and, recently, interval and fuzzy variables.
}
-\section{Preamble}
+\section*{Preamble}
Human operative knowledge is mainly composed of two components: that
of facts and that of rules between facts or between rules themselves.
\overline{B}))$ and the value it would have taken if there had been
independence between $a$ and $b$.
-\definition $$q(a,\overline{b}) = \frac{n_{a \wedge \overline{b}}- \frac{n_a.n_{\overline{b}}}{n}}{\sqrt{\frac{n_a.n_{\overline{b}}}{n}}}$$
+\definition
+\begin{equation} q(a,\overline{b}) = \frac{n_{a \wedge \overline{b}}-
+ \frac{n_a.n_{\overline{b}}}{n}}{\sqrt{\frac{n_a.n_{\overline{b}}}{n}}}
+ \label{eq2.1}
+\end{equation}
is called the implication index, the number used as an indicator of
the non-implication of $a$ to $b$.
In cases where the approximation is properly legitimized (for example
situation.
-\section{Cases of on-interval and on-interval variables}
+\section{Cases of variables-on-intervals and interval-variables}
\subsection{Variables-on-intervals}
\subsubsection{Founding situation}
optimal reciprocal involvement is satisfied for the partition of
$[b1,~ b2]$ corresponding to the second maximum.
-\section{Interval-variables}
-\subsection{Founding situation}
+\subsection{Interval-variables}
+\subsubsection{Founding situation}
Data are available from a population of $n$ individuals (who may be
each or some of the sets of individuals, e.g. a class of students)
according to variables (e.g. grades over a year in French, math,
implies $[16.40, 18]$ in mathematics.
-\subsection{Algorithm}
+\subsubsection{Algorithm}
By following the problem of E. Diday and his collaborators, if the
values taken according to the subjects by the variables $a$ and $b$
as they are with the usual topology.}, is expressed as follows by
the scalar product:
-$$dq = \frac{\partial q}{\partial n}dn + \frac{\partial q}{\partial
+\begin{equation}
+dq = \frac{\partial q}{\partial n}dn + \frac{\partial q}{\partial
n_a}dn_a + \frac{\partial q}{\partial n_b}dn_b + \frac{\partial
- q}{\partial n_{a \wedge \overline{b}}}dn_{a \wedge \overline{b}} = grad~q.dM\footnote{By a mechanistic metaphor, we will say that $dq$ is the elementary work of $q$ for a movement $dM$ (see chapter 14 of this book).}$$
+ q}{\partial n_{a \wedge \overline{b}}}dn_{a \wedge \overline{b}} =
+grad~q.dM\footnote{By a mechanistic metaphor, we will say that $dq$ is
+ the elementary work of $q$ for a movement $dM$ (see chapter 14 of
+ this book).}
+\label{eq2.2}
+\end{equation}
where $M$ is the coordinate point $(n,~ n_a,~ n_b,~ n_{a \wedge
\overline{b}})$ of the vector scalar field $C$, $dM$ is the
Let us examine the partial derivatives of $n_b$ and $n_{a \wedge
\overline{b}}$ the number of counter-examples. We get:
-$$ \frac{\partial
+\begin{equation}
+ \frac{\partial
q}{\partial n_b} = \frac{1}{2} n_{a \wedge
\overline{b}} (\frac{n_a}{n})^{-\frac{1}{2}} (n-n_b)^{-\frac{3}{2}}
-+ \frac{1}{2} (\frac{n_a}{n})^{\frac{1}{2}} (n-n_b)^{-\frac{1}{2}} > 0 $$
+ + \frac{1}{2} (\frac{n_a}{n})^{\frac{1}{2}} (n-n_b)^{-\frac{1}{2}} >
+ 0
+ \label{eq2.3}
+\end{equation}
-$$ \frac{\partial
+\begin{equation}
+ \frac{\partial
q}{\partial n_{a \wedge
\overline{b}}} = \frac{1}{\sqrt{\frac{n_a n_{\overline{b}}}{n}}}
-= \frac{1}{\sqrt{\frac{n_a (n-n_b)}{n}}} > 0 $$
+ = \frac{1}{\sqrt{\frac{n_a (n-n_b)}{n}}} > 0
+ \label{eq2.4}
+\end{equation}
+
Thus, if the increases $\Delta nb$ and $\Delta n_{a \wedge
\overline{b}}$ are positive, the increase of $q(a,\overline{b})$ is
\overline{b}}$ and minimum at values $n_b+\Delta n_b$ and $n_{a \wedge
\overline{b}}+ n_{a \wedge
\overline{b}}$.
