Prise en compte des modifs de Christophe et premier jet de la lettre

diff --git a/authors_response.tex b/authors_response.tex
index f6bcb545656fcfd48f3851c977415a7c224717f2..52f85a39172ff5bb9f07dd8c4949e479252ec927 100644 (file)
--- a/authors_response.tex
+++ b/authors_response.tex
@@ -26,9 +26,11 @@ Detailed changes and addressed issues in the revision of the paper
 "Recurrent Neural Networks and Chaos: Construction, \\
 Evaluation, and Prediction Ability" \\
 renamed in \\
 "Recurrent Neural Networks and Chaos: Construction, \\
 Evaluation, and Prediction Ability" \\
 renamed in \\

-"Chaotic Neural Networks: Construction, Evaluation, and Prediction Ability"
+Neural Networks and Chaos: Construction, Evaluation of Chaotic Networks, \\
+and Prediction of Chaos with Multilayer Feedforward Networks"

 
 

-by Jacques M. Bahi, Jean-Fran\c{c}ois Couchot, Christophe Guyeux, and Michel Salomon
+by Jacques M. Bahi, Jean-Fran\c{c}ois Couchot, \\
+Christophe Guyeux, and Michel Salomon

 
 \bigskip
 \end{center}
 
 \bigskip
 \end{center}


@@ -52,7 +54,12 @@ considered and have led to modifications in the paper.
   
   {\it Our response}
 
   
   {\it Our response}
 

-  To be completed
+  Indeed, our  contributions are twofold.  Firstly, we present  how to
+  construct chaotic neural networks, according to Devaney's definition
+  of chaos,  and we discuss  their evaluation. Secondly, we  study the
+  capacity  of  classical  multilayer  feedfoward neural  networks  to
+  predict chaotic data. To highlight clearly these contributions we
+  have changed the title of our paper, as said above.

 \item Please  explain the Line  242( "than chaotic iterations  Ff with
   initial  condition............" ) and  Line 298(  "investigate, when
   comparing neural networks and Devaney's chaos").
 \item Please  explain the Line  242( "than chaotic iterations  Ff with
   initial  condition............" ) and  Line 298(  "investigate, when
   comparing neural networks and Devaney's chaos").


@@ -71,20 +78,28 @@ considered and have led to modifications in the paper.
  
   {\it Our response}
 
  
   {\it Our response}
 

-  To be completed
+  We  have corrected  lines 402-405  (now lines  419-422)  as follows:
+  ``For all  those 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.'' More generally, in
+  the whole paper the term feedforward has been added, when needed, to
+  clarify our discussion.

 \item Please  explain the  meanings of the  percentage in TABLE  I and
   TABLE II.
 
 \item Please  explain the  meanings of the  percentage in TABLE  I and
   TABLE II.
 

- {\it Our response}
+  {\it Our response}

 
 

-  To be completed
+  We  have explained how  the success  rates (percentage)  are computed
+  lines~405-407.

 \item Except  for prediction  success rates, in  order to  reflect the
   prediction ability, please add the analysis of the prediction errors
   for data sequence and diagram it.
 
  {\it Our response}
 
 \item Except  for prediction  success rates, in  order to  reflect the
   prediction ability, please add the analysis of the prediction errors
   for data sequence and diagram it.
 
  {\it Our response}
 

-  To be completed
+ We have added two figures (Figures~3 and 4) and some comments about them
+ lines~425-431. We hope that we have answered the reviewer remark.

 \end{enumerate}
 
 \begin{itemize}
 \end{enumerate}
 
 \begin{itemize}


@@ -107,6 +122,8 @@ considered and have led to modifications in the paper.
 
   {\it Our response}
 
 
   {\it Our response}
 

+  As suggested by the reviewer the conclusion has been shortened and we
+  have presented those definitions differently.

