## Description

This dissertation is divided in two parts. In Part I, we present a method to characterize functional connectivity between sites in the cerebral cortex of primates using a novel variation of undirected graph learning based on smoothness prior. In Part II, we define and implement a novel spatiotemporal graph (STG) model for inferring causally directed graphs. Analysis of brain connectivity networks has a potential to advance our understanding of the human brain and to offer improvements in the management of various neurological disorders. In Part I, connectivity maps for two macaque monkeys are inferred from Electrocorticographic (ECoG) activity recorded while the animals were alert. The locations of ECoG electrodes are considered as nodes of a graph, the coefficients of the auto-regressive (AR) representation of the signals measured at each node are considered as the signal on the graph and the connectivity strengths between the nodes are considered as the edges of the graph. Maximization of the graph smoothness defined by minimization the Laplacian quadratic form is used to infer the connectivity map (adjacency matrix of the graph). The cortical evoked potential (CEP) map, which is the measure of underlying physiological connectivity is used as the groundtruth. The maps obtained by the graph inference and the traditional method of spectral coherence are compared with the CEP map. The results show that the proposed method provides a description of cortical connectivity that is more similar to the stimulation-based measures than spectral coherence. The results are also tested by the surrogate map analysis in which the CEP map is randomly permuted and the distribution of the errors is obtained. It is shown that error between the two maps is comfortably outside the surrogate map error distribution. This indicates that the similarity between the map calculated by the graph inference and the CEP map is statistically significant. Causally related multivariate time series appear in many applications from economical systems to brain signal analysis. Inferring the causal relationships between the signals as a directed graph is the main contribution of Part II. The directions in the inferred graph represent the causal relationships. The proposed causal graph inference method considers the time samples of each signal in the multivariate time series as nodes in an undirected spatiotemporal graph (STG). This graph expansion is named temporal node mitosis. The resultant undirected spatiotemporal graph contains synchronic and dynamic sub-graphs. The synchronic sub-graphs model the contemporaneous and the dynamic sub-graphs model the temporal relationships between the signals in the multivariate time series. By using the concept of spatiotemporal stationarity and graph smoothness, the STG can be inferred using the factor analysis method introduced by Dong et al. considering the augmented Laplacian matrix of the STG. A method is proposed to construct the causally directed graph from the dynamic sub-graphs of the STG model. The performance analysis of the proposed model using simulations on synthetic and real data demonstrates the efficiency of the proposed model.