Tracking Processes with Change Points

Abstract

In Sect. 1.2 of the Introduction we described a problem of tracking a missile in order to quickly detect significant deviation from its designed orbit. The objective is to estimate the expected value in a given time of a random process under observation, when the data available is only the past observations. The present chapter is based on the papers listed among the references. We start with the joint paper of Chernoff and Zacks (Ann Math Stat 35:999–1018, 1964). For simplification, we assume that the orbit is a given horizontal line at level μ. The observations are taken at discrete time points, and they reflect the random positions of the object (missile) around μ, and the possible changes in the horizontal trend. We assume here that as long as there are no changes in the trend, the observations are realization of independent random variables, having a common normal distribution, \(N(\mu ,\sigma _{\epsilon }^{2})\). Since the objective is to detect whether changes have occurred in the trend, we let μn = E{Xn}, where {X1, …, Xn} is the given data, and adopt the following model $$\displaystyle \begin{aligned} X_{n} =\mu _{1} +\sum _{i =1}^{n -1}J_{n -i}Z_{n -i} +\epsilon _{n},n \geq 1. \end{aligned}$$