Multivariate Time and Frequency Domain Analysis

Abstract

The time series is called multivariate here if the number $M$ of its scalar components exceeds two. Multivariate models describe linear systems with one output and $M - 1$ inputs. Studying such series in time domain can be cumbersome, and the accent moves to frequency domain analysis based upon the $M \times M$ spectral matrix obtained through Fourier transform of the time domain model. For $M > 2$, the number of quantities that characterize the system increases due to the appearance of several new quantities: partial coherence functions and coherent spectra, gain and phase factors for every tract of the system plus additional spectral densities, one multiple coherence, and one multiple coherent spectrum. These functions produce frequency domain descriptions of each tract of the system and of the net effect of all input processes characterized with multiple coherences and spectra. The analysis takes into account linear relations between the input time series. An M-variate model AR(p) contains $M^{2} p$ autoregressive coefficients. An example of time and frequency domain analysis of a simulated trivariate time series shows how to estimate the time series properties and interpret the results. Analysis of global, oceanic, and terrestrial surface temperature data as a two-input system reveals a pazzling change in system’s properties.