Robust nonparametric hypothesis tests for differences in the covariance structure of functional data
We develop a group of robust, nonparametric hypothesis tests which detect differences between the covariance operators of several populations of functional data. These tests, called FKWC tests, are based on functional data depth ranks. These tests work well even when the data is heavy tailed, which is shown both in simulation and theoretically. These tests offer several other benefits, they have a simple distribution under the null hypothesis, they are computationally cheap and they possess transformation invariance properties. We show that under general alternative hypotheses these tests are consistent under mild, nonparametric assumptions. As a result of this work, we introduce a new functional depth function called L2-root depth which works well for the purposes of detecting differences in magnitude between covariance kernels. We present an analysis of the FKWC test using L2-root depth under local alternatives. In simulation, when the true covariance kernels have strictly positive eigenvalues, we show that these tests have higher power than their competitors, while still maintaining their nominal size. We also provide a methods for computing sample size and performing multiple comparisons.
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We consider the problem of estimating high-dimensional covariance matrices of $K$-populations or classes in the setting where the samples sizes are comparable to the data dimension. We propose estimating each class covariance matrix as a distinct linear combination of all class sample covariance matrices. This approach is shown to reduce the estimation error when the sample sizes are limited, and the true class covariance matrices share a somewhat similar structure. We develop an effective method for estimating the coefficients in the linear combination that minimize the mean squared error under the general assumption that the samples are drawn from (unspecified) elliptically symmetric distributions possessing finite fourth-order moments. To this end, we utilize the spatial sign covariance matrix, which we show (under rather general conditions) to be an unbiased estimator of the normalized covariance matrix as the dimension grows to infinity. We also show how the proposed method can be used in choosing the regularization parameters for multiple target matrices in a single class covariance matrix estimation problem. We assess the proposed method via numerical simulation studies including an application in global minimum variance portfolio optimization using real stock data.
The knockoff-based multiple testing setup of Barber & Candes (2015) for variable selection in multiple regression where sample size is as large as the number of explanatory variables is considered. The method of Benjamini & Hochberg (1995) based on ordinary least squares estimates of the regression coefficients is adjusted to the setup, transforming it to a valid p-value based false discovery rate controlling method not relying on any specific correlation structure of the explanatory variables. Simulations and real data applications show that our proposed method that is agnostic to {\pi}0, the proportion of unimportant explanatory variables, and a data-adaptive version of it that uses an estimate of {\pi}0 are powerful competitors of the false discovery rate controlling method in Barber & Candes (2015).
The Wilcoxon rank-sum test is one of the most popular distribution-free procedures for testing the equality of two univariate probability distributions. One of the main reasons for its popularity can be attributed to the remarkable result of Hodges and Lehmann (1956), which shows that the asymptotic relative efficiency of Wilcoxon's test with respect to Student's $t$-test, under location alternatives, never falls below 0.864, despite the former being exactly distribution-free for all sample sizes. Even more striking is the result of Chernoff and Savage (1958), which shows that the efficiency of a Gaussian score transformed Wilcoxon's test, against the $t$-test, is lower bounded by 1. In this paper we study the two-sample problem in the multivariate setting and propose distribution-free analogues of the Hotelling $T^2$ test (the natural multidimensional counterpart of Student's $t$-test) based on optimal transport and obtain extensions of the above celebrated results over various natural families of multivariate distributions. Our proposed tests are consistent against a general class of alternatives and satisfy Hodges-Lehmann and Chernoff-Savage-type efficiency lower bounds, despite being entirely agnostic to the underlying data generating mechanism. In particular, a collection of our proposed tests suffer from no loss in asymptotic efficiency, when compared to Hotelling $T^2$. To the best of our knowledge, these are the first collection of multivariate, nonparametric, exactly distribution-free tests that provably achieve such attractive efficiency lower bounds. We also demonstrate the broader scope of our methods in optimal transport based nonparametric inference by constructing exactly distribution-free multivariate tests for mutual independence, which suffer from no loss in asymptotic efficiency against the classical Wilks' likelihood ratio test, under Konijn alternatives.
Time-to-event endpoints show an increasing popularity in phase II cancer trials. The standard statistical tool for such endpoints in one-armed trials is the one-sample log-rank test. It is widely known, that the asymptotic providing the correctness of this test does not come into effect to full extent for small sample sizes. There have already been some attempts to solve this problem. While some do not allow easy power and sample size calculations, others lack a clear theoretical motivation and require further considerations. The problem itself can partly be attributed to the dependence of the compensated counting process and its variance estimator. We provide a framework in which the variance estimator can be flexibly adopted to the present situation while maintaining its asymptotical properties. We exemplarily suggest a variance estimator which is uncorrelated to the compensated counting process. Furthermore, we provide sample size and power calculations for any approach fitting into our framework. Finally, we compare several methods via simulation studies and the hypothetical setup of a Phase II trial based on real world data.
