Seizure characterisation using frequency-dependent multivariate dynamics

The characterisation of epileptic seizures assists in the design of targeted pharmaceutical seizure prevention techniques and pre-surgical evaluations. In this paper, we expand on recent use of multivariate techniques to study the crosscorrelation dynamics between electroencephalographic (EEG) channels. The Maximum Overlap Discrete Wavelet Transform (MODWT) is applied in order to separate the EEG channels into their underlying frequencies. The dynamics of the cross-correlation matrix between channels, at each frequency, are then analysed in terms of the eigenspectrum. By examination of the eigenspectrum, we show that it is possible to identify frequency dependent changes in the correlation structure between channels which may be indicative of seizure activity. The technique is applied to EEG epileptiform data and the results indicate that the correlation dynamics vary over time and frequency, with larger correlations between channels at high frequencies. Additionally, a redistribution of wavelet energy is found, with increased fractional energy demonstrating the relative importance of high frequencies during seizures. Dynamical changes also occur in both correlation and energy at lower frequencies during seizures, suggesting that monitoring frequency dependent correlation structure can characterise changes in EEG signals during these. Future work will involve the study of other large eigenvalues and inter-frequency correlations to determine additional seizure characteristics

Multiscaled Cross-Correlation Dynamics in Financial Time-Series

The cross correlation matrix between equities comprises multiple interactions between traders with varying strategies and time horizons. In this paper, we use the Maximum Overlap Discrete Wavelet Transform to calculate correlation matrices over different timescales and then explore the eigenvalue spectrum over sliding time windows. The dynamics of the eigenvalue spectrum at different times and scales provides insight into the interactions between the numerous constituents involved. Eigenvalue dynamics are examined for both medium and high-frequency equity returns, with the associated correlation structure shown to be dependent on both time and scale. Additionally, the Epps effect is established using this multivariate method and analyzed at longer scales than previously studied. A partition of the eigenvalue time-series demonstrates, at very short scales, the emergence of negative returns when the largest eigenvalue is greatest. Finally, a portfolio optimization shows the importance of timescale information in the context of risk management

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