+
+If we examine the case where $n_a$ varies, we obtain the partial
+derivative of $q$ with respect to $n_a$ which is:
+
+\begin{equation}
+ C = \frac{ n_{a \wedge \overline{b}}}{2
+ \sqrt{\frac{n_{\overline{b}}}{n}}}
+ \left(\frac{n}{n_a}\right)^{\frac{3}{2}}
+ -\frac{1}{2}\sqrt{\frac{n_{\overline{b}}}{n_a}}<0
+ \label{eq2.5}
+ \end{equation}
+
+Thus, for variations of $n_a$ on $[0,~ nb]$, the implication index function is always decreasing (and concave) with respect to $n_a$ and is therefore minimum for $n_a= n_b$. As a result, the intensity of implication is increasing and maximum for $n_a= n_b$.
+
+Note the partial derivative of $q$ with respect to $n$:
+
+$$\frac{\partial q}{\partial n} = \frac{1}{2\sqrt{n}} \left( n_{a
+ \wedge \overline{b}}+\frac{n_a n_{\overline{b}}}{n} \right)$$
+
+Consequently, if the other 3 parameters are constant, the implication
+index decreases by $\sqrt{n}$.
+The quality of implication is therefore all the better, a specific
+property of the SIA compared to other indicators used in the
+literature~\cite{Grasab}.
+This property is in accordance with statistical and semantic
+expectations regarding the credit given to the frequency of
+observations.
+Since the partial derivatives of $q$ (at least one of them) are
+non-linear according to the variable parameters involved, we are
+dealing with a non-linear dynamic system\footnote{"Non-linear systems
+ are systems that are known to be deterministic but for which, in
+ general, nothing can be predicted because calculations cannot be
+ made"~\cite{Ekeland} p. 265.} with all the epistemological
+consequences that we will consider elsewhere.
+
+
+
+\subsection{Numerical example}
+In a first experiment, we observe the occurrences: $n = 100$, $n_a =
+20$, $n_b = 40$ (hence $n_b=60$, $ n_{a \wedge \overline{b}} = 4$).
+The application of formula (\ref{eq2.1}) gives = -2.309.
+In a 2nd experiment, $n$ and $n_a$ are unchanged but the occurrences
+of $b$ and counter-examples $n_{a \wedge \overline{b}}$ increase by one unit.
+
+At the initial point of the space of the 4 variables, the partial
+derivatives that only interest us (according to $n_b$ and $n_{a
+ \wedge \overline{b}}$) have respectively the following values when
+applying formulas (\ref{eq2.3}) and (\ref{eq2.4}): $\frac{\partial
+ q}{\partial n_b} = 0.0385$ and $\frac{\partial q}{\partial n_{a
+ \wedge \overline{b}}} = 0.2887$.
+
+As $\Delta n_b$, $\Delta n_{\overline{b}}$ and $\Delta n_{a
+ \wedge \overline{b}} $ are equal to 1, -1 and 1, then $\Delta q$ is
+equal to: $0.0385 + 0.2887 + o(\Delta q) = 0.3272 + o(\Delta q)$ and
+the approximate value of $q$ in the second experiment is $-2.309 +
+0.2887 + o(\Delta q)= -1.982 +o(\Delta q)$ using the first order
+development of $q$ (formula (\ref{eq2.2})).
+However, the calculation of the new implication index $q$ at the point
+of the 2nd experiment is, by the use of (\ref{eq2.1}): $-1.9795$, a
+value well approximated by the development of $q$.
+
+
+
+\subsection{A first differential relationship of $\varphi$ as a function of function $q$}
+Let us consider the intensity of implication $\varphi$ as a function
+of $q(a,\overline{b})$:
+$$\varphi(q)=\frac{1}{\sqrt{2\pi}}\int_q^{\infty}e^{-\frac{t^2}{2}}$$
+We can then examine how $\varphi(q)$ varies when $q$ varies in the neighberhood of a given value $(a,b)$, knowing how $q$ itself varies according to the 4 parameters that determine it. By derivation of the integration bound, we obtain:
+\begin{equation}
+ \frac{d\varphi}{dq}=-\frac{1}{\sqrt{2\pi}}e^{-\frac{q^2}{2}} < 0
+ \label{eq2.6}
+\end{equation}
+This confirms that the intensity increases when $q$ decreases, but the growth rate is specified by the formula, which allows us to study more precisely the variations of $\varphi$. Since the derivative of $\varphi$ from $q$ is always negative, the function $\varphi$ is decreasing.
+
+{\bf Numerical example}\\
+Taking the values of the occurrences observed in the 2 experiments
+mentioned above, we find for $q = -2.309$, the value of the intensity
+of implication $\varphi(q)$ is equal to 0.992. Applying formula
+(\ref{eq2.6}), the derivative of $\varphi$ with respect to $q$ is:
+-0.02775 and the negative increase in intensity is then: -0.02775,
+$\Delta q$ = 0.3272. The approximate first-order intensity is
+therefore: $0.992-\Delta q$ or 0.983. However, the actual calculation
+of this intensity is, for $q= -1.9795$, $\varphi(q) = 0.976$.
