We propose a modified algorithm - Multifractal Cross-Correlation Analysis (MFCCA) - that  is able to consistently identify and quantify  multifractal cross-correlations between two time series. Our motivation for introducing this algorithm is that the already existing methods like MF-DXA have  serious limitations for most of the signals describing complex natural processes  . The principal component of the related improvement is proper incorporation of the sign of fluctuations . 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 tool and allows a reliable quantification of the cross-correlative structure of analyzed processes. We, in particular, analyze a relation between the generalized Hurst exponent and the MFCCA parameter \lambda_q. This relation provides information about the character of potential multifractality in cross-correlations of the processes under study and thus enables selective insight into their dynamics . Using also an example of financial time series from the stock market we show that waiting times and price increments of the companies are multifractally cross-correlated but only for relatively large fluctuations, whereas the small ones could be considered mutually independent .
We propose a novel algorithm - Multifractal Cross-Correlation Analysis (MFCCA) - that constitutes a consistent extension of the Detrended Cross-Correlation Analysis (DCCA) 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 MF-DXA 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 \lambda_q. This relation provides information about 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 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 .
R
modified
novel
A
constitutes a consistent extension of the Detrended Cross-Correlation Analysis (DCCA) and
R
consistently
properly
A
subtle characteristics of
A
at best
A
and often indicate multifractal cross-correlations when there are none
R
related improvement
present extension
R
. We
to their generalized moments. Furthermore, we
R
tool and allows
and selective tool at the same time, and therefore allows for
R
We, in particular,
In particular, it allows one to identify the boundaries of the multifractal scaling and to
R
MFCCA
multifractal scaling
D
the
D
of the processes under study
R
selective insight into their dynamics . Using also an example of financial
a deeper insight into dynamics of the analyzed processes than allowed by any other related method available so far. By using examples of
R
the stock market
stock market,
R
waiting times and price increments of the companies are multifractally cross-correlated but
financial fluctuations typically cross-correlate multifractally
R
the small ones could be considered mutually independent
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