from the index $q(a,\overline{b})$ by:
\definition
-The implication intensity that measures the inductive quality of a
-over b is:
+The implication intensity that measures the inductive quality of $a$
+over $b$ is:
$$\varphi(a,b)=1-Pr[Q(a,\overline{b})\leq q(a,\overline{b})] =
\frac{1}{\sqrt{2 \pi}} \int^{\infty}_{ q(a,\overline{b})}
e^{-\frac{t^2}{2}} dt,~ if~ n_b \neq n$$
$$\psi(a,b)= \left[ i(a,b).\varphi(a,b) \right]^{\frac{1}{2}}$$
which integrates both statistical surprise and inclusive quality.
-The function $\psi$ of the variable $t$ admits a representation that has the shape indicated in Figure 4{\bf TO CHANGE}, for $n_a$ and $n_b$ fixed.
+The function $\psi$ of the variable $t$ admits a representation that has the shape indicated in Figure~\ref{chap2fig4}, for $n_a$ and $n_b$ fixed.
Note in this figure the difference in the behaviour of the function with respect to the conditional probability $P(B\mid A)$, a fundamental index of other rule measurement models, for example in Agrawal.
In addition to its linear, and therefore not very nuanced nature, this probability leads to a measure that decreases too quickly from the first counter-examples and then resists too long when they become important.
-{\bf FIGURE 4}
+\begin{figure}[htbp]
+ \centering
+\includegraphics[scale=0.5]{chap2fig4.png}
+\caption{Example of implication-inclusion.}
+\label{chap2fig4}
+\end{figure}
+
+In Figure~\ref{chap2fig4}, it can be seen that this representation of the continuous function of $t$ reflects the expected properties of the inclusion criterion:
+\begin{itemize}
+\item ``Slow reaction'' to the first counter-examples (noise resistance),
+\item ``acceleration'' of the rejection of inclusion close to the balance i.e. $\frac{n_a}{2n}$,
+\item rejection beyond $\frac{n_a}{2n}$, the intensity of implication $\varphi(a,b)$ did not ensure it.
+\end{itemize}
\noindent Example 1\\
- \begin{tabular}{|c|c|c|c|}\hline
+\begin{tabular}{|c|c|c|c|}\hline
& $b$ & $\overline{b}$ & margin\\ \hline
$a$ & 200 & 400& 600 \\ \hline
$\overline{a}$ & 600 & 2800& 3400 \\ \hline
margin & 800 & 3200& 4000 \\ \hline
- \end{tabular}
- \\
- In Example 1, implication intensity is $\varphi(a,b)=0.9999$ (with $q(a,\overline{b})=-3.65$).
+\end{tabular}
+\\
+\\
+In Example 1, implication intensity is $\varphi(a,b)=0.9999$ (with $q(a,\overline{b})=-3.65$).
The entropic values of the experiment are $h_1=h_2=0$.
The value of the moderator coefficient is therefore $i(a,b)=0$.
Hence, $\psi(a,b)=0$ whereas $P(B\mid A)=0.33$.
Thus, the "entropic" functions "moderate" the intensity of implication in this case where inclusion is poor.
+\\
+\\
+\noindent Example 2\\
+ \begin{tabular}{|c|c|c|c|}\hline
+ & $b$ & $\overline{b}$ & margin\\ \hline
+ $a$ & 400 & 200& 600 \\ \hline
+ $\overline{a}$ & 1000 & 2400& 3400 \\ \hline
+ margin & 1400 & 2600& 4000 \\ \hline
+ \end{tabular}
+ \\
+ \\
+ In Example 2, intensity of implication is 1 (for $q(a,\overline{b}) = - 8.43$).
+ The entropic values of the experiment are $h_1 = 0.918$ and $h_2 = 0.391$.
+ The value of the moderator coefficient is therefore $i(a,b) = 0.6035$.
