Refined Cross-sample Entropy based on Freedman-Diaconis Rule: ‎Application to Foreign Exchange Time Series

Document Type : Research Paper


Instituto de Estadística, Facultad de Ciencias, Universidad de Valparaíso, Valparaíso, 2360102, Chile


Shang et al. (Commun. Nonlinear Sci. 94, 105556, 2022) proposed an efficient and robust synchronization estimation between two not necessarily stationary time series, namely the refined cross-sample entropy (RCSE). This method considered the empirical cumulative distribution function of distances using histogram estimator. In contrast to classical cross-sample entropy, RCSE only depends on a fixed embedding dimension parameter. In this paper, the RCSE is revisited as Freedman-Diaconis rule was considered to estimate the number of bins for the cumulative distribution function. Results are illustrated with some simulations based on 2D Hénon maps, the sinusoidal model, and the Lorenz attractor. In addition, a practical study of foreign exchange rate time series is presented. Specifically, the Canadian/US and Singaporean/US dollar time series were considered to compute the synchrony level between the 1995-1998 (before the 1999 Asian financial crisis) and the 1999-2003 (post-crisis) periods.


Main Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Lake, D.E., Richman, J.S., Grin, M.P., Moorman, J.R., Sample entropy analysis of neonatal heart rate variability, American Journal of Physiology-Regulatory, Integrative and Comparative Physiology, 283(3), 2002, R789R797.
[2] Wang, F., Zhao, W., Jiang, S., Detecting asynchrony of two series using multiscale cross-trend sample entropy, Nonlinear Dynamics, 99(2), 2020, 1451-1465.
[3] Contreras-Reyes, J.E., Idrovo-Aguirre, B.J., Backcasting and forecasting time series using detrended cross-correlation analysis, Physica A: Statistical Mechanics and its Applications, 560, 2020, 125109.
[4] Karmakar, C., Udhayakumar, R., Palaniswami, M., Entropy Profiling: A Reduced-Parametric Measure of Kolmogorov-Sinai Entropy from Short-Term HRV Signal, Entropy, 22, 2020, 1396.
[5] Ramírez-Parietti, I., Contreras-Reyes, J.E., Idrovo-Aguirre, B.J., Cross-sample entropy estimation for time series analysis: a nonparametric approach, Nonlinear Dynamics, 105(3), 2021, 2485-2508.
[6] Udhayakumar, R. K., Karmakar, C., Palaniswami, M., Approximate entropy profile: a novel approach to comprehend irregularity of short-term HRV signal, Nonlinear Dynamics, 88(2), 2017, 823-837.
[7] Shang, D., Shang, P., Zhang, Z., Efficient synchronization estimation for complex time series using refined cross-sample entropy measure, Communications in Nonlinear Science and Numerical Simulation, 94, 2021, 105556.
[8] Mao, X., Shang, P., A new method for tolerance estimation of multivariate multiscale sample entropy and its application for short-term time series, Nonlinear Dynamics, 94(3), 2018, 1739-1752.
[9] Freedman, D., Diaconis, P., On the histogram as a density estimator: L2 theory, Probability Theory and Related Fields, 57(4), 1981, 453-476.
[10] Contreras-Reyes, J.E., Mutual information matrix based on asymmetric Shannon entropy for nonlinear interactions of time series, Nonlinear Dynamics, 104(4), 2021, 3913-3924.
[11] Richman, J.S., Moorman, J.R., Physiological time-series analysis using approximate entropy and sample entropy, American Journal of Physiology-Regulatory, Integrative and Comparative Physiology, 278(6), 2000, H2039-H2049.
[12] R Core Team, A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2020, Available at
[13] Arnold, L., Random dynamical system, Springer-Varlag, Berlin, Germany, 1998.
[14] Dlamini, P., Simelane, S., An Efficient Spectral Method-based Algorithm for Solving a High-dimensional Chaotic Lorenz System, Journal of Applied and Computational Mechanics, 7(1), 2021, 225-234.
[15] Contreras-Reyes, J.E., Chaotic systems with asymmetric and heavy-tailed noise: application to 3D attractors, Chaos, Solitons & Fractals, 2021, 145, 110820.
[16] Liu, L.Z., Qian, X.Y., Lu, H.Y., Cross-sample entropy of foreign exchange time series, Physica A: Statistical Mechanics and its Applications, 389, 2010, 4785-4792.
[17] Shi, W., Shang, P., Cross-sample entropy statistic as a measure of synchronism and cross-correlation of stock markets, Nonlinear Dynamics, 71, 2013, 539-554.
[18] Xia, J., Shang, P., Multiscale entropy analysis of financial time series, Fluctuation and Noise Letters, 11(4), 2012, 1250033.
[19] Argyris, J., Andreadis, I., Pavlos, G., Athanasiou, M., The influence of noise on the correlation dimension of chaotic attractors, Chaos, Solitons & Fractals, 9, 1998, 343-361.
[20] Abid, S.H., Hasan, H.M., About asymmetric noisy chaotic maps, International Journal of Basic and Applied Sciences, 3(2), 2014, 62-73.
[21] Contreras-Reyes, J.E., Fisher information and uncertainty principle for skew-gaussian random variables, Fluctuation and Noise Letters, 20, 2021, 21500395.
[22] Li, B., Han, G., Jiang, S., Yu, Z., Composite Multiscale Partial Cross-Sample Entropy Analysis for Quantifying Intrinsic Similarity of Two Time Series Affected by Common External Factors, Entropy, 22(9), 2020, 1003.
[23] Contreras-Reyes, J.E., Canales, T.M., Rojas, P.M., Influence of climate variability on anchovy reproductive timing off northern Chile, Journal of Marine Systems, 164, 2016, 67-75.
[24] Contreras-Reyes, J.E., Hernández-Santoro, C., Assessing Granger-causality in the southern Humboldt current ecosystem using cross-spectral methods, Entropy, 22(10), 2020, 1071.