ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus PublicationsGöttingen, Germany10.5194/angeo-36-945-2018Transfer entropy and cumulant-based cost as measures of nonlinear causal relationships in space plasmas: applications to DstCausal relationshipsJohnsonJay R.jrj@andrews.eduWingSimonhttps://orcid.org/0000-0001-9342-1813CamporealeEnricoAndrews University, Berrien Springs, MI, USAThe Johns Hopkins University, Applied Physics Laboratory, Laurel, MD, USACenter for Mathematics and Computer Science (CWI), Amsterdam, the NetherlandsJay R. Johnson (jrj@andrews.edu)2July201836494595219January201823January201825April201831May2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://angeo.copernicus.org/articles/36/945/2018/angeo-36-945-2018.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/36/945/2018/angeo-36-945-2018.pdf
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 Dst peaks at 3–12 h lags that can be
attributed to VBs, which also exhibits similar behavior.
However, the nonlinear significance that peaks at lags 25, 50, and 90 h can
be attributed to internal dynamics, which may be related to the relaxation of
the ring current. These peaks are absent in the linear and nonlinear
self-significance of VBs. Our analysis with mutual
information and transfer entropy shows that both methods can establish that
there are strong correlations and transfer of information from
Vsw to Dst at a timescale that is consistent with
that obtained from the cumulant-based analysis. However, mutual information
also shows that there is a strong correlation in the backward direction, from
Dst to Vsw, which is counterintuitive. In contrast,
transfer entropy shows that there is no or little transfer of information
from Dst to Vsw, as expected because it is the solar
wind that drives the magnetosphere, not the other way around. Our case study
demonstrates that these information theoretical tools are quite useful for
space physics studies because these tools can uncover nonlinear dynamics that
cannot be seen with the traditional analyses and models that assume linear
relationships.
Introduction
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., [a(t)] and [b(t)], is to apply linear correlation
studies such as , which showed the relationship
between the downward Poynting flux and ion outflows. Causal relationships are
typically identified by considering a time-shifted correlation function
λab(τ)≜〈a(t)b(t+τ)〉-〈a〉〈b〉〈a2〉-〈a〉2〈b2〉-〈b〉2,
where 〈…〉 is an ensemble average obtained by drawing
samples at a set of measurement times, {t0,t1,…,tN}. For
example, used such a method to identify relationships
between solar wind variables and plasma sheet variables. The causal
dependency that the plasma sheet responds to changes in the solar wind can be
identified from the time-shift of the peak of the cross-correlation
indicating a response time. From this type of analysis it can be found that
the plasma sheet generally responds from the tail to the inner magnetosphere,
consistent with the notion of earthward convection. Such analysis has been
particularly useful to help understand plasma sheet transport.
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., λab(τ)) and backward directions (i.e.,
λba(τ)=λab(-τ)). Finally, the cross-correlation
does not provide any way to clearly distinguish between two variables that
are passively correlated because of a common driver rather than causally
related.
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.
Linear vs. nonlinear dependency
It is well known that the magnetosphere responds to variation in the solar
wind parameters , and it has
been established that the magnetosphere has a significant linear response to
the solar wind. However, it is also expected that the magnetosphere has a
nonlinear response
.
The nonlinear response may be driven by internal dynamics rather than being driven
externally . For example, the internal
dynamics associated with loading and unloading of magnetic energy associated
with storms and substorms is nonlinear e.g.,and references
therein. Indeed, the data analysis of
indicated that the dynamical response of the magnetosphere to solar wind
input could not be entirely understood using linear prediction filters.
Suppose that we consider a set of variables a and
b, which could be vectors of variables
measured in time and we would like to measure their dependency. Instead of
considering the covariance matrix or correlation function, we consider a more
general measure of dependency between an input and output is obtained by
considering whether
P(a,b)=?P(a)P(b),
where P(a,b) is the joint probability of input a and
output b, while P(a) and P(b) are the probability of
a and b respectively. If the relationship holds, then the
variables a and b are independent. For all other cases, there
is some measure of dependency. In the case where the system output is
completely known given the input, P(a,b)=P(a). The
advantage of considering Eq. () is that it is possible to detect
the presence of higher order nonlinear dependencies between the input and
output even in the absence of linear dependencies .
