SpatialTemporalFeatureSelectionOptimalApproach03.tex

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\title{Optimal Neural Network Feature Selection for Forecasting of Spatial-Temporal Series}
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\author{Eurico~Covas,~\IEEEmembership{}
	    Emmanouil~Benetos~\IEEEmembership{}% <-this % stops a space
\thanks{E. Covas is with the CITEUC, Geophysical and Astronomical Observatory, University of Coimbra, 3040-004, Coimbra, Portugal, and
Queen Mary University London, 10 Godward Square, Mile End Rd, London E1 4FZ,  e-mail: (eurico.covas@mail.com)}% <-this % stops a space
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{Covas \MakeLowercase{\textit{et al.}}: Optimal Neural Network Feature Selection for forecasting of spatial-temporal series}
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\begin{abstract}
We show empirical evidence on how to construct the optimal feature selection or architecture for the input layer of a feedfoward neural 
network for the propose of forecasting spatial-temporal signals. The approach is based on results from dynamical systems theory, namely 
the non-linear embedding theorems. We demonstrate it for a variety of two-dimensional signals with one spatial dimension and one time 
dimension, and show that the optimal input layer seems to consist of a two dimensional grid, with spatial/temporal lags determined by 
the minimum of the mutual information of the spatial/temporal signals and the number of points taken in space/time decided by the 
embedding dimension of the signal. We present evidence of this conjecture by running a Monte Carlo simulation of several combinations of 
input layer architectures and show that the one predicted by the non-linear embedding theorems seems to be optimal or close of 
optimal. In total we show evidence in four unrelated systems: a series of coupled H\'{e}non maps; a series of couple Ordinary 
Differential Equations (Lorenz-96) phenomenologically modelling atmospheric dynamics; the Kuramoto-Sivashinsky equation and finally real 
data from sunspot areas in the Sun (in latitude and time) from 1870 to today.
\end{abstract}

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\section{Introduction}
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\IEEEPARstart{G}{iven} a physical data set one of the most important questions one can pose is:``Can we predict the future?'' This 
question can be put forward irrespectively of the fact that we may already have some insight or even be certain on what the exact model 
behind some or all the observed variables is. For example, for chaotic dynamical systems \cite{0813340853,2004icti.book.....M}, we may 
even have the underlying dynamics but still find it hard to predict the future, given that chaotic systems have exponential sensitivity 
to initial conditions. The more chaotic a system is (as measured by the positiveness of their largest Lyapunov exponents 
\cite{1985PhyD...16..285W,1994PhLA..185...77K}) the harder it gets to predict the future, even within very short time horizons. In the 
limit case of a random system, it is not possible to predict the future at all, although one can opine on certain future 
statistics\cite{9780486693873}. For the case of weakly chaotic systems, there is an extensive literature on forecasting methods ranging 
from linear approximations\cite{1451722}; truncated functional expansion 
series\cite{Powell:1987:RBF:48424.48433,Broomhead1988MultivariableFI}; non-linear embeddings \cite{PhysRevLett.59.845}; auto-regression 
methods\cite{0130607746}; hidden markov models \cite{1165342} to state-of-the-art neural networks and deep learning methodologies 
\cite{LANGKVIST201411}.

Most literature on forecasting chaotic signals is dedicated to a single time series, or treat a collection of related time series as a 
non-extended set, i.e. a multivariate set of discrete as opposed to continous time series. For forecasting spatial-temporal chaos we 
refer the reader to 
\cite{doi:10.1063/1.165894,PhysRevE.51.R2709,PhysRevLett.85.2300,Parlitz2000NonlinearPO,2000PhRvL..84.1890P,Xia2006APF,covas2016} and 
references therein. Even rarer are attempts to forecast spatial-temporal chaos using neural networks and deep learning methodologies 
\cite{covaspeixinhojoao,2017arXiv170805094M,2017arXiv171100636M,2017arXiv171110566R,2017arXiv171110561R,2017arXiv171009668L, 
2018JCoPh.357..125R,2018arXiv180106637R}, although this field of research is clearly growing at the moment. Nonetheless, this area of 
research is of importance, as most physical systems are spatially extended, e.g. the atmospheric system driving the weather 
\cite{9780521857291}; the solar dynamo driving the Sun's sunspots \cite{9780198512905}; and the influence of sunspots on the Earth's 
magnetic field via the solar wind, coronal mass ejections and solar flares -- the so called space weather 
\cite{1851RSPT..141..123S, 1852RSPT..142..103S, 1979P&SS...27.1001S,1983SoPh...89..195E,1965P&SS...13....9P,2000AdSpR..26...27W,2003A&AT...22..861B,2005GeoRL..3221106S,2005SpWea...3.8C01K,2006GMS...165..367T,2006GeoRL..3318101H, 2009SunGe...4...55C,2011SpWea...9.6001C,2013EGUGA..1510865W,2015SpWea..13..524S}.

