2015/08/19 by Taposh Banerjee, Banerjee, Taposh, Hamed Firouzi +3 · 1 citation
Mathematics · Medicine · #Data-Driven Disease Surveillance #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Probability (math.PR) #Random Matrices and Applications #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1508.04720
openalex publication_date 2015/08/19 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
This paper addresses the problem of quickest detection of a change in the\nmaximal coherence between columns of a n\× p random matrix based on a\nsequence of matrix observations having a single unknown change point. The\nrandom matrix is assumed to have identically distributed rows and the maximal\ncoherence is defined as the largest of the p choose 2 correlation\ncoefficients associated with any row. Likewise, the k nearest neighbor (kNN)\ncoherence is defined as the k-th largest of these correlation coefficients.\nThe forms of the pre- and post-change distributions of the observed matrices\nare assumed to belong to the family of elliptically contoured densities with\nsparse dispersion matrices but are otherwise unknown. A non-parametric stopping\nrule is proposed that is based on the maximal k-nearest neighbor sample\ncoherence between columns of each observed random matrix. This is a summary\nstatistic that is related to a test of the existence of a hub vertex in a\nsample correlation graph having a degree at least k. Performance bounds on\nthe delay and false alarm performance of the proposed stopping rule are\nobtained in the purely high dimensional regime where p\→ \∞ and\nn is fixed. When the pre-change dispersion matrix is diagonal it is shown\nthat, among all functions of the proposed summary statistic, the proposed\nstopping rule is asymptotically optimal under a minimax quickest change\ndetection (QCD) model as the stopping threshold approaches infinity. The theory\ndeveloped also applies to sequential hypothesis testing and fixed sample size\ntests.\n