Section: Overall Objectives

Overall Objectives

The scientific objectives of ASPI are the design, analysis and implementation of interacting Monte Carlo methods, also known as particle methods, with focus on

• statistical inference in hidden Markov models and particle filtering,

• risk evaluation and simulation of rare events,

• global optimization.

The whole problematic is multidisciplinary, not only because of the many scientific and engineering areas in which particle methods are used, but also because of the diversity of the scientific communities which have already contributed to establish the foundations of the field

target tracking, interacting particle systems, empirical processes, genetic algorithms (GA), hidden Markov models and nonlinear filtering, Bayesian statistics, Markov chain Monte Carlo (MCMC) methods, etc.

Intuitively speaking, interacting Monte Carlo methods are sequential simulation methods, in which particles

• explore the state space by mimicking the evolution of an underlying random process,

• learn the environment by evaluating a fitness function,

• and interact so that only the most successful particles (in view of the value of the fitness function) are allowed to survive and to get offsprings at the next generation.

The effect of this mutation / selection mechanism is to automatically concentrate particles (i.e. the available computing power) in regions of interest of the state space. In the special case of particle filtering, which has numerous applications under the generic heading of positioning, navigation and tracking, in

target tracking, computer vision, mobile robotics, ubiquitous computing and ambient intelligence, sensor networks, etc.,

each particle represents a possible hidden state, and is multiplied or terminated at the next generation on the basis of its consistency with the current observation, as quantified by the likelihood function. With these genetic–type algorithms, it becomes easy to efficiently combine a prior model of displacement with or without constraints, sensor–based measurements, and a base of reference measurements, for example in the form of a digital map (digital elevation map, attenuation map, etc.). In the most general case, particle methods provide approximations of Feynman–Kac distributions, a pathwise generalization of Gibbs–Boltzmann distributions, by means of the weighted empirical probability distribution associated with an interacting particle system, with applications that go far beyond filtering, in

simulation of rare events, simulation of conditioned or constrained random variables, interacting MCMC methods, molecular simulation, etc.

The main applications currently considered are geolocalisation and tracking of mobile terminals, terrain–aided navigation, data fusion for indoor localisation, detection in sensor networks, risk assessment in air traffic management, protection of digital documents, and credit risk estimation.

Our second research area concerns nearest neighbor estimation. This set of methods belongs to the class of supervised statistical learning algorithms. They aim at predicting some feature of a new object from a learning N –sample in the form of already known object–feature pairs. There are many different approches to this problem, and we decided to focus on the nearest neighbor approach, which only considers the k nearest objects of the new one, with k smaller than N , and for some given metric. Then we predict the unkown feature value by a mean on the features of the neighbors in the case of regression (continuous case), or a majority vote in the case of classification (discrete case). Despite its simplicity, this approach works very well in practice and has led to a large amount of both applied and theoretical literature. Nevertheless, there are still open questions, especially about the convergence of the method when the objects take values in an infinite dimensional space, or in general about the rate of convergence in different settings, or for different choices of k .