il produit exactement les même sorties que les itérations de $F_{f_u}$ avec une
condition initiale $\left((x_1^0,\dots, x_n^0),S\right) \in \mathds{B}^n \times [n]^{\mathds{N}}$.
Les itérations de $F_{f_u}$
-sont donc un modèle formel de ce genre de réseau de neurones.
-Pour vérifier s'il est chaotique, il suffit ainsi
+sont donc un modèle formel de cette classe de réseau de neurones.
+Pour vérifier si un de ces représentants est chaotique, il suffit ainsi
de vérifier si le graphe d'itérations
$\textsc{giu}(f)$ est fortement connexe.
-\section{Suitability of Feedforward Neural Networks
-for Predicting Chaotic and Non-chaotic Behaviors}
-
-In the context of computer science different topic areas have an
-interest in chaos, as for steganographic
-techniques~\cite{1309431,Zhang2005759}. Steganography consists in
-embedding a secret message within an ordinary one, while the secret
-extraction takes place once at destination. The reverse ({\it i.e.},
-automatically detecting the presence of hidden messages inside media)
-is called steganalysis. Among the deployed strategies inside
-detectors, there are support vectors
-machines~\cite{Qiao:2009:SM:1704555.1704664}, neural
-networks~\cite{10.1109/ICME.2003.1221665,10.1109/CIMSiM.2010.36}, and
-Markov chains~\cite{Sullivan06steganalysisfor}. Most of these
-detectors give quite good results and are rather competitive when
-facing steganographic tools. However, to the best of our knowledge
-none of the considered information hiding schemes fulfills the Devaney
-definition of chaos~\cite{Devaney}. Indeed, one can wonder whether
-detectors continue to give good results when facing truly chaotic
-schemes. More generally, there remains the open problem of deciding
-whether artificial intelligence is suitable for predicting topological
-chaotic behaviors.
-
-\subsection{Representing Chaotic Iterations for Neural Networks}
+\section{Un réseau de neurones peut-il approximer
+des itération unaires chaotiques?}
+
+Cette section s'intéresse à étudier le comportement d'un réseau de neurones
+face à des itérations unaires chaotiques, comme définies à
+la section~\ref{sec:TIPE12}.
+Plus précésment, on considère dans cette partie une fonction dont le graphe
+des itérations unaires est fortement connexe et une séquence dans
+$[n]^{\mathds{N}}$. On cherche à construire un réseau de neurones
+qui approximerait les itérations de la fonction $G_{f_u}$ comme définie
+à l'équation~(\ref{eq:sch:unaire}).
+
+Sans perte de généralité, on considère dans ce qui suit une instance
+de de fonction à quatre éléments.
+
+\subsection{Construction du réseau}
\label{section:translation}
-The problem of deciding whether classical feedforward ANNs are
-suitable to approximate topological chaotic iterations may then be
-reduced to evaluate such neural networks on iterations of functions
-with Strongly Connected Component (SCC)~graph of iterations. To
-compare with non-chaotic iterations, the experiments detailed in the
-following sections are carried out using both kinds of function
-(chaotic and non-chaotic). Let us emphasize on the difference between
-this kind of neural networks and the Chaotic Iterations based
-multilayer peceptron.
-
-We are then left to compute two disjoint function sets that contain
-either functions with topological chaos properties or not, depending
-on the strong connectivity of their iterations graph. This can be
-achieved for instance by removing a set of edges from the iteration
-graph $\Gamma(f_0)$ of the vectorial negation function~$f_0$. One can
-deduce whether a function verifies the topological chaos property or
-not by checking the strong connectivity of the resulting graph of
-iterations.
-
-For instance let us consider the functions $f$ and $g$ from $\Bool^4$
-to $\Bool^4$ respectively defined by the following lists:
-$$[0, 0, 2, 3, 13, 13, 6, 3, 8, 9, 10, 11, 8, 13, 14,
- 15]$$ $$\mbox{and } [11, 14, 13, 14, 11, 10, 1, 8, 7, 6, 5, 4, 3, 2,
- 1, 0] \enspace.$$ In other words, the image of $0011$ by $g$ is
-$1110$: it is obtained as the binary value of the fourth element in
-the second list (namely~14). It is not hard to verify that
-$\Gamma(f)$ is not SCC (\textit{e.g.}, $f(1111)$ is $1111$) whereas
-$\Gamma(g)$ is. The remaining of this section shows how to translate
-iterations of such functions into a model amenable to be learned by an
-ANN. Formally, input and output vectors are pairs~$((S^t)^{t \in
- \Nats},x)$ and $\left(\sigma((S^t)^{t \in
- \Nats}),F_{f}(S^0,x)\right)$ as defined in~Eq.~(\ref{eq:Gf}).
