4, p_values1_test4 = perform_test(Chen2010, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test4, p_values1_test4, 'Chen10',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test4)
# print(p_values1_test4)

p_values0_test5, p_values1_test5 = perform_test(LedoitWolf2002, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test5, p_values1_test5, 'Wolf02',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test5)
# print(p_values1_test5)
p_values0_test6, p_values1_test6 = perform_test(Fisher2012, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test6, p_values1_test6, 'Fisher12',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test6)
# print(p_values1_test6)
p_values0_test7, p_values1_test7 = perform_test(Ahmad2015, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test7, p_values1_test7, 'Ahmad15',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test7)
# print(p_values1_test7)
p_values0_test8, p_values1_test8 = perform_test(DW2023Global, Sigmad, m, n, num_matrices)
p_values0_test10, p_values1_test10 = p_values0_test8, p_values1_test8
generate_roc_curve(p_values0_test8, p_values1_test8, 'Qiu21',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test10)
# print(p_values1_test10)

p_values0_test11, p_values1_test11 = perform_test(CLRT, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test11, p_values1_test11, 'CLRT',lw=2, linestyle='-', alpha=0.8)
# print(p_values0_test11)
# print(p_values1_test11)

generate_roc_curve(p_values0_test8, p_values1_test8, 'PM1',lw=2, linestyle='--', alpha=1)
# print(p_values0_test8)
# print(p_values1_test8)
p_values0_test9, p_values1_test9 = perform_test(DW2023Local, Sigmad, m, n, num_matrices)
generate_roc_curve(p_values0_test9, p_values1_test9, 'PM2',lw=2, linestyle='--', alpha=1)
# print(p_values0_test9)
# print(p_values1_test9)




set_ticks_bold(ax0)
ax0.plot([-0.1, 1], [0, 1], color='navy', lw=2, linestyle='dotted')
ax0.set_xlim([0.0, 1.0])
ax0.set_ylim([0.0, 1.05])
ax0.set_xlabel('False Positive Rate', fontweight='bold', fontsize=14)
ax0.set_ylabel('True Positive Rate', fontweight='bold', fontsize=14)

ax0.set_title('Receiver Operating Characteristic', fontweight='bold', fontsize=16)
ax0.tick_params(axis='both', which='major', labelsize=18,  width=1, length=5)



# Second subplot for the legend
ax1 = fig.add_subplot(gs[1])
ax1.axis('off')
handles, labels = ax0.get_legend_handles_labels()
legend = ax1.legend(handles, labels, loc='center',handlelength=0.5)
bold_legend(legend)


plt.tight_layout()
plt.savefig("K4equals2treatedasGaussiann100p10000t0dot08.eps",dpi=300, bbox_inches='tight')
files.download("K4equals2treatedasGaussiann100p10000t0dot08.eps")
plt.show()

WARNING:matplotlib.backends.backend_ps:The PostScript backend does not support transparency; partially transparent artists will be rendered opaque.

From the results, we can see that in thisregime, our local statistics behave overall better than our global statistics which takes the wrong $\kappa_4$ as input and other statistics.

One-Sample Hypothesis Testing - Real Data

2009 Data

Here we use the data from https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE13528, proceeded data is also obtainable from https://www.refine.bio/experiments/GSE13528/gene-expression-profiles-of-fasting-induced-changes-in-liver-and-fat-tissues-of-pigs-expressing-the-mc4r-d298n-variant

Read Data

# Import necessary modules
from google.colab import files
import pandas as pd
import numpy as np

# Upload the TSV file
uploaded = files.upload()

# Assume the file is named "data.tsv"; replace with your file's name if different
file_name = 'GSE13528.tsv'

# Read the file into a Pandas DataFrame
df = pd.read_csv(file_name, delimiter='\t')

# Only include numeric columns in the DataFrame
df_numeric = df.select_dtypes(include=[np.number])

# Convert the DataFrame to a 2D NumPy array of float type
data_array_2d = df_numeric.to_numpy().astype(float)
data_array_2d_no_first_column = data_array_2d[:, :]

# Compute row-wise mean and standard deviation
row_means = np.mean(data_array_2d_no_first_column, axis=1, keepdims=True)
row_stds = np.std(data_array_2d_no_first_column, axis=1, keepdims=True)

# Check for zero standard deviation to avoid division by zero
if np.any(row_stds == 0):
    print("Warning: At least one row has zero standard deviation!")

# Standardize each row
standardized_data_array = (data_array_2d_no_first_column - row_means) / row_stds

# Display the shape and the standardized 2D NumPy array
print("Shape of 2D array after standardization:", standardized_data_array.shape)
print("Standardized 2D NumPy array:")
print(standardized_data_array)

Upload widget is only available when the cell has been executed in the current browser session. Please rerun this cell to enable.

Saving GSE13528.tsv to GSE13528 (3).tsv
Shape of 2D array after standardization: (7431, 47)
Standardized 2D NumPy array:
[[-1.87420155 -0.21015178  0.4600086  ...  1.64218751  1.37965243
   1.1528463 ]
 [ 0.11785913  1.89829026  0.54241833 ... -0.81702735 -0.99426793
  -0.9027692 ]
 [-0.54344755  0.57483723 -0.48740342 ... -1.35852917 -1.34812689
  -1.04097129]
 ...
 [-0.8151023  -1.04417854 -1.10897926 ...  1.17361101  0.84182685
   1.41484915]
 [-1.76321613 -2.1526713  -1.63587589 ...  0.44818769 -0.4695091
   0.2544299 ]
 [-0.98553276 -1.23452545  0.24771945 ...  0.84175022  0.67169982
   2.00111103]]

Generating the Covariance Matrix and Visualization of Eigenvalues

In this section, we try different ways of preprocessing the data.

X (Full Data Table with only row standardiztion)

X=np.transpose(standardized_data_array) ####all
# X=np.sqrt(47)*np.transpose(standardized_data_array) ####all
#X=np.sqrt(np.sqrt(46/7431))*np.transpose(standardized_data_array) ####all
# X=np.sqrt(np.sqrt(1/46/7431))*np.transpose(standardized_data_array) ####all
print("Shape of 2D array after transpose:", X.shape)

Shape of 2D array after transpose: (47, 7431)

covX=X@np.transpose(X)
E0,_=np.linalg.eigh(covX)
print(E0)
E00=np.array(E0)
E00.sort()
import matplotlib.pyplot as plt
plt.plot(E00,"*")

[-2.97835176e-12  1.20563800e+03  1.28402263e+03  1.35374932e+03
  1.35908130e+03  1.42567922e+03  1.48892615e+03  1.52946095e+03
  1.55295229e+03  1.57393577e+03  1.63507788e+03  1.65812156e+03
  1.74984556e+03  1.80346639e+03  1.82200642e+03  1.86020598e+03
  1.88580062e+03  1.91922089e+03  1.96652663e+03  2.00693436e+03
  2.08001352e+03  2.17453627e+03  2.27945108e+03  2.38972276e+03
  2.43380156e+03  2.53442025e+03  2.55386573e+03  2.69932383e+03
  2.81192456e+03  2.96855487e+03  3.05921100e+03  3.16103246e+03
  3.37266771e+03  3.62538606e+03  3.92781828e+03  4.11613805e+03
  4.26797156e+03  4.92800768e+03  5.38638453e+03  6.36547136e+03
  6.54024483e+03  8.34782265e+03  9.28588512e+03  1.04452396e+04
  1.33724403e+04  2.80772727e+04  1.74971740e+05]

[<matplotlib.lines.Line2D at 0x7e4bc14d7ca0>]

It can be seen that there are a few outliers and near zero eigenvalues.

Y (Randomly Select s Rows t Columns from the Full Data Table before Doing Row Normalization)

Other than dealing with the normalized raw data directly, we also try randomly selecting s rows (genes) and t columns (observations).

import numpy as np

# Create a sample 2D NumPy array for demonstration
# Assume its shape is (20, 10); you would replace this with your actual 2D array
sample_array = data_array_2d_no_first_column
print("Shape of 2D array after standardization:", sample_array.shape)
# Set the number of rows (s) and columns (t) you want to randomly select
s = 900
t = 30

np.random.seed(None)

# Make sure s and t are less than or equal to the number of rows and columns in the original array
if s > sample_array.shape[0] or t > sample_array.shape[1]:
    print("s and t should be less than or equal to the number of rows and columns in the original array.")
else:
    # Randomly select s rows and t columns
    selected_rows = np.random.choice(sample_array.shape[0], size=s, replace=False)
    selected_columns = np.random.choice(sample_array.shape[1], size=t, replace=False)



    ## Print out the selected rows and columns for verification
    # print(f"Selected row indices: {selected_rows}")
    # print(f"Selected column indices: {selected_columns}")



    # Extract those rows and columns to form a new array
    new_array = sample_array[selected_rows][:, selected_columns]

    # Compute row-wise mean and standard deviation for the new array
    row_means = np.mean(new_array, axis=1, keepdims=True)
    row_stds = np.std(new_array, axis=1, keepdims=True)

    # Check for zero standard deviation to avoid division by zero
    if np.any(row_stds == 0):
        print("Warning: At least one row has zero standard deviation!")

    # Standardize each row of the new array
    standardized_new_array = (new_array - row_means) / row_stds

    # Display the standardized new array
    print("Standardized new array with {} randomly selected rows and {} randomly selected columns:".format(s, t))
    # print(standardized_new_array)

Y=np.transpose(standardized_new_array)
Y_row_stds = np.std(standardized_new_array, axis=1, keepdims=True)
# print(Y_row_stds)
# Y=np.sqrt(t)*np.transpose(standardized_new_array)
#Y=np.sqrt(np.sqrt(t/s))*np.transpose(standardized_new_array)

# print(Y)
# pg=DW2023Global2(Y, Sigma="identity", c=0, K4=0, alternative='two-sided')
# print(pg)

Shape of 2D array after standardization: (7431, 47)
Standardized new array with 900 randomly selected rows and 30 randomly selected columns:

covY=Y@np.transpose(Y)/np.sqrt(s*t)
E0,_=np.linalg.eigh(covY)
print(E0)
E00=np.array(E0)
E00.sort()
import matplotlib.pyplot as plt
plt.plot(E00,"*")

[-9.49959181e-16  9.12899215e-01  9.79085549e-01  1.04822907e+00
  1.09908910e+00  1.14042529e+00  1.18066026e+00  1.28906643e+00
  1.37168942e+00  1.47455779e+00  1.59336751e+00  1.63878855e+00
  1.66393123e+00  1.73534461e+00  1.79252854e+00  1.81403750e+00
  2.00125421e+00  2.36189525e+00  2.40236797e+00  2.49868246e+00
  2.85886189e+00  3.12210005e+00  3.28842294e+00  3.75166187e+00
  3.91304058e+00  4.63865647e+00  5.61633409e+00  6.14054843e+00
  1.30078684e+01  8.79813726e+01]

[<matplotlib.lines.Line2D at 0x7e4bc130e800>]

It can be seen that there are a few outliers and near zero eigenvalues.

Z (Group and Month Deeffect)

In the preprocessing above, we treated the data as if there are no correlations and structures. In fact, the data are time series (12 months) with certain group structures (4). Since only 47 of 48 observations are downloadable, we discard 9 of them and keep 36=3*12 to de-effect the groups and months.

