# Exploring the potential of neural networks to predict statistics of solar
wind turbulence

###### Abstract

Time series datasets often have missing or corrupted entries, which need to be
handled in subsequent data analysis. For example, in the context of space
physics, calibration issues, satellite telemetry issues, and unexpected events
can make parts of a time series unusable. This causes problems for
understanding the dynamics of the heliosphere and space weather environment.
Various approaches exist to tackle this problem, including mean/median
imputation, linear interpolation, and autoregressive modeling. Here we study
the utility of artificial neural networks (ANNs) to predict statistics of
sparse time series. Our focus is not on time series prediction but on gleaning
the best possible information about the statistical behaviour of the system.
As an example application, we focus on the structure functions of turbulent
time series measured in the solar wind. Using a dataset with artificial gaps,
a neural network is trained to predict second-order structure functions and
then tested on an unseen dataset to quantify its performance. A small
feedforward ANN, with only 20 hidden neurons, can predict the large-scale
fluctuation amplitudes better than mean imputation or linear interpolation
when the percentage of missing data is high. Although they perform worse than
the other methods when it comes to capturing both the shape and fluctuation
amplitude together, their performance is better in a statistical sense for
large fractions of missing data. Caveats regarding their utility, the
optimisation procedure, and potential future improvements are discussed.

Space Weather

Victoria University of Wellington, Kelburn, Wellington, NZ 6012 Department of
Physical Sciences, Indian Institute of Science Education and Research Kolkata,
Mohanpur, 741246, India

Daniel<EMAIL_ADDRESS>
Small neural networks are able to predict structure functions for sparse solar
wind time series in a limited sense.

A network with only 20 hidden neurons statistically outperforms (in terms of
MSE) simple imputation techniques for high ($>$50%) data loss.

More work is needed to generalize the model’s architecture to improve
performance and increase applicability to other systems.

## Plain Language Summary

We explore the utility of machine learning to predict statistics of a
turbulent system such as the solar wind, in cases involving large data gaps.
It is shown that simple artificial neural networks (ANNs) are good at
estimating large-scale features of second-order structure functions even for
very large amounts of missing data. However, these simple ANNs are limited in
estimating other features of the structure functions, such as inner and outer
scales, and the inertial range slope. More sophisticated methods are required
to describe such features. Developing such a method is key to improving space
weather models (e.g. the functions that couple solar wind parameters to space
weather) in the face of incomplete data.

##  1 Introduction

Analyses of real-world time series are often hindered by incomplete datasets.
This is very common for physiological, environmental, astronomical, and
heliospheric time series [Rehfeld . (2011)]. The instrumentation used to take
measurements may be prone to failure, or variations in the environment itself
may preclude data collection for certain periods. For example, time series of
sea level and wave height based on radio signals are commonly incomplete due
to radio interference, airborne seawater spray, and the loss of line-of-sight
caused by large waves [Makarynskyy . (2005)]. In physiology, recordings of
blood flow and other processes are often contaminated with artifacts due to
movement of the subject and improper interfacing with sensors [Pavlova .
(2019)], and removal of these leaves gaps in the series. Ground-based
astronomical observations are affected by cloud cover and the maintenance and
malfunction of instruments. In the case of in situ measurements of the solar
wind —vital for models of space weather— incomplete time series result from
calibration, instrumentation, and telemetry issues . Telemetry is a particular
issue for the two Voyager spacecraft, which must align their data
transmissions with NASA’s ground-based communication facilities, the Deep
Space Network [Ludwig Taylor (2016), Gallana . (2016)].

Discontinuity in time series data represents a loss of information, affecting
the statistics and in turn polluting predictions. This includes significant
effects on frequency-domain (spectral) and scale-domain analysis. An example
is ‘spectral inheritance’, in which the gaps contaminate the rest of the data
in the form of “spurious periodicities arising from the spectral properties of
the sets of gaps” [Frick . (1998), Gallana . (2016)]. More generally, data
gaps result in dirty spectra, which lead to poor estimation of power,
particularly at high frequencies [Munteanu . (2016)]. In radio and gamma-ray
astronomy, this causes issues for calculating the periodicity of stellar
objects [VanderPlas (2018)]. In heliophysics it hinders our understanding of
the spectral properties of turbulence [Fraternale . (2019)]. For example,
Gallana2016 highlighted the importance of identifying new techniques to
extract useful spectral information from high sparsity solar wind datasets.

This is the motivation for the present study. Notably, missing data not only
affect measurements taken at great distance from the Sun, but also at the
near-Earth plasma environment, a region from which data is essential to
developing magnetic field models for space weather research. OMNI (near-Earth
magnetic field) data has 20% of plasma parameter data missing between 2000 and
2019, and 8% of interplanetary magnetic field (IMF) measurements over the same
period. These gaps are particularly significant during storm events [Kobayashi
. (2021)]. Intervals prior to 1995 are of even worse quality due to fewer
spacecraft being in operation [Qin . (2007)], leading to some gaps of more
than 150 hours [Finch Lockwood (2007)]. As well as affecting spectral
analysis, such gaps have been shown to introduce considerable errors into
combinations of near-Earth solar wind parameters known as ‘coupling functions’
[Finch Lockwood (2007), Lockwood . (2019)]. This data quality issue also
presents a problem for machine learning models, which are increasingly popular
algorithms for space weather analysis but generally require continuous inputs
[Sharpe Solly (1995), Kobayashi . (2021)]. Therefore, the development of a
technique, or series of techniques, to be able to utilise such poor-quality
datasets is essential to develop a more comprehensive space climatology, and
in turn, make more accurate predictions of space weather events.

