Estimating the population local wavelet spectrum with application to non-stationary functional magnetic resonance imaging time series.

Functional magnetic resonance imaging (fMRI) is a dynamic four-dimensional imaging modality. However, in almost all fMRI analyses, the time series elements of this data are assumed to be second-order stationary. In this paper, we examine, using time series spectral methods, whether such stationary assumptions can be made and whether estimates of non-stationarity can be used to gain understanding into fMRI experiments. A non-stationary version of replicated stationary time series analysis is proposed that takes into account the replicated time series that are available from nearby voxels in a region of interest (ROI). These are used to investigate non-stationarities in both the ROI itself and the variations within the ROI. The proposed techniques are applied to simulated data and to an anxiety-inducing fMRI experiment.

A computationally efficient nonparametric approach for changepoint detection.

In this paper we build on an approach proposed by Zou et al. (2014) for nonparametric changepoint detection. This approach defines the best segmentation for a data set as the one which minimises a penalised cost function, with the cost function defined in term of minus a non-parametric log-likelihood for data within each segment. Minimising this cost function is possible using dynamic programming, but their algorithm had a computational cost that is cubic in the length of the data set. To speed up computation, Zou et al. (2014) resorted to a screening procedure which means that the estimated segmentation is no longer guaranteed to be the global minimum of the cost function. We show that the screening procedure adversely affects the accuracy of the changepoint detection method, and show how a faster dynamic programming algorithm, pruned exact linear time (PELT) (Killick et al. 2012), can be used to find the optimal segmentation with a computational cost that can be close to linear in the amount of data. PELT requires a penalty to avoid under/over-fitting the model which can have a detrimental effect on the quality of the detected changepoints. To overcome this issue we use a relatively new method, changepoints over a range of penalties (Haynes et al. 2016), which finds all of the optimal segmentations for multiple penalty values over a continuous range. We apply our method to detect changes in heart-rate during physical activity.

Multivariate locally stationary 2D wavelet processes with application to colour texture analysis.

In this article we propose a novel framework for the modelling of non-stationary multivariate lattice processes. Our approach extends the locally stationary wavelet paradigm into the multivariate two-dimensional setting. As such the framework we develop permits the estimation of a spatially localised spectrum within a channel of interest and, more importantly, a localised cross-covariance which describes the localised coherence between channels. Associated estimation theory is also established which demonstrates that this multivariate spatial framework is properly defined and has suitable convergence properties. We also demonstrate how this model-based approach can be successfully used to classify a range of colour textures provided by an industrial collaborator, yielding superior results when compared against current state-of-the-art statistical image processing methods.