Detrended cross-correlation analysis consistently extended to multifractality

Abstract

We propose an algorithm, multifractal cross-correlation analysis (MFCCA), which constitutes a consistent extension of the detrended cross-correlation analysis and is able to properly identify and quantify subtle characteristics of multifractal cross-correlations between two time series. Our motivation for introducing this algorithm is that the already existing methods, like multifractal extension, have at best serious limitations for most of the signals describing complex natural processes and often indicate multifractal cross-correlations when there are none. The principal component of the present extension is proper incorporation of the sign of fluctuations to their generalized moments. Furthermore, we present a broad analysis of the model fractal stochastic processes as well as of the real-world signals and show that MFCCA is a robust and selective tool at the same time and therefore allows for a reliable quantification of the cross-correlative structure of analyzed processes. In particular, it allows one to identify the boundaries of the multifractal scaling and to analyze a relation between the generalized Hurst exponent and the multifractal scaling parameter λ(q). This relation provides information about the character of potential multifractality in cross-correlations and thus enables a deeper insight into dynamics of the analyzed processes than allowed by any other related method available so far. By using examples of time series from the stock market, we show that financial fluctuations typically cross-correlate multifractally only for relatively large fluctuations, whereas small fluctuations remain mutually independent even at maximum of such cross-correlations. Finally, we indicate possible utility of MFCCA to study effects of the time-lagged cross-correlations.

Detrended fluctuation analysis made flexible to detect range of cross-correlated fluctuations

“…These quantities can be used to define a family of the so-called fluctuation functions of order q [17,43]:…”

“…DCCA was also generalized in order to be applicable to signals with multifractal cross-correlations and the resulting MFDCCA/MFDXA algorithm [38] also attracted some attention [39][40][41][42]. However, this generalization raises controversy as at some stage it requires to neglect the detrended covariance sign to avoid obtaining complex values, which leads to inevitable loss of information about analyzed signals and, consequently, to incorrect results [43]. This drawback has recently been removed with a new sign-sensitive method of the multifractal detrended cross-correlation analysis with an acronym MFCCA [43,44] that is more consistent than MFDCCA as a multifractal generalization of DCCA (for more details see Section II).…”

“…The easiest way to do this is by introducing a q-dependent detrended cross-correlation coefficient ρ q (q ∈ R) based on the so-called q-dependent fluctuation functions F q from MFDFA and MFCCA [17,43]. The idea is based on the fact that from a number of different values contibuting to a sum, one can select specific values (e.g., the large, the medium, or the small ones) by rising all the values in the sum to some power (high positive, small positive, or negative one, respectively).…”

Quantitative Characteristics of Correlations of Meteorological Data

“…The generalization of the MFDFA method for two time series is MFCCA. This procedure was proposed by Oświęcimka et al in [23].…”

“…Multifractal structures were identified in systems from various areas such as physics [2][3][4][5], biology [6][7][8], chemistry [9,10], economics [11][12][13] and even music [14][15][16][17][18][19]. One of the most popular methods of the multifractal analysis is multifractal detrended correlation (MFDFA) [20][21][22] and cross-correlation analysis (MFCCA) [23,24]. In this paper, we systematically study the multifractal characteristics of meteorological data.…”