 \end{enumerate}
 
 We are very grateful to the reviewers who, by their recommendations,
 \end{enumerate}
 
 We are very grateful to the reviewers who, by their recommendations,

diff --git a/main.tex b/main.tex
index 016243a10be55ffa732d5119408e6faea4be3f41..49564db89def8615a0ad19f0d6b20ab7b6a7bc4f 100644 (file)
--- a/main.tex
+++ b/main.tex
@@ -88,7 +88,6 @@ far more difficult to learn.
 \maketitle
 
 \begin{quotation}
 \maketitle
 
 \begin{quotation}

-

 Chaotic neural  networks have received a  lot of attention  due to the
 appealing   properties  of   deterministic   chaos  (unpredictability,
 sensitivity,  and so  on). However,  such networks  are  often claimed
 Chaotic neural  networks have received a  lot of attention  due to the
 appealing   properties  of   deterministic   chaos  (unpredictability,
 sensitivity,  and so  on). However,  such networks  are  often claimed


@@ -97,7 +96,7 @@ work  a theoretical  framework based  on the  Devaney's  definition of
 chaos is  introduced.  Starting  with a relationship  between discrete
 iterations  and  Devaney's chaos,  we  firstly  show  how to  build  a
 recurrent  neural network  that is  equivalent  to a  chaotic map  and
 chaos is  introduced.  Starting  with a relationship  between discrete
 iterations  and  Devaney's chaos,  we  firstly  show  how to  build  a
 recurrent  neural network  that is  equivalent  to a  chaotic map  and

-secondly  a way  to check  whether  an already  available network  is
+secondly  a way  to  check  whether an  already  available network  is

 chaotic  or not.  We  also study  different topological  properties of
 these  truly  chaotic neural  networks.   Finally,  we  show that  the
 learning,  with   neural  networks  having   a  classical  feedforward
 chaotic  or not.  We  also study  different topological  properties of
 these  truly  chaotic neural  networks.   Finally,  we  show that  the
 learning,  with   neural  networks  having   a  classical  feedforward


@@ -110,7 +109,7 @@ chaotic maps, is far more difficult than non chaotic behaviors.
 \label{S1}
 
 Several research  works have proposed or used  chaotic neural networks
 \label{S1}
 
 Several research  works have proposed or used  chaotic neural networks

-these last  years.  The  complex dynamics of  such networks  lead to
+these  last years.   The complex  dynamics  of such  networks lead  to

 various       potential      application       areas:      associative
 memories~\cite{Crook2007267}  and  digital  security tools  like  hash
 functions~\cite{Xiao10},                                       digital
 various       potential      application       areas:      associative
 memories~\cite{Crook2007267}  and  digital  security tools  like  hash
 functions~\cite{Xiao10},                                       digital


@@ -136,8 +135,8 @@ are  suitable to model  nonlinear relationships  between data,  due to
 their            universal            approximator            capacity
 \cite{Cybenko89,DBLP:journals/nn/HornikSW89}.   Thus,   this  kind  of
 networks can  be trained  to model a  physical phenomenon known  to be
 their            universal            approximator            capacity
 \cite{Cybenko89,DBLP:journals/nn/HornikSW89}.   Thus,   this  kind  of
 networks can  be trained  to model a  physical phenomenon known  to be

-chaotic such as Chua's circuit \cite{dalkiran10}.  Sometime a neural
-network,  which is build  by combining  transfer functions  and initial
+chaotic such  as Chua's circuit \cite{dalkiran10}.   Sometime a neural
+network, which  is build by  combining transfer functions  and initial

 conditions  that are  both chaotic,  is itself  claimed to  be chaotic
 \cite{springerlink:10.1007/s00521-010-0432-2}.
 
 conditions  that are  both chaotic,  is itself  claimed to  be chaotic
 \cite{springerlink:10.1007/s00521-010-0432-2}.
 