The reasons why spatial-temporal chaos is so difficult to forecast are: first the Hausdorff dimension  of the 
attractor \cite{1985PhLA..107..101G} -- usually quite large; and second how to choose the variables to use for forecasting, i.e., is there 
enough information on the same point back in time to derive the future of that particular point, or do spatial correlations and spatial 
propagation affect it in a way that one must take into account some spatial and temporal neighbours set to forecast the future, and if 
so, can that set be defined and how can it be constructed? It is this last question that we investigate in this article, in the 
particular context of spatial-temporal forecasting using neural networks. Feature extraction and the design of the architecture of the 
input layer for a neural network is considered to be an art form, relying mostly on trial and error and domain knowledge. For 
forecasting of time series a simple approach consists of designing the input layer as a vector of previous data using a time delay, the 
time delay neural network method \cite{Waibel:1990:PRU:108235.108263, luk2000study, Frank2001, OH2002249, 1009-1963-12-6-304, 
inputlayer}. For spatial-temporal series, one can generalize it to include temporal and spatial delays 
\cite{covas2016,covaspeixinhojoao}. This is where the connection to dynamical systems can be useful. In 1981, Takens established the 
theoretical background \cite{1981LNM...898..366T} in the Takens embedding theorem for a mathemmatical method to reconstruct the dynamics 
of the underlying attractor of a chaotic dynamical system from a time ordered sequence of data observations. Notice the reconstruction 
conserves the properties of the original dynamical system up to a diffeomorphism. 
 Further developments established a series of  theorems by \cite{key1503303m}, \cite{1981LNM...898..366T, 1981LNM...898..230M} and 
\cite{1991JSP....65..579S}. These theorems serve the basis for a non-linear embedding and forecasting on the original variables. The 
theorems and related articles propose to use a time delay based on the first minima of the {\em mutual information} - see 
\cite{Fraser86, abarbanel1997analysis, opac-b1092652}) and to use the number of points using  the method of {\em method of false 
nearest neighbours} detection suggested by \cite{1992PhRvA..45.3403K} and reported in detail in \cite{1992PhRvA..45.7058M, 
1993RvMP...65.1331A, 1996PhT....49k..86A, abarbanel1997analysis}. Using this theoretical framework, we propose that this non-linear 
embedding method can be used to indicate what the best way is to contruct the input layer for a neural network in both space and time.

In this paper we show empirical evidence for an optimal architecture of the input layer of a neural network which tries
to forecast or predict a spatial-temporal signal. Here we show the empiral evidence for fours particular cases of
two-dimensional data series $s^n_m$, where byy two-dimensional we mean a scalar field which can be
defined by a $N\times M$ matrix with components $s^n_m \in \mathbb{R}$.

\subsection{Neural networks for time-series forecasting}

Describe the use of feedforward artificial neural networks, then recurrent networks and other methods (put all references).
Put all references to pure time only forecasts \cite{McCulloch1943}. DO WE REALLY NEED THIS SECTION?

\subsection{Neural networks for spatial-temporal forecasting}

There has been several attemps to forecast spatial-temporal data series in the literature using neural networks ( ADD MORE REFERENCES 
\cite{2017arXiv170805094M, 2017arXiv171100636M, 2017arXiv171205293C, ghaderi2017deepforecast}). The references above confine themselves 
to  attempts to forecast the actual full scalar field $s^n_m$, as opposed to the ongoing research on pattern recognition in moving 
images (2D and 3D), which attempt to pick particular features in images (e.g. car, pedestrian, bicycle, person) and to forecast where 
those features will be in subsequent images within the particular moving sequence (ADD REFERENCES) or research on word sequences (ADD 
REFERENCES).

\subsection{Input layer architecture for neural networks for spatial-temporal forecasting}

Some authors discuss the use of either mutual information and/or embedding dimension as a constraint on the input layer, i.e. the 
feature selection \cite{annunziato, Gkana201579, Zachilas2015, Sun2010109, HUANG20108590, 298224, Frank2001, 1998GeoRL..25..457K, 
BUHAMRA2003805, chandra2012cooperative, DBLP:journals/corr/MaslennikovaB14, sauter2010spatio, JiangS11, inputlayer, 1997IJMPC...8.1345K, 
1009-1963-12-6-304, Verdes2000, 1996SoPh..168..423F, 2007AdG....10...67L, Chandra:2012:CCE:2181341.2181747, raios, 
maass2003mathematical} used the mutual information and/or the false nearest neighbours' methods to estimate the suitable embedding 
parameters. \cite{0305-4470-28-12-012, 1995ApJ...444..916C, 1998GeoRL..25..457K, Zachilas2015, Chandra:2012:CCE:2181341.2181747, 
Gkana201579, raios} attempted to forecast the solar sunspot number using neural networks and they used the embedding dimension of the 
sunspot time series as way to define the architecture of the input layer. \cite{Simon:2007:HDS:1230147.1230294} generalize the mutual 
information approach to higher dimensions but does not connect it to the problem of the neural network input layer architecture 
optimization. There are also authors \cite{articleRagulskis} that try to use neural networks to determine the optimal embedding and time 
delay for the purposes of local reconstruction of states with a view to forecast (the opposite of what we try to empirically demonstrate 
here). Other authors \cite{Xia2006APF} try to  use Support Vector Machines (SVMs) to forecast in space and time and use time delays and 
embedding approachs to define the states vectors. In fact Parlitz and Merkwirth \cite{Parlitz2000NonlinearPO} in the
ESANN 2000 meeting proceddings paper mentioned that local reconstruction of states ``\ldots may also serve as
a starting point for deriving local mathematical models in terms of polynomials,
radial basis functions or neural networks.''. Here we attempt to show empirical evidence that this is not just
a starting point, but the optimal neural network input architecture. 

