xdfa is a software package to compute Detrended Fluctuation Analysis (DFA), and related methods, on time series. The core of the package is written in C++ for performance reasons with interfaces available for python, octave and R. The purpose of the package is to provide a uniform calling convention together with ensuring the numerical stability of the methods used.
In the context of complex systems the observable time series of several quantities are the result of many small part interactions and often exhibit long range dependency, also called long memory. A model for a long memory stochastic process is the fractional Brownian motion, where a single parameter H (Hurst exponent) controls persistence of the time series. The original work from Peng et al [1] has introduced DFA in order to estimate the Hurst exponent of a time series. This method is widely used in several scientific fields like econophysics, geophysics, biophysics and others.
In Detrended Fluctuation Analysis trends at all scales are removed from the integrated time series and the the behavior of the residuals is studied. In other variations like the Detrended Moving Average the trend is evaluated using moving averages [2]. In the previous cases the residuals are taken using the norm-2 metric. It is possible to evaluate the trend using other norms in to study the multifractal properties of the time series [3]. Other extension deal with the comparison between two time series, in a scheme similar to the covariance/correlation analysis using the Detrended Fluctuation Analysis as the basis, that correspond to the Detrended Cross Correlation Analysis [4,5].
The main goal of the xdfa software package is to have the code that deals with all these different methods in a single place and with a common interface.
References
[1] Peng C-K, Buldyrev SV, Havlin S, Simons M, Stanley HE, Goldberger AL. Mosaic organization of DNA nucleotides. Phys Rev E; 49: Pages 1685-1689 (1994).
[2] Carbone A, Castelli G, Stanley HE. Time-dependent Hurst exponent in financial time series, Physica A: Statistical Mechanics and its Applications, Volume 344, Issues 1–2, Pages 267-271 (2004).
[3] Kantelhardt, JW, Zschiegner, SA, Koscielny-Bunde, E, Havlin, S, Bunde, A, Stanley, HE. Multifractal detrended fluctuation analysis of nonstationary time series, Physica A, 316, 1–4, Pages 87-114 (2002).
[4] Podobnik, B and Stanley, HE. Detrended Cross-Correlation Analysis: A New Method for Analyzing Two Nonstationary Time Series, Phys. Rev. Lett. 100, 8, pages 084102 (2008).
[5] Qian, X-Y, Liu, Y-M, Jiang, Z-Q, Podobnik, B, Zhou, W-X, Stanley, HE. Detrended partial cross-correlation analysis of two nonstationary time series influenced by common external forces, Phys. Rev. E 91, 6062816 (2015).