## Section: New Results

### Managing the system (via probabilistic modeling)

#### Belief propagation inference with a prescribed fixed point

Participants : Cyril Furtlehner, Jean-Marc Lasgouttes.

In the context of inference with expectation constraints, we propose an approach based on the “belief propagation” algorithm (BP), as a surrogate to an exact Markov Random Field (MRF) modelling [12] . A prior information composed of correlations among a large set of N variables, is encoded into a graphical model; this encoding is optimized with respect to an approximate decoding procedure (BP), which is used to infer hidden variables from an observed subset. We focus on the situation where the underlying data have many different statistical components, representing a variety of independent patterns. Considering a single parameter family of models we show how BP may be used to encode and decode efficiently such information, without solving the NP hard inverse problem yielding the optimal MRF. Contrary to usual practice, we work in the non-convex Bethe free energy minimization framework, and manage to associate a belief propagation fixed point to each component of the underlying probabilistic mixture. The mean field limit is considered and yields an exact connection with the Hopfield model at finite temperature and steady state, when the number of mixture components is proportional to the number of variables.

In addition, we provide an enhanced learning procedure, based on a straightforward multi-parameter extension of the model in conjunction with an effective continuous optimization procedure. This is performed using the stochastic search heuristic CMAES and yields a significant improvement with respect to the single parameter basic model (joint work with Anne Auger, project-team TAO).

#### Effect of normalization in Belief propagation algorithm

Participants : Oumar Baba Diakhate, Cyril Furtlehner, Jean-Marc Lasgouttes.

We establish some general properties of BP are established concerning the effect of normalizing the messages, the relation between fixed points and their stability. In particular, we shed light on the respective effects of the factor graph topology through its spectrum on one end, and the effects of the encoded data by means of the spectral properties of a set of stochastic matrices attached to the data on the other end.

This is a work in progress. This year the focus has been set on extensions to the classical BP algorithm.

#### Belief propagation and Bethe approximation for traffic prediction

Participants : Cyril Furtlehner, Arnaud de La Fortelle, Jean-Marc Lasgouttes.

This work [64] deals with real-time prediction
of traffic conditions in a setting where the only available
information is floating car data (FCD) sent by probe vehicles. The
main focus is on finding a good way to encode some coarse information
(typically whether traffic on a segment is fluid or congested), and to
decode it in the form of real-time traffic reconstruction and
prediction. Starting from the Ising model of statistical physics, we
use a discretized space-time traffic description, on which we define
and study an inference method based on the belief propagation (BP)
algorithm. We propose a hybrid approach, by taking full advantage of
the statistical nature of the information, in combination with a
stochastic modeling of traffic patterns and a powerful message-passing
inference algorithm. The idea is to encode into a graph the *a
priori* information derived from historical data (marginal
probabilities of pairs of variables), and to use BP to estimate the
actual state from the latest FCD. Originally designed for Bayesian
inference on tree-like graphs, the BP algorithm has been widely used
in a variety of inference problems (e.g. computer vision, coding
theory, etc.), but to our knowledge it has not yet been applied in the
context of traffic prediction.

These studies are done in particular in the framework of the ANR project TRAVESTI (see 7.15 ), and of the upcoming FUI Pumas.

#### Multi-speed exclusion processes

Participants : Cyril Furtlehner, Jean-Marc Lasgouttes.

We have considered in [28] a one-dimensional stochastic reaction-diffusion generalizing the totally asymmetric simple exclusion process, and aiming at describing single lane roads with vehicles that can change speed. To each particle is associated a jump rate, and the particular dynamics that we choose (based on 3-sites patterns) ensures that clusters of occupied sites are of uniform jump rate. The basic assumption is that if a car gets in close contact to another one, it will adopt its rate. Conversely, if it arrives at a site not in contact with any other car, the new rate will be freely determined according to some random distribution. This models the acceleration or deceleration process in an admittedly crude manner. When this model is set on a circle or an infinite line, classical arguments allow to map it to a linear network of queues (a zero-range process in theoretical physics parlance) with exponential service times, but with a twist: the service rate remains constant during a busy period, but can change at renewal events.

This work has been continued this year, specifically by introducing a new type of zero-range processes with specific Markov non-reversibility and applying it to the computation of the fundamental diagram of road traffic.

#### Traffic light prediction

Participants : Samer Ammoun, Jean-Marc Lasgouttes, Diogo De Souza Dutra.

As part of the Intersafe-2 project (see 7.11 ), we are working on an algorithm for automatic prediction of the next switching time for traffic light signals on intersections. The actual traffic light controller is considered as a black box, and the algorithm is to be embedded into the traffic light hardware, which implies important restrictions in terms of speed and memory. The method we devised consists in an interpretation of traffic light state in terms of phases and programs, on which a Markov-chain-based model is built. This model automatically recognizes the parts of the behaviour which is deterministic and builds prediction tables for the rest.

