Non-parametric Quickest Change Detection for Large Scale Random Matrices

Abstract: The problem of quickest detection of a change in the distribution of a $n\times p$ random matrix based on a sequence of observations having a single unknown change point is considered. The forms of the pre- and post-change distributions of the rows of the matrices are assumed to belong to the family of elliptically contoured densities with sparse dispersion matrices but are otherwise unknown. We propose a non-parametric stopping rule that is based on a novel summary statistic related to k-nearest neighbor correlation between columns of each observed random matrix. In the large scale regime of $p\rightarrow \infty$ and $n$ fixed we show that, among all functions of the proposed summary statistic, the proposed stopping rule is asymptotically optimal under a minimax quickest change detection (QCD) model.