# Set your desired number of rows and columns
s =7200
t = 36

# Select the first s rows and t columns
selected_data = data_array_2d_no_first_column[:s, :t]

import numpy as np

# Assuming you have data of shape (p, n)
p = s  # For instance, 100 features
n = 36  # 3 groups x 12 months
data = selected_data # Generate random data for the example

num_groups = 3
num_months = 12

# De-effect for the group
for feature in range(p):
    for i in range(num_groups):
        start_idx = i * num_months
        end_idx = (i + 1) * num_months
        group_data = data[feature, start_idx:end_idx]
        group_mean = np.mean(group_data)
        data[feature, start_idx:end_idx] -= group_mean

# De-effect for the month
for feature in range(p):
    for j in range(num_months):
        month_data = data[feature, j::num_months]  # Extract every 12th value starting from j
        month_mean = np.mean(month_data)
        data[feature, j::num_months] -= month_mean

# Now the data array is adjusted for both group and month effects for each feature
print(data)

[[-4.82330689e-02  1.90971731e-02  4.00572704e-02 ... -2.55866927e-02
  -1.76563428e-02 -6.44883707e-03]
 [-1.14442543e-01  5.39168782e-01  7.33330571e-02 ...  2.93166994e-01
   3.26424548e-01  4.65611183e-01]
 [-3.05694900e-02 -2.44337836e-02 -3.01152907e-02 ...  6.21976245e-04
  -7.49606338e-03  1.21116395e-02]
 ...
 [ 2.98317947e-02 -1.63484013e-02 -5.50867121e-02 ... -2.78640157e-01
   7.55377593e-03 -4.05074979e-02]
 [-1.55844121e-01 -6.53782987e-01  3.37614125e-02 ... -1.15430889e+00
   7.45244358e-01 -3.65670184e-01]
 [-1.23627377e-02  2.45902100e-02 -3.27824996e-03 ...  5.23678867e-03
   1.77809510e-02  4.14508408e-03]]

# Compute row-wise mean and standard deviation for the new array
new_array=data
row_means = np.mean(new_array, axis=1, keepdims=True)
row_stds = np.std(new_array, axis=1, keepdims=True)

# Check for zero standard deviation to avoid division by zero
if np.any(row_stds == 0):
    print("Warning: At least one row has zero standard deviation!")

# Standardize each row of the new array
standardized_new_array = (new_array - row_means) / row_stds

Z=np.transpose(standardized_new_array)
# Z=np.sqrt(t)*np.transpose(standardized_new_array)

covZ=Z@np.transpose(Z)/np.sqrt(s*t)
E0,_=np.linalg.eigh(covZ)
print(E0)
E00=np.array(E0)
E00.sort()
import matplotlib.pyplot as plt
plt.plot(E00[1:40],"*")

[-1.27919405e-14 -8.80845806e-15 -6.18319981e-15 -3.88171290e-15
 -6.22177015e-16  3.27475452e-16  2.10965660e-15  2.16903775e-15
  5.96548026e-15  1.07079708e-14  1.36855020e-14  1.57802764e-14
  2.31366100e-14  2.58614967e-14  4.32853910e+00  4.78726276e+00
  5.11104770e+00  5.17866623e+00  5.49550562e+00  5.86972424e+00
  5.95308646e+00  6.17413525e+00  6.84579593e+00  7.12205014e+00
  7.83566572e+00  8.06098936e+00  9.05822804e+00  9.83051298e+00
  1.01095101e+01  1.22124982e+01  1.40999018e+01  1.64740587e+01
  2.01700423e+01  2.21274751e+01  4.67715861e+01  2.75500601e+02]

[<matplotlib.lines.Line2D at 0x7e4b7b931810>]

After de-effect the groups and months via simple mean substraction, we can see that the spiked structure is persistent and there are still near 0 eigenvalues.

W (ANOVA adjusted, deeffect the genotype by groups )

We follow similar scheme to the one in [Chen2010] to do the preprocessing, namely we assume the following model: $$ y_{i j k l}^g=\tau^g+\mu_i^g+\beta_j \mathbf{1}_{p_g}+\alpha_k \mathbf{1}_{p_g}+\eta_G \mathbf{1}_{p_g}+\epsilon_{i j k l}^g, $$ where $\tau^g$ and $\mu_i^g(i=1,2)$ are $p_g$ dimensional vectors denoting the intercepts and the treatment effects respectively, $\beta_j$ $(j=1, \ldots, 3), \alpha_k(k=1,2)$, and $\eta_G$ are univariate denoting the block, the genotype, and the gene-set effects respectively, $\epsilon_{i j k l}^g$ is the vector of residuals, and $\mathbf{1}_{p_g}$ is the $p_g$ dimensional vector of 1 's.

Similar to $Z$, the above grouping and data selection is partially due to the fact that in 4*12 samples, only 47 are available, so we discard 11 of them and only keep 3 groups.

The treatment and genotype effects $\mu_i^g$ and $\alpha_k$ are treated as fixed, while the block effect and the GO term effects are random. Further, assume $\mu_2^g=\alpha_2=0$. To remove the treatment, block and the genotype effects, $\mu_i^g$ and $\tau^g$ are estimated by $$ \begin{aligned} \hat{\mu}_i^g & =\frac{1}{n(i)} \sum_{j, k, l} y_{i j k l}^g-\frac{1}{n(2)} \sum_{j, k, l} y_{2 j k l}^g, \quad \\ \hat{\tau}^g & =\frac{1}{n(2)} \sum_{j, k, l} y_{2 j k l}^g, \end{aligned} $$ where $n(i)$ is the number of observations in treatment $i$. Then, estimate $\beta$ and $\alpha$ by regression $$ y_{i j k l}^g-\hat{\mu}_i^g-\hat{\tau}^g=\beta_j I_{p_g}+\alpha_k \mathbf{1}_{p_g}+e_{i j k l}^g . $$ Here $e_{i j k l}^g$ are the "residuals" representing $\eta_G \mathbf{1}_{p_g}+\epsilon_{i j k l}^g$ in model above via the least square regression.

We are interested in testing for the covariance structure of $\eta_G \mathbf{1}_{p_g}+\epsilon_{i j k l}^g$. It is not too restrictive to assume $\operatorname{var}\left(\epsilon_{i j k l}^g\right)=\sigma_\epsilon^2 I_{p_g}$ for a positive constant $\sigma_\epsilon^2$. Hence, the covariance of $\eta_G \mathbf{1}_{p_g}+\epsilon_{i j k l}^g$ is $$ \Sigma=: \sigma_\epsilon^2 I_{p_g}+\sigma_{\eta_G}^2 \mathbf{1}_{p_g} \mathbf{1}_{p_g}^T, $$ say, where $\sigma_{\eta_G}^2=\operatorname{var}\left(\eta_G\right)$. Let $\Sigma_{\mathcal{S}_g}$ be the covariance matrix corresponding to the gene set $\mathcal{S}_g$.

We want to test the identity hypothesis $H_{0}: \Sigma_{\mathcal{S}_g}=I_{p_g}$ for $g=1, \ldots, q$ after we plug in a trivial estimator for $\sigma_{ϵ}$.

Moreover, we use two different treatments:

1. We assume that $\sigma_{\eta_G}^2=0$ and directly test about $H_{0}$ .
2. We take the existence of nonvanishing $\sigma_{\eta_G}^2$ into consideration, exclude the spikes and renormalize, then test $H_{0}$ with the remaining eigenvalues.

import numpy as np
from sklearn.linear_model import LinearRegression

# Set your desired number of rows and columns
p =7200
n = 3*12

# Select the first s rows and t columns
data = data_array_2d_no_first_column[:p, :n]

treatments = 3
blocks = 12

# Calculate mu_hat and tau_hat
mu_hat = np.zeros((p, treatments))
for i in range(treatments):
    mu_hat[:, i] = np.mean(data[:, i*blocks:(i+1)*blocks], axis=1) - np.mean(data[:, blocks:2*blocks], axis=1)

tau_hat = np.mean(data[:, blocks:2*blocks], axis=1)

# Prepare residuals for linear regression
residuals_for_regression = np.zeros_like(data)
for i in range(treatments):
    residuals_for_regression[:, i*blocks:(i+1)*blocks] = data[:, i*blocks:(i+1)*blocks] - mu_hat[:, i][:, np.newaxis] - tau_hat[:, np.newaxis]

# Create design matrix for linear regression
X_beta = np.tile(np.eye(blocks), (treatments, 1))
X_alpha = np.ones((blocks * treatments, 1))
X = np.hstack([X_beta, X_alpha])
X = np.tile(X, (p, 1))

# Reshape residuals for regression
y = residuals_for_regression.T.reshape(-1)

# Fit the linear model
model = LinearRegression(fit_intercept=False)
model.fit(X, y)

# Get the predicted values
y_pred = model.predict(X)

# Calculate the residuals matrix
e_matrix = y - y_pred
residuals_matrix = e_matrix.reshape((n, p)).T

# Subtract overall mean * 1 * 1^T from residuals_matrix
overall_mean = np.mean(residuals_matrix)
residuals_matrix -= overall_mean * np.ones(residuals_matrix.shape)

# # Normalize each row
# row_norms = np.linalg.norm(residuals_matrix, axis=1, keepdims=True)
# normalized_matrix = residuals_matrix / row_norms


# Extract coefficients
beta = model.coef_[:blocks]
alpha = model.coef_[blocks:]

print("Beta coefficients:", beta)
print("Alpha coefficients:", alpha)
print("Residuals matrix:")
print(residuals_matrix)

Beta coefficients: [-3.34578427e+10 -3.34578427e+10 -3.34578427e+10 -3.34578427e+10
 -3.34578427e+10 -3.34578427e+10 -3.34578427e+10 -3.34578427e+10
 -3.34578427e+10 -3.34578427e+10 -3.34578427e+10 -3.34578427e+10]
Alpha coefficients: [3.34578427e+10]
Residuals matrix:
[[-4.82375194e-02  1.90927226e-02  4.00528199e-02 ... -2.55911432e-02
  -1.76607933e-02 -6.45328755e-03]
 [-1.14420291e-01  5.39191034e-01  7.33553095e-02 ...  2.93189247e-01
   3.26446800e-01  4.65633435e-01]
 [-3.06044580e-02 -2.44687516e-02 -3.01502588e-02 ...  5.87008187e-04
  -7.53103144e-03  1.20766714e-02]
 ...
 [ 2.98311589e-02 -1.63490371e-02 -5.50873479e-02 ... -2.78640793e-01
   7.55314015e-03 -4.05081337e-02]
 [-1.55837127e-01 -6.53775994e-01  3.37684061e-02 ... -1.15430190e+00
   7.45251351e-01 -3.65663191e-01]
 [-1.23557441e-02  2.45972036e-02 -3.27125635e-03 ...  5.24378228e-03
   1.77879446e-02  4.15207769e-03]]

s = 7200
t = 20
sample_array = residuals_matrix
np.random.seed(None)

# Make sure s and t are less than or equal to the number of rows and columns in the original array
if s > sample_array.shape[0] or t > sample_array.shape[1]:
    print("s and t should be less than or equal to the number of rows and columns in the original array.")
else:
    # Randomly select s rows and t columns
    selected_rows = np.random.choice(sample_array.shape[0], size=s, replace=False)
    selected_columns = np.random.choice(sample_array.shape[1], size=t, replace=False)



    # Print out the selected rows and columns for verification
    # print(f"Selected row indices: {selected_rows}")
    print(f"Selected column indices: {selected_columns}")



    # Extract those rows and columns to form a new array
    new_array = sample_array[selected_rows][:, selected_columns]

    # Compute row-wise mean and standard deviation for the new array
    row_means = np.mean(new_array, axis=1, keepdims=True)
    row_stds = np.std(new_array, axis=1, keepdims=True)

    # Check for zero standard deviation to avoid division by zero
    if np.any(row_stds == 0):
        print("Warning: At least one row has zero standard deviation!")

