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We propose an innovative and generic methodology to analyse individual and collective behaviour through individual trajectory data. The work is motivated by the analysis of GPS trajectories of fishing vessels collected from regulatory tracking data in the context of marine biodiversity conservation and ecosystem-based fisheries management. We build a low-dimensional latent representation of trajectories using convolutional neural networks as non-linear mapping. This is done by training a conditional variational auto-encoder taking into account covariates. The posterior distributions of the latent representations can be linked to the characteristics of the actual trajectories. The latent distributions of the trajectories are compared with the Bhattacharyya coefficient, which is well-suited for comparing distributions. Using this coefficient, we analyse the variation of the individual behaviour of each vessel during time. For collective behaviour analysis, we build proximity graphs and use an extension of the stochastic block model for multiple networks. This model results in a clustering of the individuals based on their set of trajectories. The application to French fishing vessels enables us to obtain groups of vessels whose individual and collective behaviours exhibit spatio-temporal patterns over the period 2014-2018.

We consider a broad class of random bipartite networks, the distribution of which is invariant under permutation within each type of nodes. We are interested in $U$-statistics defined on the adjacency matrix of such a network, for which we define a new type of Hoeffding decomposition. This decomposition enables us to characterize non-degenerate $U$-statistics -- which are then asymptotically normal -- and provides us with a natural and easy-to-implement estimator of their asymptotic variance. \\ We illustrate the use of this general approach on some typical random graph models and use it to estimate or test some quantities characterizing the topology of the associated network. We also assess the accuracy and the power of the proposed estimates or tests, via a simulation study., Comment: This work was partially supported by the grant ANR-18-CE02-0010-01 of the French National Research Agency ANR (project EcoNet)

Let a collection of networks represent interactions within several (social or ecological) systems. We pursue two objectives: identifying similarities in the topological structures that are held in common between the networks and clustering the collection into sub-collections of structurally homogeneous networks. We tackle these two questions with a probabilistic model based approach. We propose an extension of the Stochastic Block Model (SBM) adapted to the joint modeling of a collection of networks. The networks in the collection are assumed to be independent realizations of SBMs. The common connectivity structure is imposed through the equality of some parameters. The model parameters are estimated with a variational Expectation-Maximization (EM) algorithm. We derive an ad-hoc penalized likelihood criterion to select the number of blocks and to assess the adequacy of the consensus found between the structures of the different networks. This same criterion can also be used to cluster networks on the basis of their connectivity structure. It thus provides a partition of the collection into subsets of structurally homogeneous networks. The relevance of our proposition is assessed on two collections of ecological networks. First, an application to three stream food webs reveals the homogeneity of their structures and the correspondence between groups of species in different ecosystems playing equivalent ecological roles. Moreover, the joint analysis allows a finer analysis of the structure of smaller networks. Second, we cluster 67 food webs according to their connectivity structures and demonstrate that five mesoscale structures are sufficient to describe this collection., Comment: To appear in the Annals of Applied Statistics (2023-09-28)

The robustness of an ecological network quantifies the resilience of the ecosystem it represents to species loss. It corresponds to the proportion of species that are disconnected from the rest of the network when extinctions occur sequentially. Classically, the robustness is calculated for a given network, from the simulation of a large number of extinction sequences. The link between network structure and robustness remains an open question. Setting a joint probabilistic model on the network and the extinction sequences allows analysis of this relation. Bipartite stochastic block models have proven their ability to model bipartite networks e.g. plant-pollinator networks: species are divided into blocks and interaction probabilities are determined by the blocks of membership. Analytical expressions of the expectation and variance of robustness are obtained under this model, for different distributions of primary extinction sequences. The impact of the network structure on the robustness is analyzed through a set of properties and numerical illustrations. The analysis of a collection of bipartite ecological networks allows us to compare the empirical approach to our probabilistic approach, and illustrates the relevance of the latter when it comes to computing the robustness of a partially observed or incompletely sampled network.

