Maximum-likelihood estimation of a process with random transitions (failures)

Abstract

A process with random transitions is represented by the difference equation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x_{n} = x_{n-1}+ u_{n}</tex> where u <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</inf> is a nonlinear function of a Gaussian sequence w_{n}. The nonlinear function has a threshold such that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u_{n} =0</tex> for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|w_{n}| \leq W</tex> . This results in a finite probability of no failure at every step. Maximum likelihood estimation of the sequence <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X_{n}={x_{0},...,x_{n}}</tex> given a sequence of observations <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y_{n} = { y_{1},...,y_{n} }</tex> gives rise to a two-point boundary value (TPBV) problem, the solution of which is suggested by the analogy with a nonlinear electrical ladder network. Examples comparing the nonlinear filter that gives an approximate solution of the TPBV problem with a linear recursive filter are given, and show the advantages of the former. Directions for further investigation of the method are indicated.