Bayesian learning of continuous time dynamical systems with applications in functional magnetic resonance imaging

Temporal phenomena in a range of disciplines are more naturally modelled in continuous-time than coerced into a discrete-time formulation. Differential systems form the mainstay of such modelling, in fields from physics to economics, geoscience to neuroscience. While powerful, these are fundamentally limited by their determinism. For the purposes of probabilistic inference, their extension to stochastic differential equations permits a continuous injection of noise and uncertainty into the system, the model, and its observation. This thesis considers Bayesian filtering for state and parameter estimation in general non-linear, non-Gaussian systems using these stochastic differential models. It identifies a number of challenges in this setting over and above those of discrete time, most notably the absence of a closed form transition density. These are addressed via a synergy of diverse work in numerical integration, particle filtering and high performance distributed computing, engineering novel solutions for this class of model. In an area where the default solution is linear discretisation, the first major contribution is the introduction of higher-order numerical schemes, particularly stochastic Runge-Kutta, for more efficient simulation of the system dynamics. Improved runtime performance is demonstrated on a number of problems, and compatibility of these integrators with conventional particle filtering and smoothing schemes discussed. Finding compatibility for the smoothing problem most lacking, the major theoretical contribution of the work is the introduction of two novel particle methods, the kernel forward-backward and kernel two-filter smoothers. By harnessing kernel density approximations in an importance sampling framework, these attain cancellation of the intractable transition density, ensuring applicability in continuous time. The use of kernel estimators is particularly amenable to parallelisation, and provides broader support for smooth densities than a sample-based representation alone, helping alleviate the well known issue of degeneracy in particle smoothers. Implementation of the methods for large-scale problems on high performance computing architectures is provided. Achieving improved temporal and spatial complexity, highly favourable runtime comparisons against conventional techniques are presented. Finally, attention turns to real world problems in the domain of Functional Magnetic Resonance Imaging (fMRI), first constructing a biologically motivated stochastic differential model of the neural and hemodynamic activity underlying the observed signal in fMRI. This model and the methodological advances of the work culminate in application to the deconvolution and effective connectivity problems in this domain

Modeling brain connectivity dynamics in functional Magnetic Resonance Imaging via Particle Filtering

Interest in the studying of functional connections in the brain has grown considerably in the last decades, as many studies have pointed out that alterations in the interaction among brain areas can play a role as markers of neurological diseases. Most studies in this field treat the brain network as a system of connections stationary in time, but dynamic features of brain connectivity can provide useful information, both on physiology and pathological conditions of the brain. In this paper, we propose the application of a computational methodology, named Particle Filter (PF), to study non-stationarities in brain connectivity in functional Magnetic Resonance Imaging (fMRI). The PF algorithm estimates time-varying hidden parameters of a first-order linear time-varying Vector Autoregressive model (VAR) through a Sequential Monte Carlo strategy. On simulated time series, the PF approach effectively detected and enabled to follow time-varying hidden parameters and it captured causal relationships among signals. The method was also applied to real fMRI data, acquired in presence of periodic tactile or visual stimulations, in different sessions. On these data, the PF estimates were consistent with current knowledge on brain functioning. Most importantly, the approach enabled to detect statistically significant modulations in the cause-effect relationship between brain areas, which correlated with the underlying visual stimulation pattern presented during the acquisition

Sequential Monte Carlo methods: applications to disease surveillance and fMRI data

We present contributions to epidemic tracking and analysis of fMRI data using sequential Monte Carlo methods within a state-space modeling framework. Using a model for tracking and prediction of a disease outbreak via a syndromic surveillance system, we compare the performance of several particle filtering algorithms in terms of their abilities to efficiently estimate disease states and unknown fixed parameters governing disease transmission. In this context, we demonstrate that basic particle filters may fail due to degeneracy when estimating fixed parameters, and we suggest the use of an algorithm developed by Liu and West (2001), which incorporates a kernel density approximation to the filtered distribution of the fixed parameters to allow for their regeneration. In addition, we show that seemingly uninformative uniform priors on fixed parameters can affect posterior inferences, and we suggest the use of priors bounded only by the support of the parameter. We demonstrate the negative impact of using multinomial resampling and suggest the use of either stratified or residual resampling within the particle filter. We also run a particle MCMC algorithm and show that the performance of the Liu and West (2001) particle filter is competitive with particle MCMC in this particular syndromic surveillance model setting. Finally, the improved performance of the Liu and West (2001) particle filter enables us to relax prior assumptions on model parameters, yet still provide reasonable estimates for model parameters and disease states.We also analyze real and simulated fMRI data using a state-space formulation of a regression model with autocorrelated error structure. We demonstrate via simulation that analyzing autocorrelated fMRI data using a model with independent error structure can inflate the false positive rate of concluding significant neural activity, and we compare methods of accounting for autocorrelation in fMRI data by examining ROC curves. In addition, we show that comparing models with different autocorrelated error structures on the basis of the independence of fitted model residuals can produce misleading results. Using data collected from an fMRI experiment featuring an episodic word recognition task, we estimate parameters in dynamic regression models using maximum likelihood and identify clusters of low and high activation in specific brain regions. We compare alternative models for fMRI time series from these brain regions by approximating the marginal likelihood of the data using particle learning. Our results suggest that a regression model with a dynamic intercept is the preferred model for most fMRI time series in the episodic word recognition experiment within the brain regions we considered, while a model with a dynamic slope is preferred for a small percentage of voxels in these brain regions