@@ -151,8 +150,8 @@ a  dynamical  system:  ergodicity,   expansivity,  and  so  on.   More
 precisely, in  this paper,  which is an  extension of a  previous work
 \cite{bgs11:ip},   we  establish   the  equivalence   between  chaotic
 iterations  and  a  class  of  globally  recurrent  MLP.   The  second
 precisely, in  this paper,  which is an  extension of a  previous work
 \cite{bgs11:ip},   we  establish   the  equivalence   between  chaotic
 iterations  and  a  class  of  globally  recurrent  MLP.   The  second

-contribution is a  study of the converse problem,  indeed we investigate the
-ability  of classical  multiLayer  perceptrons to  learn a  particular
+contribution is a study of the converse problem, indeed we investigate
+the ability of classical  multilayer perceptrons to learn a particular

 family of discrete chaotic  dynamical systems.  This family is defined
 by a Boolean vector, an update function, and a sequence defining the
 component  to  update  at  each  iteration.  It  has  been  previously
 family of discrete chaotic  dynamical systems.  This family is defined
 by a Boolean vector, an update function, and a sequence defining the
 component  to  update  at  each  iteration.  It  has  been  previously


@@ -266,7 +265,8 @@ Finally,  the  dynamical system  that  iterates  $f$  is {\bf  chaotic
   according  to Devaney}  on $(\mathcal{X},\tau)$  if $f$  is regular,
 topologically transitive, and has  sensitive dependence to its initial
 conditions.   In  what follows,  iterations  are  said  to be  chaotic
   according  to Devaney}  on $(\mathcal{X},\tau)$  if $f$  is regular,
 topologically transitive, and has  sensitive dependence to its initial
 conditions.   In  what follows,  iterations  are  said  to be  chaotic

-(according to Devaney)  when the corresponding  dynamical system  is  chaotic, as it is defined in the Devaney's formulation.
+(according  to Devaney)  when  the corresponding  dynamical system  is
+chaotic, as it is defined in the Devaney's formulation.

 
 %Let us notice that for a  metric space the last condition follows from
 %the two first ones~\cite{Banks92}.
 
 %Let us notice that for a  metric space the last condition follows from
 %the two first ones~\cite{Banks92}.


@@ -373,15 +373,17 @@ X^{k+1}& = & G_{f}(X^{k})\\
 \right.
 \label{eq:Gf}
 \end{equation}
 \right.
 \label{eq:Gf}
 \end{equation}

-where  $\sigma$ is the so-called shift function that  removes the  first term  of the
-strategy  ({\it  i.e.},~$S^0$).    This  definition  allows  to  link
-asynchronous  iterations  with  classical  iterations of  a  dynamical
+where $\sigma$ is the so-called  shift function that removes the first
+term of  the strategy ({\it i.e.},~$S^0$).  This  definition allows to
+link asynchronous iterations with  classical iterations of a dynamical

 system. Note that it can be extended by considering subsets for $S^t$.
 
 To study topological properties of  these iterations, we are then left
 to  introduce a  {\bf distance}  $d$  between two  points $(S,x)$  and
 system. Note that it can be extended by considering subsets for $S^t$.
 
 To study topological properties of  these iterations, we are then left
 to  introduce a  {\bf distance}  $d$  between two  points $(S,x)$  and

-$(\check{S},\check{x})$ in $\mathcal{X} = \llbracket1;n\rrbracket^\Nats
-\times \Bool^{n}$. Let $\Delta(x,y) = 0$ if $x=y$, and $\Delta(x,y) = 1$ else, be a distance on $\mathds{B}$. The distance $d$ is defined by 
+$(\check{S},\check{x})$           in           $\mathcal{X}          =
+\llbracket1;n\rrbracket^\Nats \times \Bool^{n}$. Let $\Delta(x,y) = 0$
+if   $x=y$,  and   $\Delta(x,y)  =   1$   else,  be   a  distance   on
+$\mathds{B}$. The distance $d$ is defined by

 \begin{equation}
 d((S,x);(\check{S},\check{x}))=d_{e}(x,\check{x})+d_{s}(S,\check{S})
 \enspace ,
 \begin{equation}
 d((S,x);(\check{S},\check{x}))=d_{e}(x,\check{x})+d_{s}(S,\check{S})
 \enspace ,


@@ -460,9 +462,9 @@ we obtain  a global recurrent  neural network that behaves  as follows
 \item When  the network  is activated at  the $t^{th}$  iteration, the
   state of the system $x^t  \in \mathds{B}^n$ received from the output
   layer and  the initial  term of the  sequence $(S^t)^{t  \in \Nats}$
 \item When  the network  is activated at  the $t^{th}$  iteration, the
   state of the system $x^t  \in \mathds{B}^n$ received from the output
   layer and  the initial  term of the  sequence $(S^t)^{t  \in \Nats}$

