Physica A 484 (2017) 569–576 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa On the performance of Fisher Information Measure and Shannon entropy estimators Luciano Telesca a, *, Michele Lovallo b a National Research Council, Institute of Environmental Analysis, C.da S.Loja, 85050 Tito (PZ), Italy b ARPAB, 85100 Potenza, Italy highlights • Two estimation methods (discretization and kernel-based approach) are applied to FIM and SE. • FIM (SE) estimated by discrete approach is nearly constant with σ . • FIM (SE) estimated by discrete approach decreases (increases) with the bin number. • FIM (SE) estimated by kernel-based approach is close to the theory value for any σ . article info Article history: Received 1 November 2016 Received in revised form 16 January 2017 Available online 22 May 2017 Keywords: Fisher Information Measure Shannon entropy Estimation abstract The performance of two estimators of Fisher Information Measure (FIM) and Shannon entropy (SE), one based on the discretization of the FIM and SE formulae (discrete-based approach) and the other based on the kernel-based estimation of the probability density function (pdf) (kernel-based approach) is investigated. The two approaches are employed to estimate the FIM and SE of Gaussian processes (with different values of σ and size N), whose theoretic FIM and SE depend on the standard deviation σ . The FIM (SE) estimated by using the discrete-based approach is approximately constant with σ , but decreases (increases) with the bin number L; in particular, the discrete-based approach furnishes a rather correct estimation of FIM (SE) for L ∝ σ . Furthermore, for small values of σ , the larger the size N of the series, the smaller the mean relative error; while for large values of σ , the larger the size N of the series, the larger the mean relative error. The FIM (SE) estimated by using the kernel-based approach is very close to the theoretic value for any σ, and the mean relative error decreases with the increase of the length of the series. Comparing the results obtained using the discrete-based and kernel-based approaches, the estimates of FIM and SE by using the kernel-based approach are much closer to the theoretic values for any σ and any N and have to be preferred to the discrete-based estimates. © 2017 Elsevier B.V. All rights reserved. 1. Introduction Recently, the interest in the application of the Fisher Information Measure (FIM) and Shannon entropy (SE ) to the investigation of time series has been growing. Even though both these measures have been well known in the theoretical statistics [1,2], only in the last decade they were widely used to investigate the temporal properties of time series to gain information about dynamical systems. In particular, these two quantities play a key role in the context of analysis * Corresponding author. E-mail address: luciano.telesca@imaa.cnr.it (L. Telesca). http://dx.doi.org/10.1016/j.physa.2017.04.184 0378-4371/© 2017 Elsevier B.V. All rights reserved.