This study derives a new property of the Wishart distribution when the degree-of-freedom and the size of the matrix parameter of the distribution grow simultaneoulsy. Particularly, the asymptotic normality of the product of four independent Wishart matrices is shown under a high dimensional asymptotic regime. As an application of the result, a statistical test procedure for the common principal components hypothesis is proposed. For this problem, the proposed test statistic is asymptotically normal under the null hypothesis. In addition, the proposed test statistic diverges to positive infinity in probability under the alternative hypothesis.
We introduce a new method to qualify the goodness of fit parameter estimation of compound Wishart models. Our method based on the free deterministic equivalent Z-score, which we introduce in this paper. Furthermore, an application to two dimensional autoregressive moving-average model is provided. Our proposal method is a generalization of statistical hypothesis testing to one dimensional moving average model based on fluctuations of real compound Wishart matrices, which is a recent result by Hasegawa, Sakuma and Yoshida.
Dynamic functional brain connectivity results depend on modeling assumptions: comparing the sliding-window method and the Wishart process for dynamic hypothesis testing
Donders Institute, Radboud University（唐德斯研究所，拉德堡德大学）
In this paper we consider some hypothesis tests within a family of Wishart distributions, where both the sample space and the parameter space are symmetric cones. For such testing problems, we first derive the joint density of the ordered eigenvalues of the generalized Wishart distribution and propose a test statistic analog to that of classical multivariate statistics for testing homoscedasticity of covariance matrix. In this generalization of Bartlett's test for equality of variances to hypotheses of real, complex, quaternion, Lorentz and octonion types of covariance structures.
In this paper we consider two statistical hypotheses for the families of Wishart type distributions. These distributions are analogs of the Wishart distributions defined and parametrized over a Lorentz cone. We test these hypotheses by means of maximal invariant statistics which are explicitly derived in the paper. The testing problems, respectively, concern the hypothesis that parameters are in a sub-Lorentz-cone, and the the hypothesis that two observations have the same parameter.
A new classifier for Polarimetric SAR (PolSAR) images is proposed and assessed in this paper. Its input consists of segments, and each one is assigned the class which minimizes a stochastic distance. Assuming the complex Wishart model, several stochastic distances are obtained from the h-phi family of divergences, and they are employed to derive hypothesis test statistics that are also used in the classification process. This article also presents, as a novelty, analytic expressions for the test statistics based on the following stochastic distances between complex Wishart models: Kullback-Lei…
Extreme principal minors of Wishart and deformed GOE matrices
School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学深圳数据科学学院）；Department of Statistics and Data Science, University of Texas, Austin（德克萨斯大学奥斯汀分校统计与数据科学系）
Principal component analysis (PCA) is a well-known tool in multivariate statistics. One significant challenge in using PCA is the choice of the number of components. In order to address this challenge, we propose an exact distribution-based method for hypothesis testing and construction of confidence intervals for signals in a noisy matrix. Assuming Gaussian noise, we use the conditional distribution of the singular values of a Wishart matrix and derive exact hypothesis tests and confidence intervals for the true signals. Our paper is based on the approach of Taylor, Loftus and Tibshirani (201…
We consider the problem of testing, on the basis of a $p$-variate Gaussian random sample, the null hypothesis ${\cal H}_0: {\pmb θ}_1= {\pmb θ}_1^0$ against the alternative ${\cal H}_1: {\pmb θ}_1 \neq {\pmb θ}_1^0$, where ${\pmb θ}_1$ is the "first" eigenvector of the underlying covariance matrix and ${\pmb θ}_1^0$ is a fixed unit $p$-vector. In the classical setup where eigenvalues $λ_1>λ_2\geq \ldots\geq λ_p$ are fixed, the Anderson (1963) likelihood ratio test (LRT) and the Hallin, Paindaveine and Verdebout (2010) Le Cam optimal test for this problem are asymptotically equivalent under the…
In this paper, we study the asymptotic behavior of the extreme eigenvalues and eigenvectors of the high dimensional spiked sample covariance matrices, in the supercritical case when a reliable detection of spikes is possible. Especially, we derive the joint distribution of the extreme eigenvalues and the generalized components of the associated eigenvectors, i.e., the projections of the eigenvectors onto arbitrary given direction, assuming that the dimension and sample size are comparably large. In general, the joint distribution is given in terms of linear combinations of finitely many Gaussi…
This paper focuses on the analysis of spatially correlated functional data. The between-curve correlation is modeled by correlating functional principal component scores of the functional data. We propose a Spatial Principal Analysis by Conditional Expectation framework to explicitly estimate spatial correlations and reconstruct individual curves. This approach works even when the observed data per curve are sparse. Assuming spatial stationarity, empirical spatial correlations are calculated as the ratio of eigenvalues of the smoothed covariance surface $Cov(X_i(s),X_i(t))$ and cross-covarianc…
In the paper, we suggest three tests on the validity of a factor model which can be applied for both small dimensional and large dimensional data. Both the exact and asymptotic distributions of the resulting test statistics are derived under classical and high-dimensional asymptotic regimes. It is shown that the critical values of the proposed tests can be calibrated empirically by generating a sample from the inverse Wishart distribution with identity parameter matrix. The powers of the suggested tests are investigated by means of simulations. The results of the simulation study are consisten…
