Graham W. Pulford

Graham W. Pulford

A comprehensive review of the literature on manoeuvring target tracking for both uncluttered and cluttered measurements is presented. Various discrete-time dynamical models including non-random input, random-input and switching or hybrid system manoeuvre models are presented. The problem of manoeuvre detection is covered. Classical and current filtering methods for manoeuvre tracking are presented including multi-level process noise, input estimation, variable dimension filtering, two-stage filter, the interacting multiple model algorithm, and generalised pseudo-Bayesian algorithms. Various extensions of these algorithms to the case of cluttered measurements are also described and these include: joint manoeuvre and measurement association, probabilistic data association and multi-hypothesis tracking. Smoothing schemes, including IMM smoothing and batch expectation-maximisation using the Viterbi algorithm, are also described. The use of amplitude information for target measurement discrimination is discussed. It is noted that although many manoeuvre tacking techniques exist, the literature contains surprisingly few performance comparisons to guide the design engineer although a performance benchmark has recently been introduced.

Graham W. Pulford

Expectation maximisation (EM) is an unsupervised learning method for estimating the parameters of a finite mixture distribution. It works by introducing "hidden" or "latent" variables via Baum's auxiliary function $Q$ that allow the joint data likelihood to be expressed as a product of simple factors. The relevance of EM has increased since the introduction of the variational lower bound (VLB): the VLB differs from Baum's auxiliary function only by the entropy of the PDF of the latent variables $Z$. We first present a rederivation of the standard EM algorithm using data association ideas from the field of multiple target tracking, using $K$-valued scalar data association hypotheses rather than the usual binary indicator vectors. The same method is then applied to a little known but much more general type of supervised EM algorithm for shared kernel models, related to probabilistic radial basis function networks. We address a number of shortcomings in the derivations that have been published previously in this area. In particular, we give theoretically rigorous derivations of (i) the complete data likelihood; (ii) Baum's auxiliary function (the E-step) and (iii) the maximisation (M-step) in the case of Gaussian shared kernel models. The subsequent algorithm, called shared kernel EM (SKEM), is then applied to a digit recognition problem using a novel 7-segment digit representation. Variants of the algorithm that use different numbers of features and different EM algorithm dimensions are compared in terms of mean accuracy and mean IoU. A simplified classifier is proposed that decomposes the joint data PDF as a product of lower order PDFs over non-overlapping subsets of variables. The effect of different numbers of assumed mixture components $K$ is also investigated. High-level source code for the data generation and SKEM algorithm is provided.

Graham W. Pulford

Expectation maximisation (EM) is usually thought of as an unsupervised learning method for estimating the parameters of a mixture distribution, however it can also be used for supervised learning when class labels are available. As such, EM has been applied to train neural nets including the probabilistic radial basis function (PRBF) network or shared kernel (SK) model. This paper addresses two major shortcomings of previous work in this area: the lack of rigour in the derivation of the EM training algorithm; and the computational complexity of the technique, which has limited it to low dimensional data sets. We first present a detailed derivation of EM for the Gaussian shared kernel model PRBF classifier, making use of data association theory to obtain the complete data likelihood, Baum's auxiliary function (the E-step) and its subsequent maximisation (M-step). To reduce complexity of the resulting SKEM algorithm, we partition the feature space into $R$ non-overlapping subsets of variables. The resulting product decomposition of the joint data likelihood, which is exact when the feature partitions are independent, allows the SKEM to be implemented in parallel and at $R^2$ times lower complexity. The operation of the partitioned SKEM algorithm is demonstrated on the MNIST data set and compared with its non-partitioned counterpart. It eventuates that improved performance at reduced complexity is achievable. Comparisons with standard classification algorithms are provided on a number of other benchmark data sets.

Graham W. Pulford

Although the expectation maximisation (EM) algorithm was introduced in 1970, it remains somewhat inaccessible to machine learning practitioners due to its obscure notation, terse proofs and lack of concrete links to modern machine learning techniques like autoencoded variational Bayes. This has resulted in gaps in the AI literature concerning the meaning of such concepts like "latent variables" and "variational lower bound," which are frequently used but often not clearly explained. The roots of these ideas lie in the EM algorithm. We first give a tutorial presentation of the EM algorithm for estimating the parameters of a $K$-component mixture density. The Gaussian mixture case is presented in detail using $K$-ary scalar hidden (or latent) variables rather than the more traditional binary valued $K$-dimenional vectors. This presentation is motivated by mixture modelling from the target tracking literature. In a similar style to Bishop's 2009 book, we present variational Bayesian inference as a generalised EM algorithm stemming from the variational (or evidential) lower bound, as well as the technique of mean field approximation (or product density transform). We continue the evolution from EM to variational autoencoders, developed by Kingma & Welling in 2014. In so doing, we establish clear links between the EM algorithm and its variational counterparts, hence clarifying the meaning of "latent variables." We provide a detailed coverage of the "reparametrisation trick" and focus on how the AEVB differs from conventional variational Bayesian inference. Throughout the tutorial, consistent notational conventions are used. This unifies the narrative and clarifies the concepts. Some numerical examples are given to further illustrate the algorithms.