Regularised estimation of 2D-locally stationary wavelet processes

Locally Stationary Wavelet processes provide a flexible way of describing the time/space evolution of autocovariance structure over an ordered field such as an image/time-series. Classically, estimation of such models assume continuous smoothness of the underlying spectra and are estimated via local kernel smoothers. We propose a new model which permits spectral jumps, and suggest a regularised estimator and algorithm which can recover such structure from images. We demonstrate the effectiveness of our method in a synthetic experiment where it shows desirable estimation properties. We conclude with an application to real images which illustrate the qualitative difference between the proposed and previous methods

Semi-local scaling exponent estimation with box-penalty constraints and total-variation regularisation

We here establish and exploit the result that 2-D isotropic self-similar fields beget quasi-decorrelated wavelet coefficients and that the resulting localised log sample second moment statistic is asymptotically normal. This leads to the development of a semi-local scaling exponent estimation framework with optimally modified weights. Furthermore, recent interest in penalty methods for least squares problems and generalised Lasso for scaling exponent estimation inspires the simultaneous incorporation of both bounding box constraints and total variation smoothing into an iteratively reweighted least-squares estimator framework. Numerical results on fractional Brownian fields with global and piecewise constant, semi-local Hurst parameters illustrate the benefits of the new estimators

Regularised inference for changepoint and dependency analysis in non-stationary processes

Multivariate correlated time series are found in many modern socio-scientific domains such as neurology, cyber-security, genetics and economics. The focus of this thesis is on efficiently modelling and inferring dependency structure both between data-streams and across points in time. In particular, it is considered that generating processes may vary over time, and are thus non-stationary. For example, patterns of brain activity are expected to change when performing different tasks or thought processes. Models that can describe such behaviour must be adaptable over time. However, such adaptability creates challenges for model identification. In order to perform learning or estimation one must control how model complexity grows in relation to the volume of data. To this extent, one of the main themes of this work is to investigate both the implementation and effect of assumptions on sparsity; relating to model parsimony at an individual time- point, and smoothness; how quickly a model may change over time. Throughout this thesis two basic classes of non-stationary model are stud- ied. Firstly, a class of piecewise constant Gaussian Graphical models (GGM) is introduced that can encode graphical dependencies between data-streams. In particular, a group-fused regulariser is examined that allows for the estima- tion of changepoints across graphical models. The second part of the thesis focuses on extending a class of locally-stationary wavelet (LSW) models. Un- like the raw GGM this enables one to encode dependencies not only between data-streams, but also across time. A set of sparsity aware estimators are developed for estimation of the spectral parameters of such models which are then compared to previous works in the domain

Case study:shipping trend estimation and prediction via multiscale variance stabilisation

<p>Shipping and shipping services are a key industry of great importance to the economy of Cyprus and the wider European Union. Assessment, management and future steering of the industry, and its associated economy, is carried out by a range of organisations and is of direct interest to a number of stakeholders. This article presents an analysis of shipping credit flow data: an important and archetypal series whose analysis is hampered by rapid changes of variance. Our analysis uses the recently developed data-driven Haar–Fisz transformation that enables accurate trend estimation and successful prediction in these kinds of situation. Our trend estimation is augmented by bootstrap confidence bands, new in this context. The good performance of the data-driven Haar–Fisz transform contrasts with the poor performance exhibited by popular and established variance stabilisation alternatives: the Box–Cox, logarithm and square root transformations.</p

The locally stationary dual-tree complex wavelet model

We here harmonise two significant contributions to the field of wavelet analysis in the past two decades, namely the locally stationary wavelet process and the family of dual-tree complex wavelets. By combining these two components, we furnish a statistical model that can simultaneously access benefits from these two constructions. On the one hand, our model borrows the debiased spectrum and auto-covariance estimator from the locally stationary wavelet model. On the other hand, the enhanced directional selectivity is obtained from the dual-tree complex wavelets over the regular lattice. The resulting model allows for the description and identification of wavelet fields with significantly more directional fidelity than was previously possible. The corresponding estimation theory is established for the new model, and some stationarity detection experiments illustrate its practicality

Line-Field Based Adaptive Image Model for Blind Deblurring

Ph.DDOCTOR OF PHILOSOPH

Can we identify non-stationary dynamics of trial-to-trial variability?"

Identifying sources of the apparent variability in non-stationary scenarios is a fundamental problem in many biological data analysis settings. For instance, neurophysiological responses to the same task often vary from each repetition of the same experiment (trial) to the next. The origin and functional role of this observed variability is one of the fundamental questions in neuroscience. The nature of such trial-to-trial dynamics however remains largely elusive to current data analysis approaches. A range of strategies have been proposed in modalities such as electro-encephalography but gaining a fundamental insight into latent sources of trial-to-trial variability in neural recordings is still a major challenge. In this paper, we present a proof-of-concept study to the analysis of trial-to-trial variability dynamics founded on non-autonomous dynamical systems. At this initial stage, we evaluate the capacity of a simple statistic based on the behaviour of trajectories in classification settings, the trajectory coherence, in order to identify trial-to-trial dynamics. First, we derive the conditions leading to observable changes in datasets generated by a compact dynamical system (the Duffing equation). This canonical system plays the role of a ubiquitous model of non-stationary supervised classification problems. Second, we estimate the coherence of class-trajectories in empirically reconstructed space of system states. We show how this analysis can discern variations attributable to non-autonomous deterministic processes from stochastic fluctuations. The analyses are benchmarked using simulated and two different real datasets which have been shown to exhibit attractor dynamics. As an illustrative example, we focused on the analysis of the rat's frontal cortex ensemble dynamics during a decision-making task. Results suggest that, in line with recent hypotheses, rather than internal noise, it is the deterministic trend which most likely underlies the observed trial-to-trial variability. Thus, the empirical tool developed within this study potentially allows us to infer the source of variability in in-vivo neural recordings

Self-Adaptive Methods for PDE

[no abstract available