# Incorporating Polar Field Data for Improved Solar Flare Prediction

Mehmet Aktukmak 1, Zeyu Sun 1, Monica Bobra 2, Tamas Gombosi 3, Ward B.
Manchester 4, Yang Chen 5 and Alfred Hero 1,5,∗

###### Abstract

In this paper, we consider incorporating data associated with the sun’s north
and south polar field strengths to improve solar flare prediction performance
using machine learning models. When used to supplement local data from active
regions on the photospheric magnetic field of the sun, the polar field data
provides global information to the predictor. While such global features have
been previously proposed for predicting the next solar cycle’s intensity, in
this paper we propose using them to help classify individual solar flares. We
conduct experiments using HMI data employing four different machine learning
algorithms that can exploit polar field information. Additionally, we propose
a novel probabilistic mixture of experts model that can simply and effectively
incorporate polar field data and provide on-par prediction performance with
state-of-the-art solar flare prediction algorithms such as the Recurrent
Neural Network (RNN). Our experimental results indicate the usefulness of the
polar field data for solar flare prediction, which can improve Heidke Skill
Score (HSS2) by as much as 10.1% 111Acknowledgements: The authors would like
to thank Graham Barnes for his valuable comments and suggestions. This work
was partially supported by NASA DRIVE Science Center grant 80NSSC20K0600 and
W. Manchester was partially supported by NASA grant 80NSSC18K1208..

##  1 Keywords:

solar flare prediction, polar fields, solar cycle, mixture modeling

##  2 Introduction

Solar flares rapidly release magnetic energy stored in the Sun’s corona, which
heats large areas of the surrounding atmosphere. The terrestrial impacts of
flares can be serious, especially when they are high-intensity.  Although weak
flares have a limited impact on mankind, strong flares can disturb satellite
communication and GPS-navigation systems, in addition to causing dangerous
ionizing radiation exposure to astronauts orbiting the earth. Hence, timely
prediction of solar flares, especially the strong ones, is desirable so that
precautions can be taken. However, strong flares occur rarely as compared to
weak flares, and the underlying mechanisms and the factors that trigger and
determine the strength of the flares are not well understood. This makes the
prediction of solar flares a complex task.

As the coronal field is anchored in the photosphere, the onset of flares has
been linked to changes in the photospheric magnetic field (Wang et al., 2020;
Sun et al., 2012). Consequently, it is reasonable to expect that photospheric
magnetic fields in active regions (AR) may be useful indicators of flare-
triggering mechanisms. The Helioseismic and Magnetic Imager (HMI) (Scherrer et
al., 2012; Hoeksema et al., 2014) on Solar Dynamics Observatory (SDO) (Pesnell
et al., 2012) spacecraft has observed the photospheric magnetic field of the
sun since 2010, taking high-resolution images with 1 arcsec spatial resolution
and 720s cadence. From these magnetic field images, useful features are
computed, known as Space-weather HMI Active Region Patches (SHARP) parameters,
which are representative of the magnetic field’s scale, energy and complexity
(Bobra et al., 2014). This time-series data has been successfully used for
solar flare prediction for Solar Cycle 24 using machine learning algorithms
(Chen et al., 2019; Jonas et al., 2018; Bobra and Couvidat, 2015; Nishizuka et
al., 2017; Florios et al., 2018; Liu et al., 2017; Camporeale, 2019; Korsós et
al., 2021; Campi et al., 2019). See also (Leka et al., 2019) for a
comprehensive review.

Several supervised machine learning models have been proposed to predict solar
flares from HMI observations. NOAA Geo-stationary Operational Environmental
Satellites (GOES) flare list (Garcia, 1994) provides peak energy levels for
each flare on a logarithmic scale separated into five flare intensity classes,
labeled A, B, C, M, X, with A the weakest and X the strongest intensities.
These can be used as categorical response variables in machine learning models
for flare prediction. Such models use input features, such as HMI SHARP
parameters, to apply off-the-shelf machine learning algorithms to predict the
labels of the flares, including discriminant analysis (Leka et al., 2018),
linear regression (Jonas et al., 2018), support vector machine (Bobra and
Couvidat, 2015; Nishizuka et al., 2017; Florios et al., 2018), k-nearest
neighbor (Nishizuka et al., 2017), random forests (Liu et al., 2017; Florios
et al., 2018), multilayer perceptrons (Florios et al., 2018), and recurrent
neural networks (Chen et al., 2019).  Recently, the magnetogram images of the
active regions have been used as features in deep learning models such as
convolutional neural networks (CNN) [(Huang et al., 2018), for prediction of
flare intensity class. Ensemble models have also shown promise as a way of
combining predictions of multiple models that use different data sources
(Guerra et al., 2020; Sun et al., 2022).

Hale’s Polarity laws (Hale et al., 1919) for sunspots apprise that the
polarities of the sunspots (distinguished in pairs with a leading/following
counterpart from east to west) have a strong tendency to have the opposite
pattern in the northern and southern hemispheres of the sun in a particular
cycle. Moreover, the polarities of the spots of the present cycle are the
opposite of the polarities in the previous cycle. Additionally, polar field
reversals occur systematically at approximately the time of the solar cycle
maximum (Babcock, 1959), with peak polar field values occurring at the time of
the sunspot cycle minimum. Thus, the polar fields are 180 degrees out of phase
with the sunspot cycle. See Fig. 1 for plots over time of the polar fields
along with curves of sunspot number, unsigned AR flux, and AR latitude.
Numerous works show how the polar field is reformed each cycle from a small
fraction of active region flux, which is transported by meridional flows from
mid-latitudes to the poles (Hathaway, 2015; Wang et al., 2005; Morgan and
Taroyan, 2017).

It has been demonstrated that the strength of the sun’s polar fields can be
used to predict the next solar cycle’s sunspot intensity based on the
understanding of the sun’s magnetic dynamo (Upton and Hathaway, 2013; Dikpati
and Gilman, 2006; Hiremath, 2008)
222https://ntrs.nasa.gov/api/citations/20130013068/downloads/20130013068.pdf.
Weak polar fields at the end of Cycle 23 motivated the authors of the 2005
paper (Svalgaard et al., 2005) to forecast a weak Solar Cycle 24. Indeed,
Solar Cycle 24 has been the weakest solar cycle in a century. To graphically
illustrate the strong association between polar fields and sunspot activity,
we the polar fields and monthly mean sunspot number over four solar cycles in
the top panel of Fig. 1. It can be observed the magnitude of the difference
between the two polar fields at the solar minimum is a good indicator of the
intensity of the subsequent cycle.

