On homogeneous skewness of unimodal distributions

We introduce a new concept of skewness for unimodal continuous distributions which is built on the asymmetry of the density function around its mode. The asymmetry is captured through a skewness function. We call a distribution homogeneously skewed if this skewness function is consistently positive or negative throughout its domain, and partially homogeneously skewed if the skewness function changes its sign at most once. This type of skewness is shown to exist in many popular continuous distributions such as Triangular, Gamma, Beta, Lognormal and Weibull. Two alternative ways of partial ordering among the partially homogeneously skewed distributions are described. Extensions of the notion to broader classes of distributions including discrete distributions have also been discussed

A Two-stage Particle Filter in High Dimension

Particle Filter (PF) is a popular sequential Monte Carlo method to deal with non-linear non-Gaussian filtering problems. However, it suffers from the so-called curse of dimensionality in the sense that the required number of particle (needed for a reasonable performance) grows exponentially with the dimension of the system. One of the techniques found in the literature to tackle this is to split the high-dimensional state in to several lower dimensional (sub)spaces and run a particle filter on each subspace, the so-called multiple particle filter (MPF). It is also well-known from the literature that a good proposal density can help to improve the performance of a particle filter. In this article, we propose a new particle filter consisting of two stages. The first stage derives a suitable proposal density that incorporates the information from the measurements. In the second stage a PF is employed with the proposal density obtained in the first stage. Through a simulated example we show that in high-dimensional systems, the proposed two-stage particle filter performs better than the MPF with much fewer number of particles

Riemann-Langevin Particle Filtering in Track-Before-Detect

Track-before-detect (TBD) is a powerful approach that consists in providing the tracker with sensor measurements directly without pre-detection. Due to the measurement model non-linearities, online state estimation in TBD is most commonly solved via particle filtering. Existing particle filters for TBD do not incorporate measurement information in their proposal distribution. The Langevin Monte Carlo (LMC) is a sampling method whose proposal is able to exploit all available knowledge of the posterior (that is, both prior and measurement information). This letter synthesizes recent advances in LMC-based filtering to describe the Riemann-Langevin particle filter and introduces its novel application to TBD. The benefits of our approach are illustrated in a challenging low-noise scenario.Comment: Minor grammatical update

An analysis of the Bayesian track labelling problem

In multi-target tracking (MTT), the problem of assigning labels to tracks (track labelling) is vastly covered in literature, but its exact mathematical formulation, in terms of Bayesian statistics, has not been yet looked at in detail. Doing so, however, may help us to understand how Bayes-optimal track labelling should be performed or numerically approximated. Moreover, it can help us to better understand and tackle some practical difficulties associated with the MTT problem, in particular the so-called ``mixed labelling'' phenomenon that has been observed in MTT algorithms. In this memorandum, we rigorously formulate the optimal track labelling problem using Finite Set Statistics (FISST), and look in detail at the mixed labeling phenomenon. As practical contributions of the memorandum, we derive a new track extraction formulation with some nice properties and a statistic associated with track labelling with clear physical meaning. Additionally, we show how to calculate this statistic for two well-known MTT algorithms

SMC methods to avoid self-resolving for online Bayesian parameter estimation

Abstract—The particle filter is a powerful filtering technique that is able to handle a broad scope of nonlinear problems. However, it has also limitations: a standard particle filter is unable to handle, for instance, systems that include static variables (parameters) to be estimated together with the dynamic states. This limitation is due to the well-known “self-resolving” phenomenon, which is caused by the gradual loss of information that occurs during the resampling steps. In the context of online Bayesian parameter estimation, some approaches to handle this problem have proposed, such as adding artificial dynamics to the parameter model. However, these approaches typically both introduce new parameters (e.g. the intensity of artificial process noise) and inherent biases to the estimation problem. In this paper, we will give a give a look at two Sequential Monte Carlo techniques that do not rely on biasing the system model: the Autonomous Multiple Model particle filter and the Rao-Blackwellized Marginal particle filter. These approaches are not new, but have not been applied yet to the problem of online Bayesian parameter estimation for non-structured models. We will derive suitable adaptations of these methods for this problem and evaluate them using simulations. I

Particle filter based MAP state estimation: A comparison

MAP estimation is a good alternative to MMSE for certain applications involving nonlinear non Gaussian systems. Recently a new particle filter based MAP estimator has been derived. This new method extracts the MAP directly from the output of a running particle filter. In the recent past, a Viterbi algorithm based MAP sequence estimator has been developed. In this paper, we compare these two methods for estimating the current state and the numerical results show that the former performs better

A Bayesian analysis of the mixed labelling phenomenon in two-target tracking

In mulit-target tracking and labelling (MTTL), mixed labelling corresponds to a situation where there is ambiguity in labelling, i.e. in the assignment of labels to locations (where a "location" here means simply an unlabelled single-target state. The phenomenon is well-known in literature, and known to occur in the situation where targets move in close proximity to each other and afterwards separate. The occurrence of mixed labelling has been empirically observed using particle filter implementations of the Bayesian MTTL recursion. In this memorandum, we will instead demonstrate the occurrence of mixed labelling (in the situation of closely spaced targets) using only the Bayesian recursion itself, for a scenario containing two targets and no target births or deaths. We will also show how mixed labelling generally persists after the targets become well-separated, and how mixed labelling might not happen when the unlabelled single-target state contains non-kinematic quantities