Adaptive Fourier Analysis For Unequally-Spaced Time Series Data

Abstract

Fourier analysis, Walsh-Fourier analysis, and wavelet analysis have often been used in time series analysis. Fourier analysis can be used to detect periodic components that have sinusoidal shape; however, it might be misleading when the periodic components are not sinusoidal. Walsh-Fourier analysis is suitable for revealing the rectangular trends of time series. The flaw of the Walsh-Fourier analysis is that Walsh functions are not periodic. The resulting Walsh-Fourier analysis is more difficult to interpret than classical Fourier analysis. Wavelet analysis is very useful in analyzing and describing time series with gradual frequency changes. Wavelet analysis also has a shortcoming by giving no exact meaning to the concept of frequency because wavelets are not periodic functions. In addition, all three analysis methods above require equally-spaced time series observations.

In this dissertation, by using a sequence of periodic step functions, a new analysis method, adaptive Fourier analysis, and its theory are developed. These can be applied to time series data where patterns may take general periodic shapes that include sinusoids as special cases. Most importantly, the resulting adaptive Fourier analysis does not require equally-spaced time series observations.