The relation between polar fields and AR magnetic fields is more intricate,
involving complex physical mechanisms. This is better observed with higher
temporal resolution and in an AR-localized manner. The middle and bottom
panels of Fig. 1 shows how polar fields interact with total unsigned flux and
latitudinal distribution of ARs in the northern and the southern hemisphere,
respectively, during the period that HMI has been in operation (the shaded
area in the top panel of Fig. 1). The data are collected from the data product
hmi.meanpf_720s, available at the Joint Science Operations Center (JSOC). The
north and south polar field strengths are represented by CAPN2 and CAPS2,
respectively, which correspond to the mean radial field within latitudes
N60-N90 and the mean radial field within latitudes S60-S90. The unsigned AR
flux can be well approximated by the unsigned flux in between S40 and N40,
since flux is concentrated in ARs which are mostly located in those latitudes.
Specifically, for the northern hemisphere, we compute the unsigned AR flux
(USFLUXN) and unsigned-flux-weighted latitude (USFLATN) via

USFLUXN $\displaystyle=\textrm{FLUXN\\_P}-\textrm{FLUXN\\_N}\,,$ (1) USFLATN
$\displaystyle=\frac{\textrm{FLATN\\_P}\times\textrm{FLUXN\\_P}-\textrm{FLATN\\_N}\times\textrm{FLUXN\\_N}}{\textrm{FLUXN\\_P}-\textrm{FLUXN\\_N}}\,,$
(2)

where FLUXN_P (FLUXN_N) is positive (negative) flux between N0-N40 above a
fixed threshold, and FLATN_P (FLATN_N) is positive (negative) flux weighted
latitude between N0-N40. Similar quantities are computed for the southern
hemisphere as well. The active regions’ locations follow the well-observed
butterfly diagram of sunspots, starting approximately $\pm 40$ degrees
latitude at the beginning of a solar cycle, and gradually drifting towards the
equator as the cycle progresses. At the cycle minimum, there is an overlap of
the old (near the equator) and new (near $\pm 40$ degrees) cycle spots. These
trends can be observed from the lower two panels of Fig. 1. It is worth noting
that the rapid transition in AR latitude at the depth of solar minimum
provides a strong signature of the solar cycle when the total flux and polar
fields show the lowest variation with time. These strong and complex relations
between polar fields and AR fields suggest supplementing SHARP features with
HMI polar field strength may improve the prediction of AR-localized flare
intensity levels. This is our main motivator for applying machine learning
flare prediction approaches to the combined data.

Figure 1:  _Top_ : The north and south polar strength and the monthly mean
sunspot numbers plotted over 4 solar cycles. The shaded period is the time
during which the HMI instrument has been generating data. Polar fields data
are from the Wilcox Solar Observatory (Svalgaard et al., 1978; Hoeksema,
1995). Sunspot number data are from WDC-SILSO, Royal Observatory of Belgium,
Brussels. _Middle_ : Solar cycle variation of the north polar field strength
(CAPN2), AR unsigned flux (USFLUXN), and flux-weighted AR latitude (USFLATN)
obtained from HMI data. The unsigned flux in the northern hemisphere (USFLUXN)
is positively correlated with the north polar fields (CAPN2), suggesting the
inclusion of polar field data as additional features in a machine learning
flare prediction model can potentially improve prediction accuracy. _Bottom_ :
Similar to the Middle panel but for the Southern Hemisphere of the sun.

In summary, we validate the usefulness of polar field features in predicting
solar flares. The motivation is to assess whether combining SHARP keywords
with polar field features can improve solar flare prediction performance or
not. To this end, we design an experiment in which there are multiple possible
ways of incorporating polar field data, four different machine learning
models, and different train-test-validation splitting methods for each
algorithm. In addition, we design a novel mixture of experts model tailored
for incorporating polar field data for solar flare prediction context. Our
experimental results indicate that polar field data can improve solar flare
prediction performance. They also show that mixture of experts modeling is a
simple yet effective alternative for combining polar field data and SHARP
parameters, which can give similar or better performance to much more
complicated neural network-based algorithms.

The rest of this paper is organized as follows. In Section 2, we describe how
we incorporate multiple data sources into flare prediction algorithms, and we
introduce several data-splitting methods for assessing the performances of the
models. In Section 3, we formally define the machine learning models that are
compared in this study including the new mixture of experts model. In Section
4, we describe the experimental setup in detail and present our results.

##  3 Data Preparation

There are three distinct data sources we exploit in this study, namely, SHARP,
GOES, and SPF datasets. The SHARP dataset provides parameters computed from
the magnetograms produced by the HMI (See Table 3) instrument. These
parameters are associated with the active regions that occurred from 2010 to
2020. We use the data product hmi.sharp_cea_720s, which contains definitive
data in Cylindrical Equal-Area (CEA) coordinates and is available at the Joint
Science Operations Center (JSOC). The GOES dataset includes solar flare event
records measured by Geostationary Operational Environmental Satellites (GOES).
This dataset can be downloaded using the SunPy package 333https://sunpy.org/
and it covers the HMI time period from 2010 to 2020. The flare events are
labeled by A, B, C, M, and X, based on the peak intensity of X-ray flux
measures. The Solar Polar Fields (SPF) dataset consists of two solar polar
field features, namely, CAPS2 and CAPN2. These features correspond to the mean
radial field in latitudes N60-N90, and the mean radial field in latitudes
S60-S90, respectively. Both HMI and MDI instruments record these features,
which can also be downloaded from JSOC. The MDI instrument was operational
from 1996 to 2010 and the HMI has been operational since 2010.

Dataset |  # of instances |  B-class ratio |  MX-class ratio |  Parameters
---|---|---|---|---
SHARP |  2663 |  0.49 |  0.51 |  USFLUXL, MEANGBL, RVALUE, AREA, TOTUSJH, TOTUSJZ, SAVNCPP, ABSNJZH, TOTPOT, SIZEACR, NACR, MEANPOT, SIZE, MEANJZH, SHRGT45, MEANSHR, MEANJZD, MEANALP, MEANGBT, MEANGAM, MEANGBH, NPIX
Table 1: SHARP dataset statistics after preprocessing step. See Table 3 in the
appendix for the definitions of the keywords.

###  3.1 SPF Preprocessing

The noise levels on CAPS2 and CAPN2 features are significant, and the noise
level is different in the HMI and MDI instruments. Hence, we consider
smoothing by using a Kalman filter to filter out noise over these features.
Note that the variance over the time series data is not solely due to noise.
There is strong yearly modulation due to the Earth-solar B0 angle, where the
solar rotation axis varies by $\pm$7 degrees. Furthermore, there is strong
27-day modulation due to solar rotation.

