Integral equation methods in change-point detection problems

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NO FULL TEXT AVAILABLE. This thesis contains 3rd party copyright material. ----- In a standard formulation a Change Point Detection (CPD) problem is described by a sequence of observations whose distribution function changes abruptly at some unknown point in time. Suppose that the observations of a random process (discrete or continuous time) are received sequentially and at a certain moment (random or not but unknown) some probabilistic characteristics of this process are changing. An observer should make a decision as quickly as possible if the change-points occur or not. Instead, the observer should not make to many ’’false alarms”, i.e., decisions about detecting change-points when they are not presented. Control charts such as CUSUM (Cumulative SUM) and EWMA (Exponentially Weighted Moving Average) are procedures used to detect a change-point in sequentially observed data. In practical applications the control charts CUSUM and EWMA are widely used in many areas such as economics, finance, medicine, and engineering to mention only several. For an introduction to CUSUM and EWMA charts and their applications see Brodsky and Darkhovsky [1], [2], Basseville and Nikiforov [3]. In this thesis we analyze CUSUM and EWMA charts and their main characteristics the Average Run Length (ARL) and the Average Delay time (AD). ARL and AD are the most common characteristics used to design EWMA and CUSUM charts. In this research we used the Fredholm type integral equation methods to derive analytical closed form representations for the ARL and AD for some special cases. In particular, we derive a closed form representation for the ARL of CUSUM chart assuming that the random observations are hyperexponentially distributed, see Mititelu et al. [5]. We apply Banach’s Fixed Point theorem to analyze the existence and uniqueness for the solutions of ARL integral equations [6]. For the EWMA we solve the corresponding ARL integral equation in explicit form when the observations are Laplace distributed, and derive closed-form expression for the AD [4], Based on the same technique we derive the solution for the integral equation of the generating function [5]. Our new result extends the recent result published in the literature by Larralde [10]. For practical applications it is very important to have analytical solutions to design quick optimal EWMA and CUSUM charts. We present several numerical examples to compare the closed-form solutions with the Monte-Carlo simulations. When it is not possible to obtain close form solutions for the corresponding ARL integral equation in the case of CUSUM procedure for different distribution functions, e.g., Weibull, Gamma or Pareto, our results can be implemented to approximate the ARL and AD, approximating these distributions by hyperexponential distributions. More precisely, it is well-known that any completely monotone probability density function (pdf) can be approximated by hyperexponential distributions (see [7], and [8]). This covers a very large class of pdf functions such as Pareto, Weibull or Gamma distributions, which are completely monotone and so they can be approximated by hyperexponentials. To illustrate the applications of our results, we approximate Weibull distribution by a mixture of six exponentials with the fitting parameters obtained via the matching moments method (see [7]).
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