-
-Firstly, let us focus on how to memorize configurations. Two distinct
-translations are proposed. In the first case, we take one input in
-$\Bool$ per component; in the second case, configurations are
-memorized as natural numbers. A coarse attempt to memorize
-configuration as natural number could consist in labeling each
-configuration with its translation into decimal numeral system.
-However, such a representation induces too many changes between a
-configuration labeled by a power of two and its direct previous
-configuration: for instance, 16~(10000) and 15~(01111) are close in a
-decimal ordering, but their Hamming distance is 5. This is why Gray
-codes~\cite{Gray47} have been preferred.
-
-Secondly, let us detail how to deal with strategies. Obviously, it is
-not possible to translate in a finite way an infinite strategy, even
-if both $(S^t)^{t \in \Nats}$ and $\sigma((S^t)^{t \in \Nats})$ belong
-to $\{1,\ldots,n\}^{\Nats}$. Input strategies are then reduced to
-have a length of size $l \in \llbracket 2,k\rrbracket$, where $k$ is a
-parameter of the evaluation. Notice that $l$ is greater than or equal
-to $2$ since we do not want the shift $\sigma$~function to return an
-empty strategy. Strategies are memorized as natural numbers expressed
-in base $n+1$. At each iteration, either none or one component is
-modified (among the $n$ components) leading to a radix with $n+1$
-entries. Finally, we give an other input, namely $m \in \llbracket
-1,l-1\rrbracket$, which is the number of successive iterations that
-are applied starting from $x$. Outputs are translated with the same
-rules.
-
-To address the complexity issue of the problem, let us compute the
-size of the data set an ANN has to deal with. Each input vector of an
-input-output pair is composed of a configuration~$x$, an excerpt $S$
-of the strategy to iterate of size $l \in \llbracket 2, k\rrbracket$,
-and a number $m \in \llbracket 1, l-1\rrbracket$ of iterations that
-are executed.
-
-Firstly, there are $2^n$ configurations $x$, with $n^l$ strategies of
-size $l$ for each of them. Secondly, for a given configuration there
-are $\omega = 1 \times n^2 + 2 \times n^3 + \ldots+ (k-1) \times n^k$
-ways of writing the pair $(m,S)$. Furthermore, it is not hard to
-establish that
+On considère par exemple les deux fonctions $f$ and $g$ de $\Bool^4$
+dans $\Bool^4$ définies par:
+
+\begin{eqnarray*}
+f(x_1,x_2,x_3,x_4) &= &
+(x_1(x_2+x_4)+ \overline{x_2}x_3\overline{x_4},
+x_2,
+x_3(\overline{x_1}.\overline{x_4}+x_2x_4+x_1\overline{x_2}),
+x_4+\overline{x_2}x_3) \\
+g(x_1,x_2,x_3,x_4) &= &
+(\overline{x_1},
+\overline{x_2}+ x_1.\overline{x_3}.\overline{x_4},
+\overline{x_3}(x_1 + x_2+x_4),
+\overline{x_4}(x_1 + \overline{x_2}+\overline{x_3}))
+\end{eqnarray*}
+On peut vérifier facilement que le graphe $\textsc{giu}(f)$
+n'est pas fortement connexe car $(1,1,1,1)$ est un point fixe de $f$
+tandis que le graphe $\textsc{giu}(g)$ l'est.
+
+L'entrée du réseau est une paire de la forme
+$(x,(S^t)^{t \in \Nats})$ et sa sortie correspondante est
+de la forme $\left(F_{h_u}(S^0,x), \sigma((S^t)^{t \in
+ \Nats})\right)$ comme définie à l'équation~(\ref{eq:sch:unaire}).
+
+On s'intéresse d'abord aux différentes manières de
+mémoriser des configurations. On en considère deux principalement.
+Dans le premier cas, on considère une entrée booléenne par élément
+tandis que dans le second cas, les configurations sont mémorisées comme
+des entiers naturels. Dans ce dernier cas, une approche naïve pourrait
+consister à attribuer à chaque configuration de $\Bool^n$
+l'entier naturel naturel correspondant.
+Cependant, une telle représentation rapproche
+arbitrairement des configurations diamétralement
+opposées dans le $n$-cube comme une puissance de
+deux et la configuration immédiatement précédente: 10000 serait modélisée
+par 16 et et 01111 par 15 alros que leur distance de Hamming est 15.