We implement the Kalman filter using the local level model (Murphy, 2012)
where the state transition and observation models are defined as follows:

$a_{t}=a_{t-1}+\epsilon_{t}^{a},$ (3) $y_{t}=a_{t}+\epsilon_{t}^{y},$ (4)

where $y_{t}$ is the raw noisy observation at time $t$, $a_{t}$ is the
filtered data, and $\epsilon_{t}^{y}\sim\mathcal{N}(0,\lambda_{y})$ and
$\epsilon_{t}^{a}\sim N(0,\lambda_{a})$ are the noise processes over the
observed and hidden spaces with variances $\lambda_{y}$ and $\lambda_{a}$,
respectively. By adjusting these levels instrument-wise, we perform smoothing
for the data of both instruments. See Fig. 2 for the raw and filtered
features.

Figure 2: Raw (blue curve) and Kalman filtered (red curve) polar field
features. Polar field strengths at the north pole CAPN2, south pole CAPS2, and
the solar cycle approximation, defined as the unsigned polar field difference
$|CAPS2(t)-CAPN2(t)|$, are shown in the left, middle, and right-hand panels
respectively. The data source switched from MDI to HMI in 2010, which is
accompanied by a visually obvious reduction in noise. The HMI data source has
lower noise due to its high resolution.

###  3.2 Observation Preprocessing

In this section, we describe how we use the HMI SHARP dataset and the GOES
flare intensity labels to construct the training and test data for our
discriminative machine learning algorithms, developed in the next section. Our
Preprocessing steps for these data follow the approach in (Jiao et al., 2020;
Chen et al., 2019; Sun et al., 2022).

We only include ARs having unique NOAA AR numbers (Jiao et al., 2020) into our
training set. We eliminate low-quality observations by excluding ARs that
occur outside of range $\pm 70$ relative to the central meridian (Chen et al.,
2019). Similarly to (Bobra et al., 2014; Chen et al., 2019) we focus on
predicting whether a solar flare from a given active region will be strong
(M/X class) or weak (B class). The A-class flares are excluded as they are
almost indistinguishable from the background radiation. The C-class flares are
frequently missing their NOAA AR numbers in the GOES database, making their
locations difficult to track [3]. Exclusion of A and C flares and combining M
and X class flares together creates a binary classification problem that
enables the comprehensive comparative computational analysis we report in Sec.
4. This is a step along the way of establishing the value of polar field data
for the more complex classification task of classifying between the five
labels A, B, C, M, X, which is worthwhile of future study.

As in (Jiao et al., 2020), the class label of an AR is designated as the
strongest flare occurring in a 24-hour prediction period, according to the
GOES database. Thus an AR is assigned a B label if there are no M/X flares
within 24 hours, while it is assigned label M/X otherwise. This is
accomplished as follows. For each of the flares occurring in an AR, variables
are extracted from the time segment spanning a 24-hour time period from 48
hours to 24 hours before the occurrence of the flare. Since the cadence of the
HMI observations is one sample every 12 minutes this yields a matrix of 22
SHARP predictor variables defined over 120-time points. If, for a given flare,
the corresponding predictor variables have NaN values no class prediction is
generated and we omit the flare from our analysis. Furthermore, in our
analysis, we do not use any flare labels in the first 24 hours of the
beginning of an ARs since such flares would have no associated observed
predictor variables. Once these omissions are taken into account, the total
number of observations (predictor variables-label pairs) is 2663. See Table 1
for a summary of our set of SHARP features.

###  3.3 Train-Validation-Test Splitting

We consider three different methods for splitting the set of AR samples into
train-test sets. In all three methods, we split the AR’s into 80% train and
20% test samples. i) Instance-based splitting. The indexes of the splits are
generated randomly without replacement. This split can measure the goodness-
of-fit but is prone to information leakage when the observations of a
particular AR are scattered among the splits (Sun et al., 2022). ii) AR-based
splitting. AR numbers are used instead of the observation indexes for random
index generation. This split prevents the information leakage that happens in
Random splitting but may introduce another form of information leakage due to
sympathetic flaring between neighboring ARs (Sun et al., 2022). iii) Month-
based splitting. The timestamps are used to split the sets in terms of months
with the same ratio. This split can also measure the goodness of fit and is
closer to the operational forecasting setting. Each feature in the dataset is
normalized over the time windows of the training data. Specifically, for each
AR the mean and standard deviation of each feature is computed over the time
points of the training sample. Then for each AR and each variable, the mean is
subtracted and the result is divided by the standard deviation for both
training and test samples.

##  4 Prediction algorithms

We consider four machine learning models that can exploit polar field features
for the flare prediction task: i) Logistic Regression (LR); ii) Mixture of
Experts (MOE); iii) Multi-Layer Perceptron; iv) Recurrent Neural Network.

###  4.1 Logistic Regression

Logistic Regression (LR) is a model for which the conditional distribution of
the response variable as a Bernoulli distribution conditioned on the features
of an AR sample in the case of binary classes, and categorical (multinomial)
distribution for multiple classes. Specifically, for the binary case, LR
models the conditional distribution of the response variable as

$p(y_{i}|\bm{x}_{i},\Theta)=Ber(y_{i}|\sigma(\Theta^{T}\bm{x}_{i})),$ (5)

where $i$ indexes the data points, $y_{i}=\\{0,1\\}$ denotes the response
variable (i.e., flare class), and $\bm{x}_{i}$ corresponds to the SHARP
parameters. The model parameters $\Theta$ are estimated from the training set
by using Maximum Likelihood Estimation (MLE). In particular, the MLE objective
over the $N$ AR samples $\\{(\bm{x}_{i},y_{i})\\}_{i=1}^{n}$ in the training
set:

$\ell(\Theta)=\sum_{i=1}^{N}\log p(y_{i}|\bm{x}_{i},\Theta),$ (6)

which is convex in $\Theta$, and has a unique solution. We use LBFG-S (Liu and
Nocedal, 1989) to optimize the model parameters. In order to introduce polar
field features to LR, a simple approach is to concatenate them with the SHARP
parameters and apply the algorithm with these augmented features.

