Generalizing distance covariance to measure and test multivariate mutual dependence via complete and incomplete V-statistics

Statistical Inferences for Complex Dependence of Multimodal Imaging Data

Statistical analysis of multimodal imaging data is a challenging task, since the data involves high-dimensionality, strong spatial correlations and complex data structures. In this paper, we propose rigorous statistical testing procedures for making inferences on the complex dependence of multimodal imaging data. Motivated by the analysis of multitask fMRI data in the Human Connectome Project (HCP) study, we particularly address three hypothesis testing problems: (a) testing independence among imaging modalities over brain regions, (b) testing independence between brain regions within imaging modalities, and (c) testing independence between brain regions across different modalities. Considering a general form for all the three tests, we develop a global testing procedure and a multiple testing procedure controlling the false discovery rate. We study theoretical properties of the proposed tests and develop a computationally efficient distributed algorithm. The proposed methods and theory are general and relevant for many statistical problems of testing independence structure among the components of high-dimensional random vectors with arbitrary dependence structures. We also illustrate our proposed methods via extensive simulations and analysis of five task fMRI contrast maps in the HCP study.

An integrated model based on feedforward neural network and Taylor expansion for indicator correlation elimination

Existing correlation processing strategies make up for the defect that most evaluation algorithms do not consider the independence between indicators. However, these solutions may change the indicator system’s internal connection, affecting the final evaluation result’s interpretability and accuracy. Besides, traditional independent analysis methods cannot accurately describe the complex multivariate correlation based on the linear relationship. Aimed at these problems, we propose an indicators correlation elimination algorithm based on the feedforward neural network and Taylor expansion (NNTE). Firstly, we propose a generalized n-power correlation and a feedforward neural network to express the relationship between indicators quantitatively. Secondly, the low-order Taylor expression expanded at every sample is pointed to eliminate nonlinear relationships. Finally, to control the expansions’ accuracy, the layer-by-layer stripping method is presented to reduce the dimensionality of the correlations among multiple indicators gradually. This procedure continues to iterate until there are all simple two-dimensional correlations, eliminating multiple variables’ correlations. To compare the elimination efficiency, the ranking accuracy is proposed to measure the distance of the resulting sequence to the benchmark sequence. Under Cleveland and KDD99 two datasets, the ranking accuracy of the NNTE method is 71.64% and 96.41%, respectively. Compared with other seven common elimination methods, our proposed method’s average increase is 13.67% and 25.13%, respectively.

ASYMPTOTIC DISTRIBUTIONS OF HIGH-DIMENSIONAL DISTANCE CORRELATION INFERENCE

Distance correlation has become an increasingly popular tool for detecting the nonlinear dependence between a pair of potentially high-dimensional random vectors. Most existing works have explored its asymptotic distributions under the null hypothesis of independence between the two random vectors when only the sample size or the dimensionality diverges. Yet its asymptotic null distribution for the more realistic setting when both sample size and dimensionality diverge in the full range remains largely underdeveloped. In this paper, we fill such a gap and develop central limit theorems and associated rates of convergence for a rescaled test statistic based on the bias-corrected distance correlation in high dimensions under some mild regularity conditions and the null hypothesis. Our new theoretical results reveal an interesting phenomenon of blessing of dimensionality for high-dimensional distance correlation inference in the sense that the accuracy of normal approximation can increase with dimensionality. Moreover, we provide a general theory on the power analysis under the alternative hypothesis of dependence, and further justify the capability of the rescaled distance correlation in capturing the pure nonlinear dependency under moderately high dimensionality for a certain type of alternative hypothesis. The theoretical results and finite-sample performance of the rescaled statistic are illustrated with several simulation examples and a blockchain application.