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, wireless communications, 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, optimization of sensors location and activation, risk assessment in air traffic management, protection of digital documents.

Monte Carlo methods are numerical methods that are widely
used in situations where (i) a stochastic (usually
Markovian) model is given for some underlying process, and
(ii) some quantity of interest should be evaluated, that
can be expressed in terms of the expected value of a
functional of the process trajectory, which includes as an
important special case the probability that a given event has
occurred. Numerous examples can be found, e.g. in financial
engineering (pricing of options and derivative
securities)
, in performance evaluation of
communication networks (probability of buffer overflow), in
statistics of hidden Markov models (state estimation,
evaluation of contrast and score functions), etc. Very often
in practice, no analytical expression is available for the
quantity of interest, but it is possible to simulate
trajectories of the underlying process. The idea behind Monte
Carlo methods is to generate independent trajectories of this
process or of an alternate instrumental process, and to build
an approximation (estimator) of the quantity of interest in
terms of the weighted empirical probability distribution
associated with the resulting independent sample. By the law
of large numbers, the above estimator converges as the size
Nof the sample goes to infinity, with rate
and the asymptotic variance can be estimated using an
appropriate central limit theorem. To reduce the variance of
the estimator, many variance reduction techniques have been
proposed. Still, running independent Monte Carlo simulations
can lead to very poor results, because trajectories are
generated
*blindly*, and only afterwards are the corresponding
weights evaluated. Some of the weights can happen to be
negligible, in which case the corresponding trajectories are
not going to contribute to the estimator, i.e. computing
power has been wasted.

A recent and major breakthrough, has been the introduction
of interacting Monte Carlo methods, also known as sequential
Monte Carlo (SMC) methods, in which a whole (possibly
weighted) sample, called
*system of particles*, is propagated in time, where the
particles

*explore*the state space under the effect of a
*mutation*mechanism which mimics the evolution of
the underlying process,

and are
*replicated*or
*terminated*, under the effect of a
*selection*mechanism which automatically
concentrates the particles, i.e. the available computing
power, into regions of interest of the state space.

In full generality, the underlying process is a discrete–time Markov chain, whose state space can be

finite, continuous, hybrid (continuous / discrete), graphical, constrained, time varying, pathwise, etc.,

the only condition being that it can easily be
*simulated*. The very important case of a sampled
continuous–time Markov process, e.g. the solution of a
stochastic differential equation driven by a Wiener process
or a more general Lévy process, is also covered.

In the special case of particle filtering, originally
developed within the tracking community, the algorithms yield
a numerical approximation of the optimal Bayesian filter,
i.e. of the conditional probability distribution of the
hidden state given the past observations, as a (possibly
weighted) empirical probability distribution of the system of
particles. In its simplest version, introduced in several
different scientific communities under the name of
*bootstrap filter*
,
*Monte Carlo filter*
or
*condensation*(conditional density propagation)
algorithm
, and which historically has been
the first algorithm to include a redistribution step, the
selection mechanism is governed by the likelihood function:
at each time step, a particle is more likely to survive and
to replicate at the next generation if it is consistent with
the current observation. The algorithms also provide as a
by–product a numerical approximation of the likelihood
function, and of many other contrast functions for parameter
estimation in hidden Markov models, such as the prediction
error or the conditional least–squares criterion.

Particle methods are currently being used in many scientific and engineering areas

positioning, navigation, and tracking , , visual tracking , mobile robotics , , ubiquitous computing and ambient intelligence, sensor networks, risk evaluation and simulation of rare events , genetics, molecular simulation , etc.

Other examples of the many applications of particle
filtering can be found in the contributed volume
and in the special issue of
*IEEE Transactions on Signal Processing*devoted to
*Monte Carlo Methods for Statistical Signal
Processing*in February 2002, where the tutorial
paper
can be found, and in the
textbook
devoted to applications in target
tracking. Applications of sequential Monte Carlo methods to
other areas, beyond signal and image processing, e.g. to
genetics, can be found in
.

Particle methods are very easy to implement, since it is sufficient in principle to simulate independent trajectories of the underlying process. The whole problematic is multidisciplinary, not only because of the already mentioned diversity of the scientific and engineering areas in which particle methods are used, but also because of the diversity of the scientific communities which have 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.

