How i learned to stop worrying about a thousand and one filters and love analytic combinatorics§

Abstract

Analytic combinatorics and generating functions are powerful analytical tools for understanding measurement-to-object assignment problems in multiobject tracking filters. Analytic combinatorics encompass both the classical theory of conventional filters and the (corrected) theory of random finite sets. It provides easily parsed expressions that not only highlight the underlying filter assumptions but also link the computational complexity of an exact Bayesian filter to the number of terms in a mixed ordinary derivative. Analytic combinatorics bestows many other benefits as well, and these are also discussed in the paper. The paper is written expressly for readers working in tracking and tracking applications but who are unfamiliar with analytic combinatorics. A relaxed, colloquial style is used for easy reading.