Many complex system interactions are present throughout the disciplines of geophysics and associated solar dynamics. A number of these processes remain outside the capability of standard statistical procedures, requiring equally complex computational models in order to more accurately reflect physical relationships. Artificial neural networks (ANN) form the basis for a number of computational models based on the mammalian brain. Instead of relying on linear correlative relationships among a particular dataset, ANNs are a form of machine learning in which the system learns to recognize an output variable based on an input variable series [1] . Data is processed through a number of interconnected “neurons” which form “synaptic connections” from the input nodes through a hidden layer before converging on the output neurons (e.g., Figure 1). Each input and hidden neuron consists of statistical weights which are capable of adaptation [2] , the exact parameters which are modified by an algorithm over the course of network training procedures. These weights essentially form the synaptic connections among neurons which activate during network construction. This form of computation is capable of operating in parallel units, much like the human nervous system. Because neural networks do not rely on linear correlations for learning, ANNs are capable of nonlinear modelling and may therefore provide a useful alternative approach to a number of both theoretical and real world problems. This includes geo and solar physics [3] , medical diagnosis [4] , pattern recognition [5] , and many other areas. The nonlinear approach used in ANN computations is particularly useful in the context of highly complex or “noisy” datasets.

Backpropagation algorithms constitute a primary basis for the construction of ANNs [6] . In general backpropagation, the delta rule is employed for updating the statistical weights of each neuron in the network using a gradient descent [7] . An activation function is applied to artificial neurons in both the input and hidden layers, which in this case was a sigmoid function (e.g., Equation (1) & Figure 2).

One of the primary training algorithms employed in ANN modelling is the Levenberg Marquardt Algorithm (LMANN). This particular method is a standard backpropagation curve fitting algorithm typically employed to solve generic problems using a damped least squares process [8] . While the LMANN option is a popular choice, it is capable of obtaining a local minimum only, and is therefore not always ideal for greater generalization [9] . However, a neural network trained using a Bayesian regularization algorithm (BRANN) has been shown to possess greater generalization than other ANNs and has demonstrated more robust results overall compared to standard backpropagation networks [10] . There are a number of other benefits associated with this particular learning algorithm summarized by Burden & Winkler [10] , including reduced probability of network “overtraining” or overfitting, reduced difficulty in cross validation, and more effective stopping procedures. Overfitting refers to a resultant model which describes random noise as opposed to the intended output, which will subsequently have poor overall predictive performance by overemphasizing minute data fluctuations. By integrating Bayesian probabilities, this also allows for greater accuracy in network learning, and probabilities associated with artificial neuron weights are updated according to newly observed data using a likelihood function (Equation (2)) which compares a standard Gaussian function with the updated observations.

In the following analyses, two training algorithms were employed using a nonlinear autoregressive exogenous network (NARX). This particular form of ANN is used in time series modelling [11] in order to predict an outcome variable [y(t)] based on both d past values of the outcome variable and current and d past values of an external source of influence [x(t)], as illustrated in Equation (3). The fit of predicted output values can be compared to the original target values using simple correlation coefficients [12] , while an error term is also employed in order to further gauge predictive accuracy, typically presented as some forms of mean or summed squared error term (target-output).

A number of novel approaches to space weather dynamics have been examined using ANN models. Many of these models possess useful real world applications which enable researchers to predict future values associated with various geo and solar physical systems. The NARX form of network used in this study was previously employed by Oliveira et al. [12] in order to simulate conditions of the interplanetary magnetic field (IMF) using ground observations associated with cosmic rays as the exogenous input. Furthermore, the disturbance storm time (DST) index, which was also the focus of this study, was recently examined using various measures of solar winds and the IMF [3] . The present study focuses on daily averages of simple, untransformed values obtained from a freely available online source (http://omniweb.gsfc.nasa.gov/).

