Maximum-Likelihood Estimation of a Process with Random Transitions (Failures)

A process with random transitions is represented by the difference equation xn= xn-1 + Un where Unis a nonlinear function of a Gaussian sequence wn• The nonlinear function has a threshold such that Un= 0 for |Wn|<W. This results in a finite probability of no failure at every step. Maximum likelihood estimation of the sequence Xn = {x0., xn} given a sequence of observations Yn = {y1.,yn } 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.