Mutual information and cumulant-based cost
Mutual information and cumulant-based cost are two useful measures that
quantify Eq. (). Mutual information has the advantage that in
the limit of Gaussian joint probability distributions, it may be simply
related to the correlation coefficient Cab(τ) defined in
Eq. () . Cumulants have the advantage of good
statistics for limited datasets and noisy systems .
Moreover, for high-dimensional systems it is more efficient to compute
moments of the data rather than try to construct the probability density
function.
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 and cumulants
are useful for detecting linear as well as nonlinear
correlations. The mutual information is constructed from the probability
distribution function of the variables and may be computed using a
quantization procedure where data are binned such that the samples [a(t)]
are assigned discrete values a^∈{a1,a2,…,an} of an
alphabet ℵ1 and [b(t)] is assigned discrete values b^∈{b1,b2,…,bm} of an alphabet ℵ2. The ad hoc
time-shifted mutual entropy
Mab(τ)≜∑a^∈ℵ1,b^∈ℵ2p(a^(t+τ),b^(t))logp(a^(t+τ),b^(t))p(a^)p(b^)
has been used as an indicator of causality, but suffers from the same
problems as the time-shifted cross-correlation when it has multiple peaks and
long-range correlations.
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
DC=∑q=1∞∑i1,…,iq∈ΠqK1i2…iq2,
where
Ki=Ci=〈zi〉Kij=Cij-CiCj=〈zizj〉-〈zi〉〈zj〉Kijk=Cijk-CijCk-CjkCi-CikCj+2CiCjCkKijkl=Cijkl-CijkCl-CijlCk-CilkCj-CljkCi-CijCkl-CilCkj-CikCjl+2(CijCkCl+CikCjCl+CilCjCk+CjkCiCl+CjlCiCk+CklCiCj)-6CiCjCkCl
are the cumulants
Ci…j=∫dzP(z)zi…zj≡〈zi…zj〉
of the joint probability distribution for variables z1,…,zj.
With only two variables, a and b, defined above, we can
consider the cost function
Da,bC(τ)=DC(a(t),b(t+τ)).
The presence of nonlinear dependence has been identified by comparing the
cumulant cost for a time series with the cumulant-based cost of surrogate
time series, which are constructed to have the same linear correlations as in
. Significance measures the difference in the
discriminating statistic from the mean of the discriminating statistic of the
surrogates in terms of the spread of the surrogates, σ.
In Sect. , we will show an application of cumulant-based analysis to the disturbance storm time index (Dst). In
principle, the cross-correlation, mutual information, and cumulant-based cost
should be independent of the selection of measurement points if the system is
stationary; therefore, time stationarity can be examined by comparing these
discriminating statistics for groups of measurements drawn from different
windows of time as in and .
Transfer entropy
Another method for determining causality is the one-sided transfer entropy
, which is based upon the conditional mutual information
MC(x,y|z)≜∑x∈ℵ1∑y∈ℵ2∑z∈ℵ3p(x,y,z)logp(x,y,z)p(z)p(x,z)p(y,z).
The conditional mutual information measures the dependence of two variables,
x and y, given a conditioner variable, z. If either x or y are
dependent on z, the mutual information between x and y is reduced, and
this reduction of information provides a method to eliminate coincidental
dependence, or conversely to identify causal dependence.
Transfer entropy considers the conditional mutual information between two
variables using the past history of one of the variables as the conditioner.