###  4.2 Mixture of Logistic Regressions

A richer family of conditional distributions can be obtained by using a
mixture of experts (MOE) framework (Yuksel et al., 2012). For classification
tasks, each expert is an LR model. The predictions of the experts are mixed
probabilistically according to mixture coefficients, which are obtained by
using another linear model called the gating function. The output of the
gating function is constrained to be a simplex to support proper mixing
coefficients. This is achieved by using a Softmax function $\mathcal{S}$,
which is applied after the linear transformation of the input features of the
gating function. We propose to use polar field features as the input to the
gating function. The main intuition is that the experts will be forced to
focus on specific portions of the solar cycle.

Let $\bm{z}_{i}$ denote a $K$ dimensional categorical valued latent variable
that indicates which expert is active for data point (AR) $i$, where $K$ is
the total number of experts initialized. The categorical conditional
distribution of the latent variable $\bm{z}_{i}$ induced by the gating
function is given as

$p(z_{i}|\bm{t}_{i},\bm{V})=Cat(z_{i}|\mathcal{S}(\bm{V}^{T}\bm{t}_{i})),$ (7)

where $\bm{t}_{i}$ denotes the polar field features associated with data point
$i$, and $\bm{V}$ represents the gating function parameters that map features
to the logits of the categorical distribution. After the softmax function, we
obtain proper mixture coefficients for data point $i$. Since the experts are
logistic regression models, their conditional distributions are given as

$p(\bm{y}_{i}|\bm{x}_{i},z_{i}=k,\Theta_{k})=Ber(y_{i}|\sigma(\Theta_{k}^{T}\bm{x}_{i})),$
(8)

where $\Theta_{k}$ denotes the parameters of the expert $k$. The marginal
likelihood of this model can be computed by integrating out the latent
variables as follows

$p(\bm{y}_{i}|\bm{x}_{i},t_{i})=\sum_{k=1}^{K}p(z_{i}=k|t_{i},\bm{V})p(y_{i}|\bm{x}_{i},z_{i}=k,\Theta_{k}).$
(9)

The objective is to maximize marginal log-likelihood with respect to the
logistic regression and gating function parameters. The Expectation
Maximization algorithm is a suitable optimization method for latent variable
models. The algorithm optimizes the expected complete data log-likelihood
(ECLL) instead of marginal, where the former forms a lower bound for the
latter. There are two alternating steps, E and M steps. In the E-step of the
algorithm, the posterior distributions of the latent variables are computed
and their expectations are plugged into the complete data log-likelihood to
obtain ECLL. Subsequently, the ECLL is maximized with respect to the model
parameters in the M-step. These steps are iterated until ECLL converges. For
the mixture of experts model, the ECLL is given as

$\sum_{i=1}^{N}\sum_{k=K}^{K}r_{ik}\log[\mathcal{S}_{k}(\bm{V}^{T}t_{i})p(y_{i}|\bm{x}_{i},\Theta_{k})],$
(10)

where we replace the explicit expression of the prior distribution of mixture
coefficients. $r_{ik}$ denotes the posterior distribution of $z_{i}$, which
can be computed as

$r_{ik}=p(z_{i}=k|\bm{x}_{i},y_{i},t_{i},\Theta_{k})\propto\mathcal{S}(\bm{V}^{T}t_{i})p(y_{i}|\bm{x}_{i},z_{i}=k,\Theta_{k}),$
(11)

in the E-step of the EM algorithm. Replacing the posterior in the complete
data log-likelihood, we get parameter-specific objectives to optimize in the
M-step. Specifically, for the model parameters of the experts, the objective
becomes

$\ell(\Theta_{k})=\sum_{i=1}^{N}r_{ik}\log
p(y_{i}|\bm{x}_{i},z_{i}=k,\Theta_{k}),$ (12)

which can be computed by using any gradient-based optimizer. Note the
similarity of this objective with Eq. 6. Hence, the former corresponds to
weighted Logistic Regression. For the parameters of the gating function, the
objective takes the following form

$\ell(\bm{V})=\sum_{i=1}^{N}\sum_{k=1}^{K}r_{ik}\log\mathcal{S}_{k}(\bm{V}^{T}t_{i}),$
(13)

which is also similar to the Logistic Regression objective in Eq. 6 except
that the response variable is simplex instead of discrete. This objective can
also be maximized with any gradient-based optimizer. We use LBFG-S for all
gradient-based optimization tasks. In summary, the E-step constitutes
computing posterior distributions of the latent variables given the current
values of the parameters by using Eq. 11, the M-step constitutes maximizing
Eq. 12 and 13 with respect to $\Theta_{k}$ and $\bm{V}$. When Eq. 10
converges, the iterations are halted. If we don’t use polar field features,
mixing coefficients are unconditional as analogous to Gaussian Mixture Models.

###  4.3 Multi-layer Perceptron

The Multi-layer Perceptron (MLP) is a non-linear discriminative classification
model. MLPs can be more powerful than their linear counterparts when the
interactions between the features and the response variables can not be
explained by linear decision boundaries. An MLP can be obtained by stacking
multiple fully connected layers. Each layer performs an affine transformation,
and subsequently applies a deterministic non-linear function, e.g., softmax,
relu, tanh. Stacking multiple layers result in a complex deterministic
function of the input features. Let $f_{\Theta}$ denote this function with the
exception that the last layer does not apply any activation functions. Here,
$\Theta$ represents the set of the affine transformation parameters of all the
layers. Then, the conditional distribution for a binary classification task is
given as

$p(y_{i}|\bm{x}_{i},\Theta)=Ber(y_{i}|\sigma(f_{\Theta}(\bm{x}_{i})),$ (14)

where the loss function is the same with LR as in Eq. 6.

The maximization of this distribution can be achieved by using a back-
propagation algorithm (Hecht-Nielsen, 1992). We consider using polar field
features as the additional features by concatenating them with the SHARP
parameters.

Analogously to LR, a mixture of MLPs can be constructed by initializing
multiple MLPs and mixing them using the probabilistic objective given in Eq. 9
(Bishop, 1994). However, in the MLP case, the EM algorithm is no longer
applicable. Fortunately, back-propagation can still be used to maximize the
objective. Similar to the Mixture of Logistic regression, a separate gating
function can be initialized that maps polar field features to the mixture
coefficients. This function can be linear or non-linear (MLP). In our
experiments, we did not observe a significant performance change for Mixture
of MLPs over NN-Augmented. Hence, we omit the Mixture of MLPs in this study.