-  (\textit{i.e.}, $S^0  \in \llbracket 1;n\rrbracket$)  are used  to compute  the new
-  output vector.  This  new vector, which represents the  new state of
-  the dynamical system, satisfies:
+  (\textit{i.e.},  $S^0  \in \llbracket  1;n\rrbracket$)  are used  to
+  compute the  new output vector.   This new vector,  which represents
+  the new state of the dynamical system, satisfies:

   \begin{equation}
     x^{t+1}=F_{f_0}(S^0, x^t) \in \mathds{B}^n \enspace .
   \end{equation}
   \begin{equation}
     x^{t+1}=F_{f_0}(S^0, x^t) \in \mathds{B}^n \enspace .
   \end{equation}


@@ -477,19 +479,17 @@ we obtain  a global recurrent  neural network that behaves  as follows
 
 The behavior of the neural network is such that when the initial state
 is  $x^0~\in~\mathds{B}^n$ and  a  sequence $(S^t)^{t  \in \Nats}$  is
 
 The behavior of the neural network is such that when the initial state
 is  $x^0~\in~\mathds{B}^n$ and  a  sequence $(S^t)^{t  \in \Nats}$  is

-given  as outside  input,
-then  the sequence  of  successive published
+given  as outside  input, then  the sequence  of  successive published

 output vectors $\left(x^t\right)^{t \in \mathds{N}^{\ast}}$ is exactly
 the  one produced  by  the chaotic  iterations  formally described  in
 output vectors $\left(x^t\right)^{t \in \mathds{N}^{\ast}}$ is exactly
 the  one produced  by  the chaotic  iterations  formally described  in

-Eq.~(\ref{eq:Gf}).  It  means that  mathematically if we  use similar
+Eq.~(\ref{eq:Gf}).   It means  that mathematically  if we  use similar

 input  vectors   they  both  generate  the   same  successive  outputs
 $\left(x^t\right)^{t \in \mathds{N}^{\ast}}$,  and therefore that they
 are  equivalent  reformulations  of  the iterations  of  $G_{f_0}$  in
 $\mathcal{X}$. Finally, since the  proposed neural network is built to
 input  vectors   they  both  generate  the   same  successive  outputs
 $\left(x^t\right)^{t \in \mathds{N}^{\ast}}$,  and therefore that they
 are  equivalent  reformulations  of  the iterations  of  $G_{f_0}$  in
 $\mathcal{X}$. Finally, since the  proposed neural network is built to

-model  the  behavior  of  $G_{f_0}$,  whose iterations are
-  chaotic  according  to the
-Devaney's definition  of chaos,  we can conclude  that the  network is
-also chaotic in this sense.
+model  the  behavior  of   $G_{f_0}$,  whose  iterations  are  chaotic
+according to the  Devaney's definition of chaos, we  can conclude that
+the network is also chaotic in this sense.

 
 The previous construction scheme  is not restricted to function $f_0$.
 It can  be extended to any function  $f$ such that $G_f$  is a chaotic
 
 The previous construction scheme  is not restricted to function $f_0$.
 It can  be extended to any function  $f$ such that $G_f$  is a chaotic


@@ -526,7 +526,6 @@ to an input one.
   compute the new output one $\left(x^{t+1}_1,\dots,x^{t+1}_n\right)$.
   While  the remaining  input receives  a new  integer value  $S^t \in
   \llbracket1;n\rrbracket$, which is provided by the outside world.
   compute the new output one $\left(x^{t+1}_1,\dots,x^{t+1}_n\right)$.
   While  the remaining  input receives  a new  integer value  $S^t \in
   \llbracket1;n\rrbracket$, which is provided by the outside world.

