Detrended fluctuation analysis (DFA) was introduced in early 1990s as a method for analyzing fractal properties of underlying data. The method was later popularized in the long-range correlations and multifractal analyses. Recently, DFA has been generalized for long-range cross-correlation analysis as well as an examination of correlations between nonstationary series. The method has been applied and utilized across a wide range of disciplines ranging from physiology to cardiology, DNA analysis and neurology, to (hydro)meteorology, economics and finance, engineering, environmental studies, and many others. Here, we propose a framework based on the detrended fluctuation analysis which allows for a regression analysis of possible nonstationary and long-range dependent data at different scales. The methodology is based on the least squares framework, which is briefly recalled and translated into the language of variances and covariances. The detrended fluctuation analysis together with its bivariate generalization of the detrended cross-correlation analysis (DCCA) are described in some detail as a connecting bridge to the DFA-based regression. The DFA framework is introduced for the bivariate setting with procedures to estimate parameters, standard errors of the estimates, and the coefficient of determination (R^2), all characteristic for a specified scale. The theoretical concepts are further supported by Monte Carlo simulations. The framework is then applied to several phenomena from various disciplines: relationship between temperature and humidity, stock market betas, elasticity between corn and ethanol, and transmission between influenza outbursts and the Google Flu Trends indicator. For three out of four cases, we report a strong variability of the estimates across scales. The proposed methodology thus provides a further step in the development of DFA and related methods, here specifically from the correlation to regression framework.

References

[1] Kristoufek, L. (2015): Detrended fluctuation analysis as a regression framework: Estimating dependence at different scales, Physical Review E 91:022802.

[2] Kristoufek, L. (2014): Spectrum-based estimators of the bivariate Hurst exponent, Physical Review E 90:062802.        

[3] Kristoufek, L. (2014): Detrending moving-average cross-correlation coefficient: Measuring cross-correlations between non-stationary series, Physica A: Statistical Mechanics and Its Applications 406, pp. 169-175.

[4] Kristoufek, L. (2015): Can the bivariate Hurst exponent be higher than an average of the separate Hurst exponents?, Physica A: Statistical Mechanics and Its Applications 431, pp. 124-127.

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