All of the references above on neural network forecasting of spatial-temporal data series either use a simple delay based 
architecture for the input layer, or use a time delay based on the first minima of the {\em mutual information} as proposed in 
\cite{Fraser86, abarbanel1997analysis, opac-b1092652}) and/or use the number of points dictated by the embedding theorems by 
\cite{key1503303m}, \cite{1981LNM...898..366T, 1981LNM...898..230M} and \cite{1991JSP....65..579S} using the method of {\em method of 
false nearest neighbours} detection suggested by \cite{1992PhRvA..45.3403K} and reported in detail in \cite{1992PhRvA..45.7058M, 
1993RvMP...65.1331A, 1996PhT....49k..86A, abarbanel1997analysis}. However, as far as we are aware, all the references to using the 
embedding theorems and the related  mutual information method and the false nearest neighbours method seems to be not justified, i.e., 
the approach is explained but not proven either theoretically or empirically. 

\hfill Eurico Covas

\hfill \today

\section{Neural Network architecture}

The neural network architecture we choose to demonstrate our possible conjecture is a form of the basic feedforward neural network,
sometimes called the time-delayed neural network \cite{Waibel:1990:PRU:108235.108263}, trained using the so-called back-propagation
algorithm \cite{10.1007/BFb0006203, 1986Natur.323..533R, 58337}. We focus on spatial-temporal series, so we have extended the usual time-delayed neural network
to be a time and space delayed network. The overall architecture of the network is depicted in detail in Fig.\ \ref{architecture}.

\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{architecture.pdf}}
\caption{Forecasting method illustration. The neural network is made of an input layer, one or more hidden layer(s) and one output layer.
In this article, we use only one hidden layer and the output layer is made of a single neuron. Each input pattern  $x(i)$ is sent to the
input layer, then each of the hidden neurons values is calculated from the sum of the product of the weights by the inputs $\sum w(i,j) x(i)$
and passed via the activation function. Then the output is made by the product of the second set of
weights times the hidden node values $\sum w'(i,j) y(i)$ again passed to another (or the same) activation function.
Each input pattern  $x(i)$ is actually a matrix constructed using an embedding space of
spatial and temporal delays, calculated from the actual physical spatial-temporal data values $s(n,m)$. After many randomly chosen input patterns
are passed via the neural network, the weights hopefully converge to an optimal training value.}
\label{architecture}
\end{figure}

Under this architecture, we use the ideas proposed in \cite{Parlitz2000NonlinearPO} to construct a grid of input values
which are then fed to the neural network to produce a single output, the future state. Formally,
let $n=1,...,N$ and $m=1,...,M$.
Consider a spatial-temporal data series ${\bf s}$ which can be
defined by a $N\times M$ matrix with components
$s^n_m \in \mathbb{R}$. To these components, we will call {\em states} of the spatial-temporal series.
Consider a number $2I\in \mathbb{N}$ of neighbours in space of a given
$s^n_m$ and a number $J\in \mathbb{N}$ of temporal past neighbours relative to $s^n_m$ (see Fig.\ \ref{NeuralNetwork} for details).
For each $s^n_m$, we define the input (feature) vector ${\bf x} (s^n_m)$ with components given by
$s^n_m$, its  $2I$ spatial neighbours and its $J$ past temporal
neighbours, and with $K$ and $L$ being the spatial and temporal lags:
\begin{multline}
{\bf x}(s^n_m)=\{s^n_{m-I K }, ...,s^n_m, ..., s^n_{m+I K},
s^{n-L}_{m-I K},..., s^{n-L}_{m},..., s^{n-L}_{m+I K},
...\\ \cap
s^{n-J L}_{m-I K},..., s^{n-J L}_{m},..., s^{n-J L}_{m+I K} \}
\label{embedding}
\end{multline}

So, the input is a
$(2 I+1)(J+1)$ vector ${\bf x}(s^n_m)$ and the target (output) to train the network is the value $s^{n+1}_{m}$.
In Fig.\ \ref{NeuralNetwork} we show the details of this method. We train the network using stochastic gradient
back-propagation (using momentum and weigh decay for regularization) by running a stochastic batch
where we randomly sample pairs of inputs and outputs from the training set: ${\bf x}(s^n_m)$ and $s^{n+1}_{m}$, respectively. Then at test time
we choose inputs ${\bf x}(s^n_m)$, such that $n=N_{train}$, $N_{train}$ being the number of temporal slices on the training set.

