Fin de la relecture

diff --git a/main.tex b/main.tex
index 016243a10be55ffa732d5119408e6faea4be3f41..5bd42bc0b215cf1fc3388b8a26391d1d901b7f76 100644 (file)
--- a/main.tex
+++ b/main.tex
@@ -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}


@@ -860,8 +859,8 @@ 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
 predicted.  These values  are computed  considering  10~trainings with
 random  subsets  construction,   weights  and  biases  initialization.
 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.


@@ -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} \\


@@ -931,7 +930,7 @@ 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
 different networks are unable to learn them.
 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.


@@ -949,14 +948,14 @@ 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} \\


@@ -988,7 +987,7 @@ 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
 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 +1000,7 @@ 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 \\ 


@@ -1100,7 +1099,7 @@ 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{}