Unsupervised segmentation of triplet Markov chains hidden with long-memory noise

The hidden Markov chain (HMC) model is a couple of random sequences (X,Y), in which X is an unobservable Markov chain, and Y is its observable noisy version. Classically, the distribution p(y|x) is simple enough to ensure the Markovianity of p(x|y), that enables one to use different Bayesian restoration techniques. HMC model has recently been extended to “pairwise Markov chain” (PMC) model, in which one directly assumes the Markovianity of the pair Z=(X,Y), and which still enables one to recover X from Y. Finally, PMC has been extended to “triplet Markov chain” (TMC) model, which is obtained by adding a third chain U and considering the Markovianity of the triplet T=(X,U,Y). When U is not too complex, X can still be recovered from Y. Then U can model different situations, like non-stationarity or semi-Markovianity of (X,Y). Otherwise, PMC and TMC have been extended to pairwise “partially” Markov chains (PPMC) and triplet “partially” Markov chains (TPMC), respectively. In a PPMC Z=(X,Y) the distribution p(x|y) is a Markov distribution, but p(y|x) may not be and, similarly, in a TPMC T=(X, U, Y) the distribution p(x,u|y) is a Markov distribution, but p(y|x,u) may not be. However, both PPMC and TPMC can enable one to recover X from Y, and TPMC include different long-memory noises. The aim of this paper is to show how a particular Gaussian TPMC can be used to segment a discrete signal hidden with long-memory noise. An original parameter estimation method, based on “Iterative Conditional Estimation” (ICE) principle, is proposed and some experiments concerned with unsupervised segmentation are provided. The particular unsupervised segmentation method used in experiments can also be seen as identification of different stationarities in fractional Brownian noise, which is widely used in different problems in telecommunications, economics, finance, or hydrology.