Many different methods have been explored to deal with this issue of spectral
estimation from a time series that has gaps. A significant amount of
literature has been dedicated to estimating the power spectra and
periodicities of a gapped signal. We find that these methods can be grouped
into two broad categories:

  1. 1.

Interpolation of missing values, followed by spectral estimation from the
reconstructed signal

  2. 2.

Spectral estimation directly from the dataset with gaps

The first category of techniques is regularly used in the space plasma
literature [Lockwood . (2019), and references therein]. Often, intervals of
data that contain large gaps are simply excluded from analysis. For example,
wu_2013 and ChenApJS20 removed intervals with gaps larger than 5% and 1%
respectively. The remaining small gaps are typically filled using linear
interpolation [Burlaga (1991), J. Podesta . (2007)]. Linear interpolation has
been shown to perform better than a range of other interpolation methods for
estimating the values in small gaps in OMNI time series, as measured by root
mean-squared error (RMSE) and $R^{2}$ [Kobayashi . (2021)], where $R^{2}$
measures the association between the interpolated values and the true values.
However, linear interpolation amounts to strong smoothing of part of the
signal, which results in a loss of information at high frequencies [Frick .
(1998)]. Because this effect becomes worse with increasing data loss, linear
interpolation is only feasible for relatively small gaps [Bavassano . (1982),
Y. Chen . (2002)]. Furthermore, by excluding large segments of the data to
avoid the gaps, a considerable amount of information about the system is lost.
For this reason, interpolation methods that are more consistent with the
spectral content of the observed data segments have also been used.

For example, interpolation of sparse signals has also been achieved by
modelling the signal as a stochastic process (specifically, that of fractional
Brownian motion), and then further defining the process as a multi-point
“bridge” between the prescribed (observed) measurements [Friedrich . (2020)].
A strategy for identifying the optimal Hurst exponent required by the
fractional Brownian motion algorithm was proposed and tested by reconstructing
velocity field measurements from a superfluid helium experiment.

Singular spectrum analysis (SSA) is a non-parametric algorithm used for
forecasting from gapped time series in a number of fields, including
heliophysics [Schoellhamer (2001), Kondrashov . (2010)]. SSA involves
reconstructing a signal from its principal components, and its benefits are
that it requires no prior knowledge of the periodicities in the data, and it
accounts for noise in the signal. However, the technique is especially
sensitive to increasing gap sizes: the RMSE was shown to increase
significantly in a study investigating data gaps in soil respiration data
[Zhao . (2020)]. SSA has been used to fill the significant gaps in solar wind
data pre-1995 by Kondrashov2010. This study was able to reconstruct the
measurements with realistic variability across both storm and quiet
conditions. However, the study also made use of simultaneous continuous
measurements of geomagnetic indices, and such proxies are not always
available.

ARIMA models are the standard models for forecasting time series, and these
can be fitted to non-uniformly sampled data using a maximum likelihood
technique [Harvey Pierse (1984), Broersen (2006)]. This has been shown to
result in much better estimation of time series parameters such as level,
error variance, and slope of the time series, compared to simple mean
imputation and linear interpolation [Velicer Colby (2005)]. A similar method
of finding the best ARIMA model order based on maximising entropy has been
applied to solar oscillation data [TM. Brown Christensen-Dalsgaard (1990)].
Starting with some assumptions about the typical gap-lengths and the noise in
the signal, the authors were able to reproduce unique spectral features.

Neural networks, a prominent algorithm from machine learning, have also been
used to fill gaps in time series. Specifically, a simple feed-forward neural
network was found to accurately reproduce simulated stochastic processes and
fill gaps that matched the original power spectrum with up to 50% missing data
[Comerford . (2015)]. Generative adversarial networks [Luo . (2018)] and
convolutional neural networks [Jang . (2020)] have also been used to impute
missing intervals.

A comprehensive study of dealing with large data gaps in solar wind data used
a combination of techniques to recover the spectrum from Voyager datasets
[Gallana . (2016)]. This compared Fourier transforms of gap-free subsets;
Fourier transforms of the correlation function of the data, with and without
linear interpolation; maximum likelihood recovery; and compressed sensing
spectral estimation. All of these methods, apart from compressed sensing, fall
into the first category of gapped estimation techniques. Ultimately, this work
was able to determine spectra over a very large range of frequencies for data
with up to 50% missing and thereby extract information on various turbulent
features.

Moving now to the second category of spectral estimation methods, a continuous
wavelet transform method has been used to perform spectral estimation directly
from a gapped signal [Frick . (1998)]. Of direct relevance to our work, this
technique has been applied to magnetic field time series in the solar system
by magrini_2017 and deSouzaEcher2021. In the first study, the wavelet method
was compared with two polynomial interpolation methods for spectral analysis
of artificially-gapped OMNI data. It was found that all techniques perform
satisfactorily for small gaps, but the wavelet method better estimates the
energy of the signal at certain scales for large gaps. In the second study,
the wavelet method was used to find the dominant periodicities of magnetic
field fluctuations in the magnetosphere of Jupiter.

The present study examines a machine learning approach to the second category
of methods. Specifically, we investigate framing the estimation of high-
quality statistics from a dataset with gaps as a supervised learning
regression problem. Whereas the mapping from a complete dataset to its
statistics is generally in the form of a simple function (e.g., the equation
for the mean or standard deviation), we are interested in whether a neural
network - the ‘universal approximator’ - can learn a mapping from an
incomplete dataset to the ‘clean’ statistic that would have followed, had the
complete dataset been available. This approach is taken because the primary
goal is not to accurately reproduce the complete series itself, but rather the
statistics calculated from the complete series. Furthermore, we are not aiming
to extract the true relationship between input and output, but rather to find
an input-output mapping that achieves good performance on unseen gapped
datasets; hence the machine learning approach.

