X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/chaos1.git/blobdiff_plain/486652fa99fdbcf0992d32f910b729e648e33644..3df586d673bc4f3b32fa0dd1cb46796256744772:/main.tex?ds=sidebyside

diff --git a/main.tex b/main.tex
index 016243a..5bd42bc 100644
--- 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.
-\JFC{en dire davantage sur l'outside world}
 \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
-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
@@ -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.
 
-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}
@@ -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,
-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.
@@ -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 
-\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} \\
@@ -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
-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.
@@ -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.
-\JFC{Je n'ai pas compris le paragraphe precedent. Devrait Ãªtre 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 
-\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} \\
@@ -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
-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
@@ -1001,7 +1000,7 @@ appear to be an open issue.
 \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 \\ 
@@ -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
-will be  stated.
+will then be  stated.
 
 % \appendix{}
 