These algorithms can be interpreted as numerical
approximation schemes for Feynman–Kac distributions, a
pathwise generalization of Gibbs–Boltzmann distributions, in
terms of the weighted empirical probability distribution
associated with a system of particles. This abstract point of
view
,
, has proved to be extremely
fruitful in providing a very general framework to the design
and analysis of numerical approximation schemes, based on
systems of branching and / or interacting particles, for
nonlinear dynamical systems with values in the space of
probability distributions, associated with Feynman–Kac
distributions. Many asymptotic results have been proved as
the number
Nof particles (sample size) goes to infinity, using
techniques coming from applied probability (interacting
particle systems, empirical processes
), see e.g. the survey
article
or the recent textbook
, and references therein

convergence in
L^{p}, convergence as empirical processes indexed by
classes of functions, uniform convergence in time, see
also
,
, central limit theorem, see
also
, propagation of chaos, large
deviations principle, etc.

The objective here is to systematically study the impact of the many algorithmic variants on the convergence results.

Hidden Markov models (HMM) form a special case of partially observed stochastic dynamical systems, in which the state of a Markov process (in discrete or continuous time, with finite or continuous state space) should be estimated from noisy observations. The conditional probability distribution of the hidden state given past observations is a well–known example of a normalized (nonlinear) Feynman–Kac distribution, see . These models are very flexible, because of the introduction of latent variables (non observed) which allows to model complex time dependent structures, to take constraints into account, etc. In addition, the underlying Markovian structure makes it possible to use numerical algorithms (particle filtering, Markov chain Monte Carlo methods (MCMC), etc.) which are computationally intensive but whose complexity is rather small. Hidden Markov models are widely used in various applied areas, such as speech recognition, alignment of biological sequences, tracking in complex environment, modeling and control of networks, digital communications, etc.

Beyond the recursive estimation of a hidden state from noisy observations, the problem arises of statistical inference of HMM with general state space , including estimation of model parameters, early monitoring and diagnosis of small changes in model parameters, etc.

**Large time asymptotics** A fruitful
approach is the asymptotic study, when the observation time
increases to infinity, of an extended Markov chain, whose
state includes (i) the hidden state, (ii) the
observation, (iii) the prediction filter (i.e. the
conditional probability distribution of the hidden state
given observations at all previous time instants), and
possibly (iv) the derivative of the prediction filter
with respect to the parameter. Indeed, it is easy to express
the log–likelihood function, the conditional least–squares
criterion, and many other clasical contrast processes, as
well as their derivatives with respect to the parameter, as
additive functionals of the extended Markov chain.

The following general approach has been proposed

first, prove an exponential stability property (i.e. an exponential forgetting property of the initial condition) of the prediction filter and its derivative, for a misspecified model,

from this, deduce a geometric ergodicity property and the existence of a unique invariant probability distribution for the extended Markov chain, hence a law of large numbers and a central limit theorem for a large class of contrast processes and their derivatives, and a local asymptotic normality property,

finally, obtain the consistency (i.e. the convergence to the set of minima of the associated contrast function), and the asymptotic normality of a large class of minimum contrast estimators.

This programme has been completed in the case of a finite state space , and has been generalized under an uniform minoration assumption for the Markov transition kernel, which typically does only hold when the state space is compact. Clearly, the whole approach relies on the existence of an exponential stability property of the prediction filter, and the main challenge currently is to get rid of this uniform minoration assumption for the Markov transition kernel , , so as to be able to consider more interesting situations, where the state space is noncompact.

**Small noise asymptotics** Another
asymptotic approach can also be used, where it is rather easy
to obtain interesting explicit results, in terms close to the
language of nonlinear deterministic control theory
. Taking the simple example where
the hidden state is the solution to an ordinary differential
equation, or a nonlinear state model, and where the
observations are subject to additive Gaussian white noise,
this approach consists in assuming that covariances matrices
of the state noise and of the observation noise go
simultaneously to zero. If it is reasonable in many
applications to consider that noise covariances are small,
this asymptotic approach is less natural than the large time
asymptotics, where it is enough (provided a suitable
ergodicity assumption holds) to accumulate observations and
to see the expected limit laws (law of large numbers, central
limit theorem, etc.). In opposition, the expressions obtained
in the limit (Kullback–Leibler divergence, Fisher information
matrix, asymptotic covariance matrix, etc.) take here a much
more explicit form than in the large time asymptotics.

The following results have been obtained using this approach

the consistency of the maximum
likelihood estimator (i.e. the convergence to the set
Mof global minima of the Kullback–Leibler
divergence), has been obtained using large deviations
techniques, with an analytical approach
,

if the abovementioned set
Mdoes not reduce to the true parameter value, i.e.
if the model is not identifiable, it is still possible to
describe precisely the asymptotic behavior of the
estimators
: in the simple case where
the state equation is a noise–free ordinary differential
equation and using a Bayesian framework, it has been
shown that (i) if the rank
rof the Fisher information matrix
Iis constant in a neighborhood of the set
M, then this set is a differentiable submanifold of
codimension
r, (ii) the posterior probability distribution
of the parameter converges to a random probability
distribution in the limit, supported by the manifold
M, absolutely continuous w.r.t. the Lebesgue
measure on
M, with an explicit expression for the density, and
(iii) the posterior probability distribution of the
suitably normalized difference between the parameter and
its projection on the manifold
M, converges to a mixture of Gaussian probability
distributions on the normal spaces to the manifold
M, which generalized the usual asymptotic normality
property,