\begin{figure*}
\centering
\resizebox{\hsize}{!}{\includegraphics{NeuralNetwork.pdf}}
\caption{Forecasting method illustration. One constructs an embedding space using
space and time delays, then assemble randomly positioned grid input patterns within the training set to pass to the neural network (in this figure
we show 3 randomly selected input patterns).
The input is a
$(2 I+1)(J+1)$ vector ${\bf x}(s^n_m)$ and the target (output) to train the network is the value $s^{n+1}_{m}$.
After training with a sequence
of patterns $p(i), p(i+1), p(i+2), \ldots$ then the patterns adjacent to the forecast set are used to calculate the outputs
to compare against the forecast. To forecast the $n+2$ slice we concatenate the previously predicted $n+1$ and progress accordingly.}
\label{NeuralNetwork}
\end{figure*}

As for the back-propagation hyperparameters, we included an adaptive learning rate $\eta_n=\eta/(1+n/10000)$, where the parameter $\eta$ is the initial learning rate and $\eta_n$ is the learning rate used at time step
$n$, we included a momentum $\alpha$ and a weight decay $\rho$.
In addition we use one hidden layer with $N_h$ nodes. A further  hyperparameter is the choice of the activation function, we use either ReLu or a sigmoid function depending on the
 test case we are working with.
We also normalize the data before passing it throught the neural network, in most cases we scale it in linear fashion
$x \to \alpha_{nor} + x/\beta_{nor}$, and in the case of real physical data as we will see later, we scale it in logarithmic fashion it by $x \to \alpha_{nor}+\frac{\ln(1+x)}{\beta_{nor}}$, where $x$ is the inital data,
and $\alpha_{nor}$ and $\beta_{nor}$ are the arbitrary shift and scaling constants, respectively. For the weight (and bias)
initialization we choose random numbers with a constant distribution between $[0,1]$ and shifted by $\alpha_{rng}$ and scaled
by $\beta_{rng}$. The final hyperparameter is the number of steps taken on the stochastic gradient descent (equivalent to a mini-batch approach of $N_{\textnormal{batch}}=1$) which we denote by $N_{\textnormal{steps}}$.  All of these hyperparameters are calibrated and fixed
before we do any simulations with respect to the parameters $I$, $J$, $K$, $L$, which are autocalibrated by the below mentioned
methods derived from dynamical systems theory. In this sense these are not hyperparameters of the neural network.


We then compare the goodness of fit by first visual inspection and second by numerically calculating the so-called
structural similarity $\textnormal{SSIM}(x,y)$ which has been proposed by \cite{Wang04imagequality} and used already in the context of spatial-temporal forecasting in \cite{covas2016, covaspeixinhojoao}. It has also been used in the context of deep learning used for enhancing
resolution on two dimensional images \cite{2015arXiv150100092D} and restoring missing data in images \cite{2018arXiv180208369Z}. For details on the SSIM measure see \cite{Wang04imagequality,2009ISPM...26...98W, 2012ITIP...21.1488B}.
The SSIM index is an metric quantity used to calculate the perceived quality of digital images and videos.  It
allows two images to be compared and provides a value of their similarity - a value of $\textnormal{SSIM}=1$ corresponds to the case of two
perfectly identical images. We use it by calculating the $\textnormal{SSIM}(x,y)$ between the entire test set and the predicted set.

Although most papers using neural networks for forecasting in time, or in space and time (ADD REFERENCES) may use implicitly
(convolutional ones?) or explicitly (TDNNs) (ADD REFERENCES), as far as the authors are aware there has been no firm evidence,
either theoretical or empirical on how to choose the inputs for a neural network (see however \cite{1555956} who show how
the forecasting error for a pure time series prediction changes with the delay and the number of time delay points used as an input).

Here we propose that the optimal time delay/spatial delays ($L$ and $K$, respectively) must be the ones based on the first minima of the {\em mutual information}
\cite{Fraser86, abarbanel1997analysis, opac-b1092652} and that the optimal number of temporal/spatial points to use
($J$ and $I$, respectively) must be the
ones based on the
 the method of {\em method of false nearest neighbours} detection \cite{1992PhRvA..45.3403K, 1992PhRvA..45.7058M, 1993RvMP...65.1331A, 1996PhT....49k..86A, abarbanel1997analysis}. We conjecture that as any set of architectures ``approach'' this optimal
 architecture, then the $\textnormal{SSIM} \to 1$. In the case of finite training sets and/or noisy training sets $SSIM \to x<1$, where $x$ is the best
 forecast possible given the data set. Visually we believe that the  SSIM versus some reasonable metric constructed to
 represent the distance between any architecture and the optimal
 architecture will show a skewed bell shape as depicted in Fig.\ \ref{conjecture}. In this conjecture we use
 the most obvious candidate to represent the distance between any architecture and the optimal
 architecture, the euclidian distance between the architecture four parameters $d_e=\sqrt{(I-I^*)^2+(J-J^*)^2+(K-K^*)^2+(L-L^*)^2}$.



