A partial spectrum approach to the analysis of quasi-stationary time series

The notion of a stationary random function is generalized to include random functions whose statistical properties vary slowly with time. A criterion for quasi-stationarity is proposed, and two methods for the spectral analysis of quasi-stationary time series are presented. The first of these, the method of partial series, is equivalent to treating the series as stationary in each of several subseries. The second, the method of partial spectra, involves an expansion of the time dependent local energy spectrum in orthogonal functions of the interval of analysis. An estimate of the coefficient of the nth-order function is given by the cosine transform of the timewise cross correlation of the series with the product of the series and the nth-order function, The statistical reliability of this estimate and of the reconstructed spectral estimate is investigated, and a numerical example from a field study of the wind generation of ocean waves is presented.