    # Standardize each row of the new array
    standardized_new_array = (new_array - row_means) / row_stds

    # Display the standardized new array
    print("Standardized new array with {} randomly selected rows and {} randomly selected columns:".format(s, t))
    print(standardized_new_array)

W=np.transpose(standardized_new_array)
# print(Z)
# print(np.var(standardized_new_array))

Selected column indices: [ 9 11 34 23 35 12 13 16  8  0 26 20 33 28  6  4 30 25 22  2]
Standardized new array with 7200 randomly selected rows and 20 randomly selected columns:
[[ 0.28368638  0.81629766 -1.09616469 ...  0.34182413  1.36703654
   0.09826138]
 [ 0.43297467  0.05131877  0.10878314 ...  0.98513408 -0.49149842
   1.32142019]
 [ 0.54921776 -0.20930431 -1.16849589 ...  0.65594129  0.51737222
   0.35812078]
 ...
 [-0.77874543 -0.43510541 -0.82269104 ... -0.52772954  1.04723551
  -0.81909189]
 [-0.80025706 -0.76632935 -0.664351   ... -0.59860291  1.02109853
  -0.67525688]
 [ 0.16928291  0.59808625  0.22585904 ... -0.73151895 -1.02109905
  -1.47769774]]

covW=W@np.transpose(W)/np.sqrt(s*t)
E0,_=np.linalg.eigh(covW)
print(E0)
E00=np.array(E0)
E00.sort()
import matplotlib.pyplot as plt
plt.plot(E00,"*")

[-5.46985817e-15 -2.56619927e-15  1.06624798e-14  1.31482332e+00
  1.74656215e+00  3.73495306e+00  4.38150436e+00  4.88788117e+00
  6.34019250e+00  7.31720190e+00  8.25593790e+00  8.82974425e+00
  8.99254390e+00  1.03776563e+01  1.25216678e+01  1.52893917e+01
  1.74919702e+01  2.14243310e+01  4.19170476e+01  2.04649910e+02]

[<matplotlib.lines.Line2D at 0x7e4bc1034130>]

We can see that the spiked structure is persistent and there are still near 0 eigenvalues even for the residue matrix, which hints us to remove the largest eigenvalue which corresponds to $\sigma_{\eta_G}^2 \mathbf{1}_{p_g} \mathbf{1}_{p_g}^T$.

Global Test (assuming Gaussian)

In this section, we apply our global test DW2023Global2 which is induced by $x^2$ to data matrices with different ways of preprocessing.

X

pg=DW2023Global2(X, Sigma="identity", c=0, K4=0, alternative='two-sided')
print(pg)

0.0

Y

pg=DW2023Global2(Y, Sigma="identity", c=0, K4=0, alternative='two-sided')
print(pg)

0.0

repeat the random selection scheme for 100 times

import numpy as np

sample_array = data_array_2d_no_first_column

# Set the number of rows (s) and columns (t) you want to randomly select
s = 900
t = 40

# List to store all pg values
pg_values = []

for _ in range(200):  # repeat 100 times
    np.random.seed(None)

    # Make sure s and t are less than or equal to the number of rows and columns in the original array
    if s > sample_array.shape[0] or t > sample_array.shape[1]:
        print("s and t should be less than or equal to the number of rows and columns in the original array.")
        break
    else:
        # Randomly select s rows and t columns
        selected_rows = np.random.choice(sample_array.shape[0], size=s, replace=False)
        selected_columns = np.random.choice(sample_array.shape[1], size=t, replace=False)

        # Extract those rows and columns to form a new array
        new_array = sample_array[selected_rows][:, selected_columns]

        # Compute row-wise mean and standard deviation for the new array
        row_means = np.mean(new_array, axis=1, keepdims=True)
        row_stds = np.std(new_array, axis=1, keepdims=True)

        # Check for zero standard deviation to avoid division by zero
        if np.any(row_stds == 0):
            print("Warning: At least one row has zero standard deviation!")
            continue

        # Standardize each row of the new array
        standardized_new_array = (new_array - row_means) / row_stds


        Y = np.transpose(standardized_new_array)
        # Y = np.sqrt(t) * np.transpose(standardized_new_array)
        # Y = np.sqrt(np.sqrt(t/s)) * np.transpose(standardized_new_array)

        pg = DW2023Global2(Y, Sigma="identity", c=0, K4=0, alternative='two-sided')
        pg_values.append(pg)

print("All pg values after 100 iterations:")
print(pg_values)

All pg values after 100 iterations:
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]

Z

Other than the naive test, we also used a modified version DW2023Global2modified which excludes the spikes and near zero eigenvalues.

pg=DW2023Global2(Z, Sigma="identity", c=0, K4=0, alternative='two-sided')
print(pg)
pg=DW2023Global2modified(Z, Sigma="identity", c=0, K4=0, alternative='two-sided')
print(pg)

0.0
0.0

randomly extract submatrices and repeat 100 times

sample_array = np.transpose(Z)
# Set the number of rows (s) and columns (t) you want to randomly select
s = 900
t = 36

# List to store all pg values
pg_values = []
np.random.seed(None)
for _ in range(200):  # repeat 100 times


    # Make sure s and t are less than or equal to the number of rows and columns in the original array
    if s > sample_array.shape[0] or t > sample_array.shape[1]:
        print("s and t should be less than or equal to the number of rows and columns in the original array.")
        break
    else:
        # Randomly select s rows and t columns
        selected_rows = np.random.choice(sample_array.shape[0], size=s, replace=False)
        selected_columns = np.random.choice(sample_array.shape[1], size=t, replace=False)

        # Extract those rows and columns to form a new array
        new_array = sample_array[selected_rows][:, selected_columns]

        # Compute row-wise mean and standard deviation for the new array
        row_means = np.mean(new_array, axis=1, keepdims=True)
        row_stds = np.std(new_array, axis=1, keepdims=True)

        # Check for zero standard deviation to avoid division by zero
        if np.any(row_stds == 0):
            print("Warning: At least one row has zero standard deviation!")
            continue

        # Standardize each row of the new array
        standardized_new_array = (new_array - row_means) / row_stds

        Zt= np.transpose(standardized_new_array)
        # Zt= np.sqrt(t) * np.transpose(standardized_new_array)
        # Zt = np.sqrt(np.sqrt(t/s)) * np.transpose(standardized_new_array)

        pg = DW2023Global2(Zt, Sigma="identity", c=0, K4=0, alternative='two-sided')
        pg_values.append(pg)

print("All pg values after 100 iterations:")
print(pg_values)

All pg values after 100 iterations:
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]

sample_array = np.transpose(Z)
# Set the number of rows (s) and columns (t) you want to randomly select
s = 42
t = 36

# List to store all pg values
pg_values = []
# np.random.seed(None)
for _ in range(200):  # repeat 100 times


    # Make sure s and t are less than or equal to the number of rows and columns in the original array
    if s > sample_array.shape[0] or t > sample_array.shape[1]:
        print("s and t should be less than or equal to the number of rows and columns in the original array.")
        break
    else:
        # Randomly select s rows and t columns
        selected_rows = np.random.choice(sample_array.shape[0], size=s, replace=False)
        selected_columns = np.random.choice(sample_array.shape[1], size=t, replace=False)

        # Extract those rows and columns to form a new array
        new_array = sample_array[selected_rows][:, selected_columns]

        # Compute row-wise mean and standard deviation for the new array
        row_means = np.mean(new_array, axis=1, keepdims=True)
        row_stds = np.std(new_array, axis=1, keepdims=True)

        # Check for zero standard deviation to avoid division by zero
        if np.any(row_stds == 0):
            print("Warning: At least one row has zero standard deviation!")
            continue

        # Standardize each row of the new array
        standardized_new_array = (new_array - row_means) / row_stds

        # Zt = np.sqrt(np.sqrt(t/s)) * np.transpose(standardized_new_array)
        # Zt = np.sqrt(t) * np.transpose(standardized_new_array)
        Zt = np.transpose(standardized_new_array)


        pg = DW2023Global2(Zt, Sigma="identity", c=0, K4=0, alternative='two-sided')
        pg_values.append(pg)

print("All pg values after 100 iterations:")
print(pg_values)

All pg values after 100 iterations:
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]

W

Since we can see from the eigen plot that the existing spikes, we remove largest eigenvalues/near zero eigenvalues and normalize before testing.

n,p=W.shape
H = np.matmul(W, W.T) / np.sqrt(n * p)
TestST=np.trace(H)
# Compute eigenvalues of H
eigvals = np.linalg.eigvalsh(H)

    # Filter eigenvalues below 10^-3 and above 100
filtered_eigvals = eigvals[(eigvals < 1e-3) | (eigvals > 100)]

    # Subtract the sum of squares of filtered eigenvalues from TestST
# TestST0 = np.trace(np.matmul(H, H))
TestST = np.trace(H) - np.sum(filtered_eigvals)

normalize_constant=TestST/n/np.sqrt(p/n)
print(normalize_constant)
    # Record the number of such eigenvalues as ns
# ns = len(filtered_eigvals)

0.44979893055089815

pg=DW2023Global2(W/normalize_constant, Sigma="identity", c=0, K4=0, alternative='two-sided')
print(pg)
pg=DW2023Global2modified(W/normalize_constant, Sigma="identity", c=0, K4=0, alternative='two-sided')
print(pg)

0.0
0.0

randomly select submatrices and repeat 100 times

sample_array = np.transpose(Z)
# Set the number of rows (s) and columns (t) you want to randomly select
s = 900
t = 36

# List to store all pg values
pg_values = []
np.random.seed(None)
for _ in range(200):  # repeat 100 times


    # Make sure s and t are less than or equal to the number of rows and columns in the original array
    if s > sample_array.shape[0] or t > sample_array.shape[1]:
        print("s and t should be less than or equal to the number of rows and columns in the original array.")
        break
    else:
        # Randomly select s rows and t columns
        selected_rows = np.random.choice(sample_array.shape[0], size=s, replace=False)
        selected_columns = np.random.choice(sample_array.shape[1], size=t, replace=False)

        # Extract those rows and columns to form a new array
        new_array = sample_array[selected_rows][:, selected_columns]

        # Compute row-wise mean and standard deviation for the new array
        row_means = np.mean(new_array, axis=1, keepdims=True)
        row_stds = np.std(new_array, axis=1, keepdims=True)