As a case study, this technique is applied to time series of the fluctuating
interplanetary magnetic field measured by the NASA spacecraft Parker Solar
Probe. The interplanetary magnetic field contains many structures and
fluctuations, including Alfvén wave-like intervals [Belcher Davis Jr (1971),
B. Tsurutani . (2002)] and discontinuities [BT. Tsurutani . (1996), BT.
Tsurutani Ho (1999)]. Although historically the presence of Alfvén wave-like
structures was thought to imply that the solar wind is not turbulent, there is
overwhelming evidence that it is [Coleman Jr, P J (1968), Tu Marsch (1995),
Marsch (2006), Bruno Carbone (2013), Verscharen . (2019), and references
therein]. This turbulence is an important ingredient in determining the
evolution of the solar wind [Usmanov . (2011), Opher . (2011), Sokolov .
(2013), Oran . (2013), Burlaga . (2018), van der Holst . (2022)]. Hence, to
explore the problem of finding good statistics for sparse time series, we work
with heliospheric measurements of the turbulent magnetic field. The statistic
we attempt to estimate is the structure function.

The $n$th order structure function for a time-varying signal $a$ is defined as
[Batchelor (1953), Biskamp (2003)]

$S^{(n)}_{a}(\tau)=\langle|\delta a(t,\tau)|^{n}\rangle$ (1)

where $a(t)$ is the scalar variable of interest, $\delta
a(t,\tau)=a(t+\tau)-a(t)$ is the increment, $n$ is the order, and
$\langle\rangle$ denotes expectation over $t$. For a vector set of time series
${\bm{\mathrm{a}}}(t)$ $=(a_{x}(t),$ $a_{y}(t),$ $a_{z}(t))$, the structure
function is defined as

$S^{(n)}_{a}(\tau)=\langle|\delta{\bm{\mathrm{a}}}(t,\tau)|^{n}\rangle$ (2)

where
$\delta{\bm{\mathrm{a}}}(t,\tau)={\bm{\mathrm{a}}}(t+\tau)-{\bm{\mathrm{a}}}(t)$.

$S^{(n)}_{a}(\tau)$ gives the $n$-th moment of the probability distribution
function of the increments of $a$ at lag $\tau$. Turbulence theory predicts
that the structure functions of various orders show power-law behaviour in the
inertial range. Departures from the expected power-laws across multiple orders
are of interest because they represent the presence of intermittent structures
in the fluid and the potential need for corrections to the theory of the
inertial range [Frisch (1995)].

In this paper, as a proof-of-concept, we stick to the second-order structure
function, i.e., $n=2$ in equation (2); this will be referred to as $S^{fn}$
for brevity. This quantity is particularly important due to its close
relationship with both the autocorrelation and the energy spectrum [Chhiber .
(2018)]. For more detailed discussion of these concepts we refer the reader to
MatthaeusJGR82, BiskampBook, and BrunoLRSP13.

Machine learning algorithms are being widely applied to space weather
research, in order to take advantage of the increasing amounts of data and
computing power now available [Camporeale (2019)]. These applications have
ranged from identifying magnetic reconnection to predicting solar flares [Hu .
(2020), Bobra Couvidat (2015)]. There are two key challenges with the current
application. The first is going from a high-dimensional input space to a high-
dimensional output space. Each set of ‘features’ is an entire time series, and
each expected output is not a single label or value but rather an array of
values: the values of a statistical function. Such a task seems to be unique
in the literature, and it is therefore of interest how well a simple network
architecture can perform at such a task. The second challenge is finding an
appropriate representation of the missing data. Typically, supervised machine
learning models require complete data. For example, the training of a
feedforward neural network requires complete inputs in order for the hidden
layers to feed forward valid inputs during the forward pass and then update
the weights appropriately during the backpropagation step [ML. Brown Kros
(2003)]. Therefore, it is not immediately obvious how one would use such a
model in the second category of techniques (i.e., without interpolation of the
gaps) and this remains an open problem in the machine learning community
[Sharpe Solly (1995), Caiafa . (2021), Emmanuel . (2021)]. One solution is to
use a Cosine Neural Network [Randolph-Gips (2008)]. This architecture is able
to process and recognise missing data without any prior imputation, thereby
addressing the issue of how to represent missing data to a neural network. It
does this using ‘weighted norms’, parameters which reduce to zero when the
corresponding input feature is missing. This informs the network to ignore
these features for that instance. Another approach is to incorporate the
probability distribution of missing values into the network inputs [Smieja .
(2018)].

Alternatively, one could simply use a ‘placeholder’ value that the network
would ideally learn to ignore with exposure to the data. Generally, using zero
is advised against, due to its tendency to create a discontinuity in the
latent space formed by the features, thereby making it harder for the model to
generalise [Chollet (2021)]. Instead, it is recommended to replace the missing
values with the average or median value of the dataset. However, for the
training dataset used here, zero is the average value of each standardized
input. On the other hand, imputing an arbitrarily large value, well outside
the range of data, could instead teach the network that this value is to be
ignored. Given the desire to keep the pipeline as simple as possible for this
exploratory study, and that there is no established, universal strategy for
dealing with gaps for machine learning models, three different placeholder
values were tried: zero, one hundred (i.e., 100$\sigma$ from the mean), and a
linear interpolation across the gaps.

