A Bayesian‐Markov process for reliability prediction

Accurate reliability prediction for large-scale, long lived engineering is a crucial foundation for effective asset risk management and optimal maintenance decision making. However, a lack of failure data for assets that fail infrequently, and changing operational conditions over long periods of time, make accurate reliability prediction for such assets very challenging. To address this issue, we present a Bayesian-Marko best approach to reliability prediction using prior knowledge and condition monitoring data. In this approach, the Bayesian theory is used to incorporate prior information about failure probabilities and current information about asset health to make statistical inferences, while Markov chains are used to update and predict the health of assets based on condition monitoring data. The prior information can be supplied by domain experts, extracted from previous comparable cases or derived from basic engineering principles. Our approach differs from existing hybrid Bayesian models which are normally used to update the parameter estimation of a given distribution such as the Weibull-Bayesian distribution or the transition probabilities of a Markov chain. Instead, our new approach can be used to update predictions of failure probabilities when failure data are sparse or nonexistent, as is often the case for large-scale long-lived engineering assets.

Bayesian algorithms with applications

Advances in algorithms for approximate sampling from a multivariable target function have led to solutions to challenging statistical inference problems that would otherwise not be considered by the applied scientist. Such sampling algorithms are particularly relevant to Bayesian statistics, since the target function is the posterior distribution of the unobservables given the observables. In this thesis we develop, adapt and apply Bayesian algorithms, whilst addressing substantive applied problems in biology and medicine as well as other applications. For an increasing number of high-impact research problems, the primary models of interest are often sufficiently complex that the likelihood function is computationally intractable. Rather than discard these models in favour of inferior alternatives, a class of Bayesian "likelihoodfree" techniques (often termed approximate Bayesian computation (ABC)) has emerged in the last few years, which avoids direct likelihood computation through repeated sampling of data from the model and comparing observed and simulated summary statistics. In Part I of this thesis we utilise sequential Monte Carlo (SMC) methodology to develop new algorithms for ABC that are more efficient in terms of the number of model simulations required and are almost black-box since very little algorithmic tuning is required. In addition, we address the issue of deriving appropriate summary statistics to use within ABC via a goodness-of-fit statistic and indirect inference. Another important problem in statistics is the design of experiments. That is, how one should select the values of the controllable variables in order to achieve some design goal. The presences of parameter and/or model uncertainty are computational obstacles when designing experiments but can lead to inefficient designs if not accounted for correctly. The Bayesian framework accommodates such uncertainties in a coherent way. If the amount of uncertainty is substantial, it can be of interest to perform adaptive designs in order to accrue information to make better decisions about future design points. This is of particular interest if the data can be collected sequentially. In a sense, the current posterior distribution becomes the new prior distribution for the next design decision. Part II of this thesis creates new algorithms for Bayesian sequential design to accommodate parameter and model uncertainty using SMC. The algorithms are substantially faster than previous approaches allowing the simulation properties of various design utilities to be investigated in a more timely manner. Furthermore the approach offers convenient estimation of Bayesian utilities and other quantities that are particularly relevant in the presence of model uncertainty. Finally, Part III of this thesis tackles a substantive medical problem. A neurological disorder known as motor neuron disease (MND) progressively causes motor neurons to no longer have the ability to innervate the muscle fibres, causing the muscles to eventually waste away. When this occurs the motor unit effectively ‘dies’. There is no cure for MND, and fatality often results from a lack of muscle strength to breathe. The prognosis for many forms of MND (particularly amyotrophic lateral sclerosis (ALS)) is particularly poor, with patients usually only surviving a small number of years after the initial onset of disease. Measuring the progress of diseases of the motor units, such as ALS, is a challenge for clinical neurologists. Motor unit number estimation (MUNE) is an attempt to directly assess underlying motor unit loss rather than indirect techniques such as muscle strength assessment, which generally is unable to detect progressions due to the body’s natural attempts at compensation. Part III of this thesis builds upon a previous Bayesian technique, which develops a sophisticated statistical model that takes into account physiological information about motor unit activation and various sources of uncertainties. More specifically, we develop a more reliable MUNE method by applying marginalisation over latent variables in order to improve the performance of a previously developed reversible jump Markov chain Monte Carlo sampler. We make other subtle changes to the model and algorithm to improve the robustness of the approach.

Consistent HMM parameter estimation using Kerridge inaccuracy rates

In this paper, we propose a novel online hidden Markov model (HMM) parameter estimator based on Kerridge inaccuracy rate (KIR) concepts. Under mild identifiability conditions, we prove that our online KIR-based estimator is strongly consistent. In simulation studies, we illustrate the convergence behaviour of our proposed online KIR-based estimator and provide a counter-example illustrating the local convergence properties of the well known recursive maximum likelihood estimator (arguably the best existing solution).