        # Check for zero standard deviation to avoid division by zero
        if np.any(row_stds == 0):
            print("Warning: At least one row has zero standard deviation!")
            continue

        # Standardize each row of the new array
        standardized_new_array = (new_array - row_means) / row_stds

        Zt= np.transpose(standardized_new_array)
        # Zt= np.sqrt(t) * np.transpose(standardized_new_array)
        # Zt = np.sqrt(np.sqrt(t/s)) * np.transpose(standardized_new_array)

        nt,pt=Zt.shape
        Ht = np.matmul(Zt, Zt.T) / np.sqrt(nt * pt)
        eigvals = np.linalg.eigvalsh(Ht)  # Compute eigenvalues of H, in ascending order
        max_eigval = eigvals[-1]  # Extract the largest eigenvalue
        normalize_constantt=(np.trace(Ht)-max_eigval)/nt/np.sqrt(pt/nt)
        # print(normalize_constantt)

        pg = DW2023Global2(Zt/normalize_constantt, Sigma="identity", c=0, K4=0, alternative='two-sided')
        pg_values.append(pg)

print("All pg values after 100 iterations:")
print(pg_values)

All pg values after 100 iterations:
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]

sample_array = np.transpose(Z)
# Set the number of rows (s) and columns (t) you want to randomly select
s = 42
t = 36

# List to store all pg values
pg_values = []
# np.random.seed(None)
for _ in range(200):  # repeat 100 times


    # Make sure s and t are less than or equal to the number of rows and columns in the original array
    if s > sample_array.shape[0] or t > sample_array.shape[1]:
        print("s and t should be less than or equal to the number of rows and columns in the original array.")
        break
    else:
        # Randomly select s rows and t columns
        selected_rows = np.random.choice(sample_array.shape[0], size=s, replace=False)
        selected_columns = np.random.choice(sample_array.shape[1], size=t, replace=False)

        # Extract those rows and columns to form a new array
        new_array = sample_array[selected_rows][:, selected_columns]

        # Compute row-wise mean and standard deviation for the new array
        row_means = np.mean(new_array, axis=1, keepdims=True)
        row_stds = np.std(new_array, axis=1, keepdims=True)

        # Check for zero standard deviation to avoid division by zero
        if np.any(row_stds == 0):
            print("Warning: At least one row has zero standard deviation!")
            continue

        # Standardize each row of the new array
        standardized_new_array = (new_array - row_means) / row_stds

        # Zt = np.sqrt(np.sqrt(t/s)) * np.transpose(standardized_new_array)
        # Zt = np.sqrt(t) * np.transpose(standardized_new_array)
        Wt = np.transpose(standardized_new_array)


        pg = DW2023Global2(Wt, Sigma="identity", c=0, K4=0, alternative='two-sided')
        pg_values.append(pg)

print("All pg values after 100 iterations:")
print(pg_values)

All pg values after 100 iterations:
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]

Local Test

Prerun for local

Since real data is quite different from simulated data, we use different hyperparameters here. Therefore, we define a local function "DW2023Local" in this subsection only.

import jax
import jax.numpy as jnp
from jax import grad, jit,random
from jax.random import normal
import numpy as np
import matplotlib.pyplot as plt

# for t=0: 1: 0.1
t=0.000


n = 60
R = 1000
c=0
T = jnp.zeros(R)
# key = jax.random.PRNGKey(0)

# for k in range(R):
#     key, subkey = jax.random.split(key)
#     X = normal(subkey, shape=(m, n))
#     T = T.at[k].set(objective(X))


TestST=np.array(T)

def count_outside_interval(arr, a, b):
    outside_interval = np.logical_or(arr < a, arr > b)
    count = np.sum(outside_interval)
    return count


def extract_elements(arr, a, b):
    return arr[(arr >= a) & (arr <= b)]



K4=0

# Normalize_TestST=(TestST-(n*(1+1/(c-1/np.sqrt(phi))**2)+1/(c-1/np.sqrt(phi))**2)*np.ones(R))/np.sqrt((4+4*2*(-1/np.sqrt(phi)+1/(c-1/np.sqrt(phi))+1/n/(c-np.sqrt(phi)))**2)/((c-1/np.sqrt(phi))**4))
# plot_ecdf_and_standard_normal_cdf(Normalize_TestST)


pn=2*jnp.ones(n)

# eta=1/(n**(1/4))
eta=5

# k=int(n*eta)
R0=10000


#calculate the null distribution
phi=10
m=int(n*phi)
r=jnp.sqrt(phi)

leftb = jnp.sqrt(phi)+1/jnp.sqrt(phi)-2
    # objective_grad = jit(grad(objective))
T0 = jnp.zeros(R0)
key = jax.random.PRNGKey(4)
for r in range(R0):
    key, subkey = jax.random.split(key)
    X = random.normal(key, shape=(m, n))
    H = jnp.matmul(X.T, X) / jnp.sqrt(n * m)
    E00, _ = jnp.linalg.eigh(H)
    E0=np.array(E00)
    Et = jnp.array(extract_elements(E0, leftb-eta, leftb+eta))
    # print(E0[-k:])
    Eb = (Et - leftb*jnp.ones(Et.shape))/eta
    t0=jnp.sum(Eb)
    T0 = T0.at[r].set(t0)
TestST0=np.array(T0)
Mean_x=jnp.mean(TestST0)
Variance_x=jnp.var(TestST0)
print(Mean_x)
print(Variance_x)

WARNING:jax._src.xla_bridge:No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)

20.202404
0.07956229

Mean_x=20.202404
Variance_x=0.007956229

def DW2023Local(X, Sigma="identity", c=0, K4=0, alternative='two-sided'):
    # m=X.shape[0]
    # n=X.shape[1]
    m=X.shape[1]
    n=X.shape[0]
    phi=m/n
    r=np.sqrt(phi)
    leftb=r+1/r-2+c
    H = np.matmul(X, X.T) / np.sqrt(n * m)
    E0, _ = np.linalg.eigh(H)
    eta=5
    # print(E0)
    Et = np.array(extract_elements(E0, leftb-eta, leftb+eta))
    # print(Eb)
    # print(E0[-k:])
    Eb = (Et - leftb*np.ones(Et.shape))/eta
    # print(Eb)
    TestST=np.sum(Eb)
    z_score=(TestST-Mean_x)/np.sqrt(Variance_x)
    # print(z_score)
    if alternative == 'two-sided':
        p_value = 2 * (1 - norm.cdf(abs(z_score)))
    elif alternative == 'greater':
        p_value = 1 - norm.cdf(z_score)
    elif alternative == 'less':
        p_value = norm.cdf(z_score)
    else:
        raise ValueError("alternative must be 'two-sided', 'greater' or 'less'")

    return p_value

X

pl=DW2023Local(X, Sigma="identity", c=0, K4=0, alternative='two-sided')
print(pl)

0.0

Y

pl=DW2023Local(Y, Sigma="identity", c=0, K4=0, alternative='two-sided')
print(pl)

0.0

randomly select submatrices and repeat 100 times

import numpy as np

sample_array = data_array_2d_no_first_column

# Set the number of rows (s) and columns (t) you want to randomly select
s = 900
t = 40

# List to store all pg values
pl_values = []

for _ in range(200):  # repeat 100 times
    np.random.seed(None)

    # Make sure s and t are less than or equal to the number of rows and columns in the original array
    if s > sample_array.shape[0] or t > sample_array.shape[1]:
        print("s and t should be less than or equal to the number of rows and columns in the original array.")
        break
    else:
        # Randomly select s rows and t columns
        selected_rows = np.random.choice(sample_array.shape[0], size=s, replace=False)
        selected_columns = np.random.choice(sample_array.shape[1], size=t, replace=False)

        # Extract those rows and columns to form a new array
        new_array = sample_array[selected_rows][:, selected_columns]

        # Compute row-wise mean and standard deviation for the new array
        row_means = np.mean(new_array, axis=1, keepdims=True)
        row_stds = np.std(new_array, axis=1, keepdims=True)

        # Check for zero standard deviation to avoid division by zero
        if np.any(row_stds == 0):
            print("Warning: At least one row has zero standard deviation!")
            continue

        # Standardize each row of the new array
        standardized_new_array = (new_array - row_means) / row_stds

        Y = np.transpose(standardized_new_array)
        # Y = np.sqrt(t) * np.transpose(standardized_new_array)
        # Y = np.sqrt(np.sqrt(t/s)) * np.transpose(standardized_new_array)

        pl = DW2023Local(Y, Sigma="identity", c=0, K4=0, alternative='two-sided')
        pl_values.append(pl)

print("All pl values after 100 iterations:")
print(pl_values)

All pl values after 100 iterations:
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]

import numpy as np

sample_array = data_array_2d_no_first_column

# Set the number of rows (s) and columns (t) you want to randomly select
s = 40
t = 40

# List to store all pg values
pg_values = []

for _ in range(200):  # repeat 100 times
    np.random.seed(None)

    # Make sure s and t are less than or equal to the number of rows and columns in the original array
    if s > sample_array.shape[0] or t > sample_array.shape[1]:
        print("s and t should be less than or equal to the number of rows and columns in the original array.")
        break
    else:
        # Randomly select s rows and t columns
        selected_rows = np.random.choice(sample_array.shape[0], size=s, replace=False)
        selected_columns = np.random.choice(sample_array.shape[1], size=t, replace=False)

        # Extract those rows and columns to form a new array
        new_array = sample_array[selected_rows][:, selected_columns]

        # Compute row-wise mean and standard deviation for the new array
        row_means = np.mean(new_array, axis=1, keepdims=True)
        row_stds = np.std(new_array, axis=1, keepdims=True)

        # Check for zero standard deviation to avoid division by zero
        if np.any(row_stds == 0):
            print("Warning: At least one row has zero standard deviation!")
            continue

        # Standardize each row of the new array
        standardized_new_array = (new_array - row_means) / row_stds

        Y = np.transpose(standardized_new_array)
        # Y = np.sqrt(t) * np.transpose(standardized_new_array)
        # Y = np.sqrt(np.sqrt(t/s)) * np.transpose(standardized_new_array)


        pl = DW2023Local(Y, Sigma="identity", c=0, K4=0, alternative='two-sided')
        pl_values.append(pl)

print("All pl values after 100 iterations:")
print(pl_values)

All pl values after 100 iterations:
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]

Z

pl=DW2023Local(Z, Sigma="identity", c=0, K4=0, alternative='two-sided')
print(pl)

0.0

randomly select submatrices and repeat 100 times

sample_array = np.transpose(Z)
# Set the number of rows (s) and columns (t) you want to randomly select
s = 900
t = 36

# List to store all pg values
pl_values = []
np.random.seed(None)
for _ in range(200):  # repeat 100 times


    # Make sure s and t are less than or equal to the number of rows and columns in the original array
    if s > sample_array.shape[0] or t > sample_array.shape[1]:
        print("s and t should be less than or equal to the number of rows and columns in the original array.")
        break
    else:
        # Randomly select s rows and t columns
        selected_rows = np.random.choice(sample_array.shape[0], size=s, replace=False)
        selected_columns = np.random.choice(sample_array.shape[1], size=t, replace=False)

