On the usage of spectral matrices to enable the graphical modelling of time series
Creators: name: Zhuang, Linjie
Publisher: name: Imperial College London
Year: 2017

description: In this thesis, we look at spectral matrices estimation, combining and clustering for multidimensional time series. The above spectral matrices analysis is applied in building brain connectivity graphical models for two patient groups: those diagnosed with positive or negative syndrome schizophrenia. The 10-channel EEG dataset used here represents cortical activity and is modelled as 10-dimensional stationary time series. Multitaper spectral matrix estimation is used in the spectral clustering step. Inspired by the orthogonality of multiple data tapers used in this method, we design a data taper that is orthogonal to its shifts. This new data taper can be applied in WOSA spectral matrix estimation and reduce the variance of the WOSA estimators. The data taper is solved as the optimization of a quadratic constrained quadratic optimization problem. The resultant data taper reduces WOSA spectral estimation at the expense of side-lobe spectral leakage. So it is not worth implementing in high dynamic range time series . Then we move our focus to combining spectral matrices and increasing the degrees of freedom of individual estimated spectral matrices. Only those spectral matrices which are not statistically different from the rest will be combined to form the group representative spectral matrix. To measure the distance between spectral matrices, the Kullback-Leibler divergence measure and the Riemannian distance are examined and compared. Both disparity measures integrate the distance across frequencies. We show that the square root of the Kullback-Leibler divergence is comparable to the Riemannian distance. The distance matrices based on these disparity measures are defined and used in different clustering techniques to identify the outliers in each group. In particular, the normalized graph Laplacian is defined based on the distance matrices and is used in spectral clustering. Spectral clustering determines the number as well as the structure of the clustering. Two outlying individuals are identified via clustering and removed in each group. After excluding outlying individuals, we combine the spectral matrices while taking the geometric structure of the space of positive definite Hermitian matrices into account. The estimated spectral matrices are modelled as complex Wishart matrices. We derive several properties of a sample Riemannian mean, including the determinant, the expectation and the asymptotic properties. We compare two unbiased population mean estimators, namely the debiased sample Riemannian mean and the standard sample mean, via classical non-intrinsic statistic analysis. The risk under convex loss functions for the standard sample mean is never larger than for the debiased sample mean. In simulations, the sample mean also performs better for the estimation of partial coherence (determining the edges of graph modelling). We also derive simple expressions for the intrinsic bias for both sample means. The standard sample mean is preferable. The same is true for the asymptotic Riemannian risk. Hence, we conclude that the standard sample mean is preferred overall. For each possible connection in the graph, the partial coherence estimated from sample spectral matrices has to be zero-tested at every Nyquist frequency across the delta frequency band. We define a test statistic that averages the partial coherence across frequencies. However, the null distribution of the test statistic when the partial coherence are obtain from the standard sample mean is not known. Hence, we combine the individual graphs as follows. The p-values for individuals in a group are used in a multiple hypothesis test. Then for each edge, it is included if the proportion of rejection exceeds a given threshold. Group-specific graphs for positive and negative syndromes are constructed.