The Georgii-Nguyen-Zessin identity induces a functional differential equation for the cumulant generating functional of a point process admitting a Papangelou conditional intensity. In generating-functional form, this equation takes the structure of a Dyson-Schwinger identity. For pairwise Gibbs point processes, the equation closes: insertion of a point acts as a deterministic shift of the source field. This yields a hierarchy for cumulant densities of all orders, expressing cumulants as differences between the original law and the laws induced by point insertion. The formulation is non-perturbative and provides exact recursion relations for cumulants.

Bartlett identities link score, curvature, and Fisher information in classical likelihood theory, but no general formulation exists for spatial point processes, where likelihoods act on configuration space and parameters are functions. In this paper, we establish Bartlett identities for point processes by treating the log-likelihood as a functional and using variational derivatives. This yields a hierarchy of identities in which derivatives of the log-likelihood are represented by functions and kernels, and are expressed in terms of cumulants of the score field. As a consequence, the score, second variation, and Fisher information operator are unified through the cumulant structure of the process, placing likelihood-based inference for point processes on the same structural footing as in finite-dimensional models.

Multi-object filtering is a generalisation of stochastic filtering to deal with an unknown and time-varying number of targets, largely based on modelling with point processes. Some early works on this topic from the Soviet Union from 1960s-l980s appeared prior to well known results in the contemporary liter-ature. This article reviews some of these historical contributions.

Multi-target tracking systems typically provide sets of estimated target states as their output. It is challenging to be able to integrate these outputs as inputs to other tracking systems to gain a better picture of the area under surveillance since they do not conform to the standard observation model. Moreover, in cyclic distributed systems, there may be common information between state estimates that would mean that fused estimates may become overconfident and corrupt the system. In this paper we develop a Bayesian multi-target estimator based on the covariance intersection algorithm for multi-target track-to-track data fusion. The approach is integrated into a multitarget tracking algorithm and demonstrated in simulations. The approach is able to account for missed tracks and false tracks produced by another tracking system.

A conceptual Kalman filter for random fields is proposed for estimating multiple objects. The result exploits an approximation of point processes with Gaussian random fields. The motivation is to develop a solution for multi-object filtering problem in terms of the first two moments of a random field. Applications are discussed for a multi-target tracking model.

Tracking systems often provide sets of tracks rather than raw detections obtained from sensors. Integrating these track sets into other tracking systems is challenging because the usual sensor models do not apply. In this work we present a method for fusing track data from multiple sensors in a central fusion node. The algorithm exploits the covariance intersection algorithm as a pseudo-Kalman filter which is integrated into a multi-sensor multi-target tracker within a Bayesian paradigm. This makes it possible to (i) integrate the proposed fusion method seamlessly into any existing tracker; (ii) modify multi-target trackers to take a set of tracks as a set of measurements; and (iii) perform gating to enable data association between tracks. The described method is demonstrated in simulations using several target trackers within the Stone Soup tracking framework.

The inverse of the Fisher information, known as the Cramér-Rae bound, provides a bound on the estimator of a parameter and is fundamental for statistical analysis. It provides a minimum achievable variance or covariance for a parameter for a univariate or vector-valued parameter. However, multi-target systems often have parameters that are described by functions and the variance and covariance for point processes are themselves functions with spatial variates. Consequently, the usual formulation of the Cramér-Rae bound in these contexts is not applicable for providing a bound for the covariance of a point process. In this article we consider commonly used multi-target tracking models and determine Cramér-Rae lower bounds on the variance for different parameters, including the detection probabilities, the false alarm intensity, and the intensity of the process. This provides the first application of the Cramér-Ran bound for point processes for the analysis of practical algorithms.

A lower bound on the Kullback-Leibler divergence known as Kullback’s inequality can be determined with the Legendre transform of the cumulant generating function. The Cramer Rao bound can be derived from Kullback’s inequality as the inverse of the second order term in a Taylor expansion. Analogous forms for Kullback’s inequality and the Cram er Rao bound for point processes were recently derived using functional methods from quantum field theory. This article develops Kullback’s inequality and the Cramer Rao bound for point process parametrisations as performance bounds for models used in multi-object filtering.

This article considers the problem of guiding an unknown number of controllable interceptors to rendezvous with the same target at the same time. It is assumed that the all of the interceptors and the target are described by linear dynamics with Gaussian noise, though the theory presented does not preclude more general models. This article extends the work of Athans to consider a scenario, where the number of interceptors is unknown and time-varying. In particular, the focus is on the development of a stochastic multiobject guidance law for simulaneous rendezvous and interception. The article is presented as an homage to Athans' original work which was published nearly 50 years ago.