Ta→b(τ)=∑a^∈ℵ1∑a^(k)∈ℵ1(k)∑b^∈ℵ2p(a^(t+τ),a^(k)(t),b^(t))logp(a^(t+τ)|a^(k)(t),b^(t))p(a^(t+τ)|a^(k)(t)),
where a^(k)(t)=[a^(t),a^(t-Δ),…,a^(t-(k-1)Δ)]. The standard definition of transfer entropy takes
k=1 (no lag), but keeping a higher embedding dimension could in principle
provide a more precise measure (for example, if a has periodicity, a
dimension of 2 may provide better prediction of future values of a from its
past time series and therefore lower the transfer entropy). Transfer entropy
as a discriminating statistic has the following advantages. First, in the
absence of information flow from a to b (i.e., a(t+τ) has no
additional dependence from b(t) beyond what is known from the past history
of a(k)(t)) so that p(a^(t+τ)|a^(k)(t),b^(t))=p(a^(t+τ)|a^(k)(t)) and the transfer entropy
vanishes. The transfer entropy is also highly directional so that
Ta→b≠Tb→a. The advantage can
be clearly seen for dynamical systems in which variables are forward differenced
and the transfer entropy is clearly one-sided while mutual information and
correlation functions can even be symmetric . This
measure also accounts for static internal correlations, which can be used to
determine whether two variables are driven by a common driver or whether the
variable b is causally driving the variable a.
Both mutual information and transfer entropy require binning of data. As
mentioned in , the number of bins (nb) needs to
be chosen properly and there are some guidelines that can be followed. In
general, we would like to maximize the amount of information. Having too few
bins would lump too many points into the same bin, leading to loss of
information. Conversely, having too many bins would leave many bins with 0 or
a few number of points, which also would lead to loss of information.
proposed that for a normal distribution, optimal
nb=log2(n)+1 and bin width
w = range/nb, where n is the number of points in
the dataset and range is the maximum value minus the minimum value of the points. In
practice, there is usually a range of nb that would work.
Application to space weather: Dst analysis
Dst (disturbance storm time index) is an hourly index that gives
a measure of the strength of the symmetric ring current that, in turn,
provides a measure of the dynamics of geomagnetic storms
. Because of its global nature, Dst is
often used as one of the several indices that represent the state of the
magnetosphere. For example, used the cumulative square
amplitude of the Dst time series as a proxy for energy dissipation
rate in the magnetosphere and found that it fits a power law well with
log-periodic oscillations, which was interpreted as evidence for discrete-scale invariance in the Dst dynamics.
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) e.g.,. An increase in
the solar wind electric field, VBz, can increase the dawn–dusk
electric field in the magnetotail, which in turn determines the number of
plasma sheet particles that move to the inner magnetosphere
e.g.,. Studies have shown that the electric field,
VBs (Vsw× southward IMF Bz)
or VBz, has a strong effect on the ring current dynamics
.
For the present study, we examine the relationships between solar wind
velocity (Vsw) and VBs with Dst.
We use Dst records in the period 1974–2001 obtained from Kyoto
University World Data Center for Geomagnetism
(http://swdcwww.kugi.kyoto-u.ac.jp/index.html, last access:
18 January 2018). The corresponding solar wind
data are obtained from IMP-8, ACE, WIND, ISEE1, and ISEE3 observations. The
ACE SWEPAM and MAG data and the WIND MAG data are obtained from CDAWeb
(http://cdaweb.gsfc.nasa.gov/, last access: 18 January 2018). The WIND 3DP data are obtained from the 3DP team
directly. The ISEE1 and ISEE3 data are obtained from UCLA (these datasets are
also available at NASA NSSDC; http://nssdc.gsfc.nasa.gov/space/, last
access: 18 January 2018). The IMP8 data come
directly from the IMP teams. The solar wind is propagated with the minimum
variance technique to GSM (X, Y, Z) = (17, 0,
0) RE to produce 1 min files, from which hourly averaged solar
wind parameters are constructed.
Cumulant-based analysis
Section presents the method of cumulant-based cost. Here,
we show an application of cumulant-based cost to detect nonlinear dynamics in
Dst. We consider the forward coupling between a solar wind
variable such as VBs and Dst, which
characterizes the ring current response to the solar wind driver. We
therefore consider the nonlinear cross-correlations of the vector
c(t,τ)={VBs(t),Dst(t+τ)}={z1,z2}.