+ As a result $\psi(a,b) = 0.777$ whereas $P(B \mid A) = 0.6666$.
+ \\
+ \\
+{\bf remark}
+ \noindent The correspondence between $\varphi(a,b)$ and $\psi(a,b)$ is not monotonous as shown in the following example:
+
+\begin{tabular}{|c|c|c|c|}\hline
+ & $b$ & $\overline{b}$ & margin\\ \hline
+ $a$ & 40 & 20& 60 \\ \hline
+ $\overline{a}$ & 60 & 280& 340 \\ \hline
+ margin & 100 & 300& 400 \\ \hline
+\end{tabular}
+\\
+Thus, while $\varphi(a,b)$ decreased from the 1st to the 2nd example, $i(a,b)$ increased as well as $\psi(a,b)$. On the other hand, the opposite situation is the most frequent.
+Note that in both cases, the conditional probability does not change.
+\\
+\\
+{\bf remark}
+\noindent We refer to~\cite{Lencaa} for a very detailed comparative study of association indices for binary variables.
+In particular, the intensities of classical and entropic (inclusion) implication presented in this article are compared with other indices according to a "user" entry.
+
+\section{Implication graph}
+\subsection{Problematic}
+
+At the end of the calculations of the intensities of implication in both the classical and entropic models, we have a table $p \times p$ that crosses the $p$ variables with each other, whatever their nature, and whose elements are the values of these intensities of implication, numbers of the interval $[0,~1]$.
+It must be noted that the underlying structure of all these variables is far from explicit and remains largely unimportant.
+The user remains blind to such a square table of size $p^2$.
+It cannot simultaneously embrace the possible multiple sequences of rules that underlie the overall structure of all $p$ variables.
+In order to facilitate a clearer extraction of the rules and to examine their structure, we have associated to this table, and for a given intensity threshold, an oriented graph, weighted by the intensities of implication, without a cycle whose complexity of representation the user can control by setting himself the threshold for taking into account the implicit quality of the rules.
+Each arc in this graph represents a rule: if $n_a < n_b$, the arc $a \rightarrow b$ represents the rule $a \Rightarrow b$ ; if $n_a = n_b$, then the arc $a \leftrightarrow b$ will represent the double rule $a \Leftrightarrow b$, in other words, the equivalence between these two variables.
+By varying the threshold of intensity of implication, it is obvious that the number of arcs varies in the opposite direction: for a threshold set at $0.95$, the number of arcs is less than or equal to those that would constitute the graph at threshold $0.90$. We will discuss this further below.
+
+\subsection{Algorithm}
+
+The relationship defined by statistical implication, if it is reflexive and not symmetrical, is obviously not transitive, as is induction and, on the contrary, deduction.
+However, we want it to model the partial relationship between two variables (the successes in our initial example).
+By convention, if $a \Rightarrow b$ and $b \Rightarrow c$, we will accept the transitive closure $a \Rightarrow c$ only if $\psi(a,c) \geq 0.5$, i.e. if the implicit relationship of $a$ to $c$ is better than neutrality by emphasizing the dependence between $a$ and $c$.
+
+
+{\bf VERIFIER PHI PSI}\\
+\\
+{\bf Proposal:} By convention, if $a \Rightarrow b$ and $b \Rightarrow c$, there is a transitive closure $a \Rightarrow c$ if and only if $\psi(a,c) \geq 0.5$, i.e. if the implicit relationship of $a$ over $c$, which reflects a certain dependence between $a$ and $c$, is better than its refutation.
+Note that for any pair of variables $(x;~ y)$, the arc $x \rightarrow y$ is weighted by the intensity of involvement (x,y).
+\\
+Let us take a formal example by assuming that between the 5 variables $a$, $b$, $c$, $d$, and $e$ exist, at the threshold above $0.5$, the following rules: $c \Rightarrow a$, $c \Rightarrow e$, $c \Rightarrow b$, $d \Rightarrow a$, $d \Rightarrow e$, $a \Rightarrow b$ and $a \Rightarrow e$.