+De manière similaire, ce codage éloigne des configurations qui sont
+très proches: par exemple 10000 et 00000 ont une distance de Hamming
+de 1 et sont respectivement représentées par 16 et 0.
+Pour ces raisons, le codage retenu est celui des codes de Gray~\cite{Gray47}.
+
+Concentrons nous sur la traduction de la stratégie.
+Il n'est naturellement pas possible de traduire une stragtégie
+infinie quelconque à l'aide d'un nombre fini d'éléments.
+On se restreint donc à des stratégies de taille
+$l \in \llbracket 2,k\rrbracket$, où $k$ est un parametre défini
+initialement.
+Chaque stratégie est mémorisée comme un entier naturel exprimé en base
+$n+1$: à chaque itération, soit aucun élément n'est modifié, soit un
+élément l'est.
+Enfin, on donne une dernière entrée: $m \in \llbracket
+1,l-1\rrbracket$, qui est le nombre d'itérations successives que l'on applique
+en commençant à $x$.
+Les sorties (stratégies et configurations) sont mémorisées
+selon les mêmes règles.
+
+Concentrons nous sur la complexité du problèmew.
+Chaque entrée, de l'entrée-sortie de l'outil est un triplet
+composé d'une configuration $x$, d'un extrait $S$ de la stratégie à
+itérer de taille $l \in \llbracket 2, k\rrbracket$ et d'un nombre $m \in \llbracket 1, l-1\rrbracket$ d'itérations à exécuter.
+Il y a $2^n$ configurations $x$ et $n^l$ stratégies de
+taille $l$.
+De plus, pour une configuration donnée, il y a
+$\omega = 1 \times n^2 + 2 \times n^3 + \ldots+ (k-1) \times n^k$
+manières d'écrire le couple $(m,S)$. Il n'est pas difficile d'établir que
\begin{equation}
\displaystyle{(n-1) \times \omega = (k-1)\times n^{k+1} - \sum_{i=2}^k n^i} \nonumber
\end{equation}
-then
+donc
\begin{equation}
\omega =
\dfrac{(k-1)\times n^{k+1}}{n-1} - \dfrac{n^{k+1}-n^2}{(n-1)^2} \enspace . \nonumber
\end{equation}
-\noindent And then, finally, the number of input-output pairs for our
-ANNs is
+\noindent
+Ainsi le nombre de paire d'entrée-sortie pour les réseaux de neurones considérés
+est
$$
2^n \times \left(\dfrac{(k-1)\times n^{k+1}}{n-1} - \dfrac{n^{k+1}-n^2}{(n-1)^2}\right) \enspace .
$$
-For instance, for $4$ binary components and a strategy of at most
-$3$~terms we obtain 2304~input-output pairs.
+Par exemple, pour $4$ éléments binaires et une stratégie d'au plus
+$3$~termes on obtient 2304 couples d'entrée-sorties.
-\subsection{Experiments}
+\subsection{Expérimentations}
\label{section:experiments}
+On se focalise dans cette section sur l'entraînement d'un perceptron
+multi-couche pour apprendre des itérations chaotiques. Ce type de réseau
+ayant déjà été évalué avec succès dans la prédiction de
+séries chaotiques temporelles. En effet, les auteurs de~\cite{dalkiran10}
+ont montré qu'un MLP pouvait apprendre la dynamique du circuit de Chua.
+Ce réseau avec rétropropagation est composé de deux couches
+et entrainé à l'aide d'une propagation arrière Bayesienne.
+
+Le choix de l'achitecture du réseau ainsi que de la méthode d'apprentissage
+ont été détaillé dans~\cite{bcgs12:ij}.
+En pratique, nous avons considéré des configurations de
+quatre éléments booléens
+et une stratégie fixe de longueur 3.
+Pour le premier codage, nous avons ainsi 6~entrées et 5~sorties
+tandis que pour le second, uniquement 3 entrées et 2 sorties.
+Cela engendre ainsi 2304~combinaisons possibles comme détaillé à la
+section précédente.
+
-To study if chaotic iterations can be predicted, we choose to train
-the multilayer perceptron. As stated before, this kind of network is
-in particular well-known for its universal approximation property
-\cite{Cybenko89,DBLP:journals/nn/HornikSW89}. Furthermore, MLPs have
-been already considered for chaotic time series prediction. For
-example, in~\cite{dalkiran10} the authors have shown that a
-feedforward MLP with two hidden layers, and trained with Bayesian
-Regulation back-propagation, can learn successfully the dynamics of
-Chua's circuit.
