Modèles de Markov triplets en restauration des signaux

AbstractRestoration unsupervised statistical signal admits countless applications in fieldsas diverse economy, health, signal processing, meteorology, finance, biology, reliability,transport, environment, ... One problem base, which is central to this thesis isto estimate a sequence hidden(x n ) 1∶N from an observed sequence(y n ) 1∶N . Probabilistictreatment of the problem in these sequences are considered accomplishments,respectively, process X =(X n ) 1∶N and Y =(Y n ) 1∶N . Several techniques based onstatistical methods have been developed for solve this problem. The most commonmodel among the process is the model called og hidden Markov model fg (MMC).In this model we assume that the hidden process X is Markov and laws p(y∣x) of Yconditional on X is sufficiently simple so that the law p(x∣y) is also Markovian, thislast property is necessary for treatment. More Extensions of these models have beenproposed since 2000. In Markov models couples (MMCouples), more general thanthe MMC, the pair(X,Y) is Markovian), implying that p(x∣y) is also Markov (whenp(x) is no necessarily), which allows the same treatment as in MMC. More recently(2002), were extended to MMCouples « triplet Markov models » (MMT), in whichwe introduce a auxiliary process U and suppose that the triple T=(X,U,Y) is Markov.Again it is possible, in a more general that of MMCouples, perform treatmentswith a reasonable complexity.The objective of this thesis is to propose new modeling part of the MMT and toinvestigate their relevance and interest. We offer two types of innovations :(i) When the system is discrete and hidden when the couple(X,Y) is not stationarywith a finite number of « jumps » random parameters, the recent use of MMTin where the jumps are modeled by a discrete process U a been very convincing(Lanchantin, 2006). Our first idea is to use this approach with a process U continuous,which models non-steady "continuous" from(X,Y). We offer chains andfields and triplets present some experiments. The results obtained in the modelingof non-stationarity still seem less interesting that in the discrete case. However, newmodels may have other interests, in particular, they seem more efficient than oghidden Markov fg classic when the noise is correlated;(ii) An MMT T=(X,U,Y) such that X and Y are continuous and U is discretefinite. We are dealing with the problem of filtering, or smoothing, with randomjumps. In modeling classic the pair hidden(X,U) is Markov, but the pair(U,Y)is not, what is the cause of the impossibility of Exact calculations with linear complexityin time. It is then necessary to use various approximate methods, which thoseusing particle filtering are among the most used. In recent models MMT the pairhid(X,U) is not necessarily Markovian, but the pair(U,Y) is, what which allowsaccurate treatment with a reasonable complexity (Pieczynski 2009). Our second ideaiii