Detection of Multiple Change Points by the Filtered Derivative and False Discovery Rate

Abstract

Let $\mathbf{X}=(X_1,X_2,\ldots,X_n)$ be a time series, that is a sequence of random variable indexed by the time $ t=1,2,\ldots,n $. We assume the existence of a segmentation $\tau=(\tau_1,\tau_2,\ldots,\tau_n)$ such that $X_i$ is a family of independent identically distributed (i.i.d) random variable for i $\in (\tau_k,\tau_k+1],~and~k=0,\ldots,K$ where by convention $\tau_o$ and $\tau_{K+1}=N$. In the literature, it exist two main kinds of change points detections: The change points on-line and the change points off-line. In this work, we consider only the change point analysis (off-line), when number of change points is  unknown. The result obtained  is based on Filtered Derivative method where  we use a second step based on False Discovery Rate. We compare numerically this new method with the Filtered Derivative with p-Value.