it has been shown
that (i) the parameter
dependent probability distributions of the observations
are locally asymptotically normal (LAN)
, from which the asymptotic
normality of the maximum likelihood estimator follows,
with an explicit expression for the asymptotic covariance
matrix, i.e. for the Fisher information matrix
I, in terms of the Kalman filter associated with
the linear tangent linear Gaussian model, and
(ii) the score function (i.e. the derivative of the
log–likelihood function w.r.t. the parameter), evaluated
at the true value of the parameter and suitably
normalized, converges to a Gaussian r.v. with zero mean
and covariance matrix
I.

The estimation of the small probability of a rare but critical event, is a crucial issue in industrial areas such as

nuclear power plants, food industry, telecommunication networks, finance and insurance industry, air traffic management, etc.

In such complex systems, analytical methods cannot be used, and naive Monte Carlo methods are clearly unefficient to estimate accurately very small probabilities. Besides importance sampling, an alternate widespread technique consists in multilevel splitting , where trajectories going towards the critical set are given offsprings, thus increasing the number of trajectories that eventually reach the critical set. As shown in , the Feynman–Kac formalism of is well suited for the design and analysis of splitting algorithms for rare event simulation.

**Propagation of uncertainty** Multilevel
splitting can be used in static situations. Here, the
objective is to learn the probability distribution of an
output random variable
Y=
F(
X), where the function
Fis only defined pointwise for instance by a computer
programme, and where the probability distribution of the
input random variable
Xis known and easy to simulate from. More specifically,
the objective could be to compute the probability of the
output random variable exceeding a threshold, or more
generally to evaluate the cumulative distribution function of
the output random variable for different output values. This
problem is characterized by the lack of an analytical
expression for the function, the computational cost of a
single pointwise evaluation of the function, which means that
the number of calls to the function should be limited as much
as possible, and finally the complexity and / or
unavailability of the source code of the computer programme,
which makes any modification very difficult or even
impossible, for instance to change the model as in importance
sampling methods.

The key issue is to learn as fast as possible regions of the input space which contribute most to the computation of the target quantity. The proposed splitting methods consists in (i) introducing a sequence of intermediate regions in the input space, implicitly defined by exceeding an increasing sequence of thresholds or levels, (ii) counting the fraction of samples that reach a level given that the previous level has been reached already, and (iii) improving the diversity of the selected samples, usually using an artificial Markovian dynamics. In this way, the algorithm learns

the transition probability between successive levels, hence the probability of reaching each intermediate level,

and the probability distribution of the input random variable, conditionned on the output variable reaching each intermediate level.

A further remark, is that this conditional probability distribution is precisely the optimal (zero variance) importance distribution needed to compute the probability of reaching the considered intermediate level.

**Rare event simulation** To be specific,
consider a complex dynamical system modelled as a Markov
process, whose state can possibly contain continuous
components and finite components (mode, regime, etc.), and
the objective is to compute the probability, hopefully very
small, that a critical region of the state space is reached
by the Markov process before a final time
T, which can be deterministic and fixed, or random (for
instance the time of return to a recurrent set, corresponding
to a nominal behaviour).

The proposed splitting method consists in
(i) introducing a decreasing sequence of intermediate,
more and more critical, regions in the state space,
(ii) counting the fraction of trajectories that reach an
intermediate region before time
T, given that the previous intermediate region has been
reached before time
T, and (iii) regenerating the population at each
stage, through redistribution. In addition to the
non–intrusive behaviour of the method, the splitting methods
make it possible to learn the probability distribution of
typical critical trajectories, which reach the critical
region before final time
T, an important feature that methods based on
importance sampling usually miss. Many variants have been
proposed, whether

the branching rate (number of offsprings allocated to a successful trajectory) is fixed, which allows for depth–first exploration of the branching tree, but raises the issue of controlling the population size,

the population size is fixed, which requires a breadth–first exploration of the branching tree, with random (multinomial) or deterministic allocation of offsprings, etc.

Just as in the static case, the algorithm learns

the transition probability between successive levels, hence the probability of reaching each intermediate level,

and the entrance probability distribution of the Markov process in each intermediate region.