-\JFC{en dire davantage sur l'outside world}

 \end{itemize}
 
 The topological  behavior of these  particular neural networks  can be
 \end{itemize}
 
 The topological  behavior of these  particular neural networks  can be


@@ -557,7 +556,7 @@ condition  $\left(S,(x_1^0,\dots,  x_n^0)\right)  \in  \llbracket  1;n
 \rrbracket^{\mathds{N}}  \times \mathds{B}^n$.
 Theoretically  speaking, such iterations  of $F_f$  are thus  a formal
 model of these kind of recurrent  neural networks. In the rest of this
 \rrbracket^{\mathds{N}}  \times \mathds{B}^n$.
 Theoretically  speaking, such iterations  of $F_f$  are thus  a formal
 model of these kind of recurrent  neural networks. In the rest of this

-paper,  we will  call such  multilayer perceptrons  CI-MLP($f$), which
+paper, we will call such multilayer perceptrons ``CI-MLP($f$)'', which

 stands for ``Chaotic Iterations based MultiLayer Perceptron''.
 
 Checking  if CI-MLP($f$)  behaves chaotically  according  to Devaney's
 stands for ``Chaotic Iterations based MultiLayer Perceptron''.
 
 Checking  if CI-MLP($f$)  behaves chaotically  according  to Devaney's


@@ -639,7 +638,7 @@ of the output space can be discarded when studying CI-MLPs: this space
 is  intrinsically   complicated  and   it  cannot  be   decomposed  or
 simplified.
 
 is  intrinsically   complicated  and   it  cannot  be   decomposed  or
 simplified.
 

-Furthermore, those  recurrent neural networks  exhibit the instability
+Furthermore, these  recurrent neural networks  exhibit the instability

 property:
 \begin{definition}
 A dynamical  system $\left( \mathcal{X}, f\right)$ is {\bf unstable}
 property:
 \begin{definition}
 A dynamical  system $\left( \mathcal{X}, f\right)$ is {\bf unstable}


@@ -700,7 +699,7 @@ Figure~\ref{Fig:scheme} is a summary  of addressed neural networks and
 chaos  problems.   In  Section~\ref{S2}   we  have  explained  how  to
 construct    a    truly    chaotic    neural   networks,    $A$    for
 instance. Section~\ref{S3} has shown how  to check whether a given MLP
 chaos  problems.   In  Section~\ref{S2}   we  have  explained  how  to
 construct    a    truly    chaotic    neural   networks,    $A$    for
 instance. Section~\ref{S3} has shown how  to check whether a given MLP

-$A$ or $C$ is chaotic or not  in the sens of Devaney, and how to study
+$A$ or $C$ is chaotic or not in the sense of Devaney, and how to study

 its  topological  behavior.   The  last  thing  to  investigate,  when
 comparing neural networks and Devaney's chaos, is to determine whether
 an  artificial neural network  $C$ is  able to  learn or  predict some
 its  topological  behavior.   The  last  thing  to  investigate,  when
 comparing neural networks and Devaney's chaos, is to determine whether
 an  artificial neural network  $C$ is  able to  learn or  predict some


@@ -719,7 +718,7 @@ 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
 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
+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
 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


@@ -826,7 +825,7 @@ $3$~terms we obtain 2304~input-output pairs.
 \label{section:experiments}
 
 To study  if chaotic iterations can  be predicted, we  choose to train
 \label{section:experiments}
 
 To study  if chaotic iterations can  be predicted, we  choose to train

-the multiLayer perceptron.  As stated  before, this kind of network is
+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
 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


@@ -860,10 +859,10 @@ trainings of two data sets, one of them describing chaotic iterations,
 are compared.
 
 Thereafter we give,  for the different learning setups  and data sets,
 are compared.
 
