In this study, we use mutual information to characterise statistical dependencies of seed and relativistic electron fluxes in the Earth's radiation belts on ultra-low-frequency (ULF) wave power measured on the ground and at geostationary orbit. The benefit of mutual information, in comparison to measures such as the Pearson correlation, lies in the capacity to distinguish non-linear dependencies from linear ones. After reviewing the property of mutual information and its relationship with the Pearson correlation for Gaussian bivariates, we present a methodology to quantify and distinguish linear and non-linear statistical dependencies that can be generalised to a wide range of solar wind drivers and magnetospheric responses. We present an application of the methodology by revisiting the case events studied by

The Earth's radiation belts are non-linearly driven and weakly collisional plasma environments in which deposited energy and momentum leads to the energisation of electrons to relativistic energies

It has been known for several decades that the Earth's radiation belts were driven far from thermodynamical equilibrium as a result of variable solar wind conditions

In this study, we present an application of information theory to the search of dependencies between energetic electron fluxes measured in the Earth's radiation belts and ultra-low-frequency (ULF) wave power measured both at geostationary orbit and on the ground. Unlike more commonly used measures like the Pearson correlation, information-theoretic tools, such as mutual information, have the benefit to distinguish non-linear dependencies from linear ones. In order to demonstrate the value in the use of information-theoretic methods, we revisit the highly cited case studies of

The application of information-theoretic measures to space plasma problems is not new, but it has recently shown its utility for a wide range of methodologies and problems (see

Are the events studied by

Are non-linearities present in the instance where the dependence between ULF wave power and electron fluxes is statistically significant?

Our report is presented as follows. Section 2 presents a brief summary of the tools of information theory used for the analysis. We put a particular emphasis on the application of mutual information to the case of Gaussian random variables of arbitrary correlation, which serves as a benchmark for linear dependencies. In Sect. 3 we describe the dataset used and the associated instruments' specificities relevant to our study. In Sect. 4, we present our results for geostationary-measured seed and relativistic electron fluxes measured during the events presented by

In this section we present a definition of mutual information in terms of the Shannon entropy and the specific mutual information of Gaussian bivariate random variables. The Gaussian bivariate case with arbitrary Pearson correlation

It is preferable to introduce mutual information by first defining the Shannon entropy

The Shannon entropy is a positive definite quantity

If

The mutual information is symmetric in

Even though probability distribution functions of electromagnetic fields and particle velocity distributions in space plasmas often depart from Gaussianity, it is useful to refer to the Gaussian bivariate case to develop an appreciation of mutual information for linear systems and as a benchmark to test numerical estimates. Conveniently, there is an exact analytical relationship between the Pearson correlation and mutual information of a Gaussian bivariate in terms of the Pearson correlation

The interested reader can find a definition of mutual information for continuous random variables and the derivation of Eq. (3) for Gaussian bivariates in the Appendix. Since the mutual information is a measure of how much we know from

The procedure we follow to compute the mutual information for two time series consists in binning the data according to the Freedman–Diaconis rule

In Fig. 1 we plotted the numerical estimate and analytical solution for

Mutual information estimator for bivariate Gaussian random variables with

Comparing the theoretical and numerical value of mutual information in Fig. 1, we note that our estimator does well for low correlation values, though it gains a discrepancy as large as 10 % for correlation absolute values greater than 0.5. In order to estimate the error introduced by discretisation, we apply a shuffle test to the two time series and compute the average value of mutual information and its standard deviation for 100 shuffles. We find that the error computed with the shuffling procedure is Gaussian-distributed, and we interpret the average mutual information obtained from shuffling as the zero baseline level. This baseline for each events is plotted as a bold orange line in Figs. 4–11 for panels (a) and (c). The shaded orange area represents the 3 standard deviation range from the mean. Estimates of mutual information for electron fluxes and ULF wave power above the shaded area are therefore interpreted as significant with

The data used in this study correspond to the two events analysed by

Dependence of the mutual information and Pearson correlation for ground

Same as in Fig.

The ULF data used in this analysis were from National Aeronautics and Space Administration's (NASA) Virtual Radiation Belt Observatory (ViRBO) and the ULF indices used, Sgr and Sgeo, both describing ULF spectral power from which noise has been removed, are derived in

In order to quantify the electron fluxes we use the indices

The fluxes have been derived in

Figures (

Reducing our resolution to 24 h for a strict comparison with

Maximum values in correlation and mutual information for positive lags and associated adjusted correlation.