The generalization of cost is based on realizations of {z1,z2}. In
this case, each variable is Gaussianized with unit variance to eliminate
static nonlinearities (i.e., higher order self-correlations in
VBs and Dst are eliminated so that the cost
measures only cross-dependence between VBs and
Dst). This procedure is explained in the next paragraph.
The distributions of Dst and VBs are
generally non-Gaussian. As such, the raw distributions (e.g., distribution of
values of Dst) may have nonzero higher order cumulants (e.g.,
they can have a skew and kurtosis). This property makes it more difficult to
interpret whether the higher order cumulants in the time evolution arise from
the overall shape of the distribution of data points or from the
time-ordering of the data. To eliminate the inherent nonzero cumulants in the
overall distribution of data, we construct a rank-ordered map from the
original dataset to a proxy dataset of the same length drawn from a Gaussian
distribution . The
distribution of the proxy dataset ensures that all cumulants of the
distribution beyond second order should in principle vanish. However, the
time-ordering of the data can still lead to nonzero cumulants because the
joint probability distribution of Dst(t+τ) and
Dst(t) may be non-Gaussian even if the distribution of
Dst is Gaussian. Moreover, it is simple to construct surrogate
data from the Gaussianized data that share the same autocorrelation by using
the same power spectrum but randomly shifting the phases of the Fourier
coefficients. The surrogate data therefore have the same autocorrelation as
the original data. Any deviation from the linear statistic is apparent from
comparison with the surrogate data, and we interpret these deviations as
evidence of nonlinear dependence because we have falsified the hypothesis
that the data can be adequately described by linear statistics. This method
has been successfully employed in , in which the Kp record was
analyzed with mutual information and cumulants.
In Fig. we plot the significance obtained from the year 1999
as a function of time delay, τ. Significance extracted from
{VBs(t),Dst(t+τ)} and
{VBs(t),VBs(t+τ)} for 1999 is
plotted in panels (a) and (b), respectively. It should be noted that there is
a strong linear response at around 3 h time delay. As shown in
Fig. a, there is a clear nonlinear response with peaking
around 3–10, 25, 50, and 90 h, lasting for approximately 1 week. In contrast,
in Fig. b, the nonlinearity only has one broad peak around
3–12 h in the self-significance for VBs, suggesting
that the nonlinear and linear peaks at τ=3–12 h in
Fig. a may be associated with
VBs. We will revisit the solar wind causal relationship
with Dst using transfer entropy in
Sect. .
The absence of the nonlinear peaks at τ= 25, 50, and 90 h in the
self-significance for VBs (Fig. b)
suggests that these nonlinearities in
{VBs(t),Dst(t+τ)} are related to internal
magnetospheric dynamics. As the Dst index is thought to reflect
storm activity, it is reasonable that nonlinear significance would decay on
the order of 1 week as storms commonly last around that time. The strong
nonlinear responses at τ= 25, 50, and 90 h are likely related to
multiple modes of relaxation of the ring current following the commencement
of storms. It should also be noted that other nonlinearities detected by even
higher order cumulants may also be present; however, the calculation
demonstrates the nonlinear nature of the underlying dynamics.
Significance extracted
from (a){VBs(t),Dst(t-τ)} and
(b){VBs(t),VBs(t-τ)}
for 1999. It should be noted that there is a strong linear response at around
3 h time delay. There is a clear nonlinear response with a strong peak
around 50 h lasting for approximately 1 week. The long-term nonlinear
response is absent in the solar wind data, indicating that the long-term
nonlinear correlations between VBs and Dst
are the result of internal magnetospheric dynamics.
