Non-stationary time series analysis by Haar-like transforms: The multivariate electroencephalogram.

The multivariate time series techniques in this thesis were developed for the analysis of the electroencephalogram (EEG), characterized by its non-stationarity and multiplicity of channels. Non-stationarity is caused by bursts at approximately octave spaced frequencies. In the absence of a model with physiological parameters, EEG analysis aims to reduce the data to a representation which may be related to the subject's physiological state by statistical inference. The real-time analysis of many channels demands efficient data reduction. Some non-stationary theory is examined, in particular the concept of an evolutionary spectrum. The Haar transform is seen to provide an efficient but crude evolutionary spectral decomposition into octave frequency bands. The phase-sensitivity of Haar spectral estimates leads to two new transforms, the Fourier-Haar (FHT) and the Hadamard-Haar (HHT). The similarity between spectra produced by the EHT and the much more efficient HHT suggests that for highly non-stationary data, the choice of a sinusoid or square wave decomposition is irrelevant. Sensitive tests for bursts or changes in level can be based on the cumulative sums of Haar-like estimates for particular frequencies. However, the joint behaviour of all frequency components may be examined using the multivariate analysis of variance (MANOVA), MANOVA is performed on different samples, corresponding to different experimental conditions, of the Hadamard-Haar representation of a two channel EEG. A flexible method for deriving discriminant functions is described and is used to identify different physiological states. Although the new Haar-like representations are specialized, their usefulness may extend to a wide range of highly non-stationary time series. Their efficiency and excellent convergence properties make them well suited to data compression applications