-
-In these experiments we consider MLPs having one hidden layer of
-sigmoidal neurons and output neurons with a linear activation
-function. They are trained using the Limited-memory
-Broyden-Fletcher-Goldfarb-Shanno quasi-newton algorithm in combination
-with the Wolfe linear search. The training process is performed until
-a maximum number of epochs is reached. To prevent overfitting and to
-estimate the generalization performance we use holdout validation by
-splitting the data set into learning, validation, and test subsets.
-These subsets are obtained through random selection such that their
-respective size represents 65\%, 10\%, and 25\% of the whole data set.
-
-Several neural networks are trained for both iterations coding
-schemes. In both cases iterations have the following layout:
-configurations of four components and strategies with at most three
-terms. Thus, for the first coding scheme a data set pair is composed
-of 6~inputs and 5~outputs, while for the second one it is respectively
-3~inputs and 2~outputs. As noticed at the end of the previous section,
-this leads to data sets that consist of 2304~pairs. The networks
-differ in the size of the hidden layer and the maximum number of
-training epochs. We remember that to evaluate the ability of neural
-networks to predict a chaotic behavior for each coding scheme, the
-trainings of two data sets, one of them describing chaotic iterations,
-are compared.
-
-Thereafter we give, for the different learning setups and data sets,
-the mean prediction success rate obtained for each output. Such a rate
-represents the percentage of input-output pairs belonging to the test
-subset for which the corresponding output value was correctly
-predicted. These values are computed considering 10~trainings with
-random subsets construction, weights and biases initialization.
-Firstly, neural networks having 10 and 25~hidden neurons are trained,
-with a maximum number of epochs that takes its value in
-$\{125,250,500\}$ (see Tables~\ref{tab1} and \ref{tab2}). Secondly,
-we refine the second coding scheme by splitting the output vector such
-that each output is learned by a specific neural network
-(Table~\ref{tab3}). In this last case, we increase the size of the
-hidden layer up to 40~neurons and we consider larger number of epochs.
\begin{table}[htbp!]
-\caption{Prediction success rates for configurations expressed as boolean vectors.}
-\label{tab1}
\centering {\small
\begin{tabular}{|c|c||c|c|c|}
\hline
-\multicolumn{5}{|c|}{Networks topology: 6~inputs, 5~outputs, and one hidden layer} \\
+\multicolumn{5}{|c|}{Topologie du réseau: 6~entrées, 5~sorties, 1~couche cachée} \\
\hline
\hline
-\multicolumn{2}{|c||}{Hidden neurons} & \multicolumn{3}{c|}{10 neurons} \\
+\multicolumn{2}{|c||}{Neurones cachés} & \multicolumn{3}{c|}{10 neurones} \\
\cline{3-5}
\multicolumn{2}{|c||}{Epochs} & 125 & 250 & 500 \\
\hline
-\multirow{6}{*}{\rotatebox{90}{Chaotic}}&Output~(1) & 90.92\% & 91.75\% & 91.82\% \\
-& Output~(2) & 69.32\% & 78.46\% & 82.15\% \\
-& Output~(3) & 68.47\% & 78.49\% & 82.22\% \\
-& Output~(4) & 91.53\% & 92.37\% & 93.4\% \\
+\multirow{6}{*}{\rotatebox{90}{Chaotique $g$ }}&Entrée~(1) & 90.92\% & 91.75\% & 91.82\% \\