Same as in Fig.

Same as in Fig.

Figure

We note that the peaks in mutual information and Pearson correlation occur between 48 and 50 h time lag and have maximum values of

Dependence of the mutual information and Pearson correlation for ground

Same as in Fig.

Figure

Figure

Figure

Same as in Fig.

Same as in Fig.

Figure

Figure

We are now ready to answer the two questions stated in the Introduction. (1) Are the events studied by

Are the events evidence of strong ULF wave power and energetic electron dependence? For the two events studied, the Pearson correlation and the mutual information are both statistically significant and well above the noise level. However, the maximum correlation values for relativistic electrons range between 0.41 and 0.59, and the maximum mutual information values range between 0.36 and 0.49. For comparisons, the analyses by

Comparing between seed and relativistic electrons, the statistical dependence on ULF wave power of the 130 keV flux is significantly larger than for relativistic fluxes and ranges between 0.54 and 0.68 for the maximum Pearson correlation and 0.44 and 0.67 for the maximum mutual information. We also note that the time lag for the maximum values is comparable whether one uses the mutual information or the Pearson correlation. The 130 keV fluxes have a maximum dependence with time lags of less than a day, whereas the relativistic electrons see a maximum for time lags considerably longer between 42 and 67 h.
Moreover, the ground ULF wave power gives a larger dependence than geostationary measured ULF wave power for the 1994 event. For the 1993 event the statistical dependence is the same whether one uses ground or geostationary indices. The ground ULF index spans local daylight hours between 05:00 and 15:00, whereas the GOES ULF covers the full 24 h period. This local time difference between ground and geostationary sampling of wave power makes the latter more susceptible to be influenced by substorm activity and the former by viscous processes and pressure pulses on the dayside magnetosphere during moderate geomagnetic activity

To address the second question, we compare the values of the information-adjusted correlation with the Pearson correlation. We note that the adjusted correlation is significantly larger than the Pearson correlation for all instances. In other words, though constrained to two case studies, our results demonstrate the presence of non-linear statistical dependencies between energetic electron fluxes and ULF wave power. By using information theory we make no assumptions about the functional form of the non-linear dependence between the variables, but we can nonetheless state that non-linearities have to be accounted for. Our results are consistent with the study of

The Earth's inner magnetosphere is a non-linearly driven plasma environment in which electrons can be collectively energised to relativistic energies by ULF fluctuations

In this study, we described the use of mutual information to characterise statistical dependencies of relativistic electron fluxes on ULF wave power. The benefit of mutual information, in comparison to the Pearson correlation, lies in the capacity to distinguish non-linear dependencies from linear ones. In order to test our methodology, we revisited the case study of

For a random variable

Thus, for

We consider a bivariate

Data for the relativistic electron fluxes can be requested from Joe Borovsky. Data for ULF wave power can be found on the Augsburg website

AO designed the study, wrote most of the article, and participated in the interpretation of results. MS wrote the code and the Methodology section, performed the data analysis, and participated in the interpretation of results. EK, HK, JEB, and MK participated in the interpretation of results and literature review.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The results presented herein have been achieved under the framework of the Finnish Centre of Excellence in Research of Sustainable Space, which we gratefully acknowledge. Adnane Osmane is grateful for the enlightening conversations with Simon Wing, Jay Johnson, and Solene Lejosne on the topics of information theory and radial diffusion transport of radiation belts.

Adnane Osmane and Mikko Savola acknowledge funding from the Academy of Finland by the profiling action on Matter and Materials (grant no. 318913). Emilia Kilpua and Milla Kalliokoski acknowledge funding from FORESAIL at the Academy of Finland (grant nos. 312390 and 336809). Joseph E. Borovsky has been supported at the Space Science Institute as part of the NSF GEM Program (grant no. AGS-2027569) and by the NASA HERMES Interdisciplinary Science Program (grant no. 80NSSC21K1406). Open-access funding was provided by the Helsinki University Library.

This paper was edited by Yoshizumi Miyoshi and reviewed by two anonymous referees.