Statistical unsupervised restoration of signal can be applied in many fields such as economy, health, signal processing, meteorology, finance, biology, reliability, transportation, environment, ... the main problem treated in this thesis is to estimate a hidden sequence (Xn)1:N based on an observed sequence (Yn)1:N. In Probabilistic treatment of the problem in these sequences are considered as accomplishments of respectively, process (Xn)1:N and (Yn)1:N. Several techniques based on statistical methods have been developed to solve this problem. The most common model known for this kind of problems is the “hidden Markov model”. In this model we assume that the hidden process X is Markovian and laws p(y|x) of Y are conditional on X are sufficiently simple so that the law p(x|y) is also Markovian, this property is necessary for treatment. Many Extensions of these models have been proposed since 2000. In Markov models couples (MMCouples), more general than the MMC, the pair (X,Y) is Markovian), implying that p(x|y) is also Markovian (when p(x) is not necessarily markovian), which allows the same treatment as in MMC. More recently (2002), were extended to MMCouples are extended to Markov models Triplet (MMT), in which we introduce an auxiliary process U and suppose that the triple T=(X,U,Y) is Markovian. It’s again possible, in a general case of MMCouples, to perform treatments with a reasonable complexity. The objective of this thesis is to propose new modeling of MMT and to investigate their relevance and interest. We offer two types of innovations: (i) When the hidden system is discrete and when the couple (X,Y) is not stationary with a finite number of random “jumps” in parameters, the recent use of MMT where the jumps are modelized by a discrete process U has been very convincing (Lanchantin, 2006). Our first idea is to use this approach with a continuous process U, which models non-steady "continuous" of (X,Y). We propose chains and triplet fields and present some experiments. The results obtained in the modeling of non-stationarity still seem less interesting that in the discrete case. However, new models may have other interests, in particular, they seem more efficient than “classic hidden Markov” when the noise is correlated; (ii) Considering an MMT T=(X,U,Y) such that X and Y are continuous and U is discrete finite. We are dealing with a problem of filtering, or smoothing, with random jumps. In classic modelling the hidden pair (X,U) is Markovian, but the pair (U,Y) is not, what is the cause of the impossibility of Exact calculations with time linear complexity. It is then necessary to use various approximate methods, including methods using particle filtering which are the most common. In recent models MMT the hidden pair (X,U) is not necessarily Markovian, but the pair (U,Y) is Markovian, which allows accurate treatment with a reasonable complexity (Pieczynski 2009). Our second idea is to extend these models to triplets T=(X,U,Y) where the pairs (U,Y) are "partially" Markovian. Such a pair (U,Y) is not Markovian but U is conditionally Markovian on Y. We have in result a model with general model T=(X,U,Y) , which is no more Markovian, wherein the filtering and smoothing are accurate possible with time linear complexity. Some preliminary Simulations show the importance of new smoothing models with of jumps.