###  4.4 Recurrent Neural Network

The aforementioned algorithms do not take the sequential nature of the
observations into account. A more principled approach would model temporal
data by assuming Markov chains between the consecutive inputs or outputs.
Recurrent Neural Networks (RNN) do that by forming a first-order Markov chain
between the hidden units. RNNs have shown significant success for sequential
data modeling, and LSTM, a variant of RNNs, achieves state-of-the-art
performance in flare forecasting from time series SHARP data (Chen et al.,
2019; Liu et al., 2019; Sun et al., 2022). For this study, we use a particular
version to classify the sequences of the input observations, also known as
time series classification. The conditional distribution of the model is given
as:

$p(y_{i}|\bm{x}_{i,t=1:T},\Theta)=Ber(y_{i}|\sigma(f_{\Theta}(\bm{x}_{i,t=1:T})),$
(15)

which implies that the observed label is dependent on a sequence of input data
points. The length of the sequence is denoted as $T$ and is a hyper-parameter
of the model. The log-likelihood function associated with the RNN model is

$\ell(\Theta)=\sum_{i=1}^{N}\log p(y_{i}|\bm{x}_{i,t=1:T},\Theta),$ (16)

where $\Theta$ represents the parameters of the model and can be optimized by
using the backpropagation algorithm and $T=120$ is the number of temporal
samples in each AR training sample. The polar field features are incorporated
as additional input features by concatenating them with the SHARP parameters.

##  5 Experiments

###  5.1 Configurations

The 4 algorithms described in Sec. 4 were compared in terms of relative
performance improvements when adding the polar field to the SHARP features.
Our SHARP dataset consists of 2663 samples with 22 parameters over 120 time
points. As described above we consider 3 different data splits: instance-
based, AR-based, and month-based. For polar field features, we create two
modes: raw CAPS2 and CAPN2 used as two separate input features; and cycle,
where the absolute difference of CAPS2 and CAPN2 is used as a lower
dimensional solar cycle approximation (see Fig. 3). We also consider smoothed
versions of CAPS2 and CAPN2 by using the Kalman filter, and unfiltered
versions of these features. Hence, the total number of experimental
configurations is 12 if we don’t use the polar field feature, and 48
otherwise. We perform cross-validation by using training splits for each
configuration to select the regularization strengths of the algorithms. Also,
we perform 20 runs to assess the significance of the accuracies. Hence, the
total number of runs is around 7k.

###  5.2 Tuning

The hyper-parameters of the machine learning algorithms must be tuned for fair
performance comparison. Cross-validation on the training set is used to select
the best hyper-parameters for each experimental configuration. First, the
training dataset is divided into 5 folds. The first fold is held out for
measuring the performance of a hyper-parameter setting while the rest are used
for training the algorithm. Then the hold-out set is alternated one by one so
the total number of empirical realizations of performance for each hyper-
parameter setting is 5. The average of these 5 values is then used as the
score of that particular setting. We use the Heidke Skill Score HSS2 (Cohen,
1960; Bobra and Couvidat, 2015) for quantifying performance, which is defined
as follows:

$HSS2=\frac{2\times[(TP\times TN)-(FN\times
FP)]}{(TP+FN)(FN+TN)+(TP+FP)(FP+TN)},$ (17)

where TP, TN, FP, and FN correspond to true-positive, true-negative, false-
positive, and false-negative values computed from binary classification
predictions and ground truth, respectively.

All the algorithms studied were implemented with the standard L2-norm
regularization approach to minimize overfitting. Regularization is
accomplished by a penalty method, where the sum of the L2-norm squared of each
parameter vector is weighted by a regularization parameter and added to the
log-likelihood functions of each of the predictor models discussed in Sec. 4.
The regularization parameters are tuned to optimize the performance of each
algorithm. For LR, MLP, and RNN, the regularization strength is a single
scalar that is common for all the elements of the parameters. The grid search
range for this scalar is chosen as $[0.1,1,10]$. For MOE, we have separate
parameters for the gating function and the experts. We use one scalar
regularization parameter for the parameters of the gating function, and
another for the parameters of the experts. The grid range for these two hyper-
parameters is similarly determined as $[0.1,1,10]$.

In addition to regularization tuning, the mixture model has a hyper-parameter
associated with the total number of experts being initialized. The grid search
range for this hyper-parameter is set as $[2,3,4,5,6,7,8]$. We fix the network
structure of RNN and MLP. In particular, 32 nodes with one hidden layer are
initialized for both models.

###  5.3 Significance Assessment

To assess the significance of the results, we repeat each experimental
configuration $n=20$ times with and without using polar field data. We then
compute p-values from one-sided paired t-tests. The t-statistic is computed as
$t=\bar{d}\sqrt{n}/s$, where $\bar{d}=\frac{1}{n}\sum_{i=1}^{n}d_{i}$ is the
mean of the difference vector $\bm{d}$ whose elements are given by
$d_{i}=y_{i}-x_{i}$ where $y_{i}$ and $x_{i}$ are HSS2 scores with and without
polar field data for trial $i$, respectively. Then, under the Student-t
distribution with degrees of freedom $n-1$, we compute the right-tail p-value.

###  5.4 Results

Algorithm | Filter | Mod | Split | p-value | t-score | hss2(s+p) | hss2(s) | hss2-diff | hss2(p)
---|---|---|---|---|---|---|---|---|---
LR | Yes | Raw | Month | 0.0274 | 2.628 | 0.7745 | 0.7155 | %8.2 | 0.3010
| No | Cyc | Inst | 0.0439 | 2.1271 | 0.8143 | 0.802 | %1.5 | 0.3628
| No | Cyc | Inst | 0.0002 | 4.3055 | 0.8223 | 0.7992 | %2.9 | 0.3484
MOE | Yes | Raw | Inst | 0.0005 | 4.0369 | 0.8254 | 0.792 | %4.2 | 0.3867
| Yes | Cyc | Inst | 0.0155 | 2.6055 | 0.8205 | 0.8001 | %2.5 | 0.3689
| Yes | Raw | Month | 0.0386 | 2.4202 | 0.7502 | 0.6827 | %9.9 | 0.2485
MLP | Yes | Raw | Inst | 0.0000 | 6.8479 | 0.7992 | 0.7896 | %4.9 | 0.4384
| No | Cyc | Month | 0.0481 | 2.2856 | 0.7653 | 0.7149 | %7.0 | 0.3658
| Yes | Raw | Inst | 0.0002 | 4.4235 | 0.826 | 0.7947 | %3.9 | 0.4143
RNN | Yes | Cyc | Month | 0.0463 | -2.3092 | 0.6117 | 0.6756 | -%9.4 | 0.3627
| Yes | Raw | Month | 0.0478 | 2.2888 | 0.7194 | 0.6544 | %9.9 | 0.2791
| No | Raw | AR | 0.0221 | 2.4469 | 0.6977 | 0.6337 | %10.1 | 0.3952
Table 2: Table of statistically significant (p-value$\leq$5%) performance
improvements due to adding polar field features to SHARP features, The hss2
skill score argument (p) means only polar fields data, (s) means only SHARP
parameters, and (s+p) means both data sources are used as input to the
algorithms. (diff) is the mean hss2 score percentage difference when the polar
field is incorporated in addition to SHARP parameters. The highlight colors in
the table specify the common configurations where MOE achieves statistically
significant performance improvements due to adding the polar field. The full
tables including the statistically non-significant configurations are provided
in the supplementary.