To summarise the approach of this study, after the clean, ‘true’ structure
functions are calculated for a series of continuous intervals of the time
series, the intervals are artificially gapped in several different ways. By
creating multiple versions of inputs with gaps in different places and the
same expected outputs, we aimed to make the network more robust to missing
data. The three types of imputation described above are then applied to the
gaps so that the neural network can receive the data. These intervals are
provided as the input data to the model, with the original structure functions
as the target outputs. The model predictions are then compared with structure
functions calculated directly from the gapped intervals, and from gapped
intervals with the interpolation techniques applied. In this way we can
compare the performance of each technique in approximating the statistic of
the original ungapped interval.

##  2 Data Preparation

Parker Solar Probe is a spacecraft that was launched in 2018 to study the
physics of the inner heliosphere and the origins of the solar wind by flying
very close to the Sun (as close as 9.9 solar radii during orbit 22 in 2024).
Measurements from this spacecraft are of clear scientific interest, and their
high resolution provided us with the long continuous series needed for this
study. The data used were taken from PSP’s fluxgate magnetometer (FGM)
instrument [Fox . (2016), Bale . (2016)]. The FGM measures magnetic fields at
a native cadence of 256 samples/second. We use data from November 2018, during
and shortly after the first “encounter” (E1) of PSP. Encounters are periods
centered around each perihelion during which data is typically collected at
the highest resolution. For context, during this particular encounter, the Sun
was at solar minimum, and PSP recorded measurements at a relatively constant
longitude relative to the Sun’s rotating surface, travelling as close to the
Sun as 35.6 solar radii [Bale . (2019), Kasper . (2019)].

Two continuous, gap-free intervals were selected: one from 2018-11-01 to
2018-11-18 and another from 2018-11-21 to 2018-11-30. These intervals
contained no missing observations after performing down-sampling, justified in
the following section. Combined, these final gap-free time series consisted of
3,100,000 points for each vector component Bx, By, Bz, all three of which were
used here. These series were then split into 310 vector time series intervals
of length 10,000 (duration 125 minutes).

###  2.1 Input preparation

The data was pre-processed in order to make it easier for the ANN to process
data from different sources and intervals. We started with a set of time
series intervals with 100% of measurements available and followed the data
standardisation and augmentation procedure below. Figure 1 provides a visual
representation of both the data pre-processing and processing workflow.

Data standardisation: The timescales and magnitudes of interest vary
significantly from one system to another. For example, solar wind in the inner
heliosphere has magnetic field amplitudes in the $\sim 100nT$ range and a
correlation time of $\approx 600s$ [Parashar . (2020), C. Chen . (2020)],
whereas the solar wind at 1AU has magnetic field amplitudes in the $\approx
10nT$ range and a correlation time of $\approx 1hour$ [Isaacs . (2015),
Jagarlamudi . (2019)]. On top of this variability, the time cadences of
various instruments differ significantly. In order to train the ANN in a
system-agnostic way, we standardised both the x-values (time series) and
y-values (the fluctuation amplitudes) using the following methods. The time
series were standardised by down-sampling to have 10,000 samples across
$\approx 15t_{corr}$ so that the training series has a sampling rate of
$\delta t\sim 1.5\times 10^{-3}T_{corr}$. The amplitudes were standardised by
subtracting the mean value $\mu_{a}$ from each value of each interval and then
dividing by the standard deviation $\sigma_{a}$ (Z-score normalisation):

$a(t)_{norm}=\frac{a(t)-\mu_{a}}{\sigma_{a}}$ (3)

Gapped series preparation and augmentation: Each set of complete intervals was
augmented by duplicating each interval several times and then corrupted by
taking gaps from each new interval in different ways. By having the same
expected output for each of these copies, the aim was to make the ANN
indifferent to where and how large the gaps are, as well as giving us more
data train on. To this end, 10 copies of the training intervals and 5 copies
of the testing and validation intervals were made. Then, a percentage of data
was removed from each interval in between 3 and 20 continuous segments. Real-
world data gaps also occur as individual missing values rather than continuous
segments. However, large continuous gaps are representative of many bad
quality datasets, including OMNI [Lockwood . (2019)] and Voyager [Gallana .
(2016)], hence the decision to simulate this variety of ‘missingness’.

For the training and validation intervals, between 0 and 50% of data was
removed, whereas for the test intervals, up to 95% of data was removed. This
was done to test the algorithm’s performance on missing data in general,
rather than just the gapped percentages it was trained on. This also allows us
to test its performance ‘in the limit’, i.e. right up to only 5% data
remaining, which will help us assess overfitting of the model. (90% data loss
is also the scale of gaps in Voyager datasets from the outer heliosphere
[Gallana . (2016)].) For each interval, the exact percentage of data to
remove, the number of segments to remove it in, and the location of each
segment were chosen randomly. Finally, the resultant gapped intervals were re-
standardised.

Figure 1: Diagram showing the workflow for adding artificial gaps and
producing different structure function estimations from each interval. (M-IMP
= mean imputation, L-INT = linear interpolation.)

ANN input: The input for the ANN training program was a 3x10,000 array
representing all of the three vector components $(Bx,By,Bz)$ across a 10,000
point time series. Three versions of the inputs were created: one with gaps
filled with zeroes, another with gaps linearly interpolated, and the third
with gaps filled with 100. Note that the first two approaches were also used
as alternative estimation methods, by simply feeding these input versions into
equation (2), as shown in Figure 1.