#### Markov growing trees

Participants : Guy Fayolle, Arnaud de La Fortelle.

We analyze the height process of so-called *Markov growing
trees* . New edges appear according to a Poisson process of parameter
and leaves can be deleted at a rate . The basic model
was introduced in [63] . In the pure birth case (i.e. = 0 ,
the distribution function of the height of the tree at time t does
satisfy an interesting recursive nonlinear functional equation, which
is studied mainly from an analytic point of view. The results obtained
so far let appear interesting scalings, which apparently also hold
for more general operators, leading thus to a kind of *invariance
principle* . The next step will be to construct stationary laws from
the transient process of interest.

#### Dynamical windings of random walks and exclusion models

Participant : Guy Fayolle.

These last four years, several studies have been achieved about random walks evolving in the plane or even in and subjected to various local stochastic distortions (see activity reports of the Preval team 2004, 2005 and 2006).

In keeping with this general pattern, we pursued the work contained in reference [62] . The goal is to derive continuous limits of interacting one-dimensional diffusive systems, arising from stochastic distortions of discrete curves and involving various kinds of coding representations. These systems are essentially of a reaction-diffusion nature. In the non-reversible case, the invariant measure has generally a non Gibbs form. The corresponding steady-state regime is analyzed in detail with the help of a tagged particle and a state-graph cycle expansion of the probability currents. As a consequence, the constants appearing in Lotka-Volterra equations —which describe the fluid limits of stationary states— can be traced back directly at the discrete level to tagged particles cycles coefficients. Current fluctuations are also studied and the Lagrangian is obtained by an iterative scheme. The related Hamilton-Jacobi equation, which leads to the large deviation functional, is analyzed and solved in the reversible case, just for the sake of checking.

#### Statistical physics and hydrodynamic limits

Participants : Guy Fayolle, Cyril Furtlehner.

Having in mind a global project concerning the analysis of complex systems, we first focus on the interplay between discrete and continuous description: in some cases, this recurrent question can be addressed quite rigorously via probabilistic methods.

To attack this class of problems, in touch with many applications
domains (e.g. biology, telecommunications, transportation systems), we
started from *paradigmatic* elements, namely the discrete curves
subjected to stochastic deformations, as those mentioned in section
6.2.7 .

After convenient mappings, it appears that most problems can be set in terms of interacting exclusion processes, the ultimate goal being to derive hydrodynamic limits for these systems after proper scalings. We extend the key ideas of [61] , where the basic ASEP system on the torus was analyzed. The usual sequence of empirical measures, converges in probability to a deterministic measure, which is the unique weak solution of a Cauchy problem.

The Gordian knot is the analysis of a family of differential operators
in infinite dimension. Indeed, the values of functions at given points
play here the role of usual variables, their number becoming infinite.
The method presents some new theoretical features, involving
promeasures (as introduced by Bourbaki), variational calculus and
functional integration. In the ongoing work [50] , these
arguments are applied to various multi-type exclusion systems,
including the famous ABC model. Also, in the course of the study,
several fascinating multi-scale problems emerge quite naturally,
bringing to light quite natural connections with the so-called
*renormalization* in theoretical physics.

#### Convergence of moments in the almost sure central Limit theorem for multivariate martingales

Participant : Guy Fayolle.

Let (_{n}) be a sequence of i.i.d. random variables, with
and .
Let .

The almost sure central limit theorem (ASCLT) asserts that, for any bounded continuous function h ,

where G is a Gaussian measure . This theorem also holds for martingales.

In a joint work [10] , which started in 2006 in collaboration with Bernard Bercu (University Bordeaux 1) and Peggy Cénac (University Dijon), we investigate the almost sure asymptotic properties of vector martingale transforms. Assuming some appropriate regularity conditions both on the increasing process and on the moments of the martingale, we prove that normalized moments of any even order converge in the almost sure central limit theorem for martingales. A conjecture about almost sure upper bounds under wider hypotheses is formulated. The theoretical results are supported by examples borrowed from statistical applications, including linear autoregressive models and branching processes with immigration, for which new asymptotic properties are established on estimation and prediction errors.

#### Evaluation of collective taxi systems by event-driven simulation

Participants : Jennie Lioris, Arnaud de La Fortelle.

We completed a made-to-measure simulation tool to study and evaluate the performance of a “collective taxis” transportation mode covering an entire urban area. In addition, we proceeded with the development and implementation of a technique to treat and analyse the results provided by the simulator.

Typically, we can employ statistics to provide quantitative answers qualifying the system performance and its reliability according to various models and decisional policies. We also started to study the decentralised approach, where clients appear randomly on the network, wishing to leave as soon as possible with no advanced reservations.

After initiating a great number of simulations on multiple fictitious inputs and various strategies, by analysing and comparing the corresponding results, we were then able to propose a methodology to perform the optimal management of each agent in the respective operating mode. Once that technique is obtained, a real application of the problem can be studied and analysed by following the concepts of the proposed methodology.