Time Series Analysis by Projection. I. Statistical Properties of Fourier Analysis

Abstract

Most time series analysis methods are equivalent to treating the data as a vector in function space, then projecting the data vector onto a subspace of low dimension. By casting various Fourier methods in terms of projection, we can make their behaviors transparent and adapt them to time series with irregular time spacing. Exact statistics are derived for the discrete Fourier transform, the Lomb-Scargle modified periodogram [Lomb, APSS, 39, 447 (1976); Scargle, APJ, 263, 835 (1982)], and the date-compensated discrete Fourier transform [Ferraz-Mello; AJ, 86, 619 (1981)]. Fourier methods have an extra complication, that they are not merely projections but parametric projections. As a consequence, the standard statistical evaluation of Fourier analysis (and most period-search methods) is incomplete.