As the big data science develops, efficient methods are demanded for various data analysis. Granger causality provides the prime model for quantifying causal interactions. However, this theoretic model does not meet the requirement for real-world data analysis, because real-world time series are diverse whose models are usually unknown. Therefore, model-free measures such as information transfer measures are strongly desired. Here, we propose the multi-scale extension of conditional mutual information measures using MORLET wavelet, which are named the WM and WPM. The proposed measures are computational efficient and interpret information transfer by multi-scales. We use both synthetic data and real-world examples to demonstrate the efficiency of the new methods. The results of the new methods are robust and reliable. Via the simulation studies, we found the new methods outperform the wavelet extension of transfer entropy (WTE) in both computational efficiency and accuracy. The features and properties of the proposed measures are also discussed.
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