Summary
The assumption of stationarity, when it is appropriate, is a very useful one. Stationarity provides the basis on which one can compare ensemble averages to time averages and introduce the notion of ergodicity, which is very important for the theoretical structure of time series analysis. However, like most assumptions, stationarity is an idealization which is not always justified from the data. In some instances, small intervals of data can be considered approximately stationary, although the small interval size may affect the resolution in frequency. In ocean wave data analysis this may occur as a hurricane approaches a selected location. The wave spectrum is changing so rapidly that the customary Fourier methods are inadequate for the analysis. The same occurs in other fields, including communications and speech processing among others. As early as in 1952 this demand has led researchers to seek for methods that can be tolerated weakening of the stationarity assumption.
For this project Fourier methods for the estimation of the frequency content are combined with splines to describe the time variation, with the ultimate goal of obtaining the time varying Fourier coefficients of the wave properties. The result is a method presented which we call evolutionary Fourier analysis. Some applications of the method using separable processes, wave tank data and measurements taken during the Hurricane Camille are included.