 Thereafter we give,  for the different learning setups  and data sets,

-the mean prediction success rate obtained for each output. A such rate
-represent the  percentage of input-output pairs belonging  to the test
+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
 subset  for  which  the   corresponding  output  value  was  correctly

-predicted.  These values  are computed  considering  10~trainings with
+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
 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


@@ -879,7 +878,7 @@ hidden layer up to 40~neurons and we consider larger number of epochs.
 \centering {\small
 \begin{tabular}{|c|c||c|c|c|}
 \hline 
 \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|}{Networks topology: 6~inputs, 5~outputs, and one hidden layer} \\

 \hline
 \hline
 \multicolumn{2}{|c||}{Hidden neurons} & \multicolumn{3}{c|}{10 neurons} \\
 \hline
 \hline
 \multicolumn{2}{|c||}{Hidden neurons} & \multicolumn{3}{c|}{10 neurons} \\


@@ -931,17 +930,25 @@ 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
 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 those  feedforward network topologies and all
+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
 outputs the  obtained results for the non-chaotic  case outperform the
 chaotic  ones. Finally,  the rates  for the  strategies show  that the

-different networks are unable to learn them.
-
-\newpage
+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
 
 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.
+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
 
 Let us now compare the  two coding schemes. Firstly, the second scheme
 disturbs  the   learning  process.   In   fact  in  this   scheme  the


@@ -949,14 +956,13 @@ 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.
 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.

-\JFC{Je n'ai pas compris le paragraphe precedent. Devrait être repris}

 \begin{table}[b]
 \caption{Prediction success rates for configurations expressed with Gray code}
 \label{tab2}
 \centering
 \begin{tabular}{|c|c||c|c|c|}
 \hline 
 \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|}{Networks topology: 3~inputs, 2~outputs, and one hidden layer} \\

 \hline
 \hline
 & Hidden neurons & \multicolumn{3}{c|}{10 neurons} \\
 \hline
 \hline
 & Hidden neurons & \multicolumn{3}{c|}{10 neurons} \\


@@ -983,12 +989,26 @@ finer information on configuration evolution.
 \end{tabular}
 \end{table}
 
 \end{tabular}
 \end{table}
 

+\begin{figure}
+  \centering
+  \includegraphics[scale=0.5]{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]{non-chaotic_trace2} 
+  \caption{Second coding scheme - Predictions obtained for a non-chaotic test subset.}
+  \label{Fig:non-chaotic_predictions}
+\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
 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
+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
 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


@@ -1001,7 +1021,7 @@ appear to be an open issue.
 \centering
 \begin{tabular}{|c||c|c|c|}
 \hline 
 \centering
 \begin{tabular}{|c||c|c|c|}
 \hline 

-\multicolumn{4}{|c|}{Networks topology: 3~inputs, 1~output and one hidden layer} \\
+\multicolumn{4}{|c|}{Networks topology: 3~inputs, 1~output, and one hidden layer} \\

 \hline
 \hline
 Epochs & 125 & 250 & 500 \\ 
 \hline
 \hline
 Epochs & 125 & 250 & 500 \\ 


@@ -1063,10 +1083,6 @@ Chaotic/non chaotic & \multicolumn{3}{c|}{Output = Strategy} \\
 \end{tabular}
 \end{table}
 
 \end{tabular}
 \end{table}
 

-%
-% TO BE COMPLETED
-%
-

 \section{Conclusion}
 
 In  this paper,  we have  established an  equivalence  between chaotic
 \section{Conclusion}
 
 In  this paper,  we have  established an  equivalence  between chaotic


@@ -1100,12 +1116,10 @@ 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
 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 be  stated.
+will then be stated.

 
 % \appendix{}
 
 
 % \appendix{}
 

-
-

 % \begin{definition} \label{def2}
 % A continuous function $f$  on a topological space $(\mathcal{X},\tau)$
 % is defined  to be  {\emph{topologically transitive}}  if for any  pair of
 % \begin{definition} \label{def2}
 % A continuous function $f$  on a topological space $(\mathcal{X},\tau)$
 % is defined  to be  {\emph{topologically transitive}}  if for any  pair of


@@ -1116,7 +1130,6 @@ will be  stated.
 % $f^k(U) \cap V \neq \emptyset$.
 % \end{definition}
 
 % $f^k(U) \cap V \neq \emptyset$.
 % \end{definition}
 

-

 \bibliography{chaos-paper}% Produces the bibliography via BibTeX.
 
 \end{document}
 \bibliography{chaos-paper}% Produces the bibliography via BibTeX.
 
 \end{document}