Using the solar-wind-driven magnetosphere–ionosphere–thermosphere system, a methodology is developed to reduce a state-vector description of a time-dependent driven system to a composite scalar picture of the activity in the system. The technique uses canonical correlation analysis to reduce the time-dependent system and driver state vectors to time-dependent system and driver scalars, with the scalars describing the response in the system that is most-closely related to the driver. This reduced description has advantages: low noise, high prediction efficiency, linearity in the described system response to the driver, and compactness. The methodology identifies independent modes of reaction of a system to its driver. The analysis of the magnetospheric system is demonstrated. Using autocorrelation analysis, Jensen–Shannon complexity analysis, and permutation-entropy analysis the properties of the derived aggregate scalars are assessed and a new mode of reaction of the magnetosphere to the solar wind is found. This state-vector-reduction technique may be useful for other multivariable systems driven by multiple inputs.

In this report a methodology is described that can produce a compact
description of the behavior of a time-dependent, multivariable system driven
by a time-dependent, multivariable driver or by multiple drivers. The system
used to develop this methodology is the Earth's
magnetosphere–ionosphere–thermosphere system driven by the time-dependent
solar wind. The spatial domain wherein the Earth's magnetic field dominates
over the solar wind is known as the magnetosphere. The interaction between
the solar wind, and the magnetosphere is surprisingly complex and the
magnetosphere's evolution in response to the time-varying solar wind is rich
and diverse. The magnetospheric system is characterized by multiple
subsystems that interact with each other (cf. Lyon, 2000; Otto, 2005;
Siscoe, 2011; Eastwood et al., 2015; Borovsky and Valdivia, 2018): almost 6
orders of magnitude of spatial scales are involved in the global behavior of
the magnetosphere, from

The methodology developed here creates a compact description of a
time-dependent, multivariable system driven by a time-dependent,
multivariable driver. The diverse variables describing the system may be
intercorrelated, and the variables describing the driving may be
intercorrelated. The methodology was developed to gain an understanding of
the Earth's magnetosphere–ionosphere–thermosphere system as driven by the
solar wind. To utilize the methodology the system and its driver are
conceptualized by a time-dependent, multidimensional system state vector

The solar-wind-driven magnetospheric system very cleanly follows the

The nine time-dependent variables going into the system state vector

This report is organized as follows. In Sect. 2 the CCA approach is
applied to the magnetospheric system driven by the solar wind to derive the
first three time-dependent sets of composite variables

Using the Earth's magnetosphere–ionosphere–thermosphere system as driven by
the solar wind, the reduction of a time-dependent state-vector picture

One-hour averages of all magnetospheric and solar-wind variables are used in the years 1991–2007. No time lags are used between the solar-wind measurements and the magnetospheric measurements: most expected time lags will be about 1 h (e.g., Clauer et al., 1981; Smith et al., 1999), which is the time resolution of the data set.

Canonical correlation analysis (CCA) is applied to the time-dependent state
vectors

CCA is a matrix equation solution, non-iterative, that yields a single
unique solution (Johnson and Wichern, 2007). CCA operates on standardized variables (with the
mean value subtracted and the values then divided by the standard
deviation), denoted with an asterisk. (For each variable the mean value
and standard deviation are calculated for the entire data set.) CCA operates
most efficiently on variables that are Gaussian distributed: hence the
logarithms of some variables are used to yield more-Gaussian-like
distributions. All standardized variables

When CCA is applied to the 1991–2007

CCA applied to the time-dependent state vectors

The three sets of composite variables

In Fig. 1 the composite system variable

The aggregate system scalar

Note that whereas the correlation coefficient between

As a further note, the Pearson linear correlation coefficients between

Plots of the nine components of the coefficient vectors used to
project the system state vector

In the six panels of Fig. 2 the coefficients of the six vectors

Using the linear-regression curve in Fig. 1 as a “prediction” of the
value of

The autocorrelation functions for the system scalar

The quantity

To investigate this substorm hypothesis for

Superposed epoch averages of

Compacting the description of the system from a multidimensional state
vector to a few variables

Jensen–Shannon complexity and permutation entropy, AL (red) and am
(blue) for 500 points with no gaps sampled at 1 h interval, embedding
dimension

Jensen–Shannon complexity and permutation entropy for

The bottom left panels in Figs. 5 and 6 show the value of the permutation
entropy for AL, am,

For longer embedding delays of the order of seasonal variations ranging from
27 to 45 d (not shown), all four time series overlap and are
indistinguishable from stochastic fluctuations. In terms of
complexity–entropy plane it translates into a permutation entropy of
approximately 1 and a Jensen–Shannon complexity of almost 0. Hence, the
system variable

In Fig. 7a and b the second and third scalar pairs are plotted,

For the 1 h resolution 1991–2007 data set, the aggregate system
scalar

Figure 2c shows that mode

Figure 2e shows that

The aggregate variable

Future developments of this methodology will focus on the introduction of time lags between the driver and the system, on the introduction of integro-differential correlations rather than algebraic correlations (e.g., Borovsky, 2017), and on the use of dynamic canonical correlation analysis (e.g., Dong and Qin, 2018a, b).

The 1991–2007 data set of hourly values of

The time-dependent variables of the magnetospheric system state vector and the solar-wind driver state vector are listed in Table 1.

The magnetospheric variables measure various aspects of activity in the
magnetosphere. The auroral upper index AU (Davis and Sugiura, 1966) measures
the electrical current in the high-latitude ionosphere: this variable is taken
to be a measure of electrical currents in the dayside magnetosphere (Goertz
et al., 1993). The auroral lower index AL (Davis and Sugiura, 1966) measures
the electrical current in the high-latitude nightside ionosphere: this variable
is taken to be a measure of auroral activity in the nightside magnetosphere
(Goertz et al., 1993). The polar cap index PCI is a measure of the strength
of cross-polar-cap electrical current in the ionosphere (Troshichev et al.,
1988). The planetary

The variables going into the solar-wind driver state vector are various
measures of the time-dependent solar wind at Earth. The solar wind speed

JEB devised this study and performed the CCA analysis. AO performed the complexity and entropy analysis. Both authors are responsible for the interpretation of the results and for the writing of the manuscript.

The authors declare that they have no conflict of interest.

The authors thank Mick Denton and Juan Alejandro Valdivia for helpful discussions.

This work was supported by the NSF GEM Program via award AGS-1502947, by the NASA Heliophysics LWS TRT program via grants NNX16AB75G and NNX14AN90G, by the NSF Solar-Terrestrial Program via grant AGS-12GG13659, by the NASA Heliophysics Guest Investigator Program via grant NNX17AB71G, by the NSF SHINE program via award AGS-1723416, and by the Academy of Finland via grant no. 297688/2015.

This paper was edited by A. Surjalal Sharma and reviewed by Marina Stepanova and one anonymous referee.