Approximate inference for state-space models

This thesis is concerned with state estimation in partially observed diffusion processes with discrete time observations. This problem can be solved exactly in a Bayesian framework, up to a set of generally intractable stochastic partial differential equations. Numerous approximate inference methods exist to tackle the problem in a practical way. This thesis introduces a novel deterministic approach that can capture non normal properties of the exact Bayesian solution. The variational approach to approximate inference has a natural formulation for partially observed diffusion processes. In the variational framework, the exact Bayesian solution is the optimal variational solution and, as a consequence, all variational approximations have a universal ordering in terms of optimality. The new approach generalises the current variational Gaussian process approximation algorithm, and therefore provides a method for obtaining super optimal algorithms in relation to the current state-of-the-art variational methods. Every diffusion process is composed of a drift component and a diffusion component. To obtain a variational formulation, the diffusion component must be fixed. Subsequently, the exact Bayesian solution and all variational approximations are characterised by their drift component. To use a particular class of drift, the variational formulation requires a closed form for the family of marginal densities generated by diffusion processes with drift components from the aforementioned class. This requirement in general cannot be met. In this thesis, it is shown how this coupling can be weakened, allowing for more flexible relations between the variational drift and the variational approximations of the marginal densities of the true posterior process. Based on this revelation, a selection of novel variational drift components are proposed

Towards Bayesian System Identification: With Application to SHM of Offshore Structures

Within the offshore industry Structural Health Monitoring remains a growing area of interest. The oil and gas sectors are faced with ageing infrastructure and are driven by the desire for reliable lifetime extension, whereas the wind energy sector is investing heavily in a large number of structures. This leads to a number of distinct challenges for Structural Health Monitoring which are brought together by one unifying theme --- uncertainty. The offshore environment is highly uncertain, existing structures have not been monitored from construction and the loading and operational conditions they have experienced (among other factors) are not known. For the wind energy sector, high numbers of structures make traditional inspection methods costly and in some cases dangerous due to the inaccessibility of many wind farms. Structural Health Monitoring attempts to address these issues by providing tools to allow automated online assessment of the condition of structures to aid decision making. The work of this thesis presents a number of Bayesian methods which allow system identification, for Structural Health Monitoring, under uncertainty. The Bayesian approach explicitly incorporates prior knowledge that is available and combines this with evidence from observed data to allow the formation of updated beliefs. This is a natural way to approach Structural Health Monitoring, or indeed, many engineering problems. It is reasonable to assume that there is some knowledge available to the engineer before attempting to detect, locate, classify, or model damage on a structure. Having a framework where this knowledge can be exploited, and the uncertainty in that knowledge can be handled rigorously, is a powerful methodology. The problem being that the actual computation of Bayesian results can pose a significant challenge both computationally and in terms of specifying appropriate models. This thesis aims to present a number of Bayesian tools, each of which leverages the power of the Bayesian paradigm to address a different Structural Health Monitoring challenge. Within this work the use of Gaussian Process models is presented as a flexible nonparametric Bayesian approach to regression, which is extended to handle dynamic models within the Gaussian Process NARX framework. The challenge in training Gaussian Process models is seldom discussed and the work shown here aims to offer a quantitative assessment of different learning techniques including discussions on the choice of cost function for optimisation of hyperparameters and the choice of the optimisation algorithm itself. Although rarely considered, the effects of these choices are demonstrated to be important and to inform the use of a Gaussian Process NARX model for wave load identification on offshore structures. The work is not restricted to only Gaussian Process models, but Bayesian state-space models are also used. The novel use of Particle Gibbs for identification of nonlinear oscillators is shown and modifications to this algorithm are applied to handle its specific use in Structural Health Monitoring. Alongside this, the Bayesian state-space model is used to perform joint input-state-parameter inference for Operational Modal Analysis where the use of priors over the parameters and the forcing function (in the form of a Gaussian Process transformed into a state-space representation) provides a methodology for this output-only identification under parameter uncertainty. Interestingly, this method is shown to recover the parameter distributions of the model without compromising the recovery of the loading time-series signal when compared to the case where the parameters are known. Finally, a novel use of an online Bayesian clustering method is presented for performing Structural Health Monitoring in the absence of any available training data. This online method does not require a pre-collected training dataset, nor a model of the structure, and is capable of detecting and classifying a range of operational and damage conditions while in service. This leaves the reader with a toolbox of methods which can be applied, where appropriate, to identification of dynamic systems with a view to Structural Health Monitoring problems within the offshore industry and across engineering

