Spectral Approach to Modeling Dependence in Multivariate Time Series

Consider a multivariate time series such as prices of stocks from various sectors, amount of rainfall in many geographical locations, and brain signals from many different locations on the scalp. The goal of this paper is to present the spectral approach to modeling dependence between components of the multivariate time series. There are many measures of dependence-the most popular being cross-correlation or partial cross-correlation. This measure is easy to compute and easy to understand but it coarse in a sense that it is not able to identify the underlying frequencies that are responsible for driving the dependence. In the stock price example, two stocks may be highly correlated but it would be helpful to see if this correlation is driven by the daily fluctuations or by millisecond-level fluctuations. In the neuroscience example, when two brain regions exhibit a high level of correlation, it will be important if this synchronicity is due to low-frequency oscillations or high-frequency oscillations. Here we present an overview of the underlying principles through specific spectral models which decompose the signals into oscillations of various frequencies and then model lead-lag dependence via these oscillations.