Least Squares Wavelet Analysis

English Abstract:

The least-squares spectral analysis (LSSA), an alternative to the classical Fourier analysis, was introduced by Vanicek in order to analyze unequally spaced and non-stationary time series in their first and second statistical moments. When a time series has low or high frequency and amplitude variation over time, however, both the LSSA and Fourier analysis are not appropriate tools for analysis. On the other hand, the classical short-time Fourier transform (STFT) and the continuous wavelet transform (CWT) do not consider the covariance matrix associated with a time series nor do they consider trends or datum shifts. Both, the STFT and the CWT are not defined for unequally spaced time series.

In this talk, we present a new method called the least-squares wavelet analysis (LSWA) that can analyze a non-stationary and unequally spaced time series with high frequency and amplitude variation over time by transforming the time series to the time-frequency domain. The LSWA is a powerful method for analyzing an unequally spaced and unequally weighted time series with associated covariance matrix superseding the well-known wavelet analysis. Several examples from artificial and real time series will be presented to demonstrate the effectiveness of the method.