Model choice problems using approximate Bayesian computation with applications to pathogen transmission data sets

Analytically or computationally intractable likelihood functions can arise in complex statistical inferential problems making them inaccessible to standard Bayesian inferential methods. Approximate Bayesian computation (ABC) methods address such inferential problems by replacing direct likelihood evaluations with repeated sampling from the model. ABC methods have been predominantly applied to parameter estimation problems and less to model choice problems due to the added difficulty of handling multiple model spaces. The ABC algorithm proposed here addresses model choice problems by extending Fearnhead and Prangle (2012, Journal of the Royal Statistical Society, Series B 74, 1–28) where the posterior mean of the model parameters estimated through regression formed the summary statistics used in the discrepancy measure. An additional stepwise multinomial logistic regression is performed on the model indicator variable in the regression step and the estimated model probabilities are incorporated into the set of summary statistics for model choice purposes. A reversible jump Markov chain Monte Carlo step is also included in the algorithm to increase model diversity for thorough exploration of the model space. This algorithm was applied to a validating example to demonstrate the robustness of the algorithm across a wide range of true model probabilities. Its subsequent use in three pathogen transmission examples of varying complexity illustrates the utility of the algorithm in inferring preference of particular transmission models for the pathogens.

Faster parameter estimation using risk-sensitive filters

In this paper, we propose a risk-sensitive approach to parameter estimation for hidden Markov models (HMMs). The parameter estimation approach considered exploits estimation of various functions of the state, based on model estimates. We propose certain practical suboptimal risk-sensitive filters to estimate the various functions of the state during transients, rather than optimal risk-neutral filters as in earlier studies. The estimates are asymptotically optimal, if asymptotically risk neutral, and can give significantly improved transient performance, which is a very desirable objective for certain engineering applications. To demonstrate the improvement in estimation simulation studies are presented that compare parameter estimation based on risk-sensitive filters with estimation based on risk-neutral filters.

Short-data recursive HMM parameter estimation for rapid vision-based aircraft heading estimation

Rapid recursive estimation of hidden Markov Model (HMM) parameters is important in applications that place an emphasis on the early availability of reasonable estimates (e.g. for change detection) rather than the provision of longer-term asymptotic properties (such as convergence, convergence rate, and consistency). In the context of vision- based aircraft (image-plane) heading estimation, this paper suggests and evaluates the short-data estimation properties of 3 recursive HMM parameter estimation techniques (a recursive maximum likelihood estimator, an online EM HMM estimator, and a relative entropy based estimator). On both simulated and real data, our studies illustrate the feasibility of rapid recursive heading estimation, but also demonstrate the need for careful step-size design of HMM recursive estimation techniques when these techniques are intended for use in applications where short-data behaviour is paramount.

Towards strongly consistent online HMM parameter estimation using one-step Kerridge inaccuracy

In this paper, we propose a novel online hidden Markov model (HMM) parameter estimator based on the new information-theoretic concept of one-step Kerridge inaccuracy (OKI). Under several regulatory conditions, we establish a convergence result (and some limited strong consistency results) for our proposed online OKI-based parameter estimator. In simulation studies, we illustrate the global convergence behaviour of our proposed estimator and provide a counter-example illustrating the local convergence of other popular HMM parameter estimators.

Alive SMC^2: Bayesian model selection for low-count time series models with intractable likelihoods

In this paper we present a new method for performing Bayesian parameter inference and model choice for low count time series models with intractable likelihoods. The method involves incorporating an alive particle filter within a sequential Monte Carlo (SMC) algorithm to create a novel pseudo-marginal algorithm, which we refer to as alive SMC^2. The advantages of this approach over competing approaches is that it is naturally adaptive, it does not involve between-model proposals required in reversible jump Markov chain Monte Carlo and does not rely on potentially rough approximations. The algorithm is demonstrated on Markov process and integer autoregressive moving average models applied to real biological datasets of hospital-acquired pathogen incidence, animal health time series and the cumulative number of poison disease cases in mule deer.

On the Milito-Cruz adaptive control scheme for Markov chains

Milito and Cruz have introduced a novel adaptive control scheme for finite Markov chains when a finite parametrized family of possible transition matrices is available. The scheme involves the minimization of a composite functional of the observed history of the process incorporating both control and estimation aspects. We prove the a.s. optimality of a similar scheme when the state space is countable and the parameter space a compact subset ofR.