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 ∼ 6 h . Once the injection stops, the ring
current begins to decay and the storm enters the recovery phase. Conservation
of the magnetic moment implies that anisotropies develop in the ring current and
plasma sheet. Anisotropy drives the ring current plasma unstable to ion
cyclotron waves. The ion cyclotron waves scatter energetic ions into the loss
cone so that they are lost from the ring current. Nonlinear interaction
between waves and particles keeps the plasma near marginal stability with a
steady loss of energetic particles due to wave–particle scattering. Other
loss mechanisms include charge exchange, Coulomb scattering, and convection
of ions to the front of the magnetopause. The ring current decay can have two
stages . In the first stage, the ring current decays
rapidly and the loss mechanisms can be attributed to convective outflow,
pitch-angle scattering in the ring current, and O+ charge exchange
e.g.,. The second stage may typically
begin about 1 day from the commencement of the storm (see, for example,
Fig. 7 of ). In the second stage, the decay rate is
slower and is attributed mainly to H+ charge exchange
and can take several days to deplete the ring current
to the baseline level . We can speculate that the
multiple nonlinear response lag times that are detected with the
cumulant-based approach are likely the relaxation of the ring current due to
the complex interplay of multiple loss processes.
Comparison of mutual information and transfer entropy measures to
determine causal driving of the magnetosphere as characterized by
Dst. Note that causal driving appears to peak somewhat later
(11 h) than indicated by mutual information (2 h), indicating that internal
dynamics likely are very important initially. The backward transfer entropy
is below the noise level for all values, indicating that Dst in no
way influences the upstream solar wind velocity. Such a conclusion could not
be inferred from the mutual information measure.
Transfer entropy
As mentioned in Sect. , transfer entropy gives a measure of
how much information is transferred from one variable to another. We have
applied transfer entropy and mutual information to the relationship between
the Vsw and Dst for the period 1974–2001. The result
is shown in Fig. . Note that the mutual information
measure suggests strong correlations between prior values of Dst
and Vsw. This finding suggests that Dst could be a
driver of Vsw, which is counterintuitive. On the other hand, the
transfer entropy clearly shows that this information transfer in the backward
direction (Dst→Vsw) does not rise above the
noise level (the horizontal blue lines indicate mean and standard deviation
of 100 surrogate datasets for which the data were randomly reordered.) This
result is expected because it is the solar wind that drives the
magnetosphere, not the other way around. The transfer of information from
Vsw to Dst peaks at τ=8–11 h. The cumulant-based analysis in Sect. shows that the
response of Dst to VBs has a similar timescale. This timescale
is consistent with the 4 to 15 h transport time for the solar wind to reach
the midnight and noon regions of the geosynchronous orbit, respectively, from
the dayside magnetopause . The analysis presented here
illustrates the power of the transfer entropy for accessing causality.
Summary
We recently used mutual information, transfer entropy, and conditional mutual
information to discover the solar wind drivers of the outer radiation belt
electrons . Because Vsw anticorrelates with
solar wind density (nsw), it is hard to isolate the effects of
Vsw on radiation belt electrons, given nsw and vice
versa. However, using conditional mutual information, we were able to
determine the information transfer from nsw or any other solar
wind parameters to radiation belt electrons, given Vsw (or any
other solar wind parameters). We also showed that the triangle distribution
in the radiation belt electron vs. solar wind velocity plot
can be understood better when we consider that
Vsw and nsw transfer information to radiation belt
electrons with lags of 2 and 0 days (< 24 h), respectively. Also recently,
we used transfer entropy to better understand the causal parameters in the
solar cycle dynamo and their response lag times .
As a follow-up to , the present study
demonstrates further how information theoretical tools can be useful for
space physics and space weather studies. Cumulant-based analysis can be used
to distinguish internal vs. external driving of the system. Both mutual
information and transfer entropy give a measure of shared information between
two variables (or vectors). However, unlike mutual information, transfer
entropy is highly directional. To illustrate, we apply mutual information,
transfer entropy, and cumulant-based analysis to investigate the dynamics of
the Dst index.