The results are compiled in Table 2. We report HSS2 computed on the test set
as the performance metric. For each algorithm, we only keep the configurations
with statistically significant performance improvement due to the
incorporation of polar field data, where significance is defined as a p-value
less than 0.05 (5% level) of a one-sided paired t-test of equal mean
performance, computed from 20 runs. For the LR model, there are two
configurations that exhibit significant performance differences when polar
field features are incorporated. Both of them imply that performance improves
with the incorporation polar field. Particularly when we use unfiltered cycle
approximation for the polar field features and filtered raw features, we
obtain performance boost for the month (8.2%) and instance-based splits
(1.5%), respectively. For the MOE model, there are four significant cases with
all of them yielding performance improvements. The performance boost is up to
9.9%. The model can exploit the polar field augmented data regardless of
filtering and preprocessing options when the splits are instance and month
based to improve flare classification performance. For filtered raw features,
the MLP improves performance on instance-based splitting by 4.9% and non-
filtered cycle approximation on month-based splitting improves by 7.0%. The
RNN performance is boosted regardless of filtering, preprocessing, and
splitting options for four significant cases, achieving improvements as high
as 10.1%. In summary, out of 12 significant cases for four algorithms, 11 of
them provide performance increase, which suggests that polar field features to
have high potential to improve solar flare prediction performance.

The column hss2(p) shows the prediction performance of the algorithms when
only polar field data is used. The results indicate that sole use of polar
field data gives poor skill scores, which is less than half of those of the
SHARP data alone, for solar flare prediction. However, the hss2(p) scores are
far from zero since solar cycle intensity and polar fields are correlated, as
discussed in Section 2. Solar cycle intensity affects the expected intensities
of the flares in particular time frames. Due to this connection, one can
expect that coarse predictions about the classes of solar flares can be made
with polar field data since it captures information about the solar cycle.

For the MOE model, the solar cycle dependency of the classification accuracy
is shown in Fig. 3 by repeating five experiments using instance-based
splitting. We plot sunspot numbers, polar fields data (cycle approximation),
accuracy without polar fields data, and accuracy with polar fields data. The
classification accuracy curves are smoothed out with a 90-day moving average.
We observe that MOE can exploit polar field data to improve classification
accuracy in some parts of the solar cycle. The average classification
accuracies over the whole HMI period are 0.88 and 0.91 for without and with
polar field data, respectively. One can also observe the high negative
correlation between the sunspot number and the curve of the polar field, which
is computed as -0.86.

Finally, we compare the performance improvements of the models in Table 2.
Only those configurations for which the p-value was significant at a level of
5% are shown. Among all algorithms MOE stands out as the only one to attain
highly significant performance improvement (p-value=0.0002) when no Kalman
filtering is used, the magnitude difference north and south polar field data
are used (the solar cycle approximation), and the splitting is instance based.
MOE, MLP, and RNN, all have highly significant performance improvement when
the polar field data is smoothed using the Kalman filter, the northern and
southern polar fields are not combined into a cycle approximation (raw) and
the split is instance based. Overall, among the five common configurations for
which MOE achieves statistically significant improvement at the 5% level, MOE
exhibits better performance in three of them, comparable performance in one of
them, and slightly worse performance in the other one. We conclude that MOE is
a promising machine learning model that can incorporate polar field data to
improve flare classification accuracy.

We conclude this section by discussing some of the limitations of this study,
which create opportunities for future research. Firstly, while we have
established the value of polar field data when the train vs test data split is
randomized over time, our conclusions may not translate to operational flare
forecasting where train and test data are drawn from past and future
observations, respectively. We could not implement such a temporally defined
data splitting to train our algorithms due to the fact that HMI data is
currently available only over a single solar cycle. Secondly, it is possible
that we could strengthen our conclusions by accounting for the temporal and
longitudinal correlations of the polar field and active region magnetograms.
In particular, instead of summarizing the polar field as the difference
between both north and south polar fields (CAPN2 and CAPS2) we might be able
to improve our model by incorporating the spatial coordinates of the active
regions and independently applying CAPN2 and CAPS2 to the northern and
southern hemisphere active regions, respectively (see Fig. 1). Alternatively,
it might be beneficial to incorporate a dynamic physics-based prediction model
to account for dependencies between past active region activity and present
polar field intensity. Furthermore, the added value of using polar field data
may increase if we expanded the feature dimension to use the higher resolution
HMI image data instead of just SHARP summarized data of active regions.

Figure 3: The classification accuracy curves of MOE with and without using
polar fields data w.r.t. to the sunspot number when the splitting method is
instance-based. Cycle approximation with filtered data is used as input. The
curves are moving averaged over 90 days. The incorporation of the data
substantially improves classification accuracy in some portions of the solar
cycle.

##  6 Conclusions

In this paper, we investigated the utility of the sun’s north and south polar
field strength for improving solar flare prediction from HMI data. We
presented an extensive experimental study, that analyzes multiple ways of
preprocessing schemes for polar field data, several machine learning models
that can incorporate this additional data source, diverse experimental
conditions, and multiple splitting methods. The mixture of experts model was
introduced for this particular global data incorporation task Our experimental
results showed that polar field data is valuable for improving classification
performance in multiple configurations. Additionally, we demonstrated that the
proposed mixture of experts model can effectively supplement SHARP parameters
with polar field data with for improved flare prediction.

The results suggest a causal connection between isolated ARs captured in the
SHARP data and the global magnetic field captured in the polar field. The
connection could come through the polar field itself due to the causal
relationship between the event of decaying ARs and the subsequent drift of
enhanced field towards poles. Hence, AR activity is anti-correlated to the
polar field with a certain phase shift. In this study, we find that the
probability of individual flare events is also correlated to the total sunspot
number through the solar cycle, which closely follows the polar field
strength. To put this work in context we point out that the polar field data
has been used to inform solar cycle predictions with reasonable accuracy
(Pesnell, 2012, 2020). We believe this work represents the first time the
global field has been used to augment the prediction of solar flares.

