Time series from real data measurements are often noisy, under-sampled, irregularly sampled, and inconsistent across long-term measurements. Typically, in analyzing these time series, particularly within astronomy, it is common to use estimators such as sample entropy and multi-scale entropy that require interpolation to avoid irregular sampling. In this work, we analyze and consider a new entropy estimator that combines permutations, Fourier Coefficients, and Wasserstein distances to address the concern of irregularly sampled data.

In this work, we analyze and compare two entropy estimators applied to random walk time series. We compare the robustness of multi-scale entropy and sample entropy for different regimes of signal-to-noise ratio. We also compare multi-scale entropy and sample entropy in the case of missing data when simple linear interpolation is adopted to fill the missing data points. In the case of the signal-to-noise comparison, we show by numerical simulations and present strong mathematical arguments that multi-scale entropy is a more resistant estimator to analyze time series. We also show that multi-scale entropy provides a more resistant and accurate estimate of entropy on random walk time series in the scenario of missing data, especially when completing missing data with linear interpolation.