TERMINOLOGÍA MATERIA: ANTROPOLOGÍA SOCIAL

The research contributions of this thesis closely follow from our research objectives and span the three topics of online HMM parameter estimation, quickest change detection, and vision-based aircraft manoeuvre detection. The key contributions of this thesis are described below.

I Contribution 1: The proposal of an online HMM parameter estimator with promis- ing strong consistency and global convergence properties by exploiting ergodicity and a new information-theoretic one-step Kerridge inaccuracy (OKI) concept. In contrast to the HMM parameter estimators of [5] and [15] (which also exploit ergod- icity and information-theoretic concepts), our proposed estimator enables the online estimation of both state and observation process parameters. We are also able to develop partial convergence and strong consistency results for our proposed online HMM parameter estimator. Importantly, our proposed online HMM parameter estimator involves non-adaptive likelihoods (that are recursively calculated from the observation data), and OKI quantities (that are independent of the observation data). This structure enables our proposed HMM parameter estimator to be globally convergent online in cases where competing online estimators are only locally convergent (and fail to converge to the true unknown parameters).

I Contribution 2: The proposal and asymptotic solution of Lorden and Pollak (i.e. non-Bayesian) minimax robust quickest change detection problems with polyno- mial delay penalties in i.i.d. processes with uncertain pre-change and post-change distributions.

We identify our asymptotic solutions to these non-Bayesian minimax robust problems by exploiting new bounds on the detection delays of misspecified CUSUM rules (i.e. CUSUM rules that are designed with distributions that di↵er from those of the observed process). We also use and modify the least favourable distribution approach of [6] by introducing a new partial stochastic boundedness condition on the uncertainty sets. Importantly, our partial stochastic boundedness condition is a relaxation of the joint stochastic boundedness condition of [6]. Our partial stochastic boundedness condition therefore facilitates the introduction of new uncertainty sets into the problem of minimax robust quickest change detection. For example, we introduce uncertainty sets defined

1.5. RESEARCH CONTRIBUTIONS 11

by relative entropy tolerances (similar sets have been studied in the fields of robust control [58], robust hypothesis testing [8, 59], and robust filtering [1, 2]). Furthermore, our asymptotic Pollak results are the first Pollak results to handle uncertain pre-change distributions (the previous Pollak results of [6] assume that the pre-change distribution is known). Finally, although our asymptotic Lorden results are weaker than the exact (non-asymptotic) results of [6], they hold for a wider class of uncertainty sets and for polynomial delay penalties.

I Contribution 3: The proposal and (asymptotic) solution of Lorden, Pollak, and Bayesian minimax robust quickest change detection problems with polynomial (or higher order moment) detection delay penalties in general dependent (non-i.i.d.) processes with unknown post-change conditional density parameters.

Our investigation of quickest change detection in this thesis represents the first time that Lorden, Pollak, and Bayesian minimax robust quickest change detection problems have been successfully posed and solved (either exactly or asymptotically) in non-i.i.d. stochastic processes. Whilst there are Lorden and Pollak asymptotic optimality results for MLR rules in general dependent processes [44], and Pollak asymptotic optimality results for (adaptive) GLR rules in linear state-space and regression processes [46], these previous results are only established under linear (rather than polynomial) delay penalties. Our asymptotic robust results also allow uncertainty to be described using sets of possible post-change parameters, rather than needing to construct probability distributions on these sets (which is a prerequisite for the use of MLR rules). Further- more, our asymptotically robust rules are simpler to implement and less computationally expensive compared to MLR and GLR rules since they avoid calculating the mixture likelihoods inherent to MLR rules, and the parameter estimates inherent to GLR rules.

I Contribution 4: A performance characterisation of asymptotically minimax robust quickest change detection procedures alongside asymptotically optimal and adap- tive procedures.

We conduct our performance characterisation in theory and simulation. In particular, we derive asymptotic upper bounds on the detection delays of asymptotically minimax robust rules under our Lorden, Pollak and Bayesian criteria for general dependent processes, and under our Lorden and Pollak criteria for i.i.d. processes. Our asymptotic

12 CHAPTER 1. INTRODUCTION

upper bounds are new for the Pollak and Bayesian criteria, and are new for the Lorden criterion for (nonlinear) polynomial delays (linear delay bounds for the i.i.d. process case were previous derived in [6]). Our simulation comparisons of asymptotically optimal, adaptive, and asymptotically minimax robust rules are the first of their kind for non- i.i.d. processes (e.g. Markov chains and linear state-space systems). Significantly, our simulations suggest that asymptomatically minimax robust quickest change detection procedures o↵er comparable practical performance to (more computationally expensive) adaptive procedures (e.g., GLR rules).

I Contribution 5: The novel application of quickest change detection and HMM pa- rameter estimation techniques to the problem of vision-based aircraft manoeuvre detection.

Our use of quickest change detection and HMM parameter estimation techniques in the important application of autonomous mid-air collision avoidance highlights the signif- icance of the other (more theoretical) contributions of this thesis. Furthermore, our proposed vision-based aircraft manoeuvre detection techniques provide a previously missing capability to vision-based mid-air collision avoidance systems (since aircraft manoeuvres were not previously explicitly monitored).

I Contribution 6: The investigation and comparison of adaptive and minimax robust quickest change detection techniques in the application of vision-based aircraft manoeuvre detection (including their use on real data).

Our investigation of adaptive and robustness-inspired vision-based aircraft manoeuvre detectors appears to be one of the first application studies comparing the practical perfor- mance of adaptive and minimax robust quickest change detection procedures. Our study provides experimental evidence (from real data) that minimax robust quickest change detection procedures can o↵er comparable practical performance to competing (and more computationally expensive) adaptive quickest change detection approaches. This experimental evidence supports the extensive simulation evidence presented elsewhere in this thesis (and in [6]).