\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{conjecture-crop.pdf}}
\caption{Our main conjecture. For a infinite noiseless training set, the SSIM approaches $\textnormal{SSIM} \to 1$. For real data sets,
there is a dispersion of the SSIM versus some reasonable metric constructed to
 represent the distance between any architecture (e.g.\ $d_e\sqrt{(I-I^*)^2+(J-J^*)^2+(K-K^*)^2+(L-L^*)^2}$).
}
\label{conjecture}
\end{figure}




% needed in second column of first page if using \IEEEpubid
%\IEEEpubidadjcol

\section{Monte Carlo results}

In order to empirically substantiate our conjecture we take four examples of spatial-temporal series and attempt
to forecast using our feedforward neural network. First we calculate the optimal time delay/spatial delays ($L^*$ and $K^*$, respectively)
using the first minima of the  mutual information optimal number of temporal/spatial points to use
($J^*$ and $I^*$, respectively) using the  method of method of false nearest neighbours.
We then use a Monte Carlo simulation on each one of our four examples, sampling randomly values
of $I$, $J$, $K$, $L$ and calculating the values $d_e(I,J,K,L)$ and $\textnormal{SSIM}(I,J,K,L)$.

We first take a physical system example, a real data example, and then we progress from ``simpler'' systems (maps)
capable of generating spatial-temporal chaos
to more ``complex'' systems (Ordinary Differential Equations - ODEs) to really ``complex'' systems (Partial
Differential Equations - PDEs). This is partially motivated by results in the literature that show
that general universalities are present in different levels of simplification of physical models, from the original PDEs
to truncated ODE expansions (e.g. Galerkin expansions \cite{2001Chaos..11..404C}) to the most extreme simplications such
as maps which capture the essence of the problem. In all cases we take examples with one spatial and one temporal dimension. However we believe
that our conjecture will extend to multiple spatial and one temporal dimensional systems.

\subsection{Sunspot data - a physical system example}

% code and results in
% C:\Users\eurico\SunspotAnalysis\NeuralNetworks\SolarForecastingNeuralNetworks41.xlsm

The first example we take is a physical real data example based on a previous paper of one of us \cite{covaspeixinhojoao}, where a 
neural network using this type of architecture above (Fig.\ \ref{architecture}) was used to forecast sunspot areas $A(t,\theta)$ in our 
Sun in both space ($\theta$ latitude) and time (Carrington Rotation index). 

Several authors \cite{
1990EOSTr..71..677K,
1991PhDT.......158W,
Weigend92HubermanRumelhart,
1993AdSpR..13..447M,
1995JGR...10021735M,
1994VA.....38..351C,
0305-4470-28-12-012,
1995ApJ...444..916C,
Koskela96timeseries,
1996SoPh..168..423F,
1996AnGeo..14...20F,
1996ITNN....7..501P,
1998GeoRL..25..457K,
1998JGR...10329733C,
1998NewAR..42..343C,
Verdes2000,
2004SoPh..221..167V,
2001GMS...125..201L,
2002PhRvE..66f6701S,
2004SoPh..224..247M,
2005JASTP..67..595G,
2004SPIE.5497..542A ,
2005MmSAI..76.1030Q,
2005SoPh..227..177A,
2006JASTP..68.2061M,
2007SoPh..243..253Q ,
2013Ap&SS.344....5A,
xie2006hybrid,
2006SunGe...1a...8M,
2007IJMPC..18.1839E,
1997SPIE.3077..116P,
2006AGUFMSH21A0315L,
2008cosp...37.3467W,
gang2007sunspot,
2009JASTP..71..569U,
2010BAAA...53..241F,
1999BAAA...43...23P,
2011RAA....11..491A,
2010cosp...38.2153A,
2011CRGeo.343..433C,
JiangS11,
2012cosp...39.1194M,
Chandra:2012:CCE:2181341.2181747,
park2009prediction ,
kim2010sunspot,
moghaddam2013sunspot,
2012EPJP..127...43C,
liu2012sunspot,
Gkana201579,
DBLP:conf/ijcnn/ParsapoorBS15,
DBLP:conf/aaai/ParsapoorBS15,
raios
} have already attempted to use neural networks to forecast aspects of the sunspot cycle, although none in both space and time, having 
restricted themselves to using these neural networks to forecast mostly either the sunspot number or the sunspot areas as a function of 
time.  There are a few examples (\cite{covaspeixinhojoao}) of actual spatial-temporal forecasts using neural networks (see mine and the one on 
MDI/HDI data).

We take as a ``training set'' the data from the year 1874 to 
approximately 1997 (i.e.\ the first 1646 Carrington Rotations). We then attempt to reproduce or forecast the sunspot area butterfly 
diagram from Carrington Rotation 1921 to 2162 (the last one corresponding approximately to the year 2015); that is, we use 1646 time 
slices (\textasciitilde  122.92 years) to reproduce the next 242 time slices (\textasciitilde  18.07 years). The training set 
corresponds to around 12 solar cycles (cycle 11 to 22), while the ``forecasting set'' (or validation set) equates to around 1.5 cycles 
(cycle 23 and half of cycle 24). The entire dataset, including the training and forecasting sets, is a grid $x^i_j=x(i,j)$, with 
$i=1888$ and $j=50$. The training set is a grid $x(1646,50)$. For this case the optimal values were $I^*=2$, $J^*=6$, $K^*=9$ and 
$L^*=70$. 
The hyperparameters of the neural network were:
$N_h=70$, $\eta=0.3$, $\alpha=0.01$, $\rho=0$, a logarithmic normalization of the inputs scaled
 with $\alpha_{nor} = 10$ and $\beta_{nor} = 0$, weight initialization with $\alpha_{rng} = 10^{-2}$ and $\beta_{rng} = -0.5$ 
 and $N_{\textnormal{steps}}=$\SI{1000000}. We used the logistic sigmoid function as the activation on both the hidden and output layers.
The results are depicted in Fig.\ \ref{MonteCarloSSIMversusParameterMetricDistance}.  It shows a dispersion as conjectured and 
a convergence to the highest $\textnormal{SSIM}$ value we could obtain for this particular slicing of the training and test sets 
$\textnormal{SSIM}= 0.836876152$.