        # Extract those rows and columns to form a new array
        new_array = sample_array[selected_rows][:, selected_columns]

        # Compute row-wise mean and standard deviation for the new array
        row_means = np.mean(new_array, axis=1, keepdims=True)
        row_stds = np.std(new_array, axis=1, keepdims=True)

        # Check for zero standard deviation to avoid division by zero
        if np.any(row_stds == 0):
            print("Warning: At least one row has zero standard deviation!")
            continue

        # Standardize each row of the new array
        standardized_new_array = (new_array - row_means) / row_stds


        Zt= np.transpose(standardized_new_array)
        # Zt= np.sqrt(t) * np.transpose(standardized_new_array)
        # Zt = np.sqrt(np.sqrt(t/s)) * np.transpose(standardized_new_array)


        pl = DW2023Local(Y, Sigma="identity", c=0, K4=0, alternative='two-sided')
        pl_values.append(pl)

print("All pl values after 100 iterations:")
print(pl_values)

All pl values after 100 iterations:
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]

sample_array = np.transpose(Z)
# Set the number of rows (s) and columns (t) you want to randomly select
s = 42
t = 36

# List to store all pg values
pg_values = []
# np.random.seed(None)
for _ in range(200):  # repeat 100 times


    # Make sure s and t are less than or equal to the number of rows and columns in the original array
    if s > sample_array.shape[0] or t > sample_array.shape[1]:
        print("s and t should be less than or equal to the number of rows and columns in the original array.")
        break
    else:
        # Randomly select s rows and t columns
        selected_rows = np.random.choice(sample_array.shape[0], size=s, replace=False)
        selected_columns = np.random.choice(sample_array.shape[1], size=t, replace=False)

        # Extract those rows and columns to form a new array
        new_array = sample_array[selected_rows][:, selected_columns]

        # Compute row-wise mean and standard deviation for the new array
        row_means = np.mean(new_array, axis=1, keepdims=True)
        row_stds = np.std(new_array, axis=1, keepdims=True)

        # Check for zero standard deviation to avoid division by zero
        if np.any(row_stds == 0):
            print("Warning: At least one row has zero standard deviation!")
            continue

        # Standardize each row of the new array
        standardized_new_array = (new_array - row_means) / row_stds

        # Zt = np.sqrt(np.sqrt(t/s)) * np.transpose(standardized_new_array)
        # Zt = np.sqrt(t) * np.transpose(standardized_new_array)
        Zt = np.transpose(standardized_new_array)


        pl = DW2023Local(Y, Sigma="identity", c=0, K4=0, alternative='two-sided')
        pl_values.append(pl)

print("All pl values after 100 iterations:")
print(pl_values)

All pl values after 100 iterations:
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]

W

Similar to the treatment in Global testing, we remove largest eig and normalize before testing.

# calculate the approximated normalizing constant
n,p=W.shape
H = np.matmul(W, W.T) / np.sqrt(n * p)
eigvals = np.linalg.eigvalsh(H)  # Compute eigenvalues of H, in ascending order
max_eigval = eigvals[-1]  # Extract the largest eigenvalue
normalize_constant=(np.trace(H)-max_eigval)/n/np.sqrt(p/n) #calculating the normalization constant after removing the largest eigenvalues
print(normalize_constant)

0.4842469165931086

We can see that we reject no matter we exclude the spike or not.

pl=DW2023Local(W, Sigma="identity", c=0, K4=0, alternative='two-sided')
print(pl)

pl=DW2023Local(W/normalize_constant, Sigma="identity", c=0, K4=0, alternative='two-sided')
print(pl)

0.0
0.0

randomly select submatrices and repeat 100 times

sample_array = np.transpose(Z)
# Set the number of rows (s) and columns (t) you want to randomly select
s = 900
t = 36

# List to store all pg values
pl_values = []
np.random.seed(None)
for _ in range(200):  # repeat 100 times


    # Make sure s and t are less than or equal to the number of rows and columns in the original array
    if s > sample_array.shape[0] or t > sample_array.shape[1]:
        print("s and t should be less than or equal to the number of rows and columns in the original array.")
        break
    else:
        # Randomly select s rows and t columns
        selected_rows = np.random.choice(sample_array.shape[0], size=s, replace=False)
        selected_columns = np.random.choice(sample_array.shape[1], size=t, replace=False)

        # Extract those rows and columns to form a new array
        new_array = sample_array[selected_rows][:, selected_columns]

        # Compute row-wise mean and standard deviation for the new array
        row_means = np.mean(new_array, axis=1, keepdims=True)
        row_stds = np.std(new_array, axis=1, keepdims=True)

        # Check for zero standard deviation to avoid division by zero
        if np.any(row_stds == 0):
            print("Warning: At least one row has zero standard deviation!")
            continue

        # Standardize each row of the new array
        standardized_new_array = (new_array - row_means) / row_stds

        Wt= np.transpose(standardized_new_array)
        # Zt= np.sqrt(t) * np.transpose(standardized_new_array)
        # Zt = np.sqrt(np.sqrt(t/s)) * np.transpose(standardized_new_array)

        nt,pt=Wt.shape
        Ht = np.matmul(Wt, Wt.T) / np.sqrt(nt * pt)
        eigvals = np.linalg.eigvalsh(Ht)  # Compute eigenvalues of H, in ascending order
        max_eigval = eigvals[-1]  # Extract the largest eigenvalue
        normalize_constantt=(np.trace(Ht)-max_eigval)/nt/np.sqrt(pt/nt)
        # print(normalize_constantt)

        pl = DW2023Local(Wt/normalize_constantt, Sigma="identity", c=0, K4=0, alternative='two-sided')
        pl_values.append(pl)

print("All pl values after 100 iterations:")
print(pl_values)

All pl values after 100 iterations:
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]

sample_array = np.transpose(Z)
# Set the number of rows (s) and columns (t) you want to randomly select
s = 42
t = 36

# List to store all pg values
pl_values = []
# np.random.seed(None)
for _ in range(200):  # repeat 100 times


    # Make sure s and t are less than or equal to the number of rows and columns in the original array
    if s > sample_array.shape[0] or t > sample_array.shape[1]:
        print("s and t should be less than or equal to the number of rows and columns in the original array.")
        break
    else:
        # Randomly select s rows and t columns
        selected_rows = np.random.choice(sample_array.shape[0], size=s, replace=False)
        selected_columns = np.random.choice(sample_array.shape[1], size=t, replace=False)

        # Extract those rows and columns to form a new array
        new_array = sample_array[selected_rows][:, selected_columns]

        # Compute row-wise mean and standard deviation for the new array
        row_means = np.mean(new_array, axis=1, keepdims=True)
        row_stds = np.std(new_array, axis=1, keepdims=True)

        # Check for zero standard deviation to avoid division by zero
        if np.any(row_stds == 0):
            print("Warning: At least one row has zero standard deviation!")
            continue

        # Standardize each row of the new array
        standardized_new_array = (new_array - row_means) / row_stds

        # Zt = np.sqrt(np.sqrt(t/s)) * np.transpose(standardized_new_array)
        # Zt = np.sqrt(t) * np.transpose(standardized_new_array)
        Zt = np.transpose(standardized_new_array)


        pl = DW2023Local(Zt, Sigma="identity", c=0, K4=0, alternative='two-sided')
        pl_values.append(pl)

print("All pl values after 100 iterations:")
print(pl_values)

All pl values after 100 iterations:
[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]

We can see that we will reject $\Sigma=\operatorname{Id}$ in all four treatments above.

Two-Sample Hypothesis Testing: Theoretical and Simulated Data

In this section, we consider the two-sample hypothesis testing of sample covariance matrices in the ultra high dimensional setting.

To be more specific, suppose $\tilde{\mathbf{x}}_i\in \mathbb{R}^p$, are i.i.d. with $\mathbb{E}\tilde{\mathbf{x}}_i=\textbf{0}$, $\operatorname{Cov}(\tilde{\mathbf{x}}_i)=\Sigma_1,$ $1\le i\le n_1.$ Similarly, suppose $\tilde{\mathbf{y}}_j\in \mathbb{R}^p, $ are i.i.d. with $\mathbb{E}\tilde{\mathbf{y}}_j=\textbf{0}$, $\operatorname{Cov}(\tilde{\mathbf{y}}_j)=\Sigma_2,$ $1\le j\le n_2$.

Consider the problem of testing $$H_0: \Sigma_1=\Sigma_2.$$

Further, we reduce the problem to the case $$\tilde{X}=[\tilde{\mathbf{x}}_1, \ldots, \tilde{\mathbf{x}}_{n_1}]=\Sigma_1^{1/2}X,\quad \tilde{Y}=[\tilde{\mathbf{y}}_1, \ldots, \tilde{\mathbf{y}}_{n_2}]=\Sigma_2^{1/2}Y,$$ where $X=[\mathbf{x}_1, \ldots, \mathbf{x}_{n_1}]\in M_{p\times n_1}(\mathbb{R})$ and $X_{ij}'s$ are i.i.d. with mean $0$ and unit variance, similarly, $Y=[\mathbf{y}_1, \ldots, \mathbf{y}_{n_2}]\in M_{p\times n_2}(\mathbb{R})$ and $Y_{ij}'s$ are i.i.d. with mean $0$ and unit variance.

In order to gain some intuitions, there are two natural ways to measure the "distance" between two covariance matrices, difference and ratio

$$ \Sigma_1-\Sigma_2, \quad \Sigma_1 \Sigma_2^{-1}-\mathbf{I}_p $$

Correspondingly, there are two classes of test statistics $$ T_{\mathrm{d}}=\operatorname{Tr}\left(f(\mathbf{S}_1)-f(\mathbf{S}_2)\right), \quad T_{\mathrm{r}}=\operatorname{Tr}f\left(\mathbf{S}_1 \mathbf{S}_2^{-1}-\mathbf{I}_p\right). $$

Note that when $p>n_1,n_2$, we have that $S_1$ and $S_2$ may not be invertible.

Therefore, we focus on the difference statistics which is also a natural extension of our result in the one-sample setting.

In this subsection, we will focus on the choice of the following hyperparameters

• $n$: size of subsampling samples
• $R_2$: times of subsampling
• $\beta_n$: number of eigenvalues to be included in the local statistics.

Subsampling Scheme and Aymptotic Power of Global Test under Alternative

• In theory, if $n_1=n_2=n$, all the test statistics in our one-sample testing scheme are usable under minor modification, namely, when we have $$\left[\begin{array}{c}\sum_{s=1}^n g_1\left(\frac{\lambda_s-E}{\eta_0}\right)-n \int_{\mathbb{R}} g_1\left(\frac{x-E}{\eta_0}\right) \mathrm{d} \rho_{\phi_n, \pi}(x) \\ \vdots \\ \sum_{s=1}^n g_k\left(\frac{\lambda_s-E}{\eta_0}\right)-n \int_{\mathbb{R}} g_k\left(\frac{x-E}{\eta_0}\right) \mathrm{d} \rho_{\phi_n, \pi}(x)\end{array}\right] \sim N(\boldsymbol{\mu}, \boldsymbol{\Lambda})$$ in the one-sample setting, then we will also have

  $$ \left[\begin{array}{c} \sum_{s=1}^n g_1\left(\frac{\lambda_s^{(1)}-E}{\eta_0}\right)-\sum_{s=1}^n g_1\left(\frac{\lambda_s^{(2)}-E}{\eta_0}\right) \\ \vdots \\ \sum_{s=1}^n g_k\left(\frac{\lambda_s^{(1)}-E}{\eta_0}\right)-\sum_{s=1}^n g_k\left(\frac{\lambda_s^{(2)}-E}{\eta_0}\right) \end{array}\right] \sim N(0,2 \boldsymbol{\Lambda}) $$ under mild conditions.