+& Entrée~(2) & 69.32\% & 78.46\% & 82.15\% \\
+& Entrée~(3) & 68.47\% & 78.49\% & 82.22\% \\
+& Entrée~(4) & 91.53\% & 92.37\% & 93.4\% \\
& Config. & 36.10\% & 51.35\% & 56.85\% \\
-& Strategy~(5) & 1.91\% & 3.38\% & 2.43\% \\
+& Stratégie~(5) & 1.91\% & 3.38\% & 2.43\% \\
\hline
-\multirow{6}{*}{\rotatebox{90}{Non-chaotic}}&Output~(1) & 97.64\% & 98.10\% & 98.20\% \\
-& Output~(2) & 95.15\% & 95.39\% & 95.46\% \\
-& Output~(3) & 100\% & 100\% & 100\% \\
-& Output~(4) & 97.47\% & 97.90\% & 97.99\% \\
+\multirow{6}{*}{\rotatebox{90}{Non-chaotic $f$}}&Entrée~(1) & 97.64\% & 98.10\% & 98.20\% \\
+& Entrée~(2) & 95.15\% & 95.39\% & 95.46\% \\
+& Entrée~(3) & 100\% & 100\% & 100\% \\
+& Entrée~(4) & 97.47\% & 97.90\% & 97.99\% \\
& Config. & 90.52\% & 91.59\% & 91.73\% \\
-& Strategy~(5) & 3.41\% & 3.40\% & 3.47\% \\
+& Stratégie~(5) & 3.41\% & 3.40\% & 3.47\% \\
\hline
\hline
-\multicolumn{2}{|c||}{Hidden neurons} & \multicolumn{3}{c|}{25 neurons} \\ %& \multicolumn{3}{|c|}{40 neurons} \\
-\cline{3-5}
+\multicolumn{2}{|c||}{Neurones cachés} & \multicolumn{3}{c|}{25 neurones} \\
+\cline{3-5} \\%& \multicolumn{3}{|c|}{40 neurons} \\
\multicolumn{2}{|c||}{Epochs} & 125 & 250 & 500 \\ %& 125 & 250 & 500 \\
\hline
-\multirow{6}{*}{\rotatebox{90}{Chaotic}}&Output~(1) & 91.65\% & 92.69\% & 93.93\% \\ %& 91.94\% & 92.89\% & 94.00\% \\
-& Output~(2) & 72.06\% & 88.46\% & 90.5\% \\ %& 74.97\% & 89.83\% & 91.14\% \\
-& Output~(3) & 79.19\% & 89.83\% & 91.59\% \\ %& 76.69\% & 89.58\% & 91.84\% \\
-& Output~(4) & 91.61\% & 92.34\% & 93.47\% \\% & 82.77\% & 92.93\% & 93.48\% \\
+\multirow{6}{*}{\rotatebox{90}{Chaotique $g$}}&Entrée~(1) & 91.65\% & 92.69\% & 93.93\% \\ %& 91.94\% & 92.89\% & 94.00\% \\
+& Entrée~(2) & 72.06\% & 88.46\% & 90.5\% \\ %& 74.97\% & 89.83\% & 91.14\% \\
+& Entrée~(3) & 79.19\% & 89.83\% & 91.59\% \\ %& 76.69\% & 89.58\% & 91.84\% \\
+& Entrée~(4) & 91.61\% & 92.34\% & 93.47\% \\% & 82.77\% & 92.93\% & 93.48\% \\
& Config. & 48.82\% & 67.80\% & 70.97\% \\%& 49.46\% & 68.94\% & 71.11\% \\
-& Strategy~(5) & 2.62\% & 3.43\% & 3.78\% \\% & 3.10\% & 3.10\% & 3.03\% \\
+& Stratégie~(5) & 2.62\% & 3.43\% & 3.78\% \\% & 3.10\% & 3.10\% & 3.03\% \\
\hline
-\multirow{6}{*}{\rotatebox{90}{Non-chaotic}}&Output~(1) & 97.87\% & 97.99\% & 98.03\% \\ %& 98.16\% \\
-& Output~(2) & 95.46\% & 95.84\% & 96.75\% \\ % & 97.4\% \\
-& Output~(3) & 100\% & 100\% & 100\% \\%& 100\% \\
-& Output~(4) & 97.77\% & 97.82\% & 98.06\% \\%& 98.31\% \\
+\multirow{6}{*}{\rotatebox{90}{Non-chaotique $f$}}&Entrée~(1) & 97.87\% & 97.99\% & 98.03\% \\ %& 98.16\% \\
+& Entrée~(2) & 95.46\% & 95.84\% & 96.75\% \\ % & 97.4\% \\
+& Entrée~(3) & 100\% & 100\% & 100\% \\%& 100\% \\
+& Entrée~(4) & 97.77\% & 97.82\% & 98.06\% \\%& 98.31\% \\
& Config. & 91.36\% & 91.99\% & 93.03\% \\%& 93.98\% \\
-& Strategy~(5) & 3.37\% & 3.44\% & 3.29\% \\%& 3.23\% \\
+& Stratégie~(5) & 3.37\% & 3.44\% & 3.29\% \\%& 3.23\% \\
\hline
\end{tabular}
}
+\caption{Taux de prédiction lorsque les configurations sont exprimées comme un vecteur booléen.}
+\label{tab1}
\end{table}
+Le tableau~\ref{tab1} synthétise les résultats obtenus avec le premier
+codage. Sans surprise, la précision de la prédiction croit
+avec l'Epoch et le nombre de neurones sur la couche cachée.
