X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/hdrcouchot.git/blobdiff_plain/1863b8b84356fa645dafb42dc9fe4028d825e54f..fa0f3e5db965380aa9a72b5098cd11cfee161c6d:/chaosANN.tex?ds=inline diff --git a/chaosANN.tex b/chaosANN.tex index cdcd912..239251e 100644 --- a/chaosANN.tex +++ b/chaosANN.tex @@ -158,402 +158,272 @@ de vérifier si le graphe d'itérations $\textsc{giu}(f)$ est fortement connexe. -\section{Un réseau de neurones peut-il approximer un +\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{Representing Chaotic Iterations for Neural Networks} +\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 +à apprendre. + + +\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}