Fahed Abdallah

Brief Paper: Box particle filtering for nonlinear state estimation using interval analysis

In recent years particle filters have been applied to a variety of state estimation problems. A particle filter is a sequential Monte Carlo Bayesian estimator of the posterior density of the state using weighted particles. The efficiency and accuracy of the filter depend mostly on the number of particles used in the estimation and on the propagation function used to re-allocate weights to these particles at each iteration. If the imprecision, i.e. bias and noise, in the available information is high, the number of particles needs to be very large in order to obtain good performances. This may give rise to complexity problems for a real-time implementation. This kind of imprecision can easily be represented by interval data if the maximum error is known. Handling interval data is a new approach successfully applied to different real applications. In this paper, we propose an extension of the particle filter algorithm able to handle interval data and using interval analysis and constraint satisfaction techniques. In standard particle filtering, particles are punctual states associated with weights whose likelihoods are defined by a statistical model of the observation error. In the box particle filter, particles are boxes associated with weights whose likelihood is defined by a bounded model of the observation error. Experiments using actual data for global localization of a vehicle show the usefulness and the efficiency of the proposed approach.

Anchor-Based Localization via Interval Analysis for Mobile Ad-Hoc Sensor Networks

Location awareness is a fundamental requirement for many applications of sensor networks. This paper proposes an original technique for self-localization in mobile ad-hoc networks. This method is adapted to the limited computational and memory resources of mobile nodes. The localization problem is solved in an interval analysis framework. The propagation of the estimation errors is based on an interval formulation of a state space model, where observations consist of anchor-based connectivities. The problem is then formulated as a constraint satisfaction problem where a simple Waltz algorithm is applied in order to contract the solution. This technique yields a guaranteed and robust online estimation of the mobile node positions. Observation errors as well as anchor node imperfections are taken into consideration in a simple and computational-consistent way. Multihop anchor-based and backpropagated localizations are also made possible in our method. Simulation results on mobile node trajectories corroborate the efficiency of the proposed technique and show that it outperforms the particle filtering methods.

State Estimation Using Interval Analysis and Belief-Function Theory: Application to Dynamic Vehicle Localization

A new approach to nonlinear state estimation based on belief-function theory and interval analysis is presented. This method uses belief structures composed of a finite number of axis-aligned boxes with associated masses. Such belief structures can represent partial information on model and measurement uncertainties more accurately than can the bounded-error approach alone. Focal sets are propagated in system equations using interval arithmetics and constraint-satisfaction techniques, thus generalizing pure interval analysis. This model was used to locate a land vehicle using a dynamic fusion of Global Positioning System measurements with dead reckoning sensors. The method has been shown to provide more accurate estimates of vehicle position than does the bounded-error method while retaining what is essential: providing guaranteed computations. The performances of our method were also slightly better than those of a particle filter, with comparable running time. These results suggest that our method is a viable alternative to both bounded-error and probabilistic Monte Carlo approaches for vehicle-localization applications.

Representing uncertainty on set-valued variables using belief functions

A formalism is proposed for representing uncertain information on set-valued variables using the formalism of belief functions. A set-valued variable X on a domain @W is a variable taking zero, one or several values in @W. While defining mass functions on the frame 2^2^^^@W is usually not feasible because of the double-exponential complexity involved, we propose an approach based on a definition of a restricted family of subsets of 2^@W that is closed under intersection and has a lattice structure. Using recent results about belief functions on lattices, we show that most notions from Dempster-Shafer theory can be transposed to that particular lattice, making it possible to express rich knowledge about X with only limited additional complexity as compared to the single-valued case. An application to multi-label classification (in which each learning instance can belong to several classes simultaneously) is demonstrated.

An Introduction to Box Particle Filtering [Lecture Notes]

Resulting from the synergy between the sequential Monte Carlo (SMC) method [1] and interval analysis [2], box particle filtering is an approach that has recently emerged [3] and is aimed at solving a general class of nonlinear filtering problems. This approach is particularly appealing in practical situations involving imprecise stochastic measurements that result in very broad posterior densities. It relies on the concept of a box particle that occupies a small and controllable rectangular region having a nonzero volume in the state space. Key advantages of the box particle filter (box-PF) against the standard particle filter (PF) are its reduced computational complexity and its suitability for distributed filtering. Indeed, in some applications where the sampling importance resampling (SIR) PF may require thousands of particles to achieve accurate and reliable performance, the box-PF can reach the same level of accuracy with just a few dozen box particles. Recent developments [4] also show that a box-PF can be interpreted as a Bayes? filter approximation allowing the application of box-PF to challenging target tracking problems [5].