Contributions have been given to

minimizing the asymptotic variance, obtained through a central limit theorem, with respect to the shape of the intermediate regions (selection of the importance function), to the thresholds (levels), to the population size, etc.

controlling the probability of extinction (when not even one trajectory reaches the next intermediate level),

designing and studying variants suited for hybrid state space (resampling per mode, marginalization, mode aggregation),

and in the static case, to

minimizing the asymptotic variance, obtained through a central limit theorem, with respect to intermediate levels, to the Metropolis kernel introduced in the mutation step, etc.

A related issue is global optimization. Indeed, the
difficult problem of finding the set
Mof global minima of a real–valued function
Vcan be replaced by the apparently simpler problem of
sampling a population from a probability distribution
depending on a small parameter, and asymptotically supported
by the set
Mas the small parameter goes to zero. The usual
approach here is to use the cross–entropy method
,
, which relies on learning the
optimal importance distribution within a prescribed
parametric family. On the other hand, multilevel splitting
methods could provide an alternate nonparametric approach to
this problem.

This additional topic was not present in the initial list of objectives, and has emerged only recently.

In pattern recognition and statistical learning, also
known as macvhine learning, nearest neighbor (NN) algorithms
are amongst the simplest but also very powerful algorithms
available. Basically, given a training set of data, i.e. an
N–sample of i.i.d. object–feature pairs, with
real–valued features, the question is how to generalize, that
is how to guess the feature associated with any new object.
To achieve this, one chooses some integer
ksmaller than
N, and takes the mean–value of the
kfeatures associated with the
kobjects that are nearest to the new object, for some
given metric.

In general, there is no way to guess exactly the value of
the feature associated with the new object, and the minimal
error that can be done is that of the Bayes estimator, which
cannot be computed by lack of knowledge of the distribution
of the object–feature pair, but the Bayes estimator can be
useful to characterize the strength of the method. So the
best that can be expected is that the NN estimator converges,
say when the sample size
Ngrows, to the Bayes estimator. This is what has been
proved in great generality by Stone
for the mean square convergence,
provided that the object is a finite–dimensional random
variable, the feature is a square–integrable random variable,
and the ratio
k/
Ngoes to 0. Nearest neighbor
estimator is not the only local averaging estimator with this
property, but it is arguably the simplest.

The asymptotic behavior when the sample size grows is well understood in finite dimension, but the situation is radically different in general infinite dimensional spaces, when the objects to be classified are functions, images, etc.

**Nearest neighbor classification in infinite
dimension** In finite dimension, the
k–nearest neighbor classifier is universally
consistent, i.e. its probability of error converges to the
Bayes risk as
Ngoes to infinity, whatever the joint probability
distribution of the pair, provided that the ratio
k/
Ngoes to zero. Unfortunately, this
result is no longer valid in general metric spaces, and the
objective is to find out reasonable sufficient conditions for
the weak consistency to hold. Even in finite dimension, there
are exotic distances such that the nearest neighbor does not
even get closer (in the sense of the distance) to the point
of interest, and the state space needs to be complete for the
metric, which is the first condition. Some regularity on the
regression function is required next. Clearly, continuity is
too strong because it is not required in finite dimension,
and a weaker form of regularity is assumed. The following
consistency result has been obtained: if the metric space is
separable and if some Besicovich condition holds, then the
nearest neighbor classifier is weakly consistent. Note that
the Besicovich condition is always fulfilled in finite
dimensional vector spaces (this result is called the
Besicovich theorem), and that a counterexample
can be given in an infinite
dimensional space with a Gaussian measure (in this case, the
nearest neighbor classifier is clearly nonconsistent).
Finally, a simple example has been found which verifies the
Besicovich condition with a noncontinuous regression
function.

**Rates of convergence of the functional
k–nearest neighbor
estimator** Motivated by a broad range of
potential applications, such as regression on curves, rates
of convergence of the

This emerging topic has produced several theoretical advances in collaboration with Gérard Biau (université Pierre et Marie Curie, ENS Paris and EPI CLASSIC, INRIA Paris—Rocquencourt), and a possible target application domain has been identified in the statistical analysis of recommendation systems, that would be a source of interesting problems.

Among the many application domains of particle methods, or interacting Monte Carlo methods, ASPI has decided to focus on applications in localisation (or positioning), navigation and tracking , , which already covers a very broad spectrum of application domains. The objective here is to estimate the position (and also velocity, attitude, etc.) of a mobile object, from the combination of different sources of information, including

a prior dynamical model of typical evolutions of the mobile, such as inertial estimates and prior model for inertial errors,

measurements provided by sensors,

and possibly a digital map providing some useful feature (terrain altitude, power attenuation, etc.) at each possible position.