+Dans tous les cas, les résultats sont plus précis dans le cas non
+chaotique que dans l'autre. Enfin, le réseau ne parvient jamais à
+apprendre le comportement de la stratégie.
-Table~\ref{tab1} presents the rates obtained for the first coding
-scheme. For the chaotic data, it can be seen that as expected
-configuration prediction becomes better when the number of hidden
-neurons and maximum epochs increases: an improvement by a factor two
-is observed (from 36.10\% for 10~neurons and 125~epochs to 70.97\% for
-25~neurons and 500~epochs). We also notice that the learning of
-outputs~(2) and~(3) is more difficult. Conversely, for the
-non-chaotic case the simplest training setup is enough to predict
-configurations. For all these feedforward network topologies and all
-outputs the obtained results for the non-chaotic case outperform the
-chaotic ones. Finally, the rates for the strategies show that the
-different feedforward networks are unable to learn them.
-
-For the second coding scheme (\textit{i.e.}, with Gray Codes)
-Table~\ref{tab2} shows that any network learns about five times more
-non-chaotic configurations than chaotic ones. As in the previous
-scheme, the strategies cannot be predicted.
-Figures~\ref{Fig:chaotic_predictions} and
-\ref{Fig:non-chaotic_predictions} present the predictions given by two
-feedforward multilayer perceptrons that were respectively trained to
-learn chaotic and non-chaotic data, using the second coding scheme.
-Each figure shows for each sample of the test subset (577~samples,
-representing 25\% of the 2304~samples) the configuration that should
-have been predicted and the one given by the multilayer perceptron. It
-can be seen that for the chaotic data the predictions are far away
-from the expected configurations. Obviously, the better predictions
-for the non-chaotic data reflect their regularity.
-
-Let us now compare the two coding schemes. Firstly, the second scheme
-disturbs the learning process. In fact in this scheme the
-configuration is always expressed as a natural number, whereas in the
-first one the number of inputs follows the increase of the Boolean
-vectors coding configurations. In this latter case, the coding gives a
-finer information on configuration evolution.
\begin{table}[b]
-\caption{Prediction success rates for configurations expressed with Gray code}
-\label{tab2}
\centering
\begin{tabular}{|c|c||c|c|c|}
\hline
-\multicolumn{5}{|c|}{Networks topology: 3~inputs, 2~outputs, and one hidden layer} \\
+\multicolumn{5}{|c|}{Topologie du réseau: 3~entrées, 2~sorties, 1~couche cachée} \\
\hline
\hline
-& Hidden neurons & \multicolumn{3}{c|}{10 neurons} \\
+& Neurones cachés & \multicolumn{3}{c|}{10 neurones} \\
\cline{2-5}
& Epochs & 125 & 250 & 500 \\ %& 1000
\hline
-\multirow{2}{*}{Chaotic}& Config.~(1) & 13.29\% & 13.55\% & 13.08\% \\ %& 12.5\%
-& Strategy~(2) & 0.50\% & 0.52\% & 1.32\% \\ %& 1.42\%
+\multirow{2}{*}{Chaotique $g$}& Config.~(1) & 13.29\% & 13.55\% & 13.08\% \\ %& 12.5\%
+& Stratégie~(2) & 0.50\% & 0.52\% & 1.32\% \\ %& 1.42\%
\hline
-\multirow{2}{*}{Non-Chaotic}&Config.~(1) & 77.12\% & 74.00\% & 72.60\% \\ %& 75.81\%
-& Strategy~(2) & 0.42\% & 0.80\% & 1.16\% \\ %& 1.42\%
+\multirow{2}{*}{Non-Chaotique $f$}&Config.~(1) & 77.12\% & 74.00\% & 72.60\% \\ %& 75.81\%
+& Stratégie~(2) & 0.42\% & 0.80\% & 1.16\% \\ %& 1.42\%
\hline
\hline
-& Hidden neurons & \multicolumn{3}{c|}{25 neurons} \\
+& Neurones cachés& \multicolumn{3}{c|}{25 neurones} \\