We develop a theory to measure the variance and covariance of probability distributions defined on the nodes of a graph, which takes into account the distance between nodes. Our approach generalizes the usual (co)variance to the setting of weighted graphs and retains many of its intuitive and desired properties. Interestingly, we find that a number of famous concepts in graph theory and network science can be reinterpreted in this setting as variances and covariances of particular distributions. As a particular application, we define the maximum variance problem on graphs with respect to the effective resistance distance, and characterize the solutions to this problem both numerically and theoretically. We show how the maximum variance distribution is concentrated on the boundary of the graph, and illustrate this in the case of random geometric graphs. Our theoretical results are supported by a number of experiments on a network of mathematical concepts, where we use the variance and covariance as analytical tools to study the (co-)occurrence of concepts in scientific papers with respect to the (network) relations between these concepts.
In this paper we give a survey of methods used to calculate values of resistance distance (also known as effective resistance) in graphs. Resistance distance has played a prominent role not only in circuit theory and chemistry, but also in combinatorial matrix theory and spectral graph theory. Moreover resistance distance has applications ranging from quantifying biological structures, distributed control systems, network analysis, and power grid systems. In this paper we discuss both exact techniques and approximate techniques and for each method discussed we provide an illustrative example of the technique. We also present some open questions and conjectures.
Two-sample tests utilizing a similarity graph on observations are useful for high-dimensional data and non-Euclidean data due to their flexibility and good performance under a wide range of alternatives. Existing works mainly focused on sparse graphs, such as graphs with the number of edges in the order of the number of observations. However, the tests have better performance with denser graphs under many settings. In this work, we establish the theoretical ground for graph-based tests with graphs that are much denser than those in existing works.
Deep neural network is a state-of-art method in modern science and technology. Much statistical literature have been devoted to understanding its performance in nonparametric estimation, whereas the results are suboptimal due to a redundant logarithmic sacrifice. In this paper, we show that such log-factors are not necessary. We derive upper bounds for the $L^2$ minimax risk in nonparametric estimation. Sufficient conditions on network architectures are provided such that the upper bounds become optimal (without log-sacrifice). Our proof relies on an explicitly constructed network estimator based on tensor product B-splines. We also derive asymptotic distributions for the constructed network and a relating hypothesis testing procedure. The testing procedure is further proven as minimax optimal under suitable network architectures.
This work focuses on combining nonparametric topic models with Auto-Encoding Variational Bayes (AEVB). Specifically, we first propose iTM-VAE, where the topics are treated as trainable parameters and the document-specific topic proportions are obtained by a stick-breaking construction. The inference of iTM-VAE is modeled by neural networks such that it can be computed in a simple feed-forward manner. We also describe how to introduce a hyper-prior into iTM-VAE so as to model the uncertainty of the prior parameter. Actually, the hyper-prior technique is quite general and we show that it can be applied to other AEVB based models to alleviate the {\it collapse-to-prior} problem elegantly. Moreover, we also propose HiTM-VAE, where the document-specific topic distributions are generated in a hierarchical manner. HiTM-VAE is even more flexible and can generate topic distributions with better variability. Experimental results on 20News and Reuters RCV1-V2 datasets show that the proposed models outperform the state-of-the-art baselines significantly. The advantages of the hyper-prior technique and the hierarchical model construction are also confirmed by experiments.
Discrete random structures are important tools in Bayesian nonparametrics and the resulting models have proven effective in density estimation, clustering, topic modeling and prediction, among others. In this paper, we consider nested processes and study the dependence structures they induce. Dependence ranges between homogeneity, corresponding to full exchangeability, and maximum heterogeneity, corresponding to (unconditional) independence across samples. The popular nested Dirichlet process is shown to degenerate to the fully exchangeable case when there are ties across samples at the observed or latent level. To overcome this drawback, inherent to nesting general discrete random measures, we introduce a novel class of latent nested processes. These are obtained by adding common and group-specific completely random measures and, then, normalising to yield dependent random probability measures. We provide results on the partition distributions induced by latent nested processes, and develop an Markov Chain Monte Carlo sampler for Bayesian inferences. A test for distributional homogeneity across groups is obtained as a by product. The results and their inferential implications are showcased on synthetic and real data.
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