Weighted multifractal cross-correlation analysis based on Shannon entropy

In this paper, we propose a modification of multifractal cross-correlation analysis based on statistical moments (MFSMXA) method, called weighted MFSMXA method based on Shannon entropy (W-MFSMXA), to investigate cross-correlations and cross-multifractality between time series. Robustness of this method is verified by numerical experiments with both artificial and stock returns series. Results show that the proposed W-MFSMXA method not only keep the multifractal structure unchanged, but contains more significant information of series compared to the previous MFSMXA method. Furthermore, analytic formulas of the binomial multifractal model are generated for W-MFSMXA. Theoretical analysis and finite-size effect test demonstrate that W-MFSMXA slightly outperforms MFSMXA for relatively shorter series. We further generate the scaling exponent ratio to describe the relation of two methods, whose profile is found approximating a centrosymmetric hyperbola. Cross-multifractality is found in returns series but then destroyed after being shuffled as a consequence of the removed long memory in separate series.