A method is proposed for the analysis of time series without the assumption of stationarity. Although the developed method is for oceanographic applications, it is general and can be used for many cases of smoothly nonstationary time series or spatial data.
The data is divided into S segments of equal time length. The time dependent Fourier coefficients of the actual data are modeled using basis functions of order K of time for each segment. The segments of data are further divided into J intervals. It is shown that by solving N linear systems of J equations with K unknowns one can obtain the time-varying Fourier coefficients. The appropriate smoothing is provided using smoothing splines. A notable advantage of this method is that it provides the time-varying Fourier coefficients, which can then be used with various methods that have been developed for the analysis of stationary data. Indeed, once the time dependent Fourier coefficients have been obtained we can proceed and use them to estimate the time varying power spectrum, cross-spectrum or polyspectra.
For demonstration, some applications are provided, including an analysis of data from the Hurricane Camille and wave tank data from the Texas A&M (Offshore Technology Research Center) deep water tank.