\begin{figure}[!ht]
\centering
\resizebox{\hsize}{!}{\includegraphics[]{MonteCarloSSIMversusParameterMetricDistance-crop.pdf}}
\caption{Monte Carlo simulation of different architectures of the input layer for the neural network forecast for the sunspot data.
It shows the structural similarity (SSIM) against how far (in an Euclidean space metric) the particular parameters of a particular
run were from the supposely optimal architecture parameters (red dot).}
\label{MonteCarloSSIMversusParameterMetricDistance}
\end{figure}


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% option should be used if it is desired that the figures are to be
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\subsection{Coupled H\'{e}non maps - a discrete-time dynamical system}

% code and results in
% C:\Users\eurico\SunspotAnalysis\SpatialTemporalFeatureSelectionOptimalApproach\KuramotoSivashinsky27_HenonMaps.xlsm

Motivated by having a real case from a physical system, we then tried to investigate if this same conjecture
holds in a very simplifed example of a spatial-temporal model. Coupled maps are widely used as models of spatial-temporal
chaos and pattern/structure formation \cite{1989PThPS..99..263K,1989JSP....54.1489M,9780471937418}.

Following \cite{2000PhRvL..84.1890P,Parlitz2000NonlinearPO} we then take a lattice of $M=100$
coupled H\'{e}non maps:
\begin{IEEEeqnarray}{lCr}
\label{henon}
u_m^{n+1} = 1 - 1.45 \left[ \frac{1}{2} u^n_m + \frac{u_{m-1}^{n}+u_{m+1}^{n}}{4} \right]^2 + 0.3 v_m^n, &&\\
v_m^{n+1} = u_m^n.&& \nonumber
\end{IEEEeqnarray}
with fixed boundary conditions $u^n_1=u_M^n=\frac{1}{2}$ and $v_1^n=v_M^n=0$.

We run the simulation for $N=531$ time steps, and divided the set into $N_{\textnormal{train}}=500$ time steps for the training set
and $N_{\textnormal{test}}=31$ time steps for the test set. The other parameters of the neural network were:
$N_h=10$, $\eta=0.1$, $\alpha=0$, $\rho=0$, a linear input normalization scaling with $\alpha_{nor} = 2.947992$, $\beta_{nor} = 0.515$, $\alpha_{rng} = 10^{-3}$, $\beta_{rng} = -0.5$ and $N_{\textnormal{steps}}=1,000,000$. We used the ReLu function as the activation on both the hidden and output layers.

For this case the optimal values were $I^*=1$, $J^*=3$, $K^*=2$ and $L^*=3$. The results
are depicted in Fig.\ \ref{MonteCarloSSIMversusParameterMetricDistance}.  It shows a dispersion as conjectured and a convergence to the highest $\textnormal{SSIM}$ value we could obtain for this particular slicing of the training and test sets $\textnormal{SSIM}=
0.71139101$.


\begin{figure}[!ht]
\centering
\resizebox{\hsize}{!}{\includegraphics[]{MonteCarloSSIMversusParameterMetricDistance100HenonCoupledMaps-crop.pdf}}
\caption{Monte Carlo simulation of different architectures of the input layer for the neural network forecast for a series of 100 coupled
H\'{e}non maps.
It shows the structural similarity (SSIM) against how far (in an Euclidean space metric) the particular parameters of a particular
run were from the supposely optimal architecture parameters (red dot). The green line (trendline) seems to show that as the parameters
of a randomly choosen architecture get close to the supposely optimal architecture ones, the SSIM converges to what seems to be the
best possible forecast value given the limited dataset.}
\label{MonteCarloSSIMversusParameterMetricDistance100HenonCoupledMaps}
\end{figure}

Results seems to suggest the same structure as depicted in our conjecture diagram and in the previous results for sunspots.