Residence time distribution for a class of Gaussian Markov processes

We study the distribution of residence time or equivalently that of "mean magnetization" for a family of Gaussian Markov processes indexed by a positive parameter alpha. The persistence exponent for these processes is simply given by theta=alpha but the residence time distribution is nontrivial. The shape of this distribution undergoes a qualitative change as theta increases, indicating a sharp change in the ergodic properties of the process. We develop two alternate methods to calculate exactly but recursively the moments of the distribution for arbitrary alpha. For some special values of alpha, we obtain closed form expressions of the distribution function. [S1063-651X(99)03306-1].

A novel Q-learning algorithm with function approximation for constrained Markov decision processes

We present a novel multi-timescale Q-learning algorithm for average cost control in a Markov decision process subject to multiple inequality constraints. We formulate a relaxed version of this problem through the Lagrange multiplier method. Our algorithm is different from Q-learning in that it updates two parameters - a Q-value parameter and a policy parameter. The Q-value parameter is updated on a slower time scale as compared to the policy parameter. Whereas Q-learning with function approximation can diverge in some cases, our algorithm is seen to be convergent as a result of the aforementioned timescale separation. We show the results of experiments on a problem of constrained routing in a multistage queueing network. Our algorithm is seen to exhibit good performance and the various inequality constraints are seen to be satisfied upon convergence of the algorithm.

Bayesian parameter identification in dynamic state space models using modified measurement equations

When Markov chain Monte Carlo (MCMC) samplers are used in problems of system parameter identification, one would face computational difficulties in dealing with large amount of measurement data and (or) low levels of measurement noise. Such exigencies are likely to occur in problems of parameter identification in dynamical systems when amount of vibratory measurement data and number of parameters to be identified could be large. In such cases, the posterior probability density function of the system parameters tends to have regions of narrow supports and a finite length MCMC chain is unlikely to cover pertinent regions. The present study proposes strategies based on modification of measurement equations and subsequent corrections, to alleviate this difficulty. This involves artificial enhancement of measurement noise, assimilation of transformed packets of measurements, and a global iteration strategy to improve the choice of prior models. Illustrative examples cover laboratory studies on a time variant dynamical system and a bending-torsion coupled, geometrically non-linear building frame under earthquake support motions. (C) 2015 Elsevier Ltd. All rights reserved.

Model and parameter uncertainty in IDF relationships under climate change

Quantifying distributional behavior of extreme events is crucial in hydrologic designs. Intensity Duration Frequency (IDF) relationships are used extensively in engineering especially in urban hydrology, to obtain return level of extreme rainfall event for a specified return period and duration. Major sources of uncertainty in the IDF relationships are due to insufficient quantity and quality of data leading to parameter uncertainty due to the distribution fitted to the data and uncertainty as a result of using multiple GCMs. It is important to study these uncertainties and propagate them to future for accurate assessment of return levels for future. The objective of this study is to quantify the uncertainties arising from parameters of the distribution fitted to data and the multiple GCM models using Bayesian approach. Posterior distribution of parameters is obtained from Bayes rule and the parameters are transformed to obtain return levels for a specified return period. Markov Chain Monte Carlo (MCMC) method using Metropolis Hastings algorithm is used to obtain the posterior distribution of parameters. Twenty six CMIP5 GCMs along with four RCP scenarios are considered for studying the effects of climate change and to obtain projected IDF relationships for the case study of Bangalore city in India. GCM uncertainty due to the use of multiple GCMs is treated using Reliability Ensemble Averaging (REA) technique along with the parameter uncertainty. Scale invariance theory is employed for obtaining short duration return levels from daily data. It is observed that the uncertainty in short duration rainfall return levels is high when compared to the longer durations. Further it is observed that parameter uncertainty is large compared to the model uncertainty. (C) 2015 Elsevier Ltd. All rights reserved.

Reversible jump sampler for autoregressive time series

We use reversible jump Markov chain Monte Carlo (MCMC) methods to address the problem of model order uncertainty in autoregressive (AR) time series within a Bayesian framework. Efficient model jumping is achieved by proposing model space moves from the full conditional density for the AR parameters, which is obtained analytically. This is compared with an alternative method, for which the moves are cheaper to compute, in which proposals are made only for new parameters in each move. Results are presented for both synthetic and audio time series.

Bayesian learning of noisy Markov decision processes

This work addresses the problem of estimating the optimal value function in a Markov Decision Process from observed state-action pairs. We adopt a Bayesian approach to inference, which allows both the model to be estimated and predictions about actions to be made in a unified framework, providing a principled approach to mimicry of a controller on the basis of observed data. A new Markov chain Monte Carlo (MCMC) sampler is devised for simulation from theposterior distribution over the optimal value function. This step includes a parameter expansion step, which is shown to be essential for good convergence properties of the MCMC sampler. As an illustration, the method is applied to learning a human controller.