Gaussian processes for state space models and change point detection

This thesis details several applications of Gaussian processes (GPs) for enhanced time series modeling. We first cover different approaches for using Gaussian processes in time series problems. These are extended to the state space approach to time series in two different problems. We also combine Gaussian processes and Bayesian online change point detection (BOCPD) to increase the generality of the Gaussian process time series methods. These methodologies are evaluated on predictive performance on six real world data sets, which include three environmental data sets, one financial, one biological, and one from industrial well drilling. Gaussian processes are capable of generalizing standard linear time series models. We cover two approaches: the Gaussian process time series model (GPTS) and the autoregressive Gaussian process (ARGP). We cover a variety of methods that greatly reduce the computational and memory complexity of Gaussian process approaches, which are generally cubic in computational complexity. Two different improvements to state space based approaches are covered. First, Gaussian process inference and learning (GPIL) generalizes linear dynamical systems (LDS), for which the Kalman filter is based, to general nonlinear systems for nonparametric system identification. Second, we address pathologies in the unscented Kalman filter (UKF). We use Gaussian process optimization (GPO) to learn UKF settings that minimize the potential for sigma point collapse. We show how to embed mentioned Gaussian process approaches to time series into a change point framework. Old data, from an old regime, that hinders predictive performance is automatically and elegantly phased out. The computational improvements for Gaussian process time series approaches are of even greater use in the change point framework. We also present a supervised framework learning a change point model when change point labels are available in training.I would like to thank Rockwell Collins, formerly DataPath, Inc., which funded my studentship

The uncertainty of changepoints in time series

Analysis concerning time series exhibiting changepoints have predominantly focused on detection and estimation. However, changepoint estimates such as their number and location are subject to uncertainty which is often not captured explicitly, or requires sampling long latent vectors in existing methods. This thesis proposes efficient, flexible methodologies in quantifying the uncertainty of changepoints. The core proposed methodology of this thesis models time series and changepoints under a Hidden Markov Model framework. This methodology combines existing work on exact changepoint distributions conditional on model parameters with Sequential Monte Carlo samplers to account for parameter uncertainty. The combination of the two provides posterior distributions of changepoint characteristics in light of parameter uncertainty. This thesis also presents a methodology in approximating the posterior of the number of underlying states in a Hidden Markov Model. Consequently, model selection for Hidden Markov Models is possible. This methodology employs the use of Sequential Monte Carlo samplers, such that no additional computational costs are incurred from the existing use of these samplers. The final part of this thesis considers time series in the wavelet domain, as opposed to the time domain. The motivation for this transformation is the occurrence of autocovariance changepoints in time series. Time domain modelling approaches are somewhat limited for such types of changes, with approximations often taking place. The wavelet domain relaxes these modelling limitations, such that autocovariance changepoints can be considered more readily. The proposed methodology develops a joint density for multiple processes in the wavelet domain which can then be embedded within a Hidden Markov Model framework. Quantifying the uncertainty of autocovariance changepoints is thus possible. These methodologies will be motivated by datasets from Econometrics, Neuroimaging and Oceanography

Untangling hotel industry’s inefficiency: An SFA approach applied to a renowned Portuguese hotel chain

The present paper explores the technical efficiency of four hotels from Teixeira Duarte Group - a renowned Portuguese hotel chain. An efficiency ranking is established from these four hotel units located in Portugal using Stochastic Frontier Analysis. This methodology allows to discriminate between measurement error and systematic inefficiencies in the estimation process enabling to investigate the main inefficiency causes. Several suggestions concerning efficiency improvement are undertaken for each hotel studied.info:eu-repo/semantics/publishedVersio

Recent Advances in Signal Processing

The signal processing task is a very critical issue in the majority of new technological inventions and challenges in a variety of applications in both science and engineering fields. Classical signal processing techniques have largely worked with mathematical models that are linear, local, stationary, and Gaussian. They have always favored closed-form tractability over real-world accuracy. These constraints were imposed by the lack of powerful computing tools. During the last few decades, signal processing theories, developments, and applications have matured rapidly and now include tools from many areas of mathematics, computer science, physics, and engineering. This book is targeted primarily toward both students and researchers who want to be exposed to a wide variety of signal processing techniques and algorithms. It includes 27 chapters that can be categorized into five different areas depending on the application at hand. These five categories are ordered to address image processing, speech processing, communication systems, time-series analysis, and educational packages respectively. The book has the advantage of providing a collection of applications that are completely independent and self-contained; thus, the interested reader can choose any chapter and skip to another without losing continuity

Learning Interpretable Features of Graphs and Time Series Data

Graphs and time series are two of the most ubiquitous representations of data of modern time. Representation learning of real-world graphs and time-series data is a key component for the downstream supervised and unsupervised machine learning tasks such as classification, clustering, and visualization. Because of the inherent high dimensionality, representation learning, i.e., low dimensional vector-based embedding of graphs and time-series data is very challenging. Learning interpretable features incorporates transparency of the feature roles, and facilitates downstream analytics tasks in addition to maximizing the performance of the downstream machine learning models. In this thesis, we leveraged tensor (multidimensional array) decomposition for generating interpretable and low dimensional feature space of graphs and time-series data found from three domains: social networks, neuroscience, and heliophysics. We present the theoretical models and empirical results on node embedding of social networks, biomarker embedding on fMRI-based brain networks, and prediction and visualization of multivariate time-series-based flaring and non-flaring solar events