\subsection{Coupled Ordinary Differential Equations - Lorenz-96 model}

% code in
% C:\Users\eurico\SunspotAnalysis\SpatialTemporalFeatureSelectionOptimalApproach\lorenz4D\lorenz4Drun.m
%%% the Lorenz model is: (cyclical)
% dX[j]/dt=(X[j+1]-X[j-2])*X[j-1]-X[j]+F
%J=40;               %the number of variables
%h=0.05;             %the time step
% results in
% C:\Users\eurico\SunspotAnalysis\SpatialTemporalFeatureSelectionOptimalApproach\KuramotoSivashinsky23_Lorenz.xlsm

For the spatially extended ODEs model we used a well known 40 couple ODE dynamical system proposed by Edward Lorenz in 1996 \cite{articleLorenz96}

\begin{equation}{lCr}
\label{lorenz96equations}
\frac{d\,x_j}{d\,t}=\left( x_{j+1} - x_{j-2} \right ) x_{j-1} - x_j + F, j=1, \ldots, N=40,
\end{equation}
where $x_{-1}=x_{N-1}$, $x_0=x_N$ and $x_{N+1}=x_1$. We use the forcing $F=5$ to get some interesting behaviour in space and time. We used a time step $\Delta t=0.05$ and we have integrated this equation using Matlab (ADD REFERENCE).

\begin{figure}[!ht]
\centering
\resizebox{\hsize}{!}{\includegraphics[]{MonteCarloSSIMversusParameterMetricDistanceLorenz96-crop.pdf}}
\caption{Monte Carlo simulation of different architectures of the input layer for the neural network forecast for the 40-ODE Lorenz 96 system.
It shows the structural similarity (SSIM) against how far (in an Euclidean space metric) the particular parameters of a particular
run were from the supposely optimal architecture parameters (red dot). The green line (trendline) seems to show that as the parameters
of a randomly choosen architecture get close to the supposely optimal architecture ones, the SSIM converges to what seems to be the
best possible forecast value given the limited (and noisy) dataset.}
% https://en.wikipedia.org/wiki/Lorenz_96_model
\label{MonteCarloSSIMversusParameterMetricDistanceLorenz96}
\end{figure}

We run the simulation for $N=531$ time steps, and divided the set into $N_{\textnormal{train}}=500$ time steps for the training set
and $N_{\textnormal{test}}=31$ time steps for the test set. The other parameters of the neural network were:
$N_h=10$, $\eta=0.05$, $\alpha=0.001$, $\rho=0$, a linear normalization input scaling with $\alpha_{nor} = 10$ and $\beta_{nor} = 0.430$, weight initialization with $\alpha_{rng} = 10^{-3}$ and $\beta_{rng} = -0.5$ and $N_{\textnormal{steps}}=100,000$. We used the ReLU function as the activation on both the hidden and output layers.

For this case the optimal values obtained before the Monte Carlo simulation were $I^*=2$, $J^*=2$, $K^*=1$ and $L^*=9$. The results
are depicted in Fig.\ \ref{MonteCarloSSIMversusParameterMetricDistanceLorenz96}.  It shows a dispersion as conjectured and a convergence to the highest $\textnormal{SSIM}$ value we could obtain for this particular slicing of the training and test sets $\textnormal{SSIM}=
0.861844038$. Results seems to suggest the same structure as depicted in our conjecture diagram and in the previous results for sunspots
and the coupled H\'{e}non maps.

\subsection{Partial Differential Equations - Kuramoto-Sivashinsky model}

% code in
% C:\Users\eurico\SunspotAnalysis\SpatialTemporalFeatureSelectionOptimalApproach\KSEproject\matlabKSintrinsicSolver.m
% from http://chaosbook.org/extras/KSEproject/html/index.html
% results in
% C:\Users\eurico\SunspotAnalysis\SpatialTemporalFeatureSelectionOptimalApproach\KuramotoSivashinsky28.xlsm

Finally we take a full PDE system, the Kuramoto-Sivashinsky model \cite{1976PThPh..55..356K,1977AcAau...4.1177S},
a very well known system capable of spatiatemporal chaos
and complex spatial-temporal dynamics. It is a fourth-order nonlinear PDE introduced in the 1970s by Yoshiki
Kuramoto and Gregory Sivashinsky to model the diffusive instabilities
in a laminar flame front.

The model is described by the following equation:

\begin{equation}
\label{kuramotoSivashinskyequation}
\frac{\partial u(x,t)}{\partial t} = \frac{\partial^4 u(x,t)}{\partial x^4}+\frac{\partial^2 u(x,t)}{\partial x^2}+u(x,t)
\frac{\partial u(x,t)}{\partial x},
\end{equation}
where $x \in [-\frac{L}{2},+\frac{L}{2}]$. We have integrated this equation using Matlab (ADD REFERENCE)
using a time step of$\Delta t=0.5$ and $L=22$ fourier modes.

\begin{figure}[!ht]
\centering
\resizebox{\hsize}{!}{\includegraphics[]{MonteCarloSSIMversusParameterMetricDistanceKS_L=22-crop.pdf}}
\caption{Monte Carlo simulation of different architectures of the input layer for the neural network forecast for
Kuramoto-Sivashinsky with $L=22$ system.
It shows the structural similarity (SSIM) against how far (in an Euclidean space metric) the particular parameters of a particular
run were from the supposely optimal architecture parameters (red dot). The green line (trendline) seems to show that as the parameters
of a randomly choosen architecture get close to the supposely optimal architecture ones, the SSIM converges to what seems to be the
best possible forecast value given the limited (and noisy) dataset.}
\label{MonteCarloSSIMversusParameterMetricDistanceKS_L=22}
\end{figure}

We run the simulation for $N=531$ time steps, and divided the set into $N_{\textnormal{train}}=500$ time steps for the training set
and $N_{\textnormal{test}}=31$ time steps for the test set. The other parameters of the neural network were:
$N_h=50$, $\eta=0.1$, $\alpha=0$, $\rho=0$, a linear normalization input scaling with 
$\alpha_{nor} = 5.8472$ and $\beta_{nor} =0.5$, weight initialization 
with $\alpha_{rng} = 10^{-3}$ and $\beta_{rng} = -0.5$ and $N_{\textnormal{steps}}=1,000,000$. 
We used the ReLU function as the activation on both the hidden and output layers.