• In reality, however, $n_1=n_2=n$ is seldom the case. WLOG we may assume that we have $n_1< n_2$. In this subsection, we handle the case when $$\Sigma_1=\operatorname{Id}_p,$$ $$\Sigma_2=\operatorname{diag}\{1, \ldots,1, 1+t,\ldots,1+t\}, \text{(half 1 half 1+t)}$$

  • One of the problems is that the scaling $\frac{1}{\sqrt{n_1m}}\Sigma_1^{1/2}X_1X_1^{\top}\Sigma_1^{1/2}$ and $\frac{1}{\sqrt{n_2m}}\Sigma_2^{1/2}X_2X_2^{\top}\Sigma_2^{1/2}$ will reveal the maximum of fluctuation but they are not consistent with each other in the sense that same $\Sigma$ with different $n_1$ and $n_2$ will result in quite different behavior of $S_1$ and $S_2$ here. We use the following subsampling scheme to resolve the issue.
  • On the other hand, when we have different $n_1$ and $n_2$, the analysis for the de-centering terms $n_1 \int_{\mathbb{R}} g_k\left(\frac{x-E}{\eta_0}\right) \mathrm{d} \rho_{\phi_{n_1}, \pi}(x)$ and $n_2 \int_{\mathbb{R}} g_k\left(\frac{x-E}{\eta_0}\right) \mathrm{d} \rho_{\phi_{n_2}, \pi}(x)$ will become much more difficult since $\rho_{\phi_{n_1}, \pi}$ also depends on the factor $n_1$ in a delicate way.

We propose a subsampling scheme to resolve the problem. We introduce a hyperparameter $n$ which satisfies $n<\min\{n_1,n_2\}$ and define the partial sample covariance matrix $$ \tilde{S}_1=\frac{1}{\sqrt{n p}} \sum_{i \in I_{n}^{(1)}} \tilde{\boldsymbol{x}}_i \tilde{\boldsymbol{x}}_i^{\prime} $$ $$ \tilde{S}_2=\frac{1}{\sqrt{n p}} \sum_{i \in I_{n}^{(2)}} \tilde{\boldsymbol{y}}_i \tilde{\boldsymbol{y}}_i^{\prime} $$ where $I_n^{(1)}$ is a subset of $\left\{1,2, \ldots, n_1\right\}$ and $I_n^{(2)}$ is a subset of $\left\{1,2, \ldots, n_2\right\}$ such that $|I_n^{(1)}|=|I_n^{(2)}|=n$.

For simplicity of illustration, WLOG we assume that $n_1\le n_2$ and restrict the discussion of the case to $n=n_1$ fow now in this section.

When we are given the $p\times n_2$ matrix $X_2$ or $\Sigma^{1/2}X_2$, we can randomly choose $n_1$ columns out of the $n_2$ columns in total. We can do this random subsampling $R_2$ times to see if we will get better power as $R_2$ increases. In the simulation below, every time we get an observation from $\Sigma_1$, we subsampling $R_2$ times from $\Sigma_2$, and we will sample in total $R_1$ observations from $\Sigma_1$.

In practice, when we deal with real data, we will have only one observation (two data matrices), so $R_1=1$, then we will also need to choose $R_2$ manually.

Completely Random Trials for Subsampling and its Performance

Now we implement the above scheme to see if test with subsampling will be benificial.

We fix $R_1$ and let $R_2$ increase when $\Sigma_2$ s.t. it has half 1 half 1.05 diagonal elements and calculate the empirical powers when we have fixed $R_1=200$ and increasing $R_2$.

import jax
import jax.numpy as jnp
from jax.random import normal
import numpy as np

# Parameters
n1 = 20
n2 = 200
phi = 100
m = int(n1 * phi)
c = 3
R1 = 200
R2 = 1
center3 = (jnp.sqrt(phi) + 1/jnp.sqrt(phi) - c) * jnp.ones(n1)

# Objective function
def objective(X):
    H = jnp.matmul(jnp.transpose(X), X)
    E0 = jnp.linalg.eigh(H)[0]
    return jnp.sum(E0)

# Compute TestST11
T = jnp.zeros(R1)
key = jax.random.PRNGKey(0)
for k in range(R1):
    key, subkey = jax.random.split(key)
    X = normal(subkey, shape=(m, n1)) / jnp.sqrt(jnp.sqrt(n1 * m))
    T = T.at[k].set(objective(X))
TestST11 = np.array(T)


# Set t value
t=0.05

# Compute TestST12
T = jnp.zeros(R1 * R2)
key = jax.random.PRNGKey(1)
Sigmad = jnp.repeat(jnp.array([1, 1+t]), m//2)
for l in range(R1):
    X0 = normal(key, shape=(m, n2))
    X = X0 * jnp.sqrt(Sigmad)[:, None]
    for k in range(R2):
        key, subkey = jax.random.split(key)
        selected_columns = np.random.choice(n2, n1, replace=False)
        Xt = X[:, selected_columns]
        H = jnp.matmul(jnp.transpose(Xt), Xt) / jnp.sqrt(n1 * m)
        E0 = jnp.linalg.eigh(H)[0]
        T = T.at[l * R2 + k].set(jnp.sum(E0))
TestST12 = np.array(T)

# Combine TestST11 and TestST12
TestST1 = np.repeat(TestST11, R2) - TestST12

# Normalize TestST1
Normalize_TestST = TestST1 / np.sqrt(4)

# Check outside interval
def count_outside_interval(arr, a, b):
    outside_interval = np.logical_or(arr < a, arr > b)
    count = np.sum(outside_interval)
    return count

False_Pos = count_outside_interval(Normalize_TestST, -1.96, 1.96)
Power = False_Pos / R2 / R1

print(Power)

0.705

import jax
import jax.numpy as jnp
from jax.random import normal
import numpy as np

# Parameters
n1 = 20
n2 = 200
phi = 100
m = int(n1 * phi)
c = 3
R1 = 200
R2 = 20
center3 = (jnp.sqrt(phi) + 1/jnp.sqrt(phi) - c) * jnp.ones(n1)

# Objective function
def objective(X):
    H = jnp.matmul(jnp.transpose(X), X)
    E0 = jnp.linalg.eigh(H)[0]
    return jnp.sum(E0)

# Compute TestST11
T = jnp.zeros(R1)
key = jax.random.PRNGKey(0)
for k in range(R1):
    key, subkey = jax.random.split(key)
    X = normal(subkey, shape=(m, n1)) / jnp.sqrt(jnp.sqrt(n1 * m))
    T = T.at[k].set(objective(X))
TestST11 = np.array(T)


# Set t value
t=0.05

# Compute TestST12
T = jnp.zeros(R1 * R2)
key = jax.random.PRNGKey(1)
Sigmad = jnp.repeat(jnp.array([1, 1+t]), m//2)
for l in range(R1):
    X0 = normal(key, shape=(m, n2))
    X = X0 * jnp.sqrt(Sigmad)[:, None]
    for k in range(R2):
        key, subkey = jax.random.split(key)
        selected_columns = np.random.choice(n2, n1, replace=False)
        Xt = X[:, selected_columns]
        H = jnp.matmul(jnp.transpose(Xt), Xt) / jnp.sqrt(n1 * m)
        E0 = jnp.linalg.eigh(H)[0]
        T = T.at[l * R2 + k].set(jnp.sum(E0))
TestST12 = np.array(T)

# Combine TestST11 and TestST12
TestST1 = np.repeat(TestST11, R2) - TestST12

# Normalize TestST1
Normalize_TestST = TestST1 / np.sqrt(4)

# Check outside interval
def count_outside_interval(arr, a, b):
    outside_interval = np.logical_or(arr < a, arr > b)
    count = np.sum(outside_interval)
    return count

False_Pos = count_outside_interval(Normalize_TestST, -1.96, 1.96)
Power = False_Pos / R2 / R1

print(Power)

0.69325

import jax
import jax.numpy as jnp
from jax.random import normal
import numpy as np

# Parameters
n1 = 20
n2 = 200
phi = 100
m = int(n1 * phi)
c = 3
R1 = 200
R2 = 40
center3 = (jnp.sqrt(phi) + 1/jnp.sqrt(phi) - c) * jnp.ones(n1)

# Objective function
def objective(X):
    H = jnp.matmul(jnp.transpose(X), X)
    E0 = jnp.linalg.eigh(H)[0]
    return jnp.sum(E0)

# Compute TestST11
T = jnp.zeros(R1)
key = jax.random.PRNGKey(0)
for k in range(R1):
    key, subkey = jax.random.split(key)
    X = normal(subkey, shape=(m, n1)) / jnp.sqrt(jnp.sqrt(n1 * m))
    T = T.at[k].set(objective(X))
TestST11 = np.array(T)


# Set t value
t=0.05

# Compute TestST12
T = jnp.zeros(R1 * R2)
key = jax.random.PRNGKey(1)
Sigmad = jnp.repeat(jnp.array([1, 1+t]), m//2)
for l in range(R1):
    X0 = normal(key, shape=(m, n2))
    X = X0 * jnp.sqrt(Sigmad)[:, None]
    for k in range(R2):
        key, subkey = jax.random.split(key)
        selected_columns = np.random.choice(n2, n1, replace=False)
        Xt = X[:, selected_columns]
        H = jnp.matmul(jnp.transpose(Xt), Xt) / jnp.sqrt(n1 * m)
        E0 = jnp.linalg.eigh(H)[0]
        T = T.at[l * R2 + k].set(jnp.sum(E0))
TestST12 = np.array(T)

# Combine TestST11 and TestST12
TestST1 = np.repeat(TestST11, R2) - TestST12

# Normalize TestST1
Normalize_TestST = TestST1 / np.sqrt(4)

# Check outside interval
def count_outside_interval(arr, a, b):
    outside_interval = np.logical_or(arr < a, arr > b)
    count = np.sum(outside_interval)
    return count

False_Pos = count_outside_interval(Normalize_TestST, -1.96, 1.96)
Power = False_Pos / R2 / R1

print(Power)

0.695625

We can see that in the completely randomized design. The effect of a single factor $R_2$ on the power is not significant.

Discussion: How Many Eigenvalues $\beta_n$ Are Adequate for Local Statistics

How many eigenvalues $\beta_n$ we should choose for local statistics remains a non-negligible problem. This is a practical problem that we should answer back when we are dealing with one-sample test. (In theory, this is almost equivalent to the choice of $\eta$ since we expected to include the information of $\mathrm{O}(n^{2/3}\eta)$ for the local statistics we defined. To avoid confusion, we will only use $\beta_n$ after, while readers may keep in mind that it's adjusted through the parameter $\eta$ underneath algorithmically.)