This paper summarizes the core definitions and results regarding the chain differential for functions in locally convex topological vector spaces. In addition, it provides a few elementary calculus rules of practical interest, notably for the differentiation of characteristic functionals in various domains of physical science and engineering. 1 Functional differentiation In this section we discuss two different forms of differential, the Gâteaux differential [5] and the chain differential [2]. The chain differential, which is similar to the epiderivative [1], is adopted since it is possible to determine a chain rule, yet is not as restrictive as the Fréchet derivative. Results are stated for locally convex topological vector spaces which include Banach spaces such as Hilbert and Euclidean spaces, e.g., R, as well as spaces of test functions for the study of distributions. This type of space is therefore sufficiently general for most practical applications. 1.1 Gâteaux differential Definition 1 (Gâteaux differential). Let X and Y be locally convex topological vector spaces, and let Ω be an open subset of X and let f : Ω → Y . The Gâteaux differential at x ∈ Ω in the direction η ∈ X is δf(x; η) := lim ǫ→0 1 ǫ ( f(x+ ǫη)− f(x) ) (1) when the limit exists. If δf(x; η) exists for all η ∈ X then f is Gâteaux differentiable at x. The Gâteaux differential is homogeneous of degree one in η, so that for all real numbers α, δf(x;αη) = αδf(x; η). In Definition 1, the space X might be a function space. In this case, functions on X can be referred to as functionals. 1.2 Chain differential Due to the lack of continuity properties of the Gâteaux differential, further constraints are required in order to derive a chain rule. Bernhard [2] proposed a new form of Gâteaux differential defined with sequences, which he called the chain differential. It is not as restrictive as the Fréchet derivative though it is still possible to find a chain rule that maintains the general structure. Definition 2 (Chain differential). The function f : X → Y , where X and Y are locally convex topological vector spaces, has a chain differential δf(x; η) at point x ∈ X in the direction η ∈ X if, for any sequence ηm → η ∈ X, and any sequence of real numbers θm → 0, it holds that the following limit exists δf(x; η) := lim m→∞ 1 θm ( f(x+ θmηm)− f(x) ) . (2) If X = X1×. . .×Xn, where {Xi} n i=1 are locally convex topological vector spaces, x := (x1, . . . , xn) ∈ X, and η := (η1, . . . , ηn) ∈ X, the chain differential δf(x;η), if it exists, is also called the total chain differential of f at point x in the direction η.

The Cramér Rao bound provides a minimum achievable variance or covariance for a parameter for a univariate or vector-valued parameter. Point processes often have parameters that are described by functions and the variance and covariance for point processes are themselves functions with spatial variates. Consequently, the usual formulation of the Cramér Rao bound in these contexts is not applicable. The second-order derivative of Kullback’s inequality, which relates the Kullback-Leibler divergence to Cramér’s rate function, provides a description of the Cramér Rao bound. We follow this approach to develop a form of Cramér Rao bound for point processes and random measures derived from the second-order functional derivative of Kullback’s inequality, which relates the Kullback-Leibler divergence to Cramér’s rate functional for point processes and random measures.

Methods for information-theoretic control for networks of sensors are of interest for enabling the development of autonomous sensor systems. In this paper we revisit the fundamentals of information theoretic-based control for multi-target systems and present a systematic approach for determining information-theoretic situational awareness based on mutual information for point processes. The extension to multi-sensor systems is developed using the concept of a broadcast channel from information theory. Analytic results are presented for linear-Gaussian systems which enable low complexity solutions for determining information from multiple sensors and we consider a large number of potential sensor configurations. We consider extensions of single-target methods to multi-target scenarios and present results in simulations.

This article is focused on estimating a quantity of interest in the context of military impact assessment that we shall call adversarial risk. We formulate the adversarial risk as a function of the multiobject state describing a group of weapons, and propose two approaches to estimate it using multiobject filters. The first, optimal, approach is tailored to filters for point processes, and produces the mean estimate of the adversarial risk and its variance. The second, naïve, approach is applicable to any filter producing point estimates of the multiobject state, yet it is not capable of equipping a risk estimate with an indicator of its quality. We develop an implementation of the optimal approach for a particular multiobject filter and compare it to the naïve approach.

