Detecting common signals in multiple time series using the spectral envelope

One often collects p individual time series Y-j(t) for j = 1,..., p, where the interest is to discover whether any-and which-of the series contain common signals. Let Y(t) = (Y-1(t),...,Y-p(t))' denote the corresponding p x 1 vector-valued time series with p x p positive definite spectral matrix f(Y)(w). Models are proposed to answer the primary question of which, if any, series have common Spectral power at approximately the same frequency. These models yield a type of complex factor analytic representation for f(Y)(w). A scaling approach to the problem is taken by considering possibly complex linear combinations of the components of Y(t). The solution leads to an eigenvalue-eigenvector problem that is analogous to the spectral envelope and optimal scaling methodology first presented by Stoffer, Tyler, and McDougall. The viability of the techniques is demonstrated by analyzing data from an experiment that assessed pain perception in humans and by analyzing data from a study of ambulatory blood pressure in a cohort of preteens.