Mixture of uniform probability density functions for non linear state estimation using interval analysis

In this work, a novel approach to nonlinear non-Gaussian state estimation problems is presented based on mixtures of uniform distributions with box supports. This class of filtering methods, introduced in the light of interval analysis framework, is called Box Particle Filter (BPF). It has been shown that weighted boxes, estimating the state variables, can be propagated using interval analysis tools combined with Particle filtering ideas. In this paper, in the light of the widely used Bayesian inference, we present a different interpretation of the BPF by expressing it as an approximation of posterior probability density functions, conditioned on available measurements, using mixture of uniform distributions. This interesting interpretation is theoretically justified. It provides derivation of the BPF procedures with detailed discussions.

A Dependent Multilabel Classification Method Derived from the k -Nearest Neighbor Rule

In multilabel classification, each instance in the training set is associated with a set of labels, and the task is to output a label set whose size is unknown a priori for each unseen instance. The most commonly used approach for multilabel classification is where a binary classifier is learned independently for each possible class. However, multilabeled data generally exhibit relationships between labels, and this approach fails to take such relationships into account. In this paper, we describe an original method for multilabel classification problems derived from a Bayesian version of the k-nearest neighbor (k-NN) rule. The method developed here is an improvement on an existing method for multilabel classification, namely multilabel k-NN, which takes into account the dependencies between labels. Experiments on simulated and benchmark datasets show the usefulness and the efficiency of the proposed approach as compared to other existing methods.

An Evidence-Theoretic k-Nearest Neighbor Rule for Multi-label Classification

In multi-label learning, each instance in the training set is associated with a set of labels, and the task is to output a label set for each unseen instance. This paper describes a new method for multi-label classification based on the Dempster-Shafer theory of belief functions to classify an unseen instance on the basis of its k nearest neighbors. The proposed method generalizes an existing single-label evidence-theoretic learning method to the multi-label case. In multi-label case, the frame of discernment is not the set of all possible classes, but it is the powerset of this set. That requires an extension of evidence theory to manipulate multi-labelled data. Using evidence theory makes us able to handle ambiguity and imperfect knowledge regarding the label sets of training patterns. Experiments on benchmark datasets show the efficiency of the proposed approach as compared to other existing methods.

A Multiple-Hypothesis Map-Matching Method Suitable for Weighted and Box-Shaped State Estimation for Localization

The goal of map-matching algorithms is to identify the road taken by a vehicle and to compute an estimate of the vehicle position on that road using a digital map. In this paper, a map-matching algorithm based on interval analysis and the belief function theory is proposed. The method combines the outputs from existing bounded-error estimation techniques with piecewise rectangular roads that are selected using evidential reasoning. A set of candidate roads is first defined at each time step using the topology of the map and a similarity criterion, and a mass function on the set of candidate roads is computed. An overall estimate of the vehicle position is then derived after the most probable candidate road has been selected. This method allows multiple road junction hypotheses to efficiently be handled and can cope with missing data. In addition, the implementation of the method is quite simple, because it is based on geometrical properties of boxes and rectangular road segments. Experiments with simulated and real data demonstrate the ability of this method to handle junction situations and to compute an accurate estimate of the vehicle position.

Guaranteed Boxed Localization in MANETs by Interval Analysis and Constraints Propagation Techniques

In this contribution, we propose an original algorithm for self-localization in mobile ad-hoc networks. The proposed technique, based on interval analysis, is suited to the limited computational and memory resources of mobile nodes. The incertitude about the estimated position of each node is propagated in an interval form. The propagation is based on a state space model and formulated by a constraints satisfaction problem. Observations errors as well as anchor nodes imperfections are taken into account in a simple and computational-consistent way. A simple Waltz algorithm is then applied in order to contract the solution, yielding a guaranteed and robust online estimation of the mobile node position. Simulation results on mobile node group trajectories corroborate the efficiency of the proposed technique and show that it compares favorably to particle filtering methods.