Maximum Likelihood Estimation of Fourier Coefficients to Describe Seasonal Variations of Parameters in Stochastic Daily Precipitation Models

Abstract

Fourier series are convenient expressions for the seasonally fluctuating values of parameters in stochastic models of precipitation. Least-squares methods are often used to estimate the Fourier series coefficients, but this method has two important disadvantages. First the “data” points are in fact estimates of parameters, and because of varying sample size, they may have unequal variances and should not be given equal weight. Second, there is no statistically sound procedure to test the significance of individual harmonics. In this paper we investigate methods to obtain maximum likelihood estimates of the Fourier coefficients to describe the seasonal variability in the parameters for a stochastic rainfall model. Parameters are obtained from a two-state Markov chain model for wet and dry day occurrence, and from a mixed exponential model for distribution of depth on wet days. The procedure is demonstrated on four sample stations scattered across the continental United States. A constrained multivariate optimization scheme and a simple univariate procedure were used for maximum likelihood estimation, and these were compared with a least-squares estimate. The two seasonally varying parameters for the Markov chain are mutually independent, but the Fourier coefficients for each parameter are weakly dependent. The three seasonally varying parameters in the mixed exponential distribution are mutually dependent. However, for the four precipitation records analyzed, it appears that acceptable results can be obtained by simultaneously estimating the constant series terms and then independently estimating the harmonic amplitudes and phase angles. A likelihood ratio test can be used to test the significance of each added harmonic. It also appears from this analysis that the significant Fourier coefficients can be plotted on maps as isopleths, providing a concise description of regional precipitation climatology.