ANN output: The corresponding expected output for the ANN was a 1x2,000 array
of the values of second-order $S^{fn}$ computed from the corresponding
complete time series, up to a maximum lag of 20% of the series length
($n_{lag}=2000$). (This array can then simply be plotted to show the structure
function.)

Benchmarking the results: 70% of the intervals were used for training, 20% for
testing, and 10% for validation of the network: information about each set is
summarised in Table 1. The second-order $S^{fn}$s computed by the ANN are then
compared to the $S^{fn}$s computed three ways: i) ignoring the gaps, ii) mean
imputing the gaps, and iii) linearly interpolating the gaps, as shown in
Figure 1. For computation i), this is achieved by simply ignoring the values
for which no lagged increment is available. Finally, the input-output pairs
are arranged in random order before training.

Data source | Purpose |  Input lengths |  Output lengths ($n_{lag}$) |  % of each input removed |  No. of instances
---|---|---|---|---|---
PSP | Training set |  30000 |  2000 |  0-50 |  2170
| Validation set |  30000 |  2000 |  0-50 |  150
| Test set |  30000 |  2000 |  0-95 |  315
Table 1: Dimensions of data used to build and evaluate the neural network
model. ‘No. of instances’ refers to the count of intervals in the set after
duplication of the original unique intervals.

##  3 Model training and validation

Using a feedforward neural network, a multi-output regression model was built
in Python using the Tensorflow package [Abadi . (2015)]. The workflow to
ensure a good model fit was the following:

  1. 1.

Train the model until the early-stopping criterion is reached (see below)

  2. 2.

Evaluate the model on the validation set, checking for overfitting and
underfitting by visual inspection of the predictions

  3. 3.

Adjust the model hyperparameters, such as the number of hidden layers and the
number of nodes in each hidden layer

  4. 4.

Repeat 2-3 until a good fit is achieved

The loss function used to calculate the error for this network was the mean
squared error, or MSE. Due to each output being a vector, the overall error
for one epoch of the network $MSE_{overall}$ is calculated as the MSE for a
single instance $MSE_{i}$, averaged over all the instances. (One epoch is one
iteration through every instance in the training set.)

$\displaystyle MSE_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{n_{lag}}{\sum_{j=1}^{n_{lag}}(S^{fn}_{ij,pred}-S^{fn}_{ij,true})^{2}}$
(4)

where $S^{fn}_{ij,pred}$ is the predicted value of the second-order $S^{fn}$
for the $i^{th}$ interval at lag $j$, $S^{fn}_{ij,true}$ is the corresponding
‘ground truth’ value, and $n_{lag}$ is the number of lags for which the
$S^{fn}$ has been computed. $MSE_{overall}$ is minimised through the process
of backpropagation of error via gradient descent, and each weight and bias is
adjusted according to the learning rate and the weight’s contribution to the
overall error, calculated using partial derivatives.

For each epoch of training, the validation loss was used to check whether the
model is still improving. (A sustained increase in the validation loss
indicates that the model is beginning to overfit.) Accordingly, training was
stopped when the validation loss was reduced by no more than 0.01 over 10
epochs; this is the early-stopping criterion.

The $S^{fn}$ values decrease by a few orders of magnitude going from large to
small lags. This could potentially bias the MSE on large-scale predictions.
Hence, the MSE was not used in isolation to evaluate the final model on the
test set. It was complemented with the mean absolute percentage error (MAPE)
to quantify the model’s performance.

$\displaystyle MAPE_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{n_{lag}}\sum_{j=1}^{n_{lag}}\left|\frac{S^{fn}_{ij,true}-S^{fn}_{ij,pred}}{S^{fn}_{ij,true}}\right|$
(5)

where symbols have the same meaning as in the expression for $MSE_{i}$. This
metric is easier to interpret than the MSE in terms of relative error and is
scale-independent. It is also more stringent than MSE because it evaluates the
method’s performance in predicting both the large-scale and small-scale
values. MAPE was also trialled as a loss function for training the network;
however, this led to inferior performance and overfitting compared with MSE.
Therefore, we only present results for the model trained using MSE.

###  3.1 Optimising the network: selection of hyperparameters

Part of the challenge of supervised learning is finding the optimal
hyperparameters (‘architecture’ in the case of the dimensions of a neural
network) so as to get the best performance on unseen data. A common approach
is to use a random grid search, which involves randomly selecting a subset of
combinations of hyperparameters from a search space, and choosing the
combination that produces the minimum validation loss. This method was tried
for each of the input versions. The number of hidden layers, number of nodes
in each hidden layer, learning rate, and presence of a drop-out layer were
varied, whereas the optimizer (Adam) and activation functions (ReLU for each
hidden layer and linear for the output layer [Brownlee (2021)]) were kept the
same. However, it was found that this traditional approach of optimisation was
not suitable for this unusual prediction task. Aiming for the smallest loss
(average MSE) on the validation set led to a ‘regression to the mean’ style of
overfitting, where all predicted curves were the same. In other words, the
resultant network was only paying attention to the biases of the final layer,
rather than the actual inputs. (The networks that produced these lowest losses
(between 0.68 and 1.2 average MSE) had optimal hyperparameters of between 9
and 17 layers, depending on the input version used.) This result called for a
more careful optimisation procedure. Specifically, the best network
architecture was selected by manually increasing the size of the network and
visually inspecting the predictions of each network on a sample of validation
instances, rather than relying on the validation loss as an indicator of model
performance.

