It is well known that the magnetospheric response to the solar wind is
nonlinear. Information theoretical tools such as mutual information, transfer
entropy, and cumulant-based analysis are able to characterize the
nonlinearities in the system. Using cumulant-based cost, we show that
nonlinear significance of

One of the most practically important concepts in dynamical systems is the notion of causality. It is particularly useful to organize observational datasets according to causal relationships in order to identify variables that drive the dynamics. Understanding causal dependencies can also help to simplify descriptions of highly complex physical processes because it constrains the coupling functions between the dynamical variables. Analysis of those coupling functions can lead to simplification of the underlying physical processes that are most important for driving the system. It is particularly useful from a practical standpoint to understand causal dependencies in systems involving natural hazards because monitoring of causal variables is closely linked with warning.

A common method to establish causal dependencies in a data stream of two
variables, e.g., [

However, the procedure of detecting causal relationships based on linear
cross-correlation suffers from a number of limitations. First it should be
noted that the statistical accuracy of the correlation function is limited by
the resolution and length of the data stream. Second, the linear time series
analysis ignores nonlinear correlations, which may be important for energy
transfer in the magnetospheric system. For example, substorms are believed to
involve storage and release of energy in the magnetotail, which is a highly
nonlinear response. Similarly, magnetosphere–ionosphere coupling may also be
highly nonlinear, involving the nonlinear development of accelerating
potentials along auroral field lines and nonlinear current–voltage
relationships. Third, the cross-correlation may not be a particularly clear
measure when there are multiple peaks or if there is little or no asymmetry
in the forward (i.e.,

In the remainder of this paper, we will discuss other methods to identify causal relationships based on entropy-based discriminating statistics such as mutual information and transfer entropy. We will also discuss the cumulant-based method. We will illustrate the shortcomings and strengths of the various methods for studying causality with examples from nonlinear dynamics and space physics.

It is well known that the magnetosphere responds to variation in the solar
wind parameters

Suppose that we consider a set of variables

Mutual information and cumulant-based cost are two useful measures that
quantify Eq. (

Correlation studies also only detect linear correlations, so if the feedback
involves nonlinear processes (highly likely in this case) then their
usefulness may be seriously limited. Alternatively, entropy-based measures
such as mutual information

Similarly, examination of time-shifted cumulants could be used as an
indicator of causality in a nonlinear system. In this case, we can define a
discriminating statistic

With only two variables,

In Sect.

Another method for determining causality is the one-sided transfer entropy

Transfer entropy considers the conditional mutual information between two
variables using the past history of one of the variables as the conditioner.

Both mutual information and transfer entropy require binning of data. As
mentioned in

When plasma sheet ions are injected into the Earth's inner magnetosphere, they
drift westward around the Earth, forming the ring current. Studies have shown
that the substorm occurrence rate increases with solar wind velocity (high
speed streams)

For the present study, we examine the relationships between solar wind
velocity (

Section

The distributions of

In Fig.

The absence of the nonlinear peaks at

Significance extracted
from

A common scenario for storm–ring current interaction is the following. A
storm compresses the magnetosphere, intensifies the magnetic field in the
magnetosphere, and injects energetic particles into the ring current region.
The ring current intensifies during the main phase of the storm, which can
last

Comparison of mutual information and transfer entropy measures to
determine causal driving of the magnetosphere as characterized by

As mentioned in Sect.

We recently used mutual information, transfer entropy, and conditional mutual
information to discover the solar wind drivers of the outer radiation belt
electrons

As a follow-up to

Our analysis with mutual information and transfer entropy indicates that
there are strong linear and nonlinear correlations and transfer of
information, respectively, in the forward direction between

Using the cumulant-based significance, we have established that the
underlying dynamics of

Although linear models are useful, our results indicate that these models have to be used with caution because the solar wind–magnetosphere system is inherently nonlinear. Hence, nonlinearities generally need to be taken into account in order to describe the system accurately. Local linear models (which include slow evolution of parameters) may be able to handle some nonlinearities, but it is expected that these local linear models would have difficulties if the dynamics suddenly and rapidly change.

All the derived data products in this paper are available upon request by email (simon.wing@jhuapl.edu).

The authors declare that they have no conflict of interest.

Simon Wing acknowledges support from JHU/APL Janney Fellowship, NSF grant AGS-1058456, and NASA grants (NNX13AE12G, NNX15AJ01G, NNX16AR10G, and NNX16AQ87G). Jay R. Johnson acknowledges support from NASA grants (NNH11AR07I, NNX14AM27G, NNH14AY20I, NNX16AC39G), NSF grants (ATM0902730, AGS-1203299, AGS-1405225), and DOE contract DE-AC02-09CH11466. Enrico Camporeale is partially funded by the NWO Vidi grant no. 639.072.716. We thank James M. Weygand for the solar wind data processing. The raw solar wind data from ACE, Wind, ISEE1, and ISEE3 were obtained from NASA CDAW and NSSDC. The topical editor, Georgios Balasis, thanks one anonymous referee for help in evaluating this paper.