In some applications, another useful source of information is provided by

a map of constrained admissible displacements, for instance in the form of an indoor building map,

which particle methods can easily handle (map-matching). This Bayesian dynamical estimation problem is also called filtering, and its numerical implementation using particle methods, known as particle filtering, has been introduced by the target tracking community , , which has already contributed to many of the most interesting algorithmic improvements and is still very active, and has found applications in

target tracking, integrated navigation, points and / or objects tracking in video sequences, mobile robotics, wireless communications, ubiquitous computing and ambient intelligence, sensor networks, etc.

ASPI is contributing to several applications of particle filtering in positioning, navigation and tracking, such as geolocalisation and tracking in a wireless network, terrain–aided navigation, see , and data fusion for indoor localisation, see .

Another application domain of particle methods, or interacting Monte Carlo methods, that ASPI has decided to focus on is the estimation of the small probability of a rare but critical event, in complex dynamical systems. This is a crucial issue in industrial areas such as

nuclear power plants, food industry, telecommunication networks, finance and insurance industry, air traffic management, etc.

In such complex systems, analytical methods cannot be used, and naive Monte Carlo methods are clearly unefficient to estimate accurately very small probabilities. Besides importance sampling, an alternate widespread technique consists in multilevel splitting , where trajectories going towards the critical set are given offsprings, thus increasing the number of trajectories that eventually reach the critical set. This approach not only makes it possible to estimate the probability of the rare event, but also provides realizations of the random trajectory, given that it reaches the critical set, i.e. provides realizations of typical critical trajectories, an important feature that methods based on importance sampling usually miss.

ASPI is contributing to several applications of multilevel splitting for rare event simulation, such as risk assessment in air traffic management, see , detection in sensor networks, see , and protection of digital documents, see .

This is a collaboration with Tony Lelièvre and David Pommier (CERMICS, Ecole des Ponts ParisTech).

The numerical simulation of molecular dynamics is very important to predict the behavior of complex molecules. It is usually modeled by (overdamped) Langevin dynamics, which is a stochastic Markovian model (either a diffusion process or its integral in time). Within this framework, it is mandatory to speed up the simulation between two metastable regions (i.e. two local minima of the potential associated with the molecule). These pieces of trajectories are called reactive trajectories, and are particularly difficult to simulate at lower temperatures.

A method to generate reactive trajectories, namely equilibrium trajectories leaving a metastable state and ending in another one, is proposed , . The algorithm is based on simulating in parallel many copies of the system, and selecting the replicas which have reached the highest values along a chosen one-dimensional reaction coordinate. This reaction coordinate does not need to precisely describe all the metastabilities of the system for the method to give reliable results. An extension of the algorithm to compute transition times from one metastable state to another one is also being studied. We have demonstrated the interest of the method on two simple cases: A one-dimensional two-well potential and a two-dimensional potential exhibiting two channels to pass from one metastable state to another one.

This is a collaboration with Reuven Rubinstein and Radislav Vaisman (Technion, Israel Institute of Technology).

We have developed an enhanced version of the splitting
method, called the
*smoothed splitting method*(SSM), for counting problems
associated with complex sets, in particular for counting the
number of satisfiability assignments. A satisfiability
problem consists in several logical clauses involving several
Boolean variables (typically several hundreds or thousands
each). The goal is to find (if any) all the instances of the
variables (0 or 1) which make all the clauses true. This is
well known as a NP–hard problem if we want to solve it
exactly. We propose a new stochastic, thus approximate,
solver based on rare event simulation techniques
.

Like the conventional splitting algorithms, ours uses a sequential sampling plan to decompose a “difficult” problem into a sequence of “easy” ones. The main difference between SSM and splitting is that it works with an auxiliary sequence of continuous sets instead of the original discrete ones. The rationale of doing so is that continuous sets are easy to handle. We have shown on several examples that while the proposed method and its standard splitting counterpart are similar in their CPU time and variability, the former is more robust and more flexible than the latter. In particular, it makes it simpler for tuning the parameters.

This is a collaboration with Nicolas Hengartner (Los Alamos National Laboratories) and Eric Matzner-Løber (université Rennes 2).

Consider the output random variable obtained under some mapping from an input random vector with known probability distribution. That mapping acts as a black box, e.g., the result from some computer experiments for which no analytical expression is available. We have designed an efficient algorithm to estimate a tail probability given a quantile or a quantile given a tail probability . Our new algorithm improves upon existing multilevel splitting methods and can be analyzed using Poisson process tools that lead to exact description of the distribution of the estimated probabilities and quantiles. The performance of the algorithm is demonstrated in a problem related to digital watermarking.

See .

This is a collaboration with Christian Musso (ONERA Palaiseau).

