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Section: New Results

Markov models

Identifying Interactions between Tropical Plant Species: A Correlation Analysis of High-Throughput Environmental DNA Sequence Data based on Random Matrix Theory

Participants : Florence Forbes, Angelika Studeny.

This is joint work with: Eric Coissac and Pierre Taberlet from LECA (Laboratoire d'Ecologie Alpine) and Alain Viari from Inria team Bamboo.

The study of species cooccurence pattern has always been central to community ecology. The rise of high-throughput molecular methods and their use in ecology nowadays allows for a facilitated access to new data of an unprecedented quantity. We address the question about the identification of genuine species interactions in the light of these novel data. The statistical analysis has to be tailored to the data specifics: the large amount of available data as well as biases inherent to the data extraction methods. The latter can cause spurious interactions while the former complicates any statistical modelling approach. In addition, the resolution of the data provided is rarely on the species level. In this work, we conduct a thorough correlation analysis between MOTUs (molecular operating taxonomic unit) on different spatial scales to investigate global as well as local spatial pattern. Although this type of analysis is per se exploratory, we suggest it here in order to separate true species interaction from random pattern and to identify species subgroups for further in detail modelling. A random-matrix approach allows us to derive objective cut-off values for genuine correlations. We compare the results with those derived by the application of a model-based, sparse regression approach. Our study shows that despite their seemingly less precise nature when it comes to species identification, these data enable us to reveal mechanisms that structure an ecological community. In the light of the nowadays facilitated access to molecular data, this points the way to a novel set of efficient methods for community analysis.

Modelling multivariate counts with graphical Markov models.

Participant : Jean-Baptiste Durand.

Joint work with: Pierre Fernique (Montpellier 2 University, CIRAD and Inria Virtual Plants) and Yann Guédon (CIRAD and Inria Virtual Plants)

Multivariate count data are defined as the number of items in different states issued from sampling within a population, which individuals own items in various numbers and states. The analysis of multivariate count data is a recurrent and crucial issue in numerous modelling problems, particularly in the fields of biology and ecology (where the data can represent, for example, children counts associated with multitype branching processes), sociology and econometrics. Denoting by $K$ the number of states, multivariate count data analysis relies on modelling the joint distribution of the $K$-dimensional random vector $N=\left({N}_{0},...,{N}_{K-1}\right)$ with discrete components. Our work focused on I) Identifying states that appear simultaneously, or on the contrary that are mutually exclusive. This was achieved by identifying conditional independence relationships between the $K$ variables; II)Building parsimonious parametric models consistent with these relationships; III) Characterizing and testing the effects of covariates on the distribution of $N$, particularly on the dependencies between its components.

Our context of application was characterised by zero-inflated, often right skewed marginal distributions. Thus, Gaussian and Poisson distributions were not a priori appropriate. Moreover, the multivariate histograms typically had many cells, most of which were empty. Consequently, nonparametric estimation was not efficient.

We developed an approach based on probabilistic graphical models (Koller & Friedman, 2009 [73] ) to identify and exploit properties of conditional independence between numbers of children in different states, so as to simplify the specification of their joint distribution. The considered models are based on chain graphs. Model selection procedures are necessary to infer the graph and specify parsimonious distributions. The graph building stage was based on exploring the space of possible chain graph models, which required defining a notion of neighbourhood of these graphs. A parametric distribution was associated with each graph. It was obtained by combining families of univariate and multivariate distributions or regression models. These families were chosen by selection model procedures among different parametric families [36] . To relax the strong constraints regarding dependencies induced by using parametric distributions, mixture of graphical models were also considered [49] .

Further extensions will be considered, and particularly

• Hidden Markov tree models (see 6.4.3 ) where the hidden state process is a multitype branching process with graphical generation distributions.

• Gaussian chain graph models, where the chain components can be identified using lasso methods.

Statistical characterization of tree structures based on Markov tree models and multitype branching processes, with applications to tree growth modelling.

Participant : Jean-Baptiste Durand.

Joint work with: Pierre Fernique (Montpellier 2 University and CIRAD) and Yann Guédon (CIRAD), Inria Virtual Plants.