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 Vsw
and Dst (Vsw→Dst). However,
mutual information indicates that there is also a strong correlation in the
backward direction (Dst→Vsw), which is
puzzling and counterintuitive. In contrast, the transfer entropy indicates
that there is no information transfer in the backward direction
(Dst→Vsw), as expected because it is the
solar wind that drives the magnetosphere, not the other way around. The
transfer of information from Vsw to Dst peaks at
τ=8–11 h.
Using the cumulant-based significance, we have established that the
underlying dynamics of Dst is in general nonlinear, exhibiting a
quasiperiodicity which is detectable only if nonlinear correlations are taken
into account. The strong nonlinear responses of Dst to
VBs at τ=25, 50, and 90 h are likely related to
multiple modes of relaxation of the ring current from multiple loss
mechanisms following the commencement of storms. It is, of course, possible
that these nonlinearities are caused by solar wind drivers other than
VBs. However, the timing of these nonlinearities would
put them well in the recovery phase of a storm, and previous studies suggested
that the ring current decays in the recovery phase are strongly influenced by
VBs.
The nonlinearities at τ=3–12 h are not caused by internal dynamics
but rather by the solar wind driver, which is similar to the timescale for
the solar wind transport time from the dayside magnetopause to the inner
magnetosphere. This timescale is consistent with the timescale for the
information transfer from the solar wind to Dst obtained from
transfer entropy analysis.
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.
Acknowledgements
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.
ReferencesBaker, D. N., Zwickl, R. D., Bame, S. J., Hones, E. W., Tsurutani, B. T.,
Smith, E. J., and Akasofu, S.-I.: An ISEE 3 high time resolution study of
interplanetary parameter correlations with magnetospheric activity, J. Geophys. Res., 88, 6230, 10.1029/ja088ia08p06230, 1983.Balasis, G., Papadimitriou, C., Daglis, I. A., Anastasiadis, A.,
Athanasopoulou, L., and Eftaxias, K.: Signatures of discrete scale invariance
in Dst time series, Geophys. Res. Lett., 38, L13103, 10.1029/2011GL048019, 2011.Balikhin, M. A., Boynton, R. J., Walker, S. N., Borovsky, J. E., Billings,
S. A., and Wei, H. L.: Using the NARMAX approach to model the evolution of
energetic electrons fluxes at geostationary orbit, Geophys. Res.
Lett., 38, L18105, 10.1029/2011GL048980, 2011.
Bargatze, L. F., Baker, D. N., Hones, E. W., and McPherron, R. L.:
Magnetospheric impulse response for many levels of geomagnetic activity, J.
Geophys. Res., 90, 6387–6394, 1985.Borovsky, J. E., Thomsen, M. F., and Elphic, R. C.: The driving of the
plasma sheet by the solar wind, J. Geophys. Res., 103, 17617–17640,
10.1029/97JA02986, 1998.
Burton, R. K., McPherron, R. L., and Russell, C. T.: An Emperical
Relationship Between Interplanetary Conditions and Dst, J. Geophys. Res.,
80, 4204–4214, 1975.Clauer, C. R., McPherron, R. L., Searls, C., and Kivelson, M. G.: Solar wind
control of auroral zone geomagnetic activity, Geophys. Res. Lett.,
8, 915–918, 10.1029/gl008i008p00915, 1981.Crooker, N. U. and Gringauz, K. I.: On the low correlation between long-term
averages of solar wind speed and geomagnetic activity after 1976, J.
Geophys. Res., 98, 59–62, 10.1029/92ja01978, 1993.
Deco, G. and Schürmann, B.: Information Dynamics, Springer-Verlag, New
York,
2000.De Michelis, P., Consolini, G., Materassi, M., and Tozzi, R.: An information
theory approach to the storm-substorm relationship, J. Geophys.
