Evolutive instantaneous spectrum associated with the partial autocorrelation function for nonstationary time series

Abstract

The partial autocorrelation function (PACF) of a nonstationary time series is presented. This function characterizes the second order properties of the process but is easily identifiable by comparison with the classical autocovariance function (ACF) which must be nonnegative definite (n.n.d.). As in the stationary case, this parametrization is well adapted to the autoregressive processes. It is also an elegant tool for studying the periodically correlated processes. Nevertheless, our main result is the introduction of a new time-dependent power spectrum clearly related to the PACF. At each time, this spectrum describes a stationary situation in which the present is correlated with the past in the same way as our nonstationary process at this time. The properties of this spectrum are analyzed and the comparison with two similar other spectra is made. Sampled Brownian motion and linear "chirps" are considered for illustration.