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

Glassy swirls of active dumbbells

The dynamics of a dense binary mixture of soft dumbbells, each subject to an active propulsion force and thermal fluctuations, shows a sudden arrest, first to a translational then to a rotational glass, as one reduces temperature $T$ or the self-propulsion force $f$. Is the temperature-induced glass different from the activity-induced glass? To address this question, we monitor the dynamics along an iso-relaxation-time contour in the $(T-f)$ plane. We find dramatic differences both in the fragility and in the nature of dynamical heterogeneity which characterise the onset of glass formation - the activity-induced glass exhibits large swirls or vortices, whose scale is set by activity, and appears to diverge as one approaches the glass transition. This large collective swirling movement should have implications for collective cell migration in epithelial layers.Comment: 13 pages, 11 figure

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

Activity controls fragility: A Random First Order Transition Theory for an active glass

How does nonequilibrium activity modify the approach to a glass? This is an important question, since many experiments reveal the near-glassy nature of the cell interior, remodelled by activity. However, different simulations of dense assemblies of active particles, parametrised by a self-propulsion force, $f_0$, and persistence time, $\tau_p$, appear to make contradictory predictions about the influence of activity on characteristic features of glass, such as fragility. This calls for a broad conceptual framework to understand active glasses; here we extend the Random First-Order Transition (RFOT) theory to a dense assembly of self-propelled particles. We compute the active contribution to the configurational entropy using an effective medium approach - that of a single particle in a caging-potential. This simple active extension of RFOT provides excellent quantitative fits to existing simulation results. We find that whereas $f_0$ always inhibits glassiness, the effect of $\tau_p$ is more subtle and depends on the microscopic details of activity. In doing so, the theory automatically resolves the apparent contradiction between the simulation models. The theory also makes several testable predictions, which we verify by both existing and new simulation data, and should be viewed as a step towards a more rigorous analytical treatment of active glass

A theoretical analysis of Bayes-optimal multi-target tracking and labelling

In multi-target tracking (MTT), we are often interested not only in finding the position of the multiple objects, but also allowing individual objects to be uniquely identified with the passage of time, by placing a label on each track. While there are many MTT algorithms that produce uniquely identified tracks as output, most of them make use of certain heuristics and/or unrealistic assumptions that makes the global result suboptimal of Bayesian sense. An innovative way of performing MTT is the so-called joint multi-target tracking, where the raw output of the algorithm, rather than being already the collection of output tracks, is a multi-target density calculated by approximating the Bayesian recursion that considers the entire system to have a single multidimensional state. The raw output, i.e. the calculated multi-target density, is thereafter processed to obtain output tracks to be displayed to the operator. This elegant approach, at least in theory, would allow us to precisely represent multi-target statistics. However, most joint MTT methods in the literature handle the problem of track labelling in an ad-hoc, i.e. non-Bayesian manner. A number of methods, however, have suggested that the multi-target density, calculated using the Bayesian recursion, should contain information not only about the location of the individual objects but also their identities. This approach, that we refer as joint MTTL (joint multi-target tracking and labelling), looks intuitively advantageous. It would allow us, at least in theory, to obtain an output consisting of labelled tracks that is optimal in Bayesian sense. Moreover, it would allow us to have statistical information about the assigned labels; for instance, we would know what is the probability that track swap may have occurred after some approximation of targets (or, in simpler words, we would know how much we can believe that a target is what the display says that it is). However, the methods proposed in the still emerging joint MTTL literature do not address some problems that may considerably reduce the usefulness of the approach. These problems include: track coalescence after targets move closely to each other, gradual loss of ambiguity information when particle filters or multiple hypotheses approaches are used, and dealing with unknown/varying number of targets. As we are going to see, each of the previously proposed methods handles only a subset of these problems. Moreover, while obtaining a Bayes-optimal output of labelled tracks is one of the main motivations for joint MTTL, how such output should be obtained is a matter of debate. This work will tackle the joint MTTL problem together with a companion memorandum. In this work, we look at the problem from a theoretical perspective, i.e. we aim to provide an accurate and algorithm-independent picture of the aforementioned problems. An algorithm that actually handles these problems will be proposed in the companion memorandum. As one of the contributions of the memorandum, we clearly characterize the so-called "mixed labelling" phenomenon that leads to track coalescence and other problems, and we verify that, unlike implied in previous literature, it is a physical phenomenon inherent of the MTTL problem rather than specific to a particular approach. We also show how mixed labelling leads to nontrivial issues in practical implementations of joint MTTL. As another of the contributions of the memorandum, we propose a conceptual, algorithm-independent track extraction method for joint MTTL estimators, that gives an output with clear physical interpretation for the user

Particle filter approximations for general open loop and open loop feedback sensor management

Sensor management is a stochastic control problem where the control mechanism is directed at the generation of observations. Typically, sensor management attempts to optimize a certain statistic derived from the posterior distribution of the state, such as covariance or entropy. However, these statistics often depend on future measurements which are not available at the moment the control decision is taken, making it necessary to consider their expectation over the entire measurement space. Though the idea of computing such expectations using a particle filter is not new, so far it has been applied only to specific sensor management problems and criterions. In this memorandum, for a considerably broad class of problems, we explicitly show how particle filters can be used to approximate general sensor management criterions in the open loop and open loop feedback cases. As examples, we apply these approximations to selected sensor management criterions. As an additional contribution of this memorandum, we show that every performance metric can be used to define a corresponding estimate and a corresponding task-driven sensor management criterion, and both of them can be approximated using particle filters. This is used to propose an approximate sensor management scheme based on the OSPA metric for multi-target tracking, which is included among our examples