Unbiased detrended fluctuation analysis: Long-range correlations in very short time series

Detrended fluctuation analysis (DFA) is a standard method to evaluate long-range correlations embedded in non-stationary time series. To obtain a reliable estimation of scaling behavior, it requires the length of a time series is long enough (at least ∼10,000), which is not always the case in reality. How to evaluate long-range correlation behavior in a very short time series is still an open problem. In the present paper, we propose an improvement of DFA by correcting the bias in estimation of variance, called Unbiased Detrended Fluctuation Analysis (UDFA). Extensive calculations show its high-performance. For instance, from a fractional Brownian motion (fBm) series with length 500 the estimated long-range correlation exponent has negligible bias and acceptable confidence region (standard deviation less than 0.05). As a typical example, the proposed method is used to monitor evolution of fractal gait rhythm of a volunteer. Rich patterns are found in the evolutionary process.