\cline{2-5}
& Epochs & 125 & 250 & 500 \\ %& 1000
\hline
-\multirow{2}{*}{Chaotic}& Config.~(1) & 12.27\% & 13.15\% & 13.05\% \\ %& 15.44\%
-& Strategy~(2) & 0.71\% & 0.66\% & 0.88\% \\ %& 1.73\%
+\multirow{2}{*}{Chaotique $g$ }& Config.~(1) & 12.27\% & 13.15\% & 13.05\% \\ %& 15.44\%
+& Stratégie~(2) & 0.71\% & 0.66\% & 0.88\% \\ %& 1.73\%
\hline
-\multirow{2}{*}{Non-Chaotic}&Config.~(1) & 73.60\% & 74.70\% & 75.89\% \\ %& 68.32\%
-& Strategy~(2) & 0.64\% & 0.97\% & 1.23\% \\ %& 1.80\%
+\multirow{2}{*}{Non-Chaotique $f$}&Config.~(1) & 73.60\% & 74.70\% & 75.89\% \\ %& 68.32\%
+& Stratégie~(2) & 0.64\% & 0.97\% & 1.23\% \\ %& 1.80\%
\hline
\end{tabular}
+\caption{Taux de prédiction lorsque les configurations sont exprimées
+à l'aide de codes de Gray.}
+\label{tab2}
\end{table}
-\begin{figure}
- \centering
- \includegraphics[scale=0.5]{images/chaotic_trace2}
- \caption {Second coding scheme - Predictions obtained for a chaotic test subset.}
- \label{Fig:chaotic_predictions}
-\end{figure}
-\begin{figure}
- \centering
- \includegraphics[scale=0.5]{images/non-chaotic_trace2}
- \caption{Second coding scheme - Predictions obtained for a non-chaotic test subset.}
- \label{Fig:non-chaotic_predictions}
+
+Les résultats concernant le second codage (\textit{i.e.}, avec les codes
+de Gray) sont synthétisés dans le tableau~\ref{tab2}. On constate
+que le réseau apprend cinq fois mieux les comportement non chaotiques
+que ceux qui le sont. Ceci est est illustré au travers des
+figures~\ref{Fig:chaotic_predictions} et~\ref{Fig:non-chaotic_predictions}.
+De plus, comme dans le codage précédent, les stratégies ne peuvent pas être
+prédites.
+On constate que ce second codage réduit certe le nombre de sorties, mais est
+largement moins performant que le premier.
+On peut expliquer ceci par le fait
+que ce second codage garantit que deux entiers successifs correspondent
+à deux configurations voisines, \textit{ie.e}, qui ne diffèrent que d'un
+élément.
+La réciproque n'est cependant pas établie et deux configurations voisines
+peuvent être traduitent par des entiers très éloignés et ainsi difficils
+àapprendre.
+
+
+\begin{figure}[ht]
+ \begin{center}
+ \subfigure[Fonction chaotique $g$]{
+ \begin{minipage}{0.48\textwidth}
+ \begin{center}
+ \includegraphics[scale=0.37]{images/chaotic_trace2}
+ \end{center}
+ \end{minipage}
+ \label{Fig:chaotic_predictions}
+ }
+ \subfigure[Fonction non-chaotique $f$]{
+ \begin{minipage}{0.48\textwidth}
+ \begin{center}
+ \includegraphics[scale=0.37]{images/non-chaotic_trace2}
+ \end{center}
+ \end{minipage}
+ \label{Fig:non-chaotic_predictions}
+ }
+ \end{center}
+ \caption {Prédiction lorsque les configurations sont exprimées
+à l'aide de codes de Gray.}
\end{figure}
-Unfortunately, in practical applications the number of components is
-usually unknown. Hence, the first coding scheme cannot be used
-systematically. Therefore, we provide a refinement of the second
-scheme: each output is learned by a different ANN. Table~\ref{tab3}
-presents the results for this approach. In any case, whatever the
-considered feedforward network topologies, the maximum epoch number,
-and the kind of iterations, the configuration success rate is slightly
-improved. Moreover, the strategies predictions rates reach almost
-12\%, whereas in Table~\ref{tab2} they never exceed 1.5\%. Despite of
-this improvement, a long term prediction of chaotic iterations still
-appear to be an open issue.