+
+This set of numerical and graphical relationships can then be translated into the following table and graph:
+
+\begin{tabular}{|C{0.5cm}|c|c|c|c|c|}\hline
+\hspace{-0.5cm}\turn{45}{$\Rightarrow$} & $a$ & $b$ & $c$ & $d$ & $e$\\ \hline
+$a$ & & 0.97& & & 0.73 \\ \hline
+$b$ & & & & & \\ \hline
+ $c$ & 0.82 & 0.975& & & 0.82 \\ \hline
+ $d$ & 0.78 & & & & 0.92 \\ \hline
+ $e$ & & & & & \\ \hline
+\end{tabular}
+
+\begin{figure}[htbp]
+ \centering
+\includegraphics[scale=1]{chap2fig5.png}
+\caption{Implication graph corresponding to the previous example.}
+
+\label{chap2fig5}
+\end{figure}
+
+One of the difficulties related to the graphical representation is that the graph is not planar.
+The algorithm that allows its construction must take it into account and, in particular, must "straighten" the paths of the graph in order to allow an acceptable readability for the expert who will analyze it.
+
+The number of arcs in the graph can be reduced (or increased) if we raise (or lower) the acceptance threshold of the rules, the level of confidence in the selected rules.
+Correlatively, arcs can appear or disappear depending on the variations of the threshold.
+Let us recall that this graph is necessarily without cycle, that it is not a lattice since, for example, the variable $a$ does not imply the variable ($a$ or $\neg a$) whose support is $E$.
+A fortiori, it cannot be a Galois lattice.
+Options of the CHIC software for automatic data processing with SIA, allow to delete variables at will, to move their image in the graph in order to decrease the arcs or to focus on certain variables called vertices of a kind of "cone" whose two "plots" are made up respectively of the variables "parents" and the variables "children" of this vertex variable.
+We refer to the ends of the arcs as "nodes". A node in a given graph has a single variable or a conjunction of variables.
+The transition from a node $S_1$ to a node $S_2$ is also called "transition" which is represented by an arc in the graph.
+The upper slick of the vertex cone the variable $a$, called the nodal variable, is made up of the "fathers" of $a$, either in the "causal" sense the causes of $a$ ; the lower slick, on the other hand, is made up of the "children" of $a$ and therefore, always in the causal sense, the consequences or effects of $a$.
+The expert in the field analysed here must be particularly interested in these configurations, which are rich in information.
+See, for example~\cite{Lahanierc} and the two implicit cones below (i.e. Figures~\ref{chap2fig6} and \ref{chap2fig7}).
+
+\begin{figure}[htbp]
+ \centering
+\includegraphics[scale=0.75]{chap2fig6.png}
+\caption{Implicative cone.}
+
+\label{chap2fig6}
+\end{figure}
+
+\begin{figure}[htbp]
+ \centering
+\includegraphics[scale=0.75]{chap2fig7.png}
+\caption{Implicative cone centered on a variable.}
+
+\label{chap2fig7}
+\end{figure}
+
+
+\section{Reduction in the number of variables}
+\subsection{Motivation}
+
+
+As soon as the number of variables becomes excessive, most of the available techniques become impractical\footnote{This paragraph is strongly inspired by paper~\cite{Grask}.}.
+In particular, when an implicitive analysis is carried out by calculating association rules~\cite{Agrawal}, the number of rules discovered undergoes a combinatorial explosion with the number of variables, and quickly becomes inextricable for a decision-maker, provided that variable conjunctions are requested.
+In this context, it is necessary to make a preliminary reduction in the number of variables.
+
+Thus, ~\cite{Ritschard} proposed an efficient heuristic to reduce both the number of rows and columns in a table, using an association measure as a quasi-optimal criterion for controlling the heuristic.