Res.-Space, 116, A08225, 10.1029/2011JA016535, 2011.Dessler, A. J. and Parker, E. N.: Hydromagnetic theory of geomagnetic storms, J. Geophys. Res., 64, 2239–2252, 10.1029/JZ064i012p02239, 1959.Friedel, R. H. W., Korth, H., Henderson, M. G., Thomsen, M. F., and Scudder,
J. D.: Plasma sheet access to the inner magnetosphere, J. Geophys.
Res.-Space, 106, 5845–5858, 10.1029/2000ja003011, 2001.
Gershenfeld, N.: The Nature of Mathematical Modeling, Cambridge University
Press, Cambridge, 1998.
Hamilton, D., Gloeckler, G., Ipavich, F., Stüdemann, W., Wilken, B., and
Kremser, G.: Ring current development during the great geomagnetic storm of
February 1986, J. Geophys. Res.-Space, 93,
14343–14355, 1988.Johnson, J. R. and Wing, S.: A solar cycle dependence of nonlinearity in
magnetospheric activity, J. Geophys. Res., 110, A04211,
10.1029/2004ja010638, 2005.Johnson, J. R. and Wing, S.: External versus internal triggering of substorms:
An information-theoretical approach, Geophys. Res. Lett., 41,
5748–5754, 10.1002/2014gl060928, 2014.Johnson, J. R. and Wing, S.: The dependence of the strength and thickness of
field-aligned currents on solar wind and ionospheric parameters, J. Geophys.
Res.-Space, 120, 3987–4008, 10.1002/2014ja020312, 2015.
Kennel, M. B. and Isabelle, S.: Method to Distinguish Possible Chaos from
Colored Noise and to Determine Embedding Parameters, Phys. Rev. A, 46,
3111–3118, 1992.Kissinger, J., McPherron, R. L., Hsu, T.-S., and Angelopoulos, V.: Steady
magnetospheric convection and stream interfaces: Relationship over a solar
cycle, J. Geophys. Res.-Space, 116, A00I19,
10.1029/2010ja015763, 2011.
Klimas, A. J., Vassiliadis, D., and Baker, D. N.: Dst index prediction
using data-derived analogues of the magnetospheric dynamics, J. Geophys.
Res., 103, 20435–20448, 1998.Kozyra, J., Liemohn, M., Clauer, C., Ridley, A., Thomsen, M., Borovsky, J.,
Roeder, J., Jordanova, V., and Gonzalez, W.: Multistep Dst development and
ring current composition changes during the 4–6 June 1991 magnetic storm,
J. Geophys. Res.-Space, 107, SMP 33-1-SMP 33-22, 10.1029/2001JA000023, 2002.Li, W.: Mutual information functions versus correlation functions, J. Stat.
Phys., 60, 823, 10.1007/BF01025996, 1990.Materassi, M., Ciraolo, L., Consolini, G., and Smith, N.: Predictive Space
Weather: An information theory approach, Adv. Space Res., 47,
877–885, 10.1016/j.asr.2010.10.026, 2011.
Materassi, M., Consolini, G., Smith, N., and De Marco, R.: Information theory
analysis of cascading process in a synthetic model of fluid turbulence,
Entropy, 16, 1272–1286, 2014.Mcpherron, R. L. and O'Brien, P.: Predicting Geomagnetic Activity: The DstIndex, in: Space Weather, edited by: Song, P., Singer, H. J., and
Siscoe, G. L.,
10.1029/GM125p0339, 2001.Newell, P., Liou, K., Gjerloev, J., Sotirelis, T., Wing, S., and Mitchell, E.:
Substorm probabilities are best predicted from solar wind speed, J.
Atmos. Sol.-Terr. Phy., 146, 28–37,
10.1016/j.jastp.2016.04.019, 2016.
O'Brien, T. P. and McPherron, R. L.: An empirical phase space analysis of
ring current dynamics: Solar wind control of injection and decay, J.
