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We study the multivariate nonparametric change point detection problem, where the data are a sequence of independent $p$ -dimensional random vectors whose distributions are piecewise-constant with Lipschitz densities changing at unknown times, called change points. We quantify the size of the distributional change at any change point with the supremum norm of the difference between the corresponding densities. We are concerned with the localization task of estimating the positions of the change points. In our analysis, we allow for the model parameters to vary with the total number of time points, including the minimal spacing between consecutive change points and the magnitude of the smallest distributional change. We provide information-theoretic lower bounds on both the localization rate and the minimal signal-to-noise ratio required to guarantee consistent localization. We formulate a novel algorithm based on kernel density estimation that nearly achieves the minimax lower bound, save possibly for logarithm factors. We have provided extensive numerical evidence to support our theoretical findings. [ABSTRACT FROM AUTHOR]

We consider the problem of deciding, based on a single noisy measurement at each vertex of a given graph, whether the underlying unknown signal is constant over the graph or there exists a cluster of vertices with anomalous activation. This problem is relevant to several applications such as surveillance, disease outbreak detection, biomedical imaging, environmental monitoring, etc. Since the activations in these problems often tend to be localized to small groups of vertices in the graphs, we model such activity by a class of signals that are elevated over a (possibly disconnected) cluster with low cut size relative to its size. We analyze the corresponding generalized likelihood ratio (GLR) statistics and relate it to the problem of finding a sparsest cut in the graph. We develop a convex relaxation of the GLR statistic based on spectral graph theory, which we call the graph Fourier scan statistic (GFSS). In our main theoretical result, we show that the performance of the GFSS depends explicitly on the spectral properties of the graph. To assess the optimality of the GFSS, we prove an information theoretic lower bound for the detection of anomalous activity on graphs. Because the GFSS requires the specification of a tuning parameter, we develop an adaptive version of the GFSS. Using these results, we are able to characterize in a very explicit form the performance of the GFSS on a few notable graph topologies. We demonstrate that the GFSS can efficiently detect a simulated Arsenic contamination in groundwater. [ABSTRACT FROM AUTHOR]