Wavelet Transforms are a widely used technique for decomposing a signal into coefficient vectors that correspond to distinct frequency/scale bands while retaining time localization. This property enables an adaptive analysis of signals at different scales, capturing both temporal and spectral patterns. By examining how correlations between two signals vary across these scales, we obtain a more nuanced understanding of their relationship than what is possible from a single global correlation measure. In this work, we expand on the theory of wavelet-based correlations already used in the literature and elaborate on wavelet correlograms, partial wavelet correlations, and additive wavelet correlations using the Pearson and Kendall definitions. We use both Orthogonal and Non-decimated discrete Wavelet Transforms, and assess the robustness of these correlations under different wavelet bases. Simulation studies are conducted to illustrate these methods, and we conclude with applications to real-world datasets.

翻译：小波变换是一种广泛应用于信号处理的技术，可将信号分解为对应不同频率/尺度带的系数向量，同时保留时间局部性。这一特性支持在不同尺度上对信号进行自适应分析，捕捉其时域与频域模式。通过研究两个信号之间的相关性如何随尺度变化，我们能够获得比单一全局相关度量更精细的关系理解。本文在现有文献中小波相关理论基础上进行扩展，详细阐述了基于Pearson与Kendall定义的小波相关图、偏小波相关及可加性小波相关方法。研究采用正交与非抽取离散小波变换，评估了不同小波基下相关方法的鲁棒性。通过仿真实验验证了所提方法，最后展示了在真实数据集上的应用案例。