The results of the Monte Carlo simulation can be see below in Fig.\ \ref{MonteCarloSSIMversusParameterMetricDistanceKS_L=22}.
For this case the optimal values obtained before the Monte Carlo simulation were $I^*=1$, $J^*=2$, $K^*=2$ and $L^*=39$. It shows a dispersion as conjectured and a convergence to the highest $\textnormal{SSIM}$ value we could obtain for this particular slicing of the training and test sets a surprising high value of $\textnormal{SSIM}=0.990264382$. Results seems to suggest the same structure as depicted in our conjecture diagram and in the previous results for sunspots, the coupled H\'{e}non maps.



% Note that the IEEE typically puts floats only at the top, even when this
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\section{Conclusion}
We have shown empiral evidence for the existence of an optimal architecture of the input layer of feedforward neural networks used for 
forecasting spatial-temporal series. We believe that the selection of the features  of the input layer can be uniquely determined by 
the data itself, using two techniques from dynamical systems  embedding theory: the mutual information and the false neighbours 
methods. The former procedure determines the temporal and spatial delays to take when selecting features, while the latter procedure 
determines the number of data points in space and time to take as inputs.  We conjecture that this optimal architecture gives the best 
forecast, as measured by a standard image similarity index and  imply that the shape of the dispersion of points on a Monte Carlo 
simulation across all possible architectures in the forecast similarity versus distance to optimal architecture was a skewed bell  
shape with the highest value being the optimal architecture/maximum similarity index.
  
In order to substantiate our conjecture, we choose four independently related systems, in order of complexity: a set of spatially 
extended coupled maps; a set of spatially extended ODEs; a one-dimensional spatial PDE and a real spatial-temporal data set from 
sunspots areas in our Sun. In all four cases, we were able to first use the mutual information and  the false neighbours methods to 
determine the four parameters defining the input layer architecture\footnote{We have four parameters for the input layer architecture 
in these cases with one temporal and one spatial dimension. For higher dimensional systems, there will be more parameters, the exact 
number being double the number of dimensions of the system.}. After calibration of the hyperparameters we then were able to forecast 
reasonably the test test. We then show that for a Monte Carlo simulation across all possible architectures, the neural network did not, 
as expected, forecast as well as for the specific set of four parameters defining the input layer architecture. The Monte Carlo 
simulations show that the shape of the distribution of points in the forecast similarity versus distance to optimal architecture was a 
skewed bell  shape with the highest value being the optimal architecture/maximum similarity index (subject to noise), as conjectured.

Given how important spatial-temporal systems are and how we want to forecast the future as accurately as possible, it is quite 
important to attempt to reduce the number of hyperparameters in neural network prediction, and try to constraint the architecture from 
the data properties only. If indeed our conjecture turns out to be true, it would remove the input layer architecture as another free 
variable in the already complex process of choosing the details of the neural network to use for forecasting.

In this paper we have focused first and foremost in establishing empirical evidence for our conjecture, within a simple framework of 
feedforward neural networks with one hidden layer for the purpose of prediction in one spatial and one temporal dimensions. Naturally, 
there are many clear extensions to our research. First to include more complex architectures, with deeper networks with more hidden 
layers, to possible tackle systems which are hyperchaotic. Second, to attempt to extend the conjecture with empirical evidence in high 
dimensions, e.g. 3+1 dimensional weather systems. Third, to extend the conjecture to more complex and fashionable
neural networks models, such as recurrent neural networks, particularly echo state networks and long short-term memory networks.
 Fourth and last but not least, to prove the conjecture (HOW? Markov Chains?).

% if have a single appendix:
%\appendix[Proof of the Zonklar Equations]
% or
%\appendix  % for no appendix heading
% do not use \section anymore after \appendix, only \section*
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%\appendices
%\section{Proof of the First Zonklar Equation}
%Appendix one text goes here.

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% if you want by leaving the argument blank
%\section{}
%Appendix two text goes here.


% use section* for acknowledgment
\section*{Acknowledgment}
We would like to thank Prof. Reza Tavakol from Queen Mary University London for very useful discussions on forecasting. We also thank
Dr.\ David Hathaway from NASA's Ames Research Center for providing the sunspot data on which some of the results in this article are based upon.
CITEUC is funded by National Funds through FCT - Foundation for Science
and Technology (project: UID/Multi/00611/2013) and FEDER - European
Regional Development Fund through
COMPETE 2020 - Operational Programme Competitiveness and
Internationalization (project: POCI-01-0145-FEDER-006922).

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