The eigenvalue decomposition (EVD) parameters of the second order statistics are ubiquitous in statistical analysis and signal processing. Notably, the EVD of robust scatter $M$-estimators is a popular choice to perform robust probabilistic PCA or other dimension reduction related applications. Towards the goal of characterizing the behavior of these quantities, this paper proposes new asymptotics for the EVD parameters (i.e. eigenvalues, eigenvectors and principal subspace) of the scatter $M$-estimator in the context of complex elliptically symmetric distributions. First, their Gaussian asymp…
In this paper, we further develop the approach, originating in [GJN], to "computation-friendly" hypothesis testing via Convex Programming. Most of the existing results on hypothesis testing aim to quantify in a closed analytic form separation between sets of distributions allowing for reliable decision in precisely stated observation models. In contrast to this descriptive (and highly instructive) traditional framework, the approach we promote here can be qualified as operational -- the testing routines and their risks are yielded by an efficient computation. All we know in advance is that, un…
Images obtained from coherent illumination processes are contaminated with speckle noise, with polarimetric synthetic aperture radar (PolSAR) imagery as a prominent example. With an adequacy widely attested in the literature, the scaled complex Wishart distribution is an acceptable model for PolSAR data. In this perspective, we derive analytic expressions for the Shannon, Rényi, and restricted Tsallis entropies under this model. Relationships between the derived measures and the parameters of the scaled Wishart law (i.e., the equivalent number of looks and the covariance matrix) are discussed.…
We consider inference on the first principal direction of a $p$-variate elliptical distribution. We do so in challenging double asymptotic scenarios for which this direction eventually fails to be identifiable. In order to achieve robustness not only with respect to such weak identifiability but also with respect to heavy tails, we focus on sign-based statistical procedures, that is, on procedures that involve the observations only through their direction from the center of the distribution. We actually consider the generic problem of testing the null hypothesis that the first principal direct…
Let X be a n*p matrix and l_1 the largest eigenvalue of the covariance matrix X^{*}*X. The "null case" where X_{i,j} are independent Normal(0,1) is of particular interest for principal component analysis. For this model, when n, p tend to infinity and n/p tends to gamma in (0,\infty), it was shown in Johnstone (2001) that l_1, properly centered and scaled, converges to the Tracy-Widom law. We show that with the same centering and scaling, the result is true even when p/n or n/p tends to infinity. The derivation uses ideas and techniques quite similar to the ones presented in Johnstone (2001). …
Matrix variate beta (MVB) distributions are used in different fields of hypothesis testing, multivariate correlation analysis, zero regression, canonical correlation analysis and etc. In this approach a unified methodology is proposed to generate matrix variate distributions by combining the kernel of MVB distributions of different types with an unknown Borel measurable function of trace operator over matrix space, called generator component. The latter component is a principal element of these newly defined generator type matrix variate distributions. The matrix variate Kummer beta distributi…
Consider a standard white Wishart matrix with parameters $n$ and $p$. Motivated by applications in high-dimensional statistics and signal processing, we perform asymptotic analysis on the maxima and minima of the eigenvalues of all the $m \times m$ principal minors, under the asymptotic regime that $n,p,m$ go to infinity. Asymptotic results concerning extreme eigenvalues of principal minors of real Wigner matrices are also obtained. In addition, we discuss an application of the theoretical results to the construction of compressed sensing matrices, which provides insights to compressed sensing…
SAS Institute（SAS研究所）；Shanghai Jiao Tong University（上海交通大学）
Robust PCA methods are typically batch algorithms which requires loading all observations into memory before processing. This makes them inefficient to process big data. In this paper, we develop an efficient online robust principal component methods, namely online moving window robust principal component analysis (OMWRPCA). Unlike existing algorithms, OMWRPCA can successfully track not only slowly changing subspace but also abruptly changed subspace. By embedding hypothesis testing into the algorithm, OMWRPCA can detect change points of the underlying subspaces. Extensive simulation studies d…
Community detection in networks is a key exploratory tool with applications in a diverse set of areas, ranging from finding communities in social and biological networks to identifying link farms in the World Wide Web. The problem of finding communities or clusters in a network has received much attention from statistics, physics and computer science. However, most clustering algorithms assume knowledge of the number of clusters k. In this paper we propose to automatically determine k in a graph generated from a Stochastic Blockmodel. Our main contribution is twofold; first, we theoretically e…
This short note considers the problem of testing the null hypothesis that the mean values of two multivariate normal variables are proportional. We show that the usual likelihood ratio $χ^2$-test is valid non-asymptotically. Our proof relies on expressing the test statistic as the minimum eigenvalue of a Wishart variable and using a representation of its distribution using Legendre polynomials.