Particle filtering is a widely used Monte Carlo method to
approximate the posterior probability distribution in
non–linear filtering, with an error scaling as
in terms of the sample size
N, but otherwise independently of the underlying state
dimension. However, it has recently been observed in practice
that particle filtering can be quite inefficient when the
dimension of the system is high. The issue here is to track
the impact of the dimension on the error variance, either
non–asymptotic or asymptotic. It has been suggested that the
most important factor by which dimensionality affects the
result is the predicted likelihood, a quantitative indicator
of the consistency between the prior distribution and the
likelihood function. In a simple static linear Gaussian
model, it has been possible indeed to check that the error
increases exponentially with the dimension
. The challenge now is to extend
these preliminary results to a static non–linear /
non–Gaussian model, as part of the PhD thesis of Paul
Bui–Quang, using the Laplace method.

This is a collaboration with Valérie Monbet (université de Rennes 1).

Surprisingly, very little was known about the asymptotic
behaviour of the ensemble Kalman filter
,
,
, whereas on the other hand, the
asymptotic behaviour of many different classes of particle
filters is well understood, as the number of particles goes
to infinity. Interpreting the ensemble elements as a
population of particles with mean–field interactions, and not
only as an instrumental device producing an estimation of the
hidden state as the ensemble mean value, it has been possible
to prove the convergence of the ensemble Kalman filter, with
a rate of order
, as the number
Nof ensemble elements increases to infinity
. In addition, the limit of the
empirical distribution of the ensemble elements has been
exhibited, which differs from the usual Bayesian filter. The
next step has been to prove (by induction) the asymptotic
normality of the estimation error, i.e. to prove a central
limit theorem for the ensemble Kalman filter.

INRIA contract ALLOC 2399 — May 2007 to August 2010.

This FP6 project is coordinated by National Aerospace Laboratory (NLR) (The Netherlands), and ASPI is also collaborating with University of Twente (The Netherlands) and Direction des Services de la Navigation Aérienne (DSNA).

The objective of iFLYis to develop both an advanced airborne self separation design and a highly automated air traffic management (ATM) design for en–route traffic, which takes advantage of autonomous aircraft operation capabilities and which is aimed to manage a three to six times increase in current en–route traffic levels. The proposed research combines expertise in air transport human factors, safety and economics with analytical and Monte Carlo simulation methodologies. The contribution of ASPI to this project concerns the work package on accident risk assessment methods and their implementation using conditional Monte Carlo methods, especially for large scale stochastic hybrid systems: designing and studying variants suited for hybrid state space (resampling per mode, marginalization) are currently investigated .

See .

INRIA contract ALLOC 2857 — September 2007 to August 2010.

This collaboration with Thalès Communications is supported by DGA (Délégation Générale à l'Armement) and is related with the supervision of the CIFRE thesis of Nordine El Baraka.

The overall objective is to study innovative algorithms for terrain–aided navigation, and to demonstrate these algorithms on four different situations involving different platforms, inertial navigation units, sensors and georeferenced databases. The thesis also considers the special use of image sensors (optical, infra–red, radar, sonar, etc.) for navigation tasks, based on correlation between the observed image sequence and a reference image available on–board in the database.

Marginalized particle filters and regularized particle filters have been implemented, and several propositions have been studied to adapt the sample size, such as KLD–sampling , which could be useful in the case of a poor initial information, or if the platform flies over a poorly informative area. Besides particle methods, which are proposed as the basic navigation algorithm, simpler algorithms such as the extended Kalman filter (EKF) or the unscented Kalman filter (UKF) have also been investigated.

See and

INRIA contract ALLOC 4233 — April 2009 to March 2011.

This is a collaboration with Sébastien Paris (université Paul Cézanne), related with the supervision of the PhD thesis of Mathieu Chouchane.

The objective of this project is to optimize the position and activation times of a few sensors deployed by a platform over a search zone, so as to maximize the probability of detecting a moving target. The difficulty here is that the target can detect an activated sensor before it is detected itself, and it can then modify its own trajectory to escape from the sensor. Because of the many constraints including timing constraints involved in this optimization problem, a stochastic algorithm is preferred here over a deterministic algorithm. The underlying idea is to replace the problem of maximizing a cost function (the probability of detection) over the possible configurations (admissible position and activation times) by the apparently simpler problem of sampling a population according to a probability distribution depending on a small parameter, which asymptotically concentrates on the set of global maxima of the cost function, as the small parameter goes to zero. The usual approach here is to use the cross–entropy method , .

The contribution of ASPI has been to propose a multilevel splitting algorithm, in order to evaluate the probability of detection for a given configuration. When this probability is small, these methods are known to provide a significant reduction in the variance of the relative error.

See .

INRIA contract ALLOC 2856 — January 2008 to December 2010.

This ANR project is coordinated by Thalès Alenia Space.

