Book review: Review of Particle Filters for Random Set Models

Branko Ristic has given us a superb new book that combines two hot topics (particle filters and random sets) into a delicious feast of algorithms, Monte Carlo simulations, important applications, and mathematics that is new for normal engineers, all served up with clarity and thoroughness. The seminal research on random sets applied to tracking and data fusion problems is due to Ron Mahler [1]. It is clearly a good idea to apply random sets to such problems; in fact, it is hard to avoid using them either implicitly or explicitly. For example, the set of hypotheses for the well-known multiple hypothesis tracking (MHT) algorithms are random sets. The set of hypotheses for MHT is obviously random, because it depends on the measurements (which are random owing to noise errors and random target states). This is exactly the viewpoint adopted in [5], in which one is forced to compute equivalence classes of random sets in order to obtain a set of hypotheses that is mutually exclusive and collectively exhaustive, as required for any rational probability model of the situation (Bayesian or otherwise). It turns out that “fuzzy sets” correspond to equivalence classes of random sets, as explained by Goodman [6]. The set of particles in particle filters is also random, because it is generated from random numbers, but almost no papers explicitly use non-trivial random set theory for particle filter design. In fact, there are very few papers that explicitly talk about random sets and particle filters together, with the conspicuous exception of [20]. Nevertheless, random sets are a mature and well documented mathematical tool, as shown in [1], [2], [3], and [4]. You should not be afraid of random set mathematics; Ron Mahler has written a series of highly accessible tutorials on using random finite sets which avoid the pathologies and complexity of random unfettered sets (e.g., [26] and [28]).