The fractal dimension of state space sets is typically estimated via the scaling of either the generalized (Renyi) entropy or the correlation sum versus a size parameter. Motivated by the lack of quantitative and systematic comparisons of fractal dimension estimators in the literature, and also by new and improved methods for delay embedding, in this paper we provide a detailed and quantitative comparison for estimating the fractal dimension. We start with summarizing existing estimators and then perform an evaluation of these estimators, comparing their performance and precision using different data sets and taking into account the impact of features like length, noise, embedding dimension, non-stationarity, among many others. After comparing ten estimators, we conclude that for synthetic data the correlation based estimator is much better than the entropy one, while for real experimental data it seems to be the other way around. All other estimators perform worse. If the dynamic equations are known analytically, the Lyapunov dimension is always the most accurate. We furthermore discuss common pitfalls, like calculating the dimension of inappropriate data, automated ways to estimate the dimension, and provide an outlook of possible future research. All quantities discussed are implemented as performant and easy to use open source code via the software DynamicalSystems.jl.
