Optimal Covariance Change Point Detection in High Dimension
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We study the problem of change point detection for covariance matrices in high dimensions. We assume that Xii=1,...,n is a sequence of independent, centered p-dimensional sub-Gaussian random vectors is observed whose covariance matrices are piece-wise constant. Our task is to recover with high accuracy the number and locations the of change points, which are unknown. Our generic model setting allows for all the model parameters to change with n, including the dimension p, the minimal spacing between consecutive change points, the magnitude of smallest change and the maximal operator norm of the covariance matrices of the sample points. We introduce two procedures, one based on the binary segmentation algorithm (e.g. Vostrikova, 1981) and the other on its extension known as wild binary segmentation of Fryzlewicz (2014), and demonstrate that, under suitable conditions, both are able to consistently estimate the number and locations of change points. Our second algorithm, called Wild Binary Segmentation through Independent Projection (WBSIP) is shown to be minimax optimal in the in terms of all the relevant parameters. Our minimax analysis also reveals a phase transition effect based on our generic model setting. To the best of our knowledge, this type of results has not been established elsewhere in the change point detection literature.
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