A multilevel network is defined as the junction of two interaction networks, one level representing the interactions between individuals and the other the interactions between organizations. The levels are linked by an affiliation relationship, each individual belonging to a unique organization. A new Stochastic Block Model is proposed as a unified probalistic framework tailored for multilevel networks. This model contains latent blocks accounting for heterogeneity in the patterns of connection within each level and introducing dependencies between the levels. The sought connection patterns are not specified a priori which makes this approach flexible. Variational methods are used for the model inference and an Integrated Classified Likelihood criterion is developed for choosing the number of blocks and also for deciding whether the two levels are dependent or not. A comprehensive simulation study exhibits the benefit of considering this approach, illustrates the robustness of the clustering and highlights the reliability of the criterion used for model selection. This approach is applied on a sociological dataset collected during a television program trade fair, the inter-organizational level being the economic network between companies and the inter-individual level being the informal network between their representatives. It brings a synthetic representation of the two networks unraveling their intertwined structure and confirms the coopetition at stake.

This work is motivated by the analysis of ecological interaction networks. Poisson stochastic blockmodels are widely used in this field to decipher the structure that underlies a weighted network, while accounting for covariate effects. Efficient algorithms based on variational approximations exist for frequentist inference, but without statistical guaranties as for the resulting estimates. In absence of variational Bayes estimates, we show that a good proxy of the posterior distribution can be straightforwardly derived from the frequentist variational estimation procedure, using a Laplace approximation. We use this proxy to sample from the true posterior distribution via a sequential Monte-Carlo algorithm. As shown in the simulation study, the efficiency of the posterior sampling is greatly improved by the accuracy of the approximate posterior distribution. The proposed procedure can be easily extended to other latent variable models. We use this methodology to assess the influence of available covariates on the organization of two ecological networks, as well as the existence of a residual interaction structure.

Modeling relations between individuals is a classical question in social sciences, ecology, etc. In order to uncover a latent structure in the data, a popular approach consists in clustering individuals according to the observed patterns of interactions. To do so, Stochastic block models (SBM) and Latent Block models (LBM) are standard tools for clustering the individuals with respect to their comportment in a unique network. However, when adopting an integrative point of view, individuals are not involved in a unique network but are part of several networks, resulting into a potentially complex multipartite network. In this contribution, we propose a stochastic block model able to handle multipartite networks, thus supplying a clustering of the individuals based on their connection behavior in more than one network. Our model is an extension of the latent block models (LBM) and stochastic block model (SBM). The parameters -- such as the marginal probabilities of assignment to blocks and the matrix of probabilities of connections between blocks -- are estimated through a variational Expectation-Maximization procedure. The numbers of blocks are chosen with the Integrated Completed Likelihood criterion, a penalized likelihood criterion. The pertinence of our methodology is illustrated on two datasets issued from ecology and ethnobiology.

This paper studies nonparametric estimation of parameters of multivariate Hawkes processes. We consider the Bayesian setting and derive posterior concentration rates. First rates are derived for L1-metrics for stochastic intensities of the Hawkes process. We then deduce rates for the L1-norm of interactions functions of the process. Our results are exemplified by using priors based on piecewise constant functions, with regular or random partitions and priors based on mixtures of Betas distributions. Numerical illustrations are then proposed with in mind applications for inferring functional connec-tivity graphs of neurons.

Sequential Monte Carlo has become a standard tool for Bayesian Inference of complex models. This approach can be computationally demanding, especially when initialized from the prior distribution. On the other hand, deter-ministic approximations of the posterior distribution are often available with no theoretical guaranties. We propose a bridge sampling scheme starting from such a deterministic approximation of the posterior distribution and targeting the true one. The resulting Shortened Bridge Sampler (SBS) relies on a sequence of distributions that is determined in an adaptive way. We illustrate the robustness and the efficiency of the methodology on a large simulation study. When applied to network datasets, SBS inference leads to different statistical conclusions from the one supplied by the standard variational Bayes approximation.

