Principal Network Analysis

Many applications collect a large number of time series, for example, temperature continuously monitored by weather stations across the US or neural activity recorded by an array of electrical probes. These data are often referred to as unstructured. A first task in their analytics is often to derive a low dimensional representation { a graph or discrete manifold { that describes the inter relations among the time series and their intrarelations across time. In general, the underlying graphs can be directed and weighted, possibly capturing the strengths of causal relations, not just the binary existence of reciprocal correlations. Furthermore, the processes generating the data may be non-linear and observed in the presence of unmodeled phenomena or unmeasured agents in a complex networked system. Finally, the networks describing the processes may themselves vary through time. In many scenarios, there may be good reasons to believe that the graphs are only able to vary as linear combinations of a set of \principal graphs" that are fundamental to the system. We would then be able to characterize each principal network individually to make sense of the ensemble and analyze the behaviors of the interacting entities. This thesis acts as a roadmap of computationally tractable approaches for learning graphs that provide structure to data. It culminates in a framework that addresses these challenges when estimating time-varying graphs from collections of time series. Analyses are carried out to justify the various models proposed along the way and to characterize their performance. Experiments are performed on synthetic and real datasets to highlight their effectiveness and to illustrate their limitations.