The Earth’s magnetic field has an average static background intensity of ~50,000 nT [13] . Despite this relatively weak value, variations in geomagnetic activity have been shown to affect terrestrial systems including electronics and power systems [14] , and human health [15] , while even small perturbations of ~20 to 40 nT have been related to alterations in brain activity associated with altered states of consciousness [13] . One of many measures associated with geomagnetic activity, the DST index refers to the varying power of the ring current which surrounds the geosphere. The subsequent DST magnetic field opposes the geomagnetic field so that decreases in DST index accompany increases in geomagnetic activity. This specific measure is particularly useful in studying solar geophysical interactions given that the DST ring current is produced as a result of solar particle streams [16] , and it also has direct correlations with the geomagnetic field. Furthermore, there are a number of complex relationships previously noted between various sources of solar activity and geomagnetic fluctuations [17] . Given the large number of solar factors is found to have potential relationships with geomagnetic activity, it would be beneficial to identify an optimal statistical model for forecasting geomagnetic storms based on a simple subset of input variables. This would allow for potential early warning systems to be placed in effect at relevant institutions regarding future geomagnetic perturbations.

The results of this analysis appear to confirm those from a number of previous studies which have suggested that ANN models are more accurate compared to simple regressions for many applications [18] [19] . While the NARX models used relied on past values of the DST index, there was a significantly greater target to output fit apparently for the MLPANN network compared to regression analyses, which did not utilize DST feedback. There was a clear hierarchy of statistical models in the present analyses indicated by both fit (r) and overall accuracy (RMSE), and models from greatest to least relative fit/accuracy were BRANN, LMANN, MLPANN, LR enter, LR stepwise.

The Bayesian regularization algorithm for neural network training has previously been implicated as a superior model compared to other network algorithms [10] . This appears to be the case in the context of DST autoregressive prediction with solar wind input. It is clear that Bayesian probabilities are well suited to neural network learning for DST forecasting given the adaptive neural weights updated over the course of network construction based on new probabilities. Furthermore, regression models typically require a priori knowledge of variable interactions for accurate forecast implementation, while neural networks are capable of learning from an array of related measures and producing a highly accurate model. Standard regression models also tend to ignore non-linear effects which may be present, particularly in noisy datasets, or time series data with no clear linear relationship. It may also be important to note that the NARX model is capable of both open and closed loop performance and can be used to predict a step beyond an available dataset with associated performance measures. However, there are clear benefits to employ a range of statistical methods when interpreting complex interactions in order to gain a fuller picture of the system in question.

It is also interesting to note that the Bz component of the IMF was consistently identified as the greatest shared source of variance with geomagnetic disturbances, along with the speed and proton density of solar winds. This assemblage of contributing factors has previously been identified in the context of geomagnetic activity including Bz, particle density, and velocity of solar winds with DST index [20] . Modelling of DST variations was later examined using more complex methods of data transformation and analysis on a wider array of input measures [3] .

In the present study, the effects of solar wind variations are demonstrated using a range of input measures and statistical methods, which clearly demonstrate the enhanced potential for ANN models compared to more traditional statistical methods. Prior analyses also suggested that network performance improved with the addition of a greater range of IMF variables (e.g., Bx, By, Bz), which was also previously suggested by Boynton et al. [3] . However, network performance was reduced in magnitude with the addition of solar wind temperature in the input layer. The NARX networks still managed to outperform other models using untransformed variables of simple daily averages. Additionally, all data is freely available from an online source (http://omniweb.gsfc.nasa.gov/) open to any researcher interested in constructing models relevant to their own location.

Similarly, local forecasting with ANNs has proven effects in a number of areas and may be useful for predicting a range of events. Accurate modelling of solar and geomagnetic interactions is a particularly useful endeavor when the increasing reliance on technology civilization has reached. Given the known effects of geomagnetic storms on electrical systems, and even the significant effects of small geomagnetic perturbations on neurophysiology [13] , monitoring and forecasting of this phenomenon are increasingly vital. This is particularly prominent for satellite systems outside of the planetary atmosphere which may be quite sensitive to variations in ring current activity.

That powerful statistical software which is widely available is also a tremendous boon to statistical modelling and related research endeavors, and the complicated computations required for ANN methods are now easily employed in standard mathematical software, making relatively intimidating methods more accessible to the average researcher. Neural network predictions can be very powerful tools for both local and global predictions and can potentially be used to improve traditional models. ANN methods should be further investigated for potential applicability in better prediction of geomagnetic variations and storm level events, as well as to further elucidate the complex interconnections between solar and geophysical parameters.

The author would like to thank Dr. Michael Persinger, Nicolas Rouleau, Lucas Tessaro, David Vares, Dr. Blake Dotta, and Prof. Kevin Saroka for valuable feedback and support. Additional thanks to Dr. Natalia Papitashyili for maintaining the NASA OMNIWeb database. Funding for this research was provided by Dr. Michael Persinger.