The overall objective is to study and demonstrate information fusion algorithms for localisation of pedestrian users in an indoor environment, where GPS solution cannot be used. The sought design combines

a pedestrian dead–reckoning (PDR) unit, providing noisy estimates of the linear displacement, angular turn, and possibly of the level change through an additional pression sensor,

range and / or proximity measurements provided by beacons at fixed and known locations, and possibly indirect distance measurements to access points, through a measure of the power signal attenuation,

constraints provided by an indoor map of the building (map-matching),

collaborative localisation when two users meet and exchange their respective position estimates.

Besides particle methods, which are proposed as the basic information fusion algorithm for the centralized server–based implementation, simpler algorithms such as the extended Kalman filter (EKF) or the unscented Kalman filter (UKF) have been investigated, to be used for the local PDA–based implementation with a map of a smaller part of the building. Constraints could be taken care of automatically with the help of a Voronoi graph , but this approach implies heavy pre–computations. A more direct approach, taking care of constraints on the fly, using a simple rejection method, has been preferred. Adapting the sample size using KLD–sampling has also been investigated, which could be useful in the case of a poor initial information, or if the user walks in poorly informative area (open zone, absence of beacons). Collaboration between users has been implemented , which allows from a user with a poor localization to benefit from the more accurate localization of another user. In this implementation, the latter user is seen by the former user as a ranging beacon with uncertain position. See , for a description of the overall fusion algorithm and an illustration with simulation results.

INRIA contract ALLOC 3767 — January 2009 to December 2011.

This ANR project is coordinated by École Normale Supérieure, Paris. The other partner is Météo–France. This is a collaboration with Étienne Mémin and Anne Cuzol (INRIA Rennes Bretagne Atlantique, project–team FLUMINANCE) and Valérie Monbet (université de Rennes 1).

The contribution of ASPI to this project is to continue the comparison , of sequential data assimilation methods, such as the ensemble Kalman filter (EnKF) and the weighted ensemble Kalman filter (WEnKF), with particle filters. This comparison will be made on the basis of asymptotic variances, as the ensemble or sample size goes to infinity, and also on the impact of dimension on small sample behavior.

INRIA contract ALLOC 2229 — January 2007 to June 2010.

Arnaud Guyader is coordinator of this ANR project. This is a collaboration with Teddy Furon (INRIA Rennes Bretagne Atlantique, project–team TEMICS) and Pierre Del Moral (INRIA Bordeaux Sud–Ouest, project–team ALEA).

There are mainly two strategic axes in NEBBIANO: watermarking and independent component analysis, and watermarking and rare event simulations. To protect copyright owners, user identifiers are embedded in purchased content such as music or movie. This is basically what we mean by watermarking. This watermarking is to be “invisible” to the standard user, and as difficult to find as possible. When content is found in an illegal place (e.g. a P2P network), the right holders decode the hidden message, find a serial number, and thus they can trace the traitor, i.e. the client who has illegally broadcast their copy. However, the task is not that simple as dishonest users might collude. For security reasons, anti–collusion codes have to be employed. Yet, these solutions (also called weak traceability codes) have a non–zero probability of error defined as the probability of accusing an innocent. This probability should be, of course, extremely low, but it is also a very sensitive parameter: anti–collusion codes get longer (in terms of the number of bits to be hidden in content) as the probability of error decreases. Fingerprint designers have to strike a trade–off, which is hard to conceive when only rough estimation of the probability of error is known. The major issue for fingerprinting algorithms is the fact that embedding large sequences implies also assessing reliability on a huge amount of data which may be practically unachievable without using rare event analysis. Our task within this project is to adapt our methods for estimating rare event probabilities to this framework, and provide watermarking designers with much more accurate false detection probabilities than the bounds currently found in the literature. We have already applied these ideas to some randomized watermarking schemes and obtained much sharper estimates of the probability of accusing an innocent.

A patent
entitled
*“Computer Checking Tool”*has been submitted by INRIA
and by université de Rennes 2.

INRIA contract ALLOC 2801 — January 2008 to December 2010.

This ANR project is coordinated by Alcatel–Lucent.

The primary goal of the TCHATER project is to demonstrate
a coherent terminal operating at 40Gb/s using
*real–time*digital signal processing and efficient
polarization division multiplexing. The terminal will benefit
to next-generation high information-spectral density optical
networks, while offering straightforward compatibility with
current 10Gbit/s networks. It will require that advanced
high–speed electronic components, especially
analog–to–digital converters, are designed within the
project. Specific algorithms for polarisation demultiplexing
and forward error correction with soft decoding will also
have to be developed.

INRIA contract ALLOC 4402 — November 2009 to October 2012.

This ANR project is also coordinated by Alcatel–Lucent Bell Labs France.