+However, to our knowledge, in the various other research studies, the type of situation at the origin of the need to group rows or columns is not taken into account in the reduction criteria, whether the analyst's problem and aim are the search for similarity, dissimilarity, implication, etc., between variables.
+
+Also, to the extent that there are very similar variables in the sense of statistical implication, it might be appropriate to substitute a single variable for these variables that would be their leader in terms of representing an equivalence class of similar variables for the implicit purpose.
+We therefore propose, following the example of what is done to define the notion of quasi-implication, to define a notion of quasi-equivalence between variables, in order to build classes from which we will extract a leader.
+We will illustrate this with an example.
+Then, we will consider the possibility of using a genetic algorithm to optimize the choice of the representative for each quasi-equivalence class.
+
+\subsection{Definition of quasi-equivalence}
+
+Two binary variables $a$ and $b$ are logically equivalent for the SIA when the two quasi-implications $a \Rightarrow b$ and $b \Rightarrow a$ are simultaneously satisfied at a given threshold.
+We have developed criteria to assess the quality of a quasi-involvement: one is the statistical surprise based on the likelihood of~\cite{Lerman} relationship, the other is the entropic form of quasi-inclusion~\cite{Grash2} which is presented in this chapter (§7).
+
+According to the first criterion, we could say that two variables $a$ and $b$ are almost equivalent when the intensity of involvement $\varphi(a,b)$ of $a\Rightarrow b$ is little different from that of $b \Rightarrow a$. However, for large groups (several thousands), this criterion is no longer sufficiently discriminating to validate inclusion.
+
+According to the second criterion, an entropic measure of the imbalance between the numbers $n_{a \wedge b}$ (individuals who satisfy $a$ and $b$) and $n_{a \wedge \overline{b}} $ (individuals who satisfy $a$ and $\neg b$, counter-examples to involvement $a\Rightarrow b$) is used to indicate the quality of involvement $a\Rightarrow b$, on the one hand, and the numbers $n_{a \wedge b}$ and $n_{\overline{a} \wedge b}$ to assess the quality of mutual implication $b\Rightarrow a$, on the other.
+
+
+Here we will use a method comparable to that used in Chapter 3 to define the entropic implication index.
+
+By posing $n_a$ and $n_b$, respectively effective of $a$ and $b$, the imbalance of the rule $a\Rightarrow b$ is measured by a conditional entropy $K(b \mid a=1)$, and that of $b\Rightarrow a$ by $K(a \mid b=1)$ with:
+
+
+\begin{eqnarray*}
+ K(b\mid a=1) = - \left( 1- \frac{n_{a\wedge b}}{n_a}\right) log_2 \left( 1- \frac{n_{a\wedge b}}{n_a}\right) - \frac{n_{a\wedge b}}{n_a}log_2 \frac{n_{a\wedge b}}{n_a} & \quad if \quad \frac{n_{a \wedge b}}{n_a} > 0.5\\
+ K(b\mid a=1) = 1 & \quad if \quad \frac{n_{a \wedge b}}{n_a} \leq 0.5\\
+ K(a\mid b=1) = - \left( 1- \frac{n_{a\wedge b}}{n_b}\right) log_2 \left( 1- \frac{n_{a\wedge b}}{n_b}\right) - \frac{n_{a\wedge b}}{n_b}log_2 \frac{n_{a\wedge b}}{n_b} & \quad if \quad \frac{n_{a \wedge b}}{n_b} > 0.5\\
+ K(a\mid b=1) = 1 & \quad if \quad \frac{n_{a \wedge b}}{n_b} \leq 0.5
+\end{eqnarray*}
+
+These two entropies must be low enough so that it is possible to bet on $b$ (resp. $a$) with a good certainty when $a$ (resp. $b$) is achieved. Therefore their respective complements to 1 must be simultaneously strong.