In future work, there are several promising directions to enhance current
results. The phase shift between the solar cycle/AR activity and polar field
strengths can be estimated jointly by developing a more sophisticated model
that can estimate and take into account phase shifts to improve prediction
performance. The ARs in the northern and southern hemispheres can be modeled
separately for more focused predictions. The correlation between the polar
field strengths and other global indicators such as total active region area
and flux-weighted latitude at a certain time can be investigated and possibly
utilized for improved prediction performance. Finally, the data and the model
may be further refined by using HMI high-dimensional image data instead of
SHARP features.

## References

  * Babcock [1959] Harold D Babcock.  The sun’s polar magnetic field.  _The Astrophysical Journal_ , 130:364, 1959.
  * Bishop [1994] Christopher M Bishop.  Mixture density networks.  1994\.
  * Bobra and Couvidat [2015] Monica G Bobra and Sebastien Couvidat.  Solar flare prediction using sdo/hmi vector magnetic field data with a machine-learning algorithm.  _The Astrophysical Journal_ , 798(2):135, 2015\.
  * Bobra et al. [2014] Monica G Bobra, Xudong Sun, J Todd Hoeksema, M Turmon, Yang Liu, Keiji Hayashi, Graham Barnes, and KD Leka.  The helioseismic and magnetic imager (hmi) vector magnetic field pipeline: Sharps–space-weather hmi active region patches.  _Solar Physics_ , 289(9):3549–3578, 2014.
  * Campi et al. [2019] Cristina Campi, Federico Benvenuto, Anna Maria Massone, D Shaun Bloomfield, Manolis K Georgoulis, and Michele Piana.  Feature ranking of active region source properties in solar flare forecasting and the uncompromised stochasticity of flare occurrence.  _The Astrophysical Journal_ , 883(2):150, 2019\.
  * Camporeale [2019] Enrico Camporeale.  The challenge of machine learning in space weather: Nowcasting and forecasting.  _Space Weather_ , 17(8):1166–1207, 2019.
  * Chen et al. [2019] Yang Chen, Ward B Manchester, Alfred O Hero, Gabor Toth, Benoit DuFumier, Tian Zhou, Xiantong Wang, Haonan Zhu, Zeyu Sun, and Tamas I Gombosi.  Identifying solar flare precursors using time series of sdo/hmi images and sharp parameters.  _Space Weather_ , 17(10):1404–1426, 2019.
  * Cohen [1960] Jacob Cohen.  A coefficient of agreement for nominal scales.  _Educational and psychological measurement_ , 20(1):37–46, 1960.
  * Dikpati and Gilman [2006] Mausumi Dikpati and Peter A Gilman.  Simulating and predicting solar cycles using a flux-transport dynamo.  _The Astrophysical Journal_ , 649(1):498, 2006\.
  * Florios et al. [2018] Kostas Florios, Ioannis Kontogiannis, Sung-Hong Park, Jordan A Guerra, Federico Benvenuto, D Shaun Bloomfield, and Manolis K Georgoulis.  Forecasting solar flares using magnetogram-based predictors and machine learning.  _Solar Physics_ , 293(2):1–42, 2018.
  * Garcia [1994] Howard A Garcia.  Temperature and emission measure from goes soft x-ray measurements.  _Solar Physics_ , 154(2):275–308, 1994.
  * Guerra et al. [2020] Jordan A. Guerra, Sophie A. Murray, D. Shaun Bloomfield, and Peter T. Gallagher.  Ensemble forecasting of major solar flares: methods for combining models.  _Journal of Space Weather and Space Climate_ , 10:38, 2020\.  10.1051/swsc/2020042.
  * Hale et al. [1919] George E Hale, Ferdinand Ellerman, Seth Barnes Nicholson, and Alfred Harrison Joy.  The magnetic polarity of sun-spots.  _The Astrophysical Journal_ , 49:153, 1919.
  * Hathaway [2015] David H Hathaway.  The solar cycle.  _Living reviews in solar physics_ , 12(1):1–87, 2015.
  * Hecht-Nielsen [1992] Robert Hecht-Nielsen.  Theory of the backpropagation neural network.  In _Neural networks for perception_ , pages 65–93. Elsevier, 1992\.
  * Hiremath [2008] KM Hiremath.  Prediction of solar cycle 24 and beyond.  _Astrophysics and Space Science_ , 314(1):45–49, 2008.
  * Hoeksema et al. [2014] J. T. Hoeksema, Y. Liu, K. Hayashi, X. Sun, J. Schou, S. Couvidat, A. Norton, M. Bobra, R. Centeno, K. D. Leka, G. Barnes, and M. Turmon.  The Helioseismic and Magnetic Imager (HMI) vector magnetic field pipeline: Overview and performance.  _Solar Physics_ , 289:3483–3530, 2014.  10.1007/s11207-014-0516-8.
  * Hoeksema [1995] J Todd Hoeksema.  The large-scale structure of the heliospheric current sheet during the ulysses epoch.  _The High Latitude Heliosphere_ , pages 137–148, 1995.
  * Huang et al. [2018] Xin Huang, Huaning Wang, Long Xu, Jinfu Liu, Rong Li, and Xinghua Dai.  Deep learning based solar flare forecasting model. i. results for line-of-sight magnetograms.  _The Astrophysical Journal_ , 856(1):7, 2018.
  * Jiao et al. [2020] Zhenbang Jiao, Hu Sun, Xiantong Wang, Ward Manchester, Tamas Gombosi, Alfred Hero, and Yang Chen.  Solar flare intensity prediction with machine learning models.  _Space Weather_ , 18(7):e2020SW002440, 2020.
  * Jonas et al. [2018] Eric Jonas, Monica Bobra, Vaishaal Shankar, J Todd Hoeksema, and Benjamin Recht.  Flare prediction using photospheric and coronal image data.  _Solar Physics_ , 293(3):1–22, 2018.
  * Korsós et al. [2021] MB Korsós, R Erdélyi, Jiajia Liu, and H Morgan.  Testing and validating two morphological flare predictors by logistic regression machine learning.  _Frontiers in Astronomy and Space Sciences_ , 7:571186, 2021\.
  * Leka et al. [2018] K. D. Leka, G. Barnes, and E. L. Wagner.  The nwra classification infrastructure: description and extension to the discriminant analysis flare forecasting system (daffs).  _J. Space Weather Space Clim._ , 8:A25, 2018.  10.1051/swsc/2018004.
  * Leka et al. [2019] KD Leka, Sung-Hong Park, Kanya Kusano, Jesse Andries, Graham Barnes, Suzy Bingham, D Shaun Bloomfield, Aoife E McCloskey, Veronique Delouille, David Falconer, et al.  A comparison of flare forecasting methods. ii. benchmarks, metrics, and performance results for operational solar flare forecasting systems.  _The Astrophysical Journal Supplement Series_ , 243(2):36, 2019.
  * Liu et al. [2017] Chang Liu, Na Deng, Jason TL Wang, and Haimin Wang.  Predicting solar flares using sdo/hmi vector magnetic data products and the random forest algorithm.  _The Astrophysical Journal_ , 843(2):104, 2017\.
  * Liu and Nocedal [1989] Dong C Liu and Jorge Nocedal.  On the limited memory bfgs method for large scale optimization.  _Mathematical programming_ , 45(1):503–528, 1989\.
  * Liu et al. [2019] Hao Liu, Chang Liu, Jason TL Wang, and Haimin Wang.  Predicting solar flares using a long short-term memory network.  _The Astrophysical Journal_ , 877(2):121, 2019\.
  * Morgan and Taroyan [2017] Huw Morgan and Youra Taroyan.  Global conditions in the solar corona from 2010 to 2017.  _Science advances_ , 3(7):e1602056, 2017.
  * Murphy [2012] Kevin P Murphy.  _Machine learning: a probabilistic perspective_.  MIT press, 2012.
  * Nishizuka et al. [2017] N Nishizuka, K Sugiura, Y Kubo, M Den, S Watari, and M Ishii.  Solar flare prediction model with three machine-learning algorithms using ultraviolet brightening and vector magnetograms.  _The Astrophysical Journal_ , 835(2):156, 2017\.
  * Pesnell et al. [2012] W. D. Pesnell, B. J. Thompson, and P. C. Chamberlin.  The solar dynamics observatory (sdo).  _Solar Physics_ , 275:3–15, 2012.  10.1007/s11207-011-9841-3.
  * Pesnell [2012] W. Dean Pesnell.  Solar Cycle Predictions (Invited Review).  _Solar Physics_ , 281(1):507–532, November 2012\.  10.1007/s11207-012-9997-5.
  * Pesnell [2020] W. Dean Pesnell.  Lessons learned from predictions of Solar Cycle 24.  _Journal of Space Weather and Space Climate_ , 10:60, October 2020.  10.1051/swsc/2020060.
  * Scherrer et al. [2012] P. H. Scherrer, J. Schou, R. I. Bush, A. G. Kosovichev, R. S. Bogart, J. T. Hoeksema, Y. Liu, T. L. Duvall, J. Zhao, A. M. Title, C. J. Schrijver, T. D. Tarbell, and S. Tomczyk.  The Helioseismic and Magnetic Imager (HMI) investigation for the Solar Dynamics Observatory (SDO).  _Solar Physics_ , 275:207–227, 2012.  10.1007/s11207-011-9834-2.
  * Sun et al. [2012] Xudong Sun, J. Todd Hoeksema, Yang Liu, Thomas Wiegelmann, Keiji Hayashi, Qingrong Chen, and Julia Thalmann.  Evolution of Magnetic Field and Energy in a Major Eruptive Active Region Based on SDO/HMI Observation.  _The Astrophysical Journal_ , 748(2):77, April 2012.  10.1088/0004-637X/748/2/77.
  * Sun et al. [2022] Zeyu Sun, Monica G Bobra, Xiantong Wang, Yu Wang, Hu Sun, Tamas Gombosi, Yang Chen, and Alfred Hero.  Predicting solar flares using cnn and lstm on two solar cycles of active region data.  _The Astrophysical Journal_ , 931(2):163, 2022\.
  * Svalgaard et al. [1978] Leif Svalgaard, Thomas L Duvall, and Philip H Scherrer.  The strength of the sun’s polar fields.  _Solar Physics_ , 58(2):225–239, 1978.
  * Svalgaard et al. [2005] Leif Svalgaard, Edward W Cliver, and Yohsuke Kamide.  Sunspot cycle 24: Smallest cycle in 100 years?  _Geophysical Research Letters_ , 32(1), 2005.
  * Upton and Hathaway [2013] Lisa Upton and David H Hathaway.  Predicting the sun’s polar magnetic fields with a surface flux transport model.  _The Astrophysical Journal_ , 780(1):5, 2013.
  * Wang et al. [2020] Xiantong Wang, Yang Chen, Gabor Toth, Ward B Manchester, Tamas I Gombosi, Alfred O Hero, Zhenbang Jiao, Hu Sun, Meng Jin, and Yang Liu.  Predicting solar flares with machine learning: Investigating solar cycle dependence.  _The Astrophysical Journal_ , 895(1):3, 2020.
  * Wang et al. [2005] Y-M Wang, JL Lean, and NR Sheeley Jr.  Modeling the sun’s magnetic field and irradiance since 1713.  _The Astrophysical Journal_ , 625(1):522, 2005\.
  * Yuksel et al. [2012] Seniha Esen Yuksel, Joseph N Wilson, and Paul D Gader.  Twenty years of mixture of experts.  _IEEE transactions on neural networks and learning systems_ , 23(8):1177–1193, 2012.