Geophys. Res., 105, 7707–7720, 2000.Papitashvili, V. O., Papitashvili, N. E., and King, J. H.: Solar cycle effects
in planetary geomagnetic activity: Analysis of 36-year long OMNI dataset,
Geophys. Res. Lett., 27, 2797–2800, 10.1029/2000gl000064,
2000.Prichard, D. and Theiler, J.: Generalized redundancies for time series
analysis, Phys. D, 84, 476–493,
10.1016/0167-2789(95)00041-2, 1995.Reeves, G. D., Morley, S. K., Friedel, R. H. W., Henderson, M. G., Cayton,
T. E., Cunningham, G., Blake, J. B., Christensen, R. A., and Thomsen, D.: On
the relationship between relativistic electron flux and solar wind velocity:
Paulikas and Blake revisited, J. Geophys. Res.-Space,
116, A02213, 10.1029/2010ja015735, 2011.Schreiber, T.: Measuring Information Transfer, Phys. Rev. Lett., 85, 461–464,
10.1103/PhysRevLett.85.461, 2000.
Schreiber, T. and Schmitz, A.: Improved Surrogate Data for Nonlinearity Tests,
Phys. Rev. Lett., 77, 635–639, 1996.Smith, P. H., Hoffman, R. A., and Fritz, T. A.: Ring current proton decay by
charge exchange, J. Geophys. Res., 81, 2701–2708,
10.1029/JA081i016p02701, 1976.Strangeway, R., Ergun, J. R. E., Su, Y.-J., Carlson, C. W., and Elphic, R. C.:
Factors controlling ionospheric outflows as observed at intermediate
altitudes, J. Geophys. Res., 110, A03221, 10.1029/2004ja010829,
2005.Sturges, H. A.: The choice of class interval, J. Am. Stat. Assoc., 21, 65–66,
10.1080/01621459.1926.10502161, 1926.
Tsurutani, B. T., Sugiura, M., Iyemori, T., Goldstein, B. E., Gonzalez, W. D.,
Akasofu, S. I., and Smith, E. J.: The nonlinear response of AE to the IMF Bs
driver: A spectral break at 5 hours, Geophys. Res. Lett., 17,
279–282, 1990.Valdivia, J. A., Rogan, J., Muñoz, V., Toledo, B. A., and Stepanova, M.:
The magnetosphere as a complex system, Adv. Space Res., 51,
1934–1941, 10.1016/j.asr.2012.04.004, 2013.
Vassiliadis, D. V., Sharma, A. S., Eastman, T. E., and Papadopoulos,
K.: Low-dimensional chaos in magnetospheric activity from AE time series,
Geophys. Res. Lett., 17, 1841–1844, 1990.Weimer, D. R., Ober, D. M., Maynard, N. C., Collier, M. R., McComas, D. J.,
Ness, N. F., Smith, C. W., and Watermann, J.: Predicting interplanetary
magnetic field (IMF) propagation delay times using the minimum variance
technique, J. Geophys. Res., 108, 1026, 10.1029/2002ja009405,
2003.
Weygand, J. M. and McPherron, R. L.: Dependence of ring current asymmetry on
storm phase, J. Geophys. Res.-Space, 111, A11221,
10.1029/2006JA011808, 2006.Wing, S. and Johnson, J. R.: Theory and observations of upward field-aligned
currents at the magnetopause boundary layer, Geophys. Res. Lett.,
42, 9149–9155, 10.1002/2015gl065464, 2015.Wing, S., Johnson, J. R., Jen, J., Meng, C.-I., Sibeck, D. G., Bechtold, K.,
Freeman, J., Costello, K., Balikhin, M., and Takahashi, K.: Kp forecast
models, J. Geophys. Res., 110, A04203, 10.1029/2004ja010500, 2005.Wing, S., Johnson, J. R., Camporeale, E., and Reeves, G. D.: Information
theoretical approach to discovering solar wind drivers of the outer radiation
belt, J. Geophys. Res.-Space, 121, 9378–9399,
10.1002/2016ja022711, 2016.Wing, S., Johnson, J. R., and Vourlidas, A.: Information Theoretic Approach to
Discovering Causalities in the Solar Cycle, Astrophys. J., 854, 2,
10.3847/1538-4357/aaa8e7, 2018.