Scale-independence and long-range correlations have been observed in many time-series ranging from financial time series and internet traffic to human heart rate variability. Presumed fractal behavior implied by the scale independence is taken by some as the defining characteristic of complex, highly nonlinear systems. Because of the scale-independence (i.e. self-similarity under magnification), there is a power law relationship between arbitrary time scales and variation of the time series at these scales Detrended Fluctuation Analysis is widely used to determine the scaling exponent, k in noisy time series, and has been shown to identify long-range correlations.
However, it should be noted that DFA merely measures the the degree of dispersion in the time series as the segment length increases. Spectral analysis in the frequency domain provides a direct measure of cyclical variations at different time scales, and is a more intuitive measure of the data. This work completes the observations by Francis that DFA is equivalent to spectral weighting by deriving explicit filters equivalent to DFA. Here we directly derive the frequency-domain analogue of DFA, show the exact mathematical equivalence between the scaling exponent estimated using DFA and that estimated from a weighed power spectrum. The filter that corresponds to the DFA procedure is derived in time and frequency domains. A number of filters are derived from this analysis and the performance of some of these are close to optimal for estimating Hurst exponents.
Finally, we study the error inherent in numerical implimentation of discrete wavelet transforms. Error in the filter coefficients leads to muliplicative error in the estimates. To obtain a 1% error in the estimation of the wavelet coefficients at the coarsest scales used, an accuracy of 5 significant figures in wavelet coefficients is necessary at minumum.
Keywords: Detrended fluctuation analysis; Hurst exponent; Numerical approximation
Biography: I was born and Graduated from High School in Queens New York City in 1964; obtained my Bachelors Degree from M.I.T. in Pure Mathematics in 1969; and my Ph.D in Experimental Psychology from Harvard in 1979. From 1972–1975 I worked as a Professional Programmer in Newton Massachusetts and Cambridge MA. Since July 1980. I have been at Boston University first. a Research Assistant Professor and eventually as an Associate Professor at the Department of Cognitive and Neural Systems. From 1980 to 1995 as well as being a researcher, I was director of their computer systems. I have worked in many diverse fields involving psychology, physiology, and mathematics including child language, ordinary differential equations, and cardiovascular physiology and statistics. I am married and live in Brookline Massachusetts.