After running several different iterations it was found that 10 or more layers
always lead to regression to the mean, where virtually the same curve was
predicted for every input interval. The best network configuration from those
that were tried - i.e., that which produced smooth curves with shapes that at
least partly matched the different shapes of the expected outputs - was one
with 2 hidden layers, each with 10 nodes; a learning rate of 0.001; no drop-
out layer; and trained for 500 epochs. A schematic diagram of this
configuration is shown in Figure 2. We note that the input and the output
layers require the largest number of parameters, 300,010 and 22,000,
respectively in this case. Any additional hidden layers having order 10
neurons contribute only order 100 parameters to the list of trainable
parameters. Finally, the input version that produced the best predictions was
the linearly interpolated dataset.

Figure 2: Schematic diagram of the final ANN architecture. The inputs consist
of 3 x 10,000 stacked vector components, and the outputs consist of 1 x 2,000
second-order structure functions. The number of hidden layers and nodes were
chosen by iteratively increasing the size of each hyperparameter and visually
inspecting a sample of validation predictions

made by the trained network for each architecture.

##  4 Results

The favoured network architecture and input version were then used to produce
predictions on the test set and ultimately evaluate the model’s performance.
Its approximation ability was compared with the alternative interpolation
methods discussed previously, as well as naive calculation of the structure
functions directly from the gapped intervals (see Figure 1). We start with a
case study in Figure 3 of four versions of the same original complete
interval, each with increasing data loss. This is one example of how the
network responds to the input data; an overall evaluation of its ability
follows.

Figure 3: Results of the four different $S^{fn}$ approximation methods for a
standardised PSP interval that has been gapped in four different ways. (For
simplicity, only one of three vector components for each input interval has
been shown, but this still illustrates the number and size of the gaps, which
were consistent between components.) Note that the log-log plot emphasises
differences between the curves at small lags (i.e., small spatial scales), and
is the standard way of presenting structure functions, in order to facilitate
comparison with theoretical power-laws.

Performance of ANN on case study: For input interval 1a, with 35% data
missing, the ANN does not accurately reproduce the flattening of the true
structure function above about lag 500. The prediction interval 1b, at 59%
data loss, gets closer to the true shape. The log-log plot shows how the ANN
particularly has better predictions at larger scales (higher lags). At 66%
data loss (interval 1c), the ANN shows excellent performance, relative to
other methods. In this case it predicts the true values well across the range
of scales - this is particularly evident in the log-log plot. Finally, at 69%
data loss (interval 1d), the performance of the ANN poorly follows the shape
of the true curve, though does still get closer than other methods, especially
at large scales. Given that the $S^{fn}$ decreases by a few orders of
magnitude from large to small lags, small lags contribute very little to MSE.
Hence, it is expected that, given this loss function, values at larger lags
would be better predicted by the neural network. This foreshadows the ultimate
conclusion on the utility of the neural network, which seems to perform best,
relative to other methods, when dealing with large-scale values at high data
loss.

Performance of simple imputation on case study: At 35% data loss, M-IMP and
L-INT, do show the correct shape, albeit shifted downwards. At 59% missing,
these methods noticeably degrade in their estimation of this structure
function, and at 66% missing they are clearly inferior to the ANN. With just
slightly more missing data (interval 1d), M-IMP and L-INT show little change,
retaining their severe under-estimation of the true curve. In contrast to ANN,
L-INT and M-IMP show a predictable consistent underestimation, due to the
decreased variability in magnetic field increments caused by the smoothing
effect of the imputation. This effect worsens with increasing gap percentage.

Overall statistical performance: The overall performance of each method,
including the $S^{fn}$s calculated from the un-filled gapped intervals, was
evaluated using the following measures:

  * •

Average MSE ($MSE_{overall}=\frac{1}{n}{\sum_{i=1}^{n}MSE_{i}}$) across all
expected-observed $S^{fn}$ pairs (recall this was the loss function used to
train the neural network - see equation (4))).

  * •

Average MAPE ($MAPE_{overall}=\frac{1}{n}\sum_{i=1}^{n}MAPE_{i}$) across all
expected-observed $S^{fn}$ pairs. Both $MSE_{overall}$ and $MAPE_{overall}$
are given in Table 2.

  * •

Scatterplots and corresponding linear regression lines of MSE and MAPE against
% data missing of individual test intervals. This shows us the how each method
performs with increasing data loss shown in Figures 4 and 5.

Spacecraft | Performance |  $S^{fn}$calculated from |  $S^{fn}$estimated using
---|---|---|---
| measure | GAPS | M-IMP | L-INT | ANN
PSP | MSE | 16536.91 | 7.02 | 2.82 | 2.71
| MAPE | 5.13 | 0.44 | 0.27 | 1.97
Table 2: Calculated performance measures for each $S^{fn}$ approximation
method. Bolded figures are the lowest of each row. GAPS: Gapped interval with
no imputation. M-IMP: Gapped interval with mean (0) imputation. L-INT: Gapped
interval with linear interpolation of gaps. ANN: Artificial neural network
model.

The overall results shown in Table 2, suggest that, for overall average
performance, the GAPS (calculation from gapped series) method is the worst
$S^{fn}$ approximation method across the board with $MSE_{overall}\approx
17,000$ and $MAPE_{overall}\approx 5.1$. The relative performance of the other
methods differ depending on the evaluation metric used. This occurs because
MAPE is lower for predictions that better estimate the true curve as a
proportion at each lag, whereas MSE is lower for predictions that simply have
the smaller average squared distance from the true curve. Because of this
difference, according to MSE, the ANN has the best overall performance,
followed by L-INT. According to MAPE, L-INT has the best performance, followed
by M-IMP, and the ANN is only the third best.