Modeling relations between individuals is a classical question in social sciences and clustering individuals according to the observed patterns of interactions allows to uncover a latent structure in the data. Stochastic block model (SBM) is a popular approach for grouping the individuals with respect to their social comportment. When several relationships of various types can occur jointly between the individuals, the data are represented by multiplex networks where more than one edge can exist between the nodes. In this paper, we extend the SBM to multiplex networks in order to obtain a clustering based on more than one kind of relationship. We propose to estimate the parameters --such as the marginal probabilities of assignment to groups (blocks) and the matrix of probabilities of connections between groups-- through a variational Expectation-Maximization procedure. Consistency of the estimates as well as statistical properties of the model are obtained. The number of groups is chosen thanks to the Integrated Completed Likelihood criteria, a penalized likelihood criterion. Multiplex Stochastic Block Model arises in many situations but our applied example is motivated by a network of French cancer researchers. The two possible links (edges) between researchers are a direct connection or a connection through their labs. Our results show strong interactions between these two kinds of connections and the groups that are obtained are discussed to emphasize the common features of researchers grouped together.

We provide general conditions to derive posterior concentration rates for Aalen counting processes. The conditions are designed to resemble those proposed in the literature for the problem of density estimation, for instance in Ghosal et al. (2000), so that existing results on density estimation can be adapted to the present setting. We apply the general theorem to some prior models including Dirichlet process mixtures of uniform densities to estimate monotone non-increasing intensities and log-splines., Comment: 26 pages

In this paper we provide general conditions to check on the model and the prior to derive posterior concentration rates for data-dependent priors (or empirical Bayes approaches). We aim at providing conditions that are close to the conditions provided in the seminal paper by Ghosal and van der Vaart (2007a). We then apply the general theorem to two different settings: the estimation of a density using Dirichlet process mixtures of Gaussian random variables with base measure depending on some empirical quantities and the estimation of the intensity of a counting process under the Aalen model. A simulation study for inhomogeneous Poisson processes also illustrates our results. In the former case we also derive some results on the estimation of the mixing density and on the deconvolution problem. In the latter, we provide a general theorem on posterior concentration rates for counting processes with Aalen multiplicative intensity with priors not depending on the data., Comment: With supplementary material

In most relevant cases in the Bayesian analysis of ODE inverse problems, a numerical solver needs to be used. Therefore, we cannot work with the exact theoretical posterior distribution but only with an approximate posterior deriving from the error in the numerical solver. To compare a numerical and the theoretical posterior distributions we propose to use Bayes Factors (BF), considering both of them as models for the data at hand. We prove that the theoretical vs a numerical posterior BF tends to 1, in the same order (of the step size used) as the numerical forward map solver does. For higher order solvers (eg. Runge-Kutta) the Bayes Factor is already nearly 1 for step sizes that would take far less computational effort. Considerable CPU time may be saved by using coarser solvers that nevertheless produce practically error free posteriors. Two examples are presented where nearly 90% CPU time is saved while all inference results are identical to using a solver with a much finer time step., Comment: 28 pages, 6 figures

A new methodology for model determination in decomposable graphical Gaussian models is developed. The Bayesian paradigm is used and, for each given graph, a hyper inverse Wishart prior distribution on the covariance matrix is considered. This prior distribution depends on hyper-parameters. It is well-known that the models's posterior distribution is sensitive to the specification of these hyper-parameters and no completely satisfactory method is registered. In order to avoid this problem, we suggest adopting an empirical Bayes strategy, that is a strategy for which the values of the hyper-parameters are determined using the data. Typically, the hyper-parameters are fixed to their maximum likelihood estimations. In order to calculate these maximum likelihood estimations, we suggest a Markov chain Monte Carlo version of the Stochastic Approximation EM algorithm. Moreover, we introduce a new sampling scheme in the space of graphs that improves the add and delete proposal of Armstrong et al. (2009). We illustrate the efficiency of this new scheme on simulated and real datasets.