The focus of our project is to reduce the impact of nonlinear effect. The objective is twofold: specify, design, realize and evaluate fibres of reduced nonlinear effects by firstly increasing the effective area to unprecedented values and secondly, by splitting optical power along two modes, using bimodal propagation. While the first step is ambitious but primarily relies in the evolution of current fibre technologies, the second is disruptive and requires not only deep changes in fibre technologies but also new advanced transmitter / receiver equipment, preferably based on coherent detection. Naturally, bimodal propagation also brings another key advantage, namely a twofold increase of system capacity.

Jointly with the team Processus Stochastiques of IRMAR, ASPI organizes a working group on the Freidlin–Wentzell theoryand its applications. One of the main goals of these talks is to study the theory of large deviations which describe how a metastable diffusion process evolves. Moreover, several talks are dedicated to simulation algorithms and applications (molecular dynamics, turbulence modelling)

François Le Gland organizes at ONERA Palaiseau a working group on particle methods and their applications to Bayesian filtering and to rare event simulation.

François Le Gland has reported on the PhD thesis of Anissa Rabhi (université Pierre et Marie Curie, advisor: Yury Kutoyants). He was also a member of the committee for the PhD thesis of Joe Youssef (université Joseph Fourier, advisor: Suzanne Lesecq).

Arnaud Guyader is a member of the “comité de sélection” in applied mathematics (section 26) of université d'Angers. François Le Gland is a member of the “comité de sélection” in mathematics (sections 25–26) of INSA (institut national de sciences appliquées) Rennes, and he is a member of the “conseil d'UFR” of the department of mathematics of université de Rennes 1.

François Le Gland gives a course on Kalman filtering and hidden Markov models, at université de Rennes 1, within the Master SISEA (signal, image, systèmes embarqués, automatique, école doctorale MATISSE), a 3rd year course on Bayesian filtering and particle approximation, at ENSTA (école nationale supérieure de techniques avancées), Paris, within the systems and control module, a 3rd year course on linear and nonlinear filtering, linear and nonlinear filtering, at ENSAI (école nationale de la statistique et de l'analyse de l'information), Ker Lann, within the statistical engineering track, and a 3rd year course on hidden Markov models and particle filtering, at Télécom Bretagne, Brest.

Arnaud Guyader is a member of the committee of “oraux blancs d'agrégation de mathématiques” for ENS Cachan at Ker Lann.

In addition to presentations with a publication in the proceedings, and which are listed at the end of the document, members of ASPI have also given the following presentations.

Arnaud Guyader has given talks on iterative Monte Carlo for extreme quantiles and extreme probabilities at 42ème Journées de Statistiques in Marseille, at RESIM 2010 in Cambridge in June 2010 and at IWAP 2010 in Madrid in July 2010 (invited). He has been invited to give a talk in the local seminar on statistics in Montpellier and at the Rare Event Simulation workshop held in Bordeaux in October. He has also been invited by Nicolas Hengartner to visit Los Alamos National Laboratories in January 2010 and in April 2010.

Frédéric Cerou has given a talk on an adaptive replica approach to simulate reactive trajectories at RESIM 2010 in Cambridge in June 2010, on rare event simulation for a static distribution at IWAP 2010 (organizer of an invited session) in Madrid in July 2010, and has been invited to give a talk about importance splitting for rare event simulation at the CEA–EDF–INRIA school on Simulation of Hybrid Dynamical Systems and Applications to Molecular Dynamics held in Paris in September 2010.

François Le Gland has been invited to give a survey lecture on nonlinear filtering at the forum TISIC (traitement de l'information signal image et connaissance) of INRETS held in Paris in June 2010, a talk on asymptotic normality of the ensemble Kalman filter at the workshop on Numerical Methods for Filtering and for Parabolic PDE's held at Imperial College in September 2010, and a talk on marginalization for rare event simulation in switching diffusions at the Rare Event Simulation workshop held in Bordeaux in October 2010.

Florient Malrieu has defended his habilitation thesis on
functional inequalities and long time behavior of some Markov
processes, in Rennes in November 2010. He has given a
talk on ergodicity of piecewise deterministic Markov
processes, at the meeting of the MAS (modélisation aléatoire
et statistique) thematic group of SMAI (société de
mathématiques appliquées et industrielles) held in Bordeaux
in September 2010. He has been invited to give seminar
talks on long time behavior of McKean–Vlasov equations in
Nice in October 2010, on Markov switched
Ornstein–Uhlenbeck processes at INRIA Sophia–Antipolis in
October 2010, on ergodicity of modulated flows, in the
working group
*Mathématiques et Neurosciences*, at IHP Paris, on
functional inequalities for mixtures, in Lille in
December 2010, and on a piecewise determinisitic Markov
process for bacteria movements, in Nancy in
December 2010.