Robust q-negative Multifractal Detrended Cross-Correlation Coefficient

The multifractal detrended cross-correlation coefficient $\rho_q(n)$ is widely used to investigate scale-dependent interactions, but its application to negative fluctuation orders is affected by numerical instabilities, unbounded values, and interpretational difficulties.

We propose a Signed Multifractal Detrended Cross-Correlation Coefficient, $\rho_{\mathrm{SMFDCCA}}(n,q)$, an amplitude-conditioned correlation observable for multifractal detrended analysis, based on locally normalized detrended correlations and regularized fluctuation amplitudes.

The proposed coefficient preserves the sign of local interactions, remains strictly bounded within $[-1,1]$ for both positive and negative values of $q$, and eliminates the corrective procedures required by previous approaches.

Validation using independent fractional Gaussian noise confirms the absence of spurious cross-correlations and the numerical stability of the method.

Applications demonstrate that the proposed observable resolves how cross-correlations evolve jointly with temporal scale and fluctuation amplitude, revealing scale- and amplitude-dependent correlation structures, including stronger synchronization during large fluctuations in stock-market indices and heterogeneous coupling patterns in temperature records.