## Appendix A: Definitions of the SHARP parameters

Parameters |  Description
---|---
USFLUXL |  Total unsigned flux
MEANGBL |  Mean gradient of the line-of-sight field
RVALUE |  Total line-of-sight unsigned flux
AREA |  De-projected area of patch on sphere in microhemisphere
TOTUSJH |  Total unsigned current helicity
TOTUSJZ |  Total unsigned vertical current
SAVNCPP |  Sum of the modulus of the net current per polarity
ABSNJZH |  Absolute value of the net current helicity
TOTPOT |  Proxy for total photospheric magnetic free energy density
SIZEACR |  Deprojected area of active pixels (Bz magnitude larger than noise threshold) on image in microhemisphere (defined as one millionth of half the surface of the Sun)
NACR |  The number of strong LOS magnetic field pixels in the patch
MEANPOT |  Proxy for mean photospheric excess magnetic energy density
SIZE |  Projected area of the image in microhemispheres
SHRGT45 |  Fraction of area with shear $>$ 45°
MEANSHR |  Mean shear angle
MEANJZD |  Vertical current density
MEANALP |  Characteristic twist parameter, $\alpha$
MEANGBT |  Horizontal gradient of total field
MEANGAM |  Mean angle of field from radial
MEANGBH |  Horizontal gradient of horizontal field
NPIX |  The number of pixels in a SHARP image
Table 3: List of SHARP Parameters and Brief Descriptions