+
+\begin{figure}[htbp]
+ \centering
+\includegraphics[scale=0.5]{chap2fig8.png}
+\caption{Illustration of the functions $K$ et $1-K^2$ on $[0; 1]$ .}
+
+\label{chap2fig7}
+\end{figure}
+
+
+\definition A first entropic index of equivalence is given by:
+$$e(a,b) = \left (\left[ 1 - K^2(b \mid a = 1)\right ]\left[ 1 - K^2(a \mid b = 1) \right]\right)^{\frac{1}{4}}$$
+
+When this index takes values in the neighbourhood of $1$, it reflects a good quality of a double implication.
+In addition, in order to better take into account $a \wedge b$ (the examples), we integrate this parameter through a similarity index $s(a,b)$ of the variables, for example in the sense of I.C. Lerman~\cite{Lermana}.
+The quasi-equivalence index is then constructed by combining these two concepts.
+
+\definition A second entropic equivalence index is given by the formula
+
+$$\sigma(a,b)= \left [ e(a,b).s(a,b)\right ]^{\frac{1}{2}}$$
+
+From this point of view, we then set out the quasi-equivalence criterion that we use.
+
+\definition The pair of variables $\{a,b\}$ is said to be almost equivalent for the selected quality $\beta$ if $\sigma(a,b) \geq \beta$.
+For example, a value $\beta=0.95$ could be considered as a good quasi-equivalence between $a$ and $b$.
+
+\subsection{Algorithm of construction of quasi-equivalence classes}
+
+Let us assume a set $V = \{a,b,c,...\}$ of $v$ variables with a valued relationship $R$ induced by the measurement of quasi-equivalence on all pairs of $V$.
+We will assume the pairs of variables classified in a decreasing order of quasi-equivalence.
+If we have set the quality threshold for quasi-equivalence at $\beta$, only the first of the pairs $\{a,b\}$ checking for inequality $\sigma(a,b)\ge \beta$ will be retained.
+In general, only a part $V'$, of cardinal $v'$, of the variables of $V$ will verify this inequality.
+If this set $V'$ is empty or too small, the user can reduce his requirement to a lower threshold value.
+The relationship being symmetrical, we will have at most pairs to study.
+As for $V-V'$, it contains only non-reducible variables.
+
+We propose to use the following greedy algorithm:
+\begin{enumerate}
+\item A first potential class $C_1^0= \{e,f\}$ is constituted such that $\sigma(e,f)$ represents the largest of the $\beta$-equivalence values.
+ If possible, this class is extended to a new class $C_1$ by taking from $V'$ all the elements $x$ such that any pair of variables within this class allows a quasi-equivalence greater than or equal to $\beta$;
+
+\item We continue with:
+ \begin{enumerate}
+ \item If $o$ and $k$ forming the pair $(o,k)$ immediately below $(e,f)$ according to the index $\sigma$, belong to $C_1$, then we move to the pair immediately below (o,k) and proceed as in 1.;
+ \item If $o$ and $k$ do not belong to $C_1$, proceed as in 1. from the pair they constitute by forming the basis of a new class;
+ \item If $o$ or $k$ does not belong to $C_1$, one of these two variables can either form a singleton class or belong to a future class. On this one, we will of course practice as above.
+ \end{enumerate}
+ \end{enumerate}
+
+After a finite number of iterations, a partition of $V$ is available in $r$ classes of $\sigma$-equivalence: $\{C_1, C_2,..., C_r\}$.
+The quality of the reduction may be assessed by a gross or proportional index of $\beta^{\frac{p}{k}}$.
+However, we prefer the criterion defined below, which has the advantage of integrating the choice of representative.
+
+In addition, $k$ variables representing the $k$ classes of $\sigma$-equivalence could be selected on the basis of the following elementary criterion: the quality of connection of this variable with those of its class.
+However, this criterion does not optimize the reduction since the choice of representative is relatively arbitrary and may be a sign of triviality of the variable.
+
