# Team : mistis

## Section: New Results

### Markovian models

#### Convergence properties of EM like algorithms for inference in Hidden Markov Random Fields

Participants : Florence Forbes, Gersende Fort.

Outside simple cases, the EM algorithm is seldom tractable analytically. In practice, difficulties arise due to the dependence structure in the models and approximations are required. A heuristic solution using mean field approximation principle has been proposed in [31]. Using ideas from this principle, we proposed [3] a class of EM-like algorithms generalizing [31]. The mean field approach consists of neglecting fluctuations from the mean in the environment of each variable. More generally, we talk about mean field-like approximations when the value at node i does not depend on the values at other nodes which are all set to constants (not necessarily the means) independently of the value at node i ([3]). The following computation then reduces to dealing with systems of independent variables, which is much simpler.

This approach is very flexible in that many ways to set the neighboring nodes are possible and lead to as many different algorithms. We investigated some of these choices [3][6] which led to promising procedures. Their behavior is satisfying in practice but no theoretical study as regards convergence properties is available yet.

To investigate such convergence properties, we propose to consider a particular way to set the neighbors which induces the increase of a function of interest. The function is chosen so as to facilitate the the convergence study of the subsequent algorithm. After implementing and assessing the performance of this algorithm in practice, a second step is to consider techniques developped in [26] to link the properties of the algorithm to the other algorithms originally developped in [3].

#### Factorial Hidden Markov Models for time series in finance

Participants : Christian Lavergne, Mohamed Saidane.

The purpose of our work is the development of dynamic factor models for multivariate financial time series, and the incorporation of stochastic volatility components for latent factor processes. The models are direct generalizations of univariate stochastic volatility models, and represent specific varieties of models recently discussed in the growing multivariate stochastic volatility literature.

#### Stastistical tools for the analysis of bacterial genomes organisation

Participants : Florence Forbes, Matthieu Vignes.

We investigated a part of the exploratory analysis of bacterial genomes, beyond gene detection. We aim at detecting relationships among genes based on different kinds of information: nucleotide sequence, gene position, functional annotation,... The ideal goal is to link proximities among genes on the chromosome with genetic mechanisms of the cell. In fact, the cell machinery is thought to be coded inside the genome. We reviewed the main work in progress on the subject in order to suggest an appropriate formalism. We focused on the notion of neighborhood, related to intrinsic properties among entities (genes) considered. Neighborhood must be understood in a broad sense which leads to some specific mathematical tools and processes. Our investigation is based on tools from mixture models and markovian models. We consider various classification methods.

#### Markov Random Fields for recognizing textures

Participants : Florence Forbes, Juliette Blanchet.

We present a new probabilistic framework for recognizing textures in images. Images are described by local affine-invariant descriptors and by spatial relationships between these descriptors. A graph is associated to an image with the nodes representing feature vectors describing image regions and the edges joining spatially related regions. Incorporating information about the spatial organization of the descriptors leads to better recognition results. Current approaches consist in augmenting the data with information coming from the spatial relationships, for instance by using co-occurence statistics, but without modeling explicitly the dependencies between neighboring descriptors. In such approaches the underlying model is one where the descriptors are statistically independent variables. Our claim is that recognition results can be further improved by considering that descriptors are statistically dependent. We propose to introduce in texture recognition the use of statistical parametric models of the dependence between descriptors. In this work, we chose Hidden Markov Models (HMM) which are both well statistically-based and appropriate models for such a task. They are parametric models and their use requires non trivial parameter estimation. We propose to use recent estimation procedures based on the mean field principle of statistical physics. Using sample images, textures are then learned as HMM's and a set of estimated parameters is associated to each texture. At recognition time, another HMM is used to compute, for each feature vector, the membership probabilities to the different texture classes. Preliminary experiments show promising results [22] .