Algorithmic issues in hidden Markov tree models were considered by Durand et al. (2004) [68] . This family of models was used to represent local dependencies and heterogeneity within tree-structured data. It relied on a tree-structured hidden state process, where the children states were assumed independent given their parent state. The latter assumption has been relaxed in an extension of these models and new algorithmic solutions for model inference have been proposed in Pierre Fernique's PhD [70] . An application to the study of the cell lineage in biological tissues responsible for the plant growth has been considered. In this setting, the number of children is small (between 0 and 2) and a saturated model has been considered to model transitions between parent and configurations of children states. Extensions will be proposed, based on the parametric discrete multivariate distributions developed in Section 6.4.2 .

Change-point models for tree-structured data

Participant : Jean-Baptiste Durand.

Joint work with: Pierre Fernique (Montpellier 2 University and CIRAD) and Yann Guédon (CIRAD), Inria Virtual Plants.

As an alternative to the hidden Markov tree models discussed in Section 6.4.3 , subtrees with similar attributes can be identified using multiple change-point models. These approaches are well-developed in the context of sequence analysis, but their extensions to tree-structured data are not straightforward. Their advantage on hidden Markov models is to relax the strong constraints regarding dependencies induced by parametric distributions and local parent-children dependencies. Heuristic approaches for change-point detection in trees were proposed and applied to the analysis of patchiness patterns (consisting of canopies made of clumps of either vegetative or flowering botanical units) in mango trees [70] .

Hidden Markov models for the analysis of eye movements

Participant : Jean-Baptiste Durand.

Joint work with: Anne Guérin-Dugué (GIPSA-lab) and Benoit Lemaire (Laboratoire de Psychologie et Neurocognition)

In the last years, GIPSA-lab has developed computational models of information search in web-like materials, using data from both eye-tracking and electroencephalograms (EEGs). These data were obtained from experiments, in which subjects had to make some kinds of press reviews. In such tasks, reading process and decision making are closely related. Statistical analysis of such data aims at deciphering underlying dependency structures in these processes. Hidden Markov models (HMMs) have been used on eye movement series to infer phases in the reading process that can be interpreted as steps in the cognitive processes leading to decision. In HMMs, each phase is associated with a state of the Markov chain. The states are observed indirectly through eye-movements. Our approach was inspired by Simola et al. (2008) [76] , but we used hidden semi-Markov models for better characterization of phase length distributions. The estimated HMM highlighted contrasted reading strategies (i.e., state transitions), with both individual and document-related variability.

However, the characteristics of eye movements within each phase tended to be poorly discriminated. As a result, high uncertainty in the phase changes arose, and it could be difficult to relate phases to known patterns in EEGs.

As a perspective, we aim at developing an integrated model coupling EEG and eye movements within one single HMM for better identification of the phases. Here, the coupling should incorporate some delay between the transitions in both (EEG and eye-movement) chains, since EEG patterns associated to cognitive processes occur lately with respect to eye-movement phases. Moreover, EEGs and scanpaths were recorded with different time resolutions, so that some resampling scheme must be added into the model, for the sake of synchronizing both processes. Probabilistic graphical models (see Section 6.4.2 ) will be inferred from the channel correlations to represent interactions between brain zones. The variability of these graphs is partly explained by individual differences in text exploration, which will have to be quantified.

Hyper-Spectral Image Analysis with Partially-Latent Regression and Spatial Markov Dependencies

Participant : Florence Forbes.

Joint work with: Antoine Deleforge, Sileye Ba and Radu Horaud from the Inria Perception team.

Hyper-spectral data can be analyzed to recover physical properties at large planetary scales. This involves resolving inverse problems which can be addressed within machine learning, with the advantage that, once a relationship between physical parameters and spectra has been established in a data-driven fashion, the learned relationship can be used to estimate physical parameters for new hyper-spectral observations. Within this framework, we propose a spatially-constrained and partially-latent regression method which maps high-dimensional inputs (hyper-spectral images) onto low-dimensional responses (physical parameters). The proposed regression model comprises two key features. Firstly, it combines a Gaussian mixture of locally-linear mappings (GLLiM) with a partially-latent response model described in [17] . While the former makes high-dimensional regression tractable, the latter enables to deal with physical parameters that cannot be observed or, more generally, with data contaminated by experimental artifacts that cannot be explained with noise models. Secondly, spatial constraints are introduced in the model through a Markov random field (MRF) prior which provides a spatial structure to the Gaussian-mixture hidden variables. Experiments conducted on a database composed of remotely sensed observations collected from the Mars planet by the Mars Express orbiter demonstrate the effectiveness of the proposed model. A preliminary version of the work can be found in [31] .