Figure 4: Scatter plot of MSE against proportion of data removed for the PSP
test intervals with overlaid linear regression lines and confidence regions
from other panels. The line for the GAPS method (no imputation) is not shown
here as it quickly disappears from the plotting area, and Table 2 shows it is
clearly inferior to the other methods. Figure 5: Scatter plot of MAPE against
proportion of data removed for the PSP test intervals with overlaid linear
regression lines and confidence regions. The line for the GAPS method (no
imputation) is not shown here as it quickly disappears from the plotting area
shows it is clearly inferior to the other methods.

However, these overall measures do not take into account the variation in
their performance as a function of the degree of sparsity. Hence, we take a
statistical approach to quantify the performance with increasing sparsity.
Figure 4 shows the scatter plots of MSE versus percentage of missing data for
each set of test intervals for each method, overlain with linear regression
lines of best fit. As seen in the middle two panels, the MSE tends to increase
with increasing sparsity for M-IMP and L-INT. This is to be expected: as the
amount of data missing increases, the $S^{fn}$ estimation gets worse for these
methods. However, there is distinct funnelling on the plots, representing
heteroskedasticity or unequal variance in MSE for different proportions
missing. For low % missing values, around less than 20% missing, there is a
very small range of MSE values for the imputation methods. This means that the
accuracy of the $S^{fn}$ estimations do not vary much for small percentages of
missing data. On the other hand, as the amount of missing data increases, not
only does the average error increase, but also the variation in error. This
shows that intervals with large amounts of data missing can, in some cases,
produce $S^{fn}$s quite similar to the true $S^{fn}$s, if imputation is
performed. This is especially true for intervals that have been linearly
interpolated - we can see in the L-INT scatterplot that there are intervals
with up to 90% missing that have very low MSE values. This is due to the
importance of not only the size of the gaps, but where the gaps are in the
interval: if the removed data does not significantly depart from the overall
trend, linear interpolation will result in a $S^{fn}$ not very different from
the expected curve. On the other hand, M-IMP does not show similarly low MSE
values beyond about 45% missing, though there is still distinctly increasing
variance.

In stark contrast to the M-IMP and L-INT methods, the MSE of the ANN model
predictions are largely indifferent to the proportion of data missing. There
is a constant band of MSE values across this scatterplot, and the Pearson
correlation coefficient, a measure of the linear association between two
variables, is very close to 0 for this method (-0.06).

As a way of establishing the comparative usefulness of each method, linear
regression lines were fit to the data. Although fitting a linear model is
inappropriate for this data due to the unequal variance present in the M-IMP
and L-INT methods, it still provides a useful indicator of the typical values
of MSE for different proportions missing. The final panel in Figure 4 shows a
clear distinction in the slopes of the regression lines. MSE increases the
fastest for the M-IMP method, and this association is also that with the
highest correlation between MSE and % missing (0.87). The next largest slope
is that of the L-INT method, and this association has the next highest
correlation (0.70). Finally, the ANN has a very flat slope, with an
aforementioned correlation of close to 0. This result shows that the ability
of the neural network to approximate the true $S^{fn}$ is much less affected
by increasing amounts of missing data than the other two methods. However,
this does not make it the best $S^{fn}$ approximation method for any bad
dataset. What we can see in Figure 4 is that up to about 50% missing data, the
simple imputation methods have lower typical values of MSE than the neural
network. At values greater than 50%, the neural network, on average, produces
the best estimations of the three approaches, according to the MSE metric.
This, however, could be a result of the dominance of large lag values of the
$S^{fn}$ controlling the MSE.

To quantify the performance of a method to predict not only the large lag
value but also the small lag values, we use the MAPE. The MAPE measure, shown
in Figure 5, shows much less heteroskedasticity for the L-INT and M-IMP
methods, with M-IMP in particular showing a much stronger linear relationship
between % missing and MAPE, producing a correlation of 0.98. The ANN method
has a small positive slope and a relatively constant band of scatter, with
correlation of 0.04. In the final panel of Figure 5, we see that the neural
network linear regression line remains above that of both other methods for
all gap percentages.

Overall, we find that the ANN is largely indifferent to the proportion of data
missing in its approximations of the true $S^{fn}$. However, this means that
the worst predictions for inputs with little or no data missing are as bad as
those for inputs with at least 90% data missing. This means that while this
small neural network can produce good predictions at lower data loss, it
should not be relied upon to do so, unlike the simpler imputation methods.
Based on these findings, it seems reasonable for this task to use linear
interpolation for data gaps as large as $\sim 20\%$.

##  5 Summary and Conclusions

Gaps are a common problem in almost all fields that deal with time series. The
field is mature with many ways of filling the gaps, including mean-imputation,
linear interpolation, maximum likelihood estimation of ARIMA models [Velicer
Colby (2005)], singular spectrum analysis [Schoellhamer (2001)], and
artificial neural networks (ANNs) [Comerford . (2015)]. Our interest here is
not in prediction, but in gleaning the best estimates of scale-domain
statistics of a system without having to reconstruct the time series. To this
end, we studied the potential of simple feedforward artificial neural networks
to predict turbulent statistics of solar wind magnetic field measurements. In
space plasma physics, an accurate description of the inertial range of the
structure function is desirable. This is particularly important to estimate
not only the slope of the structure function in this range, but also to
estimate the inner and outer scales of turbulence.

Starting with complete time series with 100% coverage, we created gapped time
series for which the second-order structure functions were estimated in four
ways: i) direct computation ignoring gaps, ii) mean imputation of the gaps,
iii) linear interpolation across the gap, and iv) a trained ANN.

