稳健的q负多重分形去趋势互相关系数

中文摘要 AI 辅助

多重分形去趋势互相关系数$\rho_q(n)$被广泛用于研究尺度依赖的相互作用，但其在负波动阶数上的应用受到数值不稳定性、无界值和解释困难的影响。我们提出了一种符号多重分形去趋势互相关系数$\rho_{\mathrm{SMFDCCA}}(n,q)$，它基于局部归一化去趋势相关性和正则化波动幅度，是一种用于多重分形去趋势分析的幅度条件相关可观测量。该系数保留了局部相互作用的符号，对于$q$的正负值都严格限制在$[-1,1]$内，并且消除了先前方法所需的校正程序。使用独立分数高斯噪声的验证证实了该方法不存在虚假互相关且具有数值稳定性。应用表明，所提出的可观测量解决了互相关如何随时间尺度和波动幅度共同演变的问题，揭示了尺度和幅度依赖的相关结构。

英文摘要

The multifractal detrended cross-correlation coefficient $ρ_q(n)$ 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, $ρ_{\mathrm{SMFDCCA}}(n,q)$, 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.

发表机构

• Universidade SENAI CIMATEC（SENAI CIMATEC大学）
• Universidade de Aveiro（阿威罗大学）
• UNEB（巴伊亚联邦大学）

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