Evolutionary Fourier Analysis

The convenience of the assumption of stationarity in FFT spectral analysis encourages the routine use of that assumption in processing data. The time series is taken to be ``pseudo-stationary'' even though changes in time are probably present to some extent. The possible existence of nonstationarity is examined by looking for differences between the spectra in one time interval as compared to the next.

In a statistical sense, this is incredibly crude! It would be like fitting a curve y(x_i) through a set of values $(x_1, x_2, x_3, \ldots, x_n)$ by subdividing the range of x-values into discrete intervals, averaging the y(x_i) values in each interval, plotting the average at the midpoint of the interval, and connecting the means to get the predictor curve for y(x). Modern least-square regression gives a much more sophisticated estimation process. The interval or bin method can be quite useful as a nonparametric procedure for data exploration, but even there, other procedures based on kernals or the technique of kriging are preferable.

There is a very simple way to bring curve fitting methods into spectral estimation. This can be applied in so-called ``interval-averaging'' analysis. In this mode of analysis, the data sequence introduced in (1) is subdivided into K equal length increments of length N_K = N/K. For example, N = 1024 might be broken up into 8 intervals of length 128. Let $A_k(m\Delta f)$ be the FFT of the data in the k-th interval for $1 \leq k \leq K$,according to Eqs. (2) and (3) with N replaced by N_K. The usual spectral estimate at frequency $m \Delta f$ is

 

$$\hat{S}(m\Delta f) = \frac1K \sum^K_{k=1} \frac{\vert A_k (m\Delta f)\vert^2}{N_K \Delta t}$$ (5)

(Earle and McGehee [9]).

Curve fitting methods for nonstationarity can be introduced by setting

 

$$x_k = m\Delta f = \frac{m}{N_K\Delta t}$$ (6)

 

$$y_k = \frac{\left\vert A_k (m\Delta f) \right\vert^2}{N_K \Delta t}$$ (7)

and then fitting a straight line (or other selected polynomial) through $[(x_k, y_k); k= 1,2,3,\ldots, K]$. The fit for successive intervals of length N, could be improved by making them into first-order splines and requiring that the lines patch together at the ``knot'' points between the intervals to maintain continuity. This would give a piecewise linear continuous representation of the evolutionary spectra at each frequency, and allow nonstationarity on a minute-to-minute scale.

More sophisticated methods for estimating evolutionary spectra have been proposed based on filter theory [15] and splines [5,11,14].