Tests for harmonic components in the spectra of categorical time series
Ensor, Katherine B.
Doctor of Philosophy thesis
The main purpose of spectral analysis in time series is to determine what patterns exist in a particular set of data. To accomplish this, one often calculates the Fourier periodogram of the data and inspects it for peaks. However, since the periodogram is not a consistent estimate of the true spectral density, peaks can be obscured. Therefore, it is necessary to test for significance of the peaks. One of the most widely used tests for significance of periodicities in the periodogram is Fisher's test. The test statistic for Fisher's test is the quotient of the maximum periodogram ordinate and the sum of all the ordinates. If the test statistic is too large, we reject the null hypothesis that the data are white noise. In this thesis, I develop a test for categorical time series which is an analog of Fisher's test for continuous parameter time series. The test involves finding the Walsh-Fourier periodogram of the data and then calculating the test statistic for Fisher's test. I explain the theory behind Walsh-Fourier analysis and compare the theory to that of Fourier analysis. Asymptotic results for the distribution of Fisher's test for Walsh-Fourier spectra are presented and compared with a simulated distribution. I also perform power studies in order to assess the detection capability of the test. In the presence of multiple peaks, this test tends to loose power. Therefore, I also explore several alternatives to Fisher's test for Walsh-Fourier spectra and apply all of the alternative methods to several real data sets.