### The Parameters Optimization of Filtered Derivative for Change Points Analysis

#### Abstract

Let $\mathbf{X} = ( X_1,X_2,\ldots,X_N )$ be a time series. That is a sequence random variable indexed by the time $t=(1,2,\ldots,N)$, we suppose that the parameters of $\mathbf{X}$ are piecewise constant. In other words, it exists a subdivision $\tau=(\tau_1< \tau_2<\ldots < \tau_K )$ such that $ X_i$ is a family of independent and identically distributed (i.i.d) random variables for $i \in (\tau_k,\tau_{k+1}] $, and $k = 0,1,\ldots,K$ where by convention $\tau_o=0$ and $\tau_{K+1}=N $. The preceding works such that (Bertrand, 2000) control the probability of false alarms for minimizing the probability of type I error of change point analysis. The novelty in this work is to control the number of false alarms. We give an bound of number of false alarms and the necessary condition for number of no detection. In other hand, we know the filtered derivative (Basseville \& Nikirov, 1993) depends the parameters such that the threshold and the window, we give in order to choose the optimal parameters. We compare the results of Filtered Derivative optimized parameters and the Penalized Square Error methods in particulary the adaptive method of (Lavielle \& Teyssi\`ere, 2006).

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International Journal of Statistics and Probability ISSN 1927-7032(Print) ISSN 1927-7040(Online)

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