-
-\begin{table}
-\caption{Prediction success rates for split outputs.}
-\label{tab3}
-\centering
-\begin{tabular}{|c||c|c|c|}
-\hline
-\multicolumn{4}{|c|}{Networks topology: 3~inputs, 1~output, and one hidden layer} \\
-\hline
-\hline
-Epochs & 125 & 250 & 500 \\
-\hline
-\hline
-Chaotic & \multicolumn{3}{c|}{Output = Configuration} \\
-\hline
-10~neurons & 12.39\% & 14.06\% & 14.32\% \\
-25~neurons & 13.00\% & 14.28\% & 14.58\% \\
-40~neurons & 11.58\% & 13.47\% & 14.23\% \\
-\hline
-\hline
-Non chaotic & \multicolumn{3}{c|}{Output = Configuration} \\
-\cline{2-4}
-%Epochs & 125 & 250 & 500 \\
-\hline
-10~neurons & 76.01\% & 74.04\% & 78.16\% \\
-25~neurons & 76.60\% & 72.13\% & 75.96\% \\
-40~neurons & 76.34\% & 75.63\% & 77.50\% \\
-\hline
-\hline
-Chaotic/non chaotic & \multicolumn{3}{c|}{Output = Strategy} \\
-\cline{2-4}
-%Epochs & 125 & 250 & 500 \\
-\hline
-10~neurons & 0.76\% & 0.97\% & 1.21\% \\
-25~neurons & 1.09\% & 0.73\% & 1.79\% \\
-40~neurons & 0.90\% & 1.02\% & 2.15\% \\
-\hline
-\multicolumn{4}{c}{} \\
-\hline
-Epochs & 1000 & 2500 & 5000 \\
-\hline
-\hline
-Chaotic & \multicolumn{3}{c|}{Output = Configuration} \\
-\hline
-10~neurons & 14.51\% & 15.22\% & 15.22\% \\
-25~neurons & 16.95\% & 17.57\% & 18.46\% \\
-40~neurons & 17.73\% & 20.75\% & 22.62\% \\
-\hline
-\hline
-Non chaotic & \multicolumn{3}{c|}{Output = Configuration} \\
-\cline{2-4}
-%Epochs & 1000 & 2500 & 5000 \\
-\hline
-10~neurons & 78.98\% & 80.02\% & 79.97\% \\
-25~neurons & 79.19\% & 81.59\% & 81.53\% \\
-40~neurons & 79.64\% & 81.37\% & 81.37\% \\
-\hline
-\hline
-Chaotic/non chaotic & \multicolumn{3}{c|}{Output = Strategy} \\
-\cline{2-4}
-%Epochs & 1000 & 2500 & 5000 \\
-\hline
-10~neurons & 3.47\% & 9.98\% & 11.66\% \\
-25~neurons & 3.92\% & 8.63\% & 10.09\% \\
-40~neurons & 3.29\% & 7.19\% & 7.18\% \\
-\hline
-\end{tabular}
-\end{table}
\section{Conclusion}
-
-In this paper, we have established an equivalence between chaotic
-iterations, according to the Devaney's definition of chaos, and a
-class of multilayer perceptron neural networks. Firstly, we have
-described how to build a neural network that can be trained to learn a
-given chaotic map function. Secondly, we found a condition that allow
-to check whether the iterations induced by a function are chaotic or
-not, and thus if a chaotic map is obtained. Thanks to this condition
-our approach is not limited to a particular function. In the dual
-case, we show that checking if a neural network is chaotic consists in
-verifying a property on an associated graph, called the graph of
-iterations. These results are valid for recurrent neural networks
-with a particular architecture. However, we believe that a similar
-work can be done for other neural network architectures. Finally, we
-have discovered at least one family of problems with a reasonable
-size, such that artificial neural networks should not be applied in
-the presence of chaos, due to their inability to learn chaotic
-behaviors in this context. Such a consideration is not reduced to a
-theoretical detail: this family of discrete iterations is concretely
-implemented in a new steganographic method \cite{guyeux10ter}. As
-steganographic detectors embed tools like neural networks to
-distinguish between original and stego contents, our studies tend to
-prove that such detectors might be unable to tackle with chaos-based
-information hiding schemes.
-
-In future work we intend to enlarge the comparison between the
-learning of truly chaotic and non-chaotic behaviors. Other
-computational intelligence tools such as support vector machines will
-be investigated too, to discover which tools are the most relevant
-when facing a truly chaotic phenomenon. A comparison between learning
-rate success and prediction quality will be realized. Concrete
-consequences in biology, physics, and computer science security fields
-will then be stated.
-
+Dans ce chapitre, nous avons établi une simlilitude entre les itérations
+chaotiques et une famille de perceptrons multicouches.
+Nous avons d'abord montré comment construire un réseau de neurones
+ayant un comportement chaotique.
+Nous avons présenté ensuite comment vérifier si un réseau de neurones
+établi était chaotique.
+Nous avons enfin montré en pratique qu'il est difficile pour un
+réseau de neurones d'apprendre le comportement global d'itérations
+chaotiques.
% \appendix{}
% \begin{Def} \label{def2}
