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- Wasserstein F-tests for Fréchet regression on Bures-Wasserstein manifolds(arXiv)
Author : Haoshu Xu, Hongzhe Li
Abstract : This paper considers the problem of regression analysis with random covariance matrix as outcome and Euclidean covariates in the framework of Fréchet regression on the Bures-Wasserstein manifold. Such regression problems have many applications in single cell genomics and neuroscience, where we have covariance matrix measured over a large set of samples. Fréchet regression on the Bures-Wasserstein manifold is formulated as estimating the conditional Fréchet mean given covariates x. A non-asymptotic n−−√-rate of convergence (up to logn factors) is obtained for our estimator Q^n(x) uniformly for ∥x∥≲logn−−−−√, which is crucial for deriving the asymptotic null distribution and power of our proposed statistical test for the null hypothesis of no association. In addition, a central limit theorem for the point estimate Q^n(x) is obtained, giving insights to a test for covariate effects. The null distribution of the test statistic is shown to converge to a weighted sum of independent chi-squares, which implies that the proposed test has the desired significance level asymptotically. Also, the power performance of the test is demonstrated against a sequence of contiguous alternatives. Simulation results show the accuracy of the asymptotic distributions. The proposed methods are applied to a single cell gene expression data set that shows the change of gene co-expression network as people age.
2. A Type of Nonlinear Fréchet Regressions(arXiv)
Abstract : The existing Fréchet regression is actually defined within a linear framework, since the weight function in the Fréchet objective function is linearly defined, and the resulting Fréchet regression function is identified to be a linear model when the random object belongs to a Hilbert space. Even for nonparametric and semiparametric Fréchet regressions, which are usually nonlinear, the existing methods handle them by local linear (or local polynomial) technique, and the resulting Fréchet regressions are (locally) linear as well. We in this paper introduce a type of nonlinear Fréchet regressions. Such a framework can be utilized to fit the essentially nonlinear models in a general metric space and uniquely identify the nonlinear structure in a Hilbert space. Particularly, its generalized linear form can return to the standard linear Fréchet regression through a special choice of the weight function. Moreover, the generalized linear form possesses methodological and computational simplicity because the Euclidean variable and the metric space element are completely separable. The favorable theoretical properties (e.g. the estimation consistency and presentation theorem) of the nonlinear Fréchet regressions are established systemically. The comprehensive simulation studies and a human mortality data analysis demonstrate that the new strategy is significantly better than the competitors.