Modern tracking problems require fast, scalable, and robust solutions for tracking multiple targets from noisy sensor data. In this article, an algorithm that has linear computational complexity with respect to the number of targets and measurements is presented. The method is based on the propagation of the first two factorial cumulants of a point process. The algorithm is demonstrated for tracking a million targets in cluttered environments in the fastest time yet for any such solution. A low-computational-complexity solution to the problem of joint multitarget tracking and parameter estimation is also presented. The multitarget filtering approach utilizes a single-cluster point process method for joint multiobject estimation and parameter estimation and is shown to be more computationally efficient and robust than previous implementations.

This article develops a systematic approach for stochastic multi-target control based on the integration of the linear-quadratic regulator into multi-target tracking models. Using the assumption that possible targets are independent, follow linear-Gaussian trajectories, and can be described with Gaussians, it is shown that the linear-quadratic regulator can be applied to each target individually which enables them to be controlled with the classical linear-quadratic regulator. The result is demonstrated in a simple multi-target scenario which has the effect of controlling all of the targets to the same state over a given time-frame.

The number of nodes in sensor networks is continually increasing, and maintaining accurate track estimates inside their common surveillance region is a critical necessity. Modern sensor platforms are likely to carry a range of different sensor modalities, all providing data at differing rates, and with varying degrees of uncertainty. These factors complicate the fusion problem as multiple observation models are required, along with a dynamic prediction model. However, the problem is exacerbated when sensors are not registered correctly with respect to each other, i.e., if they are subject to a static or dynamic bias. In this case, measurements from different sensors may correspond to the same target, but do not correlate with each other when in the same Frame of Reference (FoR), which decreases track accuracy. This paper presents a method to jointly estimate the state of multiple targets in a surveillance region, and to correctly register a radar and an Infrared Search and Track (IRST) system onto the same FoR to perform sensor fusion. Previous work using this type of parent-offspring process has been successful when calibrating a pair of cameras, but has never been attempted on a heterogeneous sensor network, or in a maritime environment. This article presents results on both simulated scenarios and a segment of real data that show a significant increase in track quality in comparison to using incorrectly calibrated sensors or single-radar only.

A representation of heterogeneous stochastic populations that are composed of sub-populations with different levels of distinguishability is introduced together with an analysis of its properties. It is demonstrated that any instance of this representation where individuals are independent can be related to a point process on the set of probability measures on the individual state space. The introduction of the proposed representation is fully constructive which ensures the meaningfulness of the approach.

This paper presents a low computational complexity solution to the problem of joint multi-target tracking and parameter estimation. The multi-target filtering approach is based on the linear-complexity factorial cumulant filter presented in this conference last year, and the parameter estimation approach is based on the single-cluster point process method for joint multi-object estimation and parameter estimation. The joint multitarget and measurement process is approximated at each step with a Panjer point process, which permits the low cost solution. It is shown in simulated studies the paper that the approach is more robust than using a Poisson point process prior since second-order information is retained and propagated.

Point processes are often described with functionals, such as the probability generating functional, the Laplace functional, and the factorial cumulant generating functional. These are used to facilitate modelling of different processes and to determine important statistics via functional differentiation. In information theory, generating functions have also been defined for probability densities to determine information quantities such as the Shannon information and Kullback-Leibler divergence, though as yet there are no such analogues for point processes. The purpose of this article is to exploit the advantages of both types of generating function to facilitate the derivation of information statistics for point processes. In particular, a generating functional for point processes is introduced for determining statistics related to entropy and relative entropy based on Golomb’s information function and Moyal’s probability generating functional. It is shown that the information generating functional permits the derivation of a suite of statistics, including localised Shannon entropy and Kullback-Leibler divergence calculations.

A recent trend in distributed multisensor fusion is to use random finite-set filters at the sensor nodes and fuse the filtered distributions algorithmically using their exponential mixture densities (EMDs). Fusion algorithms that extend covariance intersection and consensus-based approaches are such examples. In this paper, we analyze the variational principle underlying EMDs and show that the EMDs of finite-set distributions do not necessarily lead to consistent fusion of cardinality distributions. Indeed, we demonstrate that these inconsistencies may occur with overwhelming probability in practice, through examples with Bernoulli, Poisson, and independent identically distributed cluster processes. We prove that pointwise consistency of EMDs does not imply consistency in global cardinality and vice versa. Then, we redefine the variational problems underlying fusion and provide iterative solutions thereby establishing a framework that guarantees cardinality consistent fusion.