arXiv subjects
Publications and source records attributed to Jeremie Houssineau.
This paper determines the general formula for describing differentials of composite functions in terms of differentials of their factor functions. This generalises the formula commonly attributed to Faa di Bruno to functions in locally convex topological vector spaces. The result highlights the general structure of the higher-order chain rule in terms of partitions of the directions.
Recent progress in multi-object filtering has led to algorithms that compute the first-order moment of multi-object distributions based on sensor measurements. The number of targets in arbitrarily selected regions can be estimated using the first-order moment. In this work, we introduce explicit formulae for the computation of the second-order statistic on the target number. The proposed concept of regional variance quantifies the level of confidence on target number estimates in arbitrary regions and facilitates information-based decisions. We provide algorithms for its computation for the Probability Hypothesis Density (PHD) and the Cardinalized Probability Hypothesis Density (CPHD) filters. We demonstrate the behaviour of the regional statistics through simulation examples.
The problem of estimating interacting systems of multiple objects is important to a number of different fields of mathematics, physics, and engineering. Drawing from a range of disciplines, including statistical physics, variational calculus, point process theory, and statistical sensor fusion, we develop a unified probabilistic framework for modelling systems of this nature. In order to do this, we derive a new result in variational calculus, Faa di Bruno's formula for Gateaux differentials. Using this result, we derive the Chapman-Kolmogorov equation and Bayes' rule for stochastic population processes with interactions and hierarchies. We illustrate the general approach through case studies in multi-target tracking, branching processes and renormalization.