Abstract
In this paper, we consider the estimation of a change-point for possibly high-dimensional data in a Gaussian model, using a maximum likelihood method. We are interested in how dimension reduction can affect the performance of the method. We provide an estimator of the change-point that has a minimax rate of convergence, up to a logarithmic factor. The minimax rate is in fact composed of a fast rate —dimension-invariant— and a slow rate —increasing with the dimension. Moreover, it is proved that considering the case of sparse data, with a Sobolev regularity, there is a bound on the separation of the regimes above which there exists an optimal choice of dimension reduction, leading to the fast rate of estimation. We propose an adaptive dimension reduction procedure based on Lepski’s method and show that the resulting estimator attains the fast rate of convergence. Our results are then illustrated by a simulation study. In particular, practical strategies are suggested to perform dimension reduction.
Highlights
We will address a framework where the change between classes occurs on a time scale, which casts the problem into the change-point estimation issue
We do not have this opportunity, which adds a difficulty to the problem. Another related reference is the paper by [59], who proposed a two-stage procedure based on a projection followed by a univariate change point estimation algorithm applied to the projected data, providing rates of convergence for the estimator of the change-point location
We present the problem of dimension reduction and the maximum likelihood estimator of the change-point
Summary
An important problem in the vast domain of statistical learning is the question of unsupervised classification of high-dimensional data. For high-dimensional data, from a computational point of view, there is an obvious need for dimension reduction when estimating τ. Without such a step, the segmentation algorithm might be unstable or even not work at all. We will consider the dimension reduction problem from a theoretical point of view (as opposed to the algorithmic point of view). From a theoretical point of view, one might suspect that it should always be better to keep the whole data, to get the best precision on the estimation of the change-point. We show that this intuition is not correct Addressing this dimension reduction problem can require sophisticated tools directly connected to smoothing questions in nonparametric estimation. (d) Does on-line (signal by signal) dimension reduction perform as well as offline (using a preprocessing involving all the signals)?
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