We have multiple motivations for this question. We start from a tradeoff on the size of $\beta_n$.

• On the one hand, we want to include enough number of eigenvalues such that the local statistics are stable to certain extent. The large $\beta_n$ is, the better.

• On the other hand, when $\beta_n$ approaches $n$, the local statistics will become more "global" which results in the occurence of $\kappa_4$ in the related formula, which is something we don't desire.

This motivates us to find a safe region for $\beta_n$ according to the separation of ECDFs for gaussian and 2PN.

Besides, there is also transition of leading eigenvalues from Tracy Widom fluctuation to Gaussian fluctuation. Therefore, the second motivation comes from observing whether there is a visually significant separation of normalized ECDFs for gaussian and 2PN from the ECDF of standard normal.

The above scheme will narrow down the choice of $\beta_n$ to a safe region. However, it is not yet clear that inside the safe region which $\beta$ is better. Therefore, the first subsection will be devoted to the selection of $\beta_n$ inside the safe region with comparison of power via simulated data.

In the second subsection, we will briefly discuss how we could possibly made a joint selection of $\beta_n$ and $K$ according to the data where $K$ is the number of subsampling mentioned in the previous section Subsampling Scheme.

In theory, the number of eigenvalues included in the local statistics could be as small as $n^{ɛ}$ for any $ɛ>0$. However, in reality, roughly $n^{ɛ}$ cannot guarantee the stability of local statistics when $n$ and $ɛ$ are too small.

In this subsection, we simply replicate what we have done in Figure 2(b). Further, we adjust the parameter $\eta$ until either

1. a visually significant separation of ECDFs for gaussian and 2PN settings can be observed. or
2. a visually significant separation of normalized ECDFs for gaussian and 2PN from the ECDF of standard normal the can be observed.

We showed previously in Section Figure 4 that when $n=800$ our test statistics behave fine for different $\beta_n$. However, when we deal with real data in practice, $n$ is seldom as large as 800. Therefore, we conduct the following experiment for relatively small $n=50$.

%%time
import jax
import jax.numpy as jnp
from jax import grad, jit, random
from jax.random import normal, choice, split, PRNGKey
import numpy as np
import matplotlib.pyplot as plt

# Constants
t = 0.000
n = 50
R0 = 1000
c = 1
phi = 100
m = int(n * phi)
r = jnp.sqrt(phi)

### you may choose between left edge leftb, right edge rightb, middle point midb and any other points in the spectra
leftb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi) - 2)
midb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi))
rightb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi) + 2)


eta1 = 1/(n**(3/5))

## Parameter for 2PN
values = jnp.array([np.sqrt(2), -1/np.sqrt(2)])
probabilities = jnp.array([1/3, 2/3])



def generate_matrix_normal(key, m, n):
    return normal(key, shape=(m, n)) / jnp.sqrt(jnp.sqrt(n * m))


def generate_matrix_choice(key, m, n, values, probabilities):
    return choice(key, a=values, shape=(m, n), p=probabilities) / jnp.sqrt(jnp.sqrt(n * m))

def compute_local_test_statistic(E0, midb, eta1):
    Et = jnp.array(extract_elements(E0, midb-2*eta1, midb+2*eta1))
    Eb = (Et - midb * jnp.ones(Et.shape)) / eta1
    return jnp.sum(Eb)

def simulate_local_test_statistics_normal(R0):
    T0 = jnp.zeros(R0)
    key = PRNGKey(0)
    for r in range(R0):
        key, subkey = split(key)
        X = generate_matrix_normal(subkey, m, n)
        H = jnp.matmul(X.T, X)
        E00, _ = jnp.linalg.eigh(H)
        E0 = np.array(E00)
        t0 = compute_local_test_statistic(E0, leftb, eta1)
        T0 = T0.at[r].set(t0)
    return np.array(T0)

def simulate_local_test_statistics_choice(R0):
    T0 = jnp.zeros(R0)
    key = PRNGKey(0)
    for r in range(R0):
        key, subkey = split(key)
        X=generate_matrix_choice(key, m, n, values, probabilities)
        H = jnp.matmul(X.T, X)
        E00, _ = jnp.linalg.eigh(H)
        E0 = np.array(E00)
        t0 = compute_local_test_statistic(E0, leftb, eta1)
        T0 = T0.at[r].set(t0)
    return np.array(T0)



# Simulate test statistics for normal and choice methods
TestST1 = simulate_local_test_statistics_normal(R0)
Mean_x1 = jnp.mean(TestST1)
Variance_x1 = jnp.var(TestST1)
print(Mean_x1)
print(Variance_x1)
Normalize_TestST_1 = (TestST1 - Mean_x1) / np.sqrt(Variance_x1)

TestST2 = simulate_local_test_statistics_choice(R0)
Mean_x2 = jnp.mean(TestST2)
Variance_x2 = jnp.var(TestST2)
print(Mean_x2)
print(Variance_x2)
Normalize_TestST_2 = (TestST2 - Mean_x1) / np.sqrt(Variance_x1)

# # Plot the ECDFs
# labels = [f"$\mathbf{{\phi={phi}}}$" for phi in phi_values]
# plot_ecdf_and_standard_normal_cdf(*normalized_data_sets, labels=labels)
labels=['$\mathbf{\kappa_4\ =\ 0}$', r'$\mathbf{\kappa_4\ \neq\ 0}$']
plot_ecdf2(TestST1, TestST2, r'$\mathbf{\kappa_4\ =\ 0}$', r'$\mathbf{\kappa_4\ \neq\ 0}$')
plot_ecdf_and_standard_normal_cdf(Normalize_TestST_1,Normalize_TestST_2,labels=labels )

After quite a number of trials we see that $\beta_n={n^{2/3-1/6}}$ is still generally fine, but we do prefer that the lower bound for $\beta$ lifted to $n^{2/3-1/3}=n^{1/3}$.

To conclude, we think the generally the safe region for $\beta_n$ is between $[n^{1/3}, n^{1/2}]$ in practice.

Empirical Power for different $\beta_n$'s

In this subsection, we will assume that the selection of $\beta_n$ is relatively independent of the selection of other parameters and compare the empirical power for different $\beta_n$'s.

import jax
import jax.numpy as jnp
from jax import random
import numpy as np
import matplotlib.pyplot as plt

n = 200
R = 100
c = 1
phi = 100
m = int(n * phi)
repeat = 20

def count_outside_interval(arr, a, b):
    outside_interval = np.logical_or(arr < a, arr > b)
    count = np.sum(outside_interval)
    return count

def extract_elements(arr, a, b):
    return arr[(arr >= a) & (arr <= b)]

def calculate_null_distribution(eta, R0=1000):
    leftb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi) - 2)
    T0 = jnp.zeros(R0)
    key = jax.random.PRNGKey(0)
    for r in range(R0):
        key, subkey = jax.random.split(key)
        X = random.normal(key, shape=(m, n))
        H = jnp.matmul(X.T, X) / jnp.sqrt(n * m)
        E00, _ = jnp.linalg.eigh(H)
        E0 = np.array(E00)
        Et = np.array(extract_elements(E0, leftb - eta, leftb + eta))
        Eb = (Et - leftb * jnp.ones(Et.shape)) / eta
        t0 = jnp.sum(Eb)
        T0 = T0.at[r].set(t0)
    return jnp.mean(T0), jnp.var(T0)

def power_phi(n, phi, R, c, t, eta):
    m = int(n * phi)
    Sigmad = jnp.repeat(jnp.array([1, 1 + t]), m // 2)
    leftb = (jnp.sqrt(phi) + 1/jnp.sqrt(phi) - 2)
    T = jnp.zeros(R)
    key = jax.random.PRNGKey(0)
    num_in = np.zeros(R)
    for r in range(R):
        key, subkey = jax.random.split(key)
        X = random.normal(subkey, shape=(m, n))
        X = X * jnp.sqrt(Sigmad)[:, None]
        H = jnp.matmul(X.T, X) / jnp.sqrt(n * m)
        E00, _ = jnp.linalg.eigh(H)
        E0 = np.array(E00)
        Et = np.array(extract_elements(E0, leftb - eta, leftb + eta))
        Eb = (Et - leftb * jnp.ones(Et.shape)) / eta
        t1 = jnp.sum(Eb)
        T = T.at[r].set(t1)
        # num_in[r]=count_inside_interval(E0, leftb - eta, leftb + eta)
    # print(np.mean(num_in))


    TestST = np.array(T)
    Mean_x, Variance_x = calculate_null_distribution(eta)
    Normalize_TestST = (TestST - Mean_x * jnp.ones(R)) / np.sqrt(Variance_x)
    True_Pos = count_outside_interval(Normalize_TestST, -1.96, 1.96)
    return True_Pos / R

eta1= 1/(n**(1/3))

L1 = [power_phi(n, phi, R, c, 0.01 * s, eta1) for s in range(repeat)]

x_values = [0.01 * s for s in range(repeat)]
plt.plot(x_values, L1, label=r'$\beta^{(1)} = 5$')

plt.xlabel('t value (separation from Identity)')
plt.ylabel('Power')
plt.title(r'Power of Statistical Test for $\beta=5, n=200$')
plt.legend()
plt.grid(True)
plt.show()

eta2= 1/(n**(1/6))

L2 = [power_phi(n, phi, R, c, 0.01 * s, eta2) for s in range(repeat)]

x_values = [0.01 * s for s in range(repeat)]

plt.plot(x_values, L2, label=r'$\beta^{(2)} = 14$')

plt.xlabel('t value (separation from Identity)')
plt.ylabel('Power')
plt.title(r'Power of Statistical Test for $\beta=14$, $n=200$')
plt.legend()
plt.grid(True)
plt.show()

We can see that different $\beta_n$'s lead to somehow similar power curves and power levels. Further, we may try different parameters and plot the different power curves with the block below. The results are essentially similar. We omit the diagrams here for compactness.

n = 100
R = 100
c = 1
phi = 100
m = int(n * phi)
repeat = 20
eta1= 1/(n**(1/3))
eta2= 1/(n**(1/4))
eta3= 1/(n**(1/5))
eta4= 1/(n**(1/6))
L1 = [power_phi(n, phi, R, c, 0.01 * s, eta1) for s in range(repeat)]
L2 = [power_phi(n, phi, R, c, 0.01 * s, eta2) for s in range(repeat)]
L3 = [power_phi(n, phi, R, c, 0.01 * s, eta3) for s in range(repeat)]
L4 = [power_phi(n, phi, R, c, 0.01 * s, eta4) for s in range(repeat)]
x_values = [0.01 * s for s in range(repeat)]
plt.plot(x_values, L1, label=r'\beta^{(1)} = n^{1/3}')
plt.plot(x_values, L2, label=r'\beta^{(2)} = n^{2/3-1/4}')
plt.plot(x_values, L3, label=r'\beta^{(3)} = n^{2/3-1/5}')
plt.plot(x_values, L4, label=r'\beta^{(4)} = n^{1/2}')
plt.xlabel('t value (separation from Identity)')
plt.ylabel('Power')
plt.title(r'Power of Statistical Test for Different $\beta$ Values')
plt.legend()
plt.grid(True)
plt.show()

Overall, our conclusion is that when we consider $\beta_n$ alone, the performance differences for different $\beta_n$'s are relatively minor. In other words, our observation is that the performance is rather stable under different selections of $\beta_n$'s within the safe region.

