Detrended cross-correlation analysis for non-stationary time series with periodic trends

Noisy signals in many real-world systems display long-range autocorrelations and long-range cross-correlations. Due to periodic trends, these correlations are difficult to quantify. We demonstrate that one can accurately quantify power-law cross-correlations between different simultaneously recorded time series in the presence of highly non-stationary sinusoidal and polynomial overlying trends by using the new technique of detrended cross-correlation analysis with varying order ! of the polynomial. To demonstrate the utility of this new method —which we call DCCA-!(n), where n denotes the scale— we apply it to meteorological data. Copyright c © EPLA, 2011 Data on many real-world systems, ranging from geophysics to physiology, exhibit two important properties, correlations and periodicity. Depending on whether we are studying correlations in a single signal or between a pair of signals, we can use autocorrelation functions or cross-correlation functions to gain insight into the correlation dynamics. But like many other techniques, these methods were devised to identify correlations in data that are stationary and linear. For example, the power spectrum S(f) assumes stationarity in data and is defined as a positive real function of a frequency variable f that describes how the energy of a signal is distributed with frequency. In the case of power-law correlations, using the Fourier transform the Khinchin-Kolmogorov theorem [1] relates the power spectral density S(f)∼ f−β of a wide-sense-stationary random process to the corresponding autocorrelation function C(n)∼ n−γ , where the exponents are related as β = 1− γ. Despite their widespread popularity and applicability, power spectral density and correlation analysis have limitations when applied to the real-world data associated with physical, biological, hydrological, and social systems. These are commonly non-stationary and they often exhibit periodicity [2,3]. When a time series is non-stationary, the limitations of methods that assume stationarity are clear. Suppose (a)E-mail: [email protected] a time series Xt has a large upward trend. Then a large value of Xt is more likely to be followed by a large value of Xt+1 implying strong autocorrelations, not because autocorrelations are actually present, but because the autocorrelation function is being used for a non-stationary time series —which is inappropriate. Similarly, a US market index time series may strongly cross-correlate with the population of, say, Pakistan, simply because each time series has a characteristic strong upward trend. Thus, detrending is essential to properly analyze many time series for at least two reasons: i) detrending prevents a time series from being correlated if correlations are not present, and ii) if correlations do exist, detrending reveals a genuine correlation functional dependence—in case of power-law correlations, for example, we expect to obtain a genuine correlation exponent. Most methods using detrending start with the assumption that the functional form of a trend is predetermined [4]. The application of detrending to original data can be either local or global. When done locally, detrended fluctuation analysis (DFA) [4–8] quantifies not periodicity but a single scaling parameter that represents the longrange autocorrelation properties of a signal. DFA has been used in fields ranging from cardiac dynamics [9], bioinformatics [4], and economics [10,11] to meteorology [12,13]. In physiology, the DFA method can be used diagnostically —it can help identify different states of the same system according to their different scaling behaviors,