ANNs do not seem to be the panacea that one might naively hope for in such a
situation. Our results showed a narrow range of conditions in which the
supervised learning model could be useful. As reflected by the error functions
of MSE and MAPE, the ANN seems to somewhat learn to estimate the large-scale
values of the structure function. This is not very surprising as the large lag
$S^{fn}$ values approach the mean-squared value of the fluctuation amplitudes.
Given the trend in error with increasing data loss, the ANN is more useful for
large portions of missing data, but over the entire range of data loss it
tends to perform worse than simpler methods.

However, it is worth noting that with only 20 neurons (and about 322,000
trainable parameters), ANN performs comparably to L-INT or M-IMP methods, and
with MSE as the cost function it even outperforms these two methods for large
gaps in the data. However, when using MAPE to evaluate the network we find the
ANN consistently falling short compared with other methods, indicating its
lack of ability to predict the structure function accurately at different
scales, according to this proportional measure of accuracy. In addition, the
model was only evaluated on data from the same system as it was trained on
(Encounter 1 of PSP). While we did take the step of re-sampling the input data
based on the correlation time of PSP data, we cannot yet make claims to the
model’s ability to generalise to other space weather domains, for example, the
near-Earth environment. Future iterations could train the model on data from a
range of spacecraft and solar conditions to attempt to improve its performance
on diverse unseen data.

Our results indicate that to achieve a reasonable description of turbulent
statistics for gapped time series, one needs to go beyond simple-minded
feedforward ANNs. Possible improvements to ANNs could include more advanced
architectures such as LSTM networks or autoencoders, and grey box modeling
with turbulence physics incorporated into the input. This study has taken a
reasonably physics-agnostic approach to training the model, but a grey box
model that incorporates additional physical parameters and constraints could
potentially improve its performance. Such an approach could also be applied to
the adjacent problem of identifying waves in the solar wind. The presence and
nature of wave-like helical structures are measured by the magnetic helicity,
and this helps to identify the nature of turbulence [BT. Tsurutani . (2018)].
Magnetic helicity spectra have been used by Podesta_2011, He_2011, and
Klein_2014 to identify the presence of ion cyclotron waves and kinetic Alfvén
waves in the solar wind. This represents an ideal problem for grey box
modeling, where the helicity signatures guide the machine learning algorithms
to quantitatively describe solar wind plasma properties. Another example of
helicity, the total unsigned current helicity, has been shown to be an
important feature in multiple solar flare forecasting models, including a CNN
model [Yi . (2021)]. CNNs have also been used to distinguish between sub-
Alfvenic and super-Alfvenic MHD turbulence using snapshots of simulations
[Peek Burkhart (2019)].

It may also be the case that even the performance of this simple feedforward
ANN structure could be improved through better optimisation of the model
weights, biases, and hyperparameters (in particular, the number of hidden
layers and nodes). To this end, a reliable method of avoiding over-fitting
when trying to predict the shape of a curve, rather than a scalar output, is
an important issue to address. The conventionally used loss function
($MSE_{overall}$) is a very limited measure because it can cause the neural
network to regress to the mean and essentially ignore the input values. This
will be explored in follow-up studies, along with other ways of representing
the missing data when feeding it into the network. On this note, our results
also suggest that not only does linear interpolation remain a reliable method
of dealing with relatively small data gaps, but it is also a useful and simple
way of preparing a gapped series for input into a machine learning model.
Imputing 100$\sigma$ and 0 in place of the gaps failed in this regard, likely
due to the network’s failure to both detect and discount the effect of these
‘placeholder’ values during the processing of the input vector. This is a
significant challenge for any mapping technique that builds in local
linearity, as neural nets do.

The recent advances in applying machine learning to space physics provide
exciting avenues for exploration and discovery. The literature is expanding
rapidly and many interesting applications remain, such as identifying
discontinuities [BT. Tsurutani Ho (1999)] and helical structures [BT.
Tsurutani . (2018)] in the solar wind. Progress has already been made in
classifying states of the wind [Camporeale . (2017)], as well as improving the
resolution of solar images through deep learning [Kim . (2019)] and extracting
features from these images for flare prediction [Jiao . (2020)]. The
playground is big and largely unexplored, and could lead to breakthroughs in
understanding and predicting the space weather environment. In order to do
this, we need a technique for dealing with significant data sparsity in time
series and the attendant degradation of models and statistics. This study has
examined a machine learning approach to this issue, and in doing so aimed to
make the literature less ‘sparse’.

##  6 Author Contributions

TNP came up with the project idea, DW performed the analysis and created the
figures, RR, MF, and RKS provided guidance on ANNs. All authors discussed the
results and contributed to manuscript writing.

##  7 Data Availability Statement

The raw data can be downloaded from the Space Physics Data Facility website:
https://spdf.gsfc.nasa.gov/pub/data/psp/fields/l2/mag_rtn/. The processed data
and software are available on Zenodo and Github:
https://doi.org/10.5281/zenodo.6960975. This work used computational and
storage services associated with the Rāpoi cluster provided by Victoria
University of Wellington.

###### Acknowledgements.

We would like to acknowledge the PSP instrument teams for high quality
measurements in the inner heliosphere and the Space Physics Data Facility
(SPDF) at the Goddard Space Flight Center for providing access to the data
used for this study. This research was seeded by funding from a Summer
Research Scholarship provided by Victoria University of Wellington. The work
of RKS is partially supported by SERB, DST, Government of India through the
project EMR/2017/002778.

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