Possible joint selection of $\beta_n$ and $R_2$

We can try to select $\beta_n$ and $R_2$ at the same time. First, with the above two parameters fixed, we use the following scheme to calculate the power for local statistics.

import jax
import jax.numpy as jnp
from jax.random import normal
import numpy as np

# Parameters
n1 = 20
phi = 100
m = int(n1 * phi)
c = 3
R1 = 100
center3 = (jnp.sqrt(phi) + 1/jnp.sqrt(phi) - c) * jnp.ones(n1)
leftb=np.sqrt(phi)+1/np.sqrt(phi)-2
eta= n1**(-1/3)
# Objective function
def objective(X):
    H = jnp.matmul(jnp.transpose(X), X)
    E00, _ = jnp.linalg.eigh(H)
    E0 = np.array(E00)
    Et = np.array(extract_elements(E0, leftb - eta, leftb + eta))
    Eb = (Et - leftb * jnp.ones(Et.shape)) / eta
    return jnp.sum(Eb)

# Compute TestST11
T = jnp.zeros(R1)
key = jax.random.PRNGKey(0)
for k in range(R1):
    key, subkey = jax.random.split(key)
    X = normal(subkey, shape=(m, n1)) / jnp.sqrt(jnp.sqrt(n1 * m))
    T = T.at[k].set(objective(X))
TestST11 = np.array(T)

mean_X = np.mean(TestST11)
var_X  = np.var(TestST11)


t=0.00
# Parameters for TestST12
n2 = 200
R2 = 200
Sigmad = jnp.repeat(jnp.array([1, 1+t]), m//2)

# Compute TestST12
T = jnp.zeros(R1 * R2)
key = jax.random.PRNGKey(1)
for l in range(R1):
    X0 = normal(key, shape=(m, n2))
    X = X0 * jnp.sqrt(Sigmad)[:, None]
    for k in range(R2):
        key, subkey = jax.random.split(key)
        selected_columns = np.random.choice(n2, n1, replace=False)
        Xt = X[:, selected_columns]
        H = jnp.matmul(jnp.transpose(Xt), Xt) / jnp.sqrt(n1 * m)
        E00, _ = jnp.linalg.eigh(H)
        E0 = np.array(E00)
        Et = np.array(extract_elements(E0, leftb - eta, leftb + eta))
        Eb = (Et - leftb * jnp.ones(Et.shape)) / eta
        T = T.at[l * R2 + k].set(jnp.sum(Eb))
TestST12 = np.array(T)


# Combine TestST11 and TestST12
TestST1 = np.repeat(TestST11, R2) - TestST12


# Normalize TestST1
Normalize_TestST = (TestST1-2*mean_X*np.ones(np.size(TestST1))) / np.sqrt(2*var_X)

# Check outside interval
def count_outside_interval(arr, a, b):
    outside_interval = np.logical_or(arr < a, arr > b)
    count = np.sum(outside_interval)
    return count

False_Pos = count_outside_interval(Normalize_TestST, -1.96, 1.96)
Power = False_Pos / R2 / R1
print(Power)

0.3975

To optimize the hyperparameter $\beta_n$ and $R_2$ for maximizing the power, we first implement a toy Grid Search with Cross-Validation Scheme:

• Define a range of values for $\beta_n$ and $R_2$.

• For each combination of $\beta_n$ and $R_2$, compute the power using cross-validation.

• Select the combination that gives the highest average power across the cross-validation folds.

def compute_power(eta, R2, TestST11_train):
    # Parameters
    center3 = (jnp.sqrt(phi) + 1/jnp.sqrt(phi) - c) * jnp.ones(n1)
    leftb = np.sqrt(phi) + 1/np.sqrt(phi) - 2

    # Compute TestST11 for the training data
    T = jnp.zeros(len(TestST11_train))
    key = jax.random.PRNGKey(0)
    for k in range(len(TestST11_train)):
        key, subkey = jax.random.split(key)
        X = normal(subkey, shape=(m, n1)) / jnp.sqrt(jnp.sqrt(n1 * m))
        T = T.at[k].set(objective(X))
    TestST11_train = np.array(T)

    # setting parameter t here
    t=0.05

    # Compute TestST12
    T = jnp.zeros(len(TestST11_train) * R2)
    key = jax.random.PRNGKey(1)
    Sigmad = jnp.repeat(jnp.array([1, 1+t]), m//2)
    for l in range(len(TestST11_train)):
        X0 = normal(key, shape=(m, n2))
        X = X0 * jnp.sqrt(Sigmad)[:, None]
        for k in range(R2):
            key, subkey = jax.random.split(key)
            selected_columns = np.random.choice(n2, n1, replace=False)
            Xt = X[:, selected_columns]
            H = jnp.matmul(jnp.transpose(Xt), Xt) / jnp.sqrt(n1 * m)
            E00, _ = jnp.linalg.eigh(H)
            E0 = np.array(E00)
            Et = np.array(extract_elements(E0, leftb - eta, leftb + eta))
            Eb = (Et - leftb * jnp.ones(Et.shape)) / eta
            T = T.at[l * R2 + k].set(jnp.sum(Eb))
    TestST12 = np.array(T)

    # Combine TestST11 and TestST12
    TestST1 = np.repeat(TestST11_train, R2) - TestST12

    # Normalize TestST1
    Normalize_TestST = (TestST1 - 2 * np.mean(TestST11_train) * np.ones(np.size(TestST1))) / np.sqrt(2 * np.var(TestST11_train))

    # Check outside interval
    False_Pos = count_outside_interval(Normalize_TestST, -1.96, 1.96)
    Power = False_Pos / R2 / len(TestST11_train)

    return Power



from sklearn.model_selection import KFold

# Define a range of values for eta and R2
eta_values = [n1**(-1/2), n1**(-1/3), n1**(-1/4), n1**(-1/5)]
R2_values = [50, 100, 150, 200]

# Cross-validation
kf = KFold(n_splits=5)

best_eta = None
best_R2 = None
best_power = 0

for eta in eta_values:
    for R2 in R2_values:
        powers = []
        for train_index, test_index in kf.split(TestST11):
            # Split the data into training and test sets
            TestST11_train, TestST11_test = TestST11[train_index], TestST11[test_index]

            # Compute the power for this combination of eta and R2 using the training data
            power = compute_power(eta, R2, TestST11_train)
            powers.append(power)

        # Average power across the cross-validation folds
        avg_power = np.mean(powers)

        if avg_power > best_power:
            best_power = avg_power
            best_eta = eta
            best_R2 = R2

print("Best beta:", r"$n^{2/3}\times$" best_eta)
print("Best R2:", best_R2)
print("Best Power:", best_power)

Two-Sample Hypothesis Testing: Real Data

2009 Data

We use the same data as the one in our one-sample hypothesis testing https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE13528. We split the data into two disjoint halves. It's natural that we expect $$H_0: \ \Sigma_1=\Sigma_2$$ to be true.

Read Data and Split

# Import necessary modules
from google.colab import files
import pandas as pd
import numpy as np

# # Upload the TSV file
# uploaded = files.upload()

# Assume the file is named "data.tsv"; replace with your file's name if different
file_name = 'GSE13528.tsv'

# Read the file into a Pandas DataFrame
df = pd.read_csv(file_name, delimiter='\t')

# Only include numeric columns in the DataFrame
df_numeric = df.select_dtypes(include=[np.number])

# Convert the DataFrame to a 2D NumPy array of float type
data_array_2d = df_numeric.to_numpy().astype(float)
data_array_2d_no_first_column = data_array_2d[:, :]

# Compute row-wise mean and standard deviation
row_means = np.mean(data_array_2d_no_first_column, axis=1, keepdims=True)

# De-mean each row
demean_data_array = (data_array_2d_no_first_column - row_means)

######## We have in total 47 columns
k = 20

# Take the first 20 columns as data1
raw_data1= data_array_2d[:, :k]

# Compute row-wise mean and standard deviation
row_means1 = np.mean(raw_data1, axis=1, keepdims=True)

# Standardize each row
data1 = np.transpose((raw_data1 - row_means1))

# Take the last 20 columns as data2
raw_data2= data_array_2d[:, -k:]

# Compute row-wise mean and standard deviation
row_means2 = np.mean(raw_data2, axis=1, keepdims=True)

# Standardize each row
data2 = np.transpose((raw_data2 - row_means2))

# cov1=data1@np.transpose(data1)
# cov2=data2@np.transpose(data2)
# E01,_=np.linalg.eigh(cov1)
# E02,_=np.linalg.eigh(cov2)
# # print(E01)
# # print(E02)
# E1=np.array(E01)
# E2=np.array(E02)
# E1.sort()
# E2.sort()
# # import matplotlib.pyplot as plt
# # plt.plot(E1,"*",color='blue')
# # plt.plot(E2,"*",color='red',alpha=0.5)

Testing via Local Statistics

Now we have one sample from $\Sigma_1$ and one sample from $\Sigma_2$, note that under null, the means of two test statistics will cancel each other in the difference statistics, as indicated in our Section Subsampling Scheme and Aymptotic Power of Global Test under Alternative. For the variance of the local statistics, we refer to the Corollary 3.9 of our paper arXiv: 2308.08646. The variance for a single local statistics induced by $g$ is either (in the bulk) $$ \operatorname{Var}\left(g\right)=\frac{1}{2 \pi^2} \int_{\mathbb{R}} \int_{\mathbb{R}} \frac{\left(g\left(x_1\right)-g\left(x_2\right)\right)^2}{\left(x_1-x_2\right)^2} \mathrm{~d} x_1 \mathrm{~d} x_2 . $$ or (at the edge) $$ \frac{1}{4 \pi^2} \int_{\mathbb{R}} \int_{\mathbb{R}} \frac{\left(g\left(-x_1^2\right)-g\left(-x_2^2\right)\right)^2}{\left(x_1-x_2\right)^2} \mathrm{~d} x_1 \mathrm{~d} x_2 $$

We select the local statistics such that the test function is approximately $\frac{x-E}{\eta}\mathcal{K}_{0.05,1}\left(\frac{x-E}{\eta}\right)$ (we refer to the discussion subsection under Figure 2(b) for details), which yields variance either $\frac{4}{2\pi^2}$ or $\frac{2}{3\pi^2}$.

Therefore, the distribution of difference statistics of the local statistics induced by $$\frac{x-E}{\eta}\mathcal{K}_{0.05,1}\left(\frac{x-E}{\eta}\right)$$ will asymptotically follow $$\mathcal{N}\left(0,\frac{4}{\pi^2}\right)\text{ (in the bulk) or } \mathcal{N}\left(0,\frac{4}{3\pi^2}\right)\text{ (at the edge).}$$

Now we use our two-sample local test function DW2023Local_two_sample to test whether $\Sigma_1=\Sigma_2$.

pl=DW2023Local_two_sample(data1,data2)
print(pl)

1.1492749483099882
True

Therefore, we cannot reject the null hypothesis that $\Sigma_1=\Sigma_2$.