Two Fractional Order Cumulative Residual Time Series Measures Based on Rényi Entropy

In the study of time series analysis, Kullback-Leibler divergence can measure the difference between the distribution diverges of time series, and transfer entropy is usually used to quantify the causal influence among time series. In order to analyze the relationship between time series more comprehensively, two novel fractional order times series measures, i.e., R & eacute;nyi cumulative residual Kullback-Leibler (RCRKL) and R & eacute;nyi cumulative residual transfer entropy (RCRTE), are proposed in this paper. Some mathematical properties of RCRKL are proved as a distance metric, and the zeros and bounds of RCRTE are studied. Then, some examples on synthetic data are provided to study the effect of parameter changes on metric values. Finally, the proposed two fractional-order time series measures are applied to pavement rutting time series, for which a RCRKL-based clustering is designed and the causal relationships between influence factors and rutting are detected by RCETE. The results demonstrate that the RCRKL-based clustering algorithm performs better than those based on the baseline distance measures. Meanwhile, RCRTE can also detect the true causality between rutting and its influence factors and reject the false causality. In summary, the proposed RCRKL and RCRTE take the advantages of both cumulative residual entropy and R & eacute;nyi entropy, and provide a more comprehensive insight for time series analysis.