Wavelet-based entropy rate measure for characterizing stochastic, deterministic, and chaotic dynamics: An application on cryptocurrencies

In this article, we introduce the Wavelet-Based Entropy-Rate Measure, designed to characterise the complexity and predictability of time series by combining wavelet decomposition with ordinal-pattern and entropy-rate estimation. The measure weights scale-specific entropy-rate contributions according to the relative wavelet energy, thus jointly accounting for local dynamical structure and global multiscale organization. We show that WERM effectively distinguishes among purely stochastic processes, persistent and anti-persistent noises, periodic deterministic components, and chaotic dynamics. In particular, the measure captures systematic differences between random, regular, and nonlinear deterministic regimes, while remaining sensitive to the interaction between stochastic memory and deterministic structure. We then apply the proposed framework to cryptocurrency. The empirical results indicate that price levels are mainly associated with persistent stochastic dynamics, whereas returns are better characterized by short-memory or anti-persistent regimes combined with chaotic components. WERM offers a useful and flexible tool for the analysis of complex multiscale signals in financial systems and other applications.