We consider the five classes of multivariate statistical problems identified by James (1964), which together cover much of classical multivariate analysis, plus a simpler limiting case, symmetric matrix denoising. Each of James' problems involves the eigenvalues of $E^{-1}H$ where $H$ and $E$ are proportional to high dimensional Wishart matrices. Under the null hypothesis, both Wisharts are central with identity covariance. Under the alternative, the non-centrality or the covariance parameter of $H$ has a single eigenvalue, a spike, that stands alone. When the spike is smaller than a case-spec…
Current statistical inference problems in areas like astronomy, genomics, and marketing routinely involve the simultaneous testing of thousands -- even millions -- of null hypotheses. For high-dimensional multivariate distributions, these hypotheses may concern a wide range of parameters, with complex and unknown dependence structures among variables. In analyzing such hypothesis testing procedures, gains in efficiency and power can be achieved by performing variable reduction on the set of hypotheses prior to testing. We present in this paper an approach using data-adaptive multiple testing t…
We develop new tools in the theory of nonlinear random matrices and apply them to study the performance of the Sum of Squares (SoS) hierarchy on average-case problems. The SoS hierarchy is a powerful optimization technique that has achieved tremendous success for various problems in combinatorial optimization, robust statistics and machine learning. It's a family of convex relaxations that lets us smoothly trade off running time for approximation guarantees. In recent works, it's been shown to be extremely useful for recovering structure in high dimensional noisy data. It also remains our best…
Detecting complex interactions among risk factors in case-control studies is a fundamental task in clinical and population research. However, though hypothesis testing using logistic regression (LR) is a convenient solution, the LR framework is poorly powered and ill-suited under several common circumstances in practice including missing or unmeasured risk factors, imperfectly correlated "surrogates", and multiple disease sub-types. The weakness of LR in these settings is related to the way in which the null hypothesis is defined. Here we propose the Asymmetric Independence Model (AIM) as a bi…
For planar landmark based shapes, taking into account the non-Euclidean geometry of the shape space, a statistical test for a common mean first geodesic principal component (GPC) is devised. It rests on one of two asymptotic scenarios, both of which are identical in a Euclidean geometry. For both scenarios, strong consistency and central limit theorems are established, along with an algorithm for the computation of a Ziezold mean geodesic. In application, this allows to verify the geodesic hypothesis for leaf growth of Canadian black poplars and to discriminate genetically different trees by o…
We study the problem of detecting the presence of an underlying high-dimensional geometric structure in a random graph. Under the null hypothesis, the observed graph is a realization of an Erdős-Rényi random graph $G(n,p)$. Under the alternative, the graph is generated from the $G(n,p,d)$ model, where each vertex corresponds to a latent independent random vector uniformly distributed on the sphere $\mathbb{S}^{d-1}$, and two vertices are connected if the corresponding latent vectors are close enough. In the dense regime (i.e., $p$ is a constant), we propose a near-optimal and computationally e…
Principal component analysis (PCA) is a widely used method for dimension reduction. In high dimensional data, the "signal" eigenvalues corresponding to weak principal components (PCs) do not necessarily separate from the bulk of the "noise" eigenvalues. Therefore, popular tests based on the largest eigenvalue have little power to detect weak PCs. In the special case of the spiked model, certain tests asymptotically equivalent to linear spectral statistics (LSS)---averaging effects over all eigenvalues---were recently shown to achieve some power. We consider a nonparametric, non-Gaussian genera…
This article develops a statistical test for the null hypothesis of strict stationarity of a discrete time stochastic process in the frequency domain. When the null hypothesis is true, the second order cumulant spectrum is zero at all the discrete Fourier frequency pairs in the principal domain. The test uses a window averaged sample estimate of the second order cumulant spectrum to build a test statistic with an asymptotic complex standard normal distribution. We derive the test statistic, study the properties of the test and demonstrate its application using 137Cs gamma ray decay data. Futur…
