Power Spectrum and Detrended Fluctuation Analysis: Application to Daily Temperatures

The variability measures of fluctuation analysis (FA) and detrended fluctuation analysis (DFA) are expressed in terms of the power spectral density and of the autocovariance of a given process. The diagnostic potential of these methods is tested on several model power spectral densities. In particular we find that both FA and DFA reveal an algebraic singularity of the power spectral density at small frequencies corresponding to an algebraic decay of the autocovariance. A scaling behavior of the power spectral density in an intermediate frequency regime is better reflected by DFA than by FA. We apply FA and DFA to ambient temperature data from the 20th century with the primary goal to resolve the controversy in literature whether the low frequency behavior of the corresponding power spectral densities are better described by a power law or a stretched exponential. As a third possible model we suggest a Weibull distribution. However, it turns out that neither FA nor DFA can reliably distinguish between the proposed models.