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

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 Henon 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.