+
+
+
+\subsection{Examination of other indices}
+Unlike the core index $q$ and the intensity of implication, which
+measures quality through probability (see definition 2.3), the other
+most common indices are intended to be direct measures of quality.
+We will examine their respective sensitivities to changes in the
+parameters used to define these indices.
+We keep the ratings adopted in paragraph 2.2 and select indices that
+are recalled in~\cite{Grasm},~\cite{Lencaa} and~\cite{Grast2}.
+
+\subsubsection{The Loevinger Index}
+
+It is an "ancestor" of the indices of
+implication~\cite{Loevinger}. This index, rated $H(a,b)$, varies from
+1 to $-\infty$. It is defined by: $H(a,b) =1-\frac{n n_{a \wedge
+ b}}{n_a n_b}$. Its partial derivative with respect to the variable number of counter-examples is therefore:
+$$\frac{\partial H}{\partial n_{a \wedge \overline{b}}}=-\frac{n}{n_a n_b}$$
+Thus the implication index is always decreasing with $n_{a \wedge
+ \overline{b}}$. If it is "close" to 1, implication is "almost"
+satisfied. But this index has the disadvantage, not referring to a
+probability scale, of not providing a probability threshold and being
+invariant in any dilation of $E$, $A$, $B$ and $A \cap \overline{B}$.
+
+
+\subsubsection{The Lift Index}
+
+It is expressed by: $l =\frac{n n_{a \wedge b}}{n_a n_b}$.
+This expression, linear with respect to the examples, can still be
+written to highlight the number of counter-examples:
+$$l =\frac{n (n_a - n_{a \wedge \overline{b}})}{n_a n_b}$$
+To study the sensitivity of the $l$ to parameter variations, we use:
+$$\frac{\partial l}{\partial n_{a \wedge \overline{b}} } =
+-\frac{1}{n_a n_b}$$
+Thus, the variation of the Lift index is independent of the variation
+of the number of counter-examples.
+It is a constant that depends only on variations in the occurrences of $a$ and $b$. Therefore, $l$ decreases when the number of counter-examples increases, which semantically is acceptable, but the rate of decrease does not depend on the rate of growth of $n_{a \wedge \overline{b}}$.
+
+\subsubsection{Confidence}
+
+This index is the best known and most widely used thanks to the sound
+box available in an Anglo-Saxon publication~\cite{Agrawal}.
+It is at the origin of several other commonly used indices which are only variants satisfying this or that semantic requirement... Moreover, it is simple and can be interpreted easily and immediately.
+$$c=\frac{n_{a \wedge b}}{n_a} = 1-\frac{n_{a \wedge \overline{b}}}{n_a}$$
+
+The first form, linear with respect to the examples, independent of
+$n_b$, is interpreted as a conditional frequency of the examples of
+$b$ when $a$ is known.
+The sensitivity of this index to variations in the occurrence of
+counter-examples is read through the partial derivative:
+$$\frac{\partial c}{\partial n_{a \wedge \overline{b}} } =
+-\frac{1}{n_a }$$
+
+
+Consequently, confidence increases when $n_{a \wedge \overline{b}}$
+decreases, which is semantically acceptable, but the rate of variation
+is constant, independent of the rate of decrease of this number, of
+the variations of $n$ and $n_b$.
+This property seems not to satisfy intuition.
+The gradient of $c$ is expressed only in relation to $n_{a \wedge
+ \overline{b}}$ and $n_a$:(). {\bf CHECK FORMULA}
+This may also appear to be a restriction on the role of parameters in
+expressing the sensitivity of the index.
+
+\section{Gradient field, implicative field}
+We highlight here the existence of fields generated by the variables
+of the corpus.
+
+\subsection{Existence of a gradient field}
+Like our Newtonian physical space, where a gravitational field emitted
+by each material object acts, we can consider that it is the same
+around each variable.
+For example, the variable $a$ generates a scalar field whose value in
+$b$ is maximum and equal to the intensity of implication or the
+implicition index $q(a,\overline{b})$.
+Its action spreads in V according to differential laws as J.M. Leblond
+says, in~\cite{Leblond} p.242.
+
+Let us consider the space $E$ of dimension 4 where the coordinates of
+the points $M$ are the parameters relative to the binary variables $a$
+and $b$, i.e. ($n$, $n_a$, $n_b$, $n_{a\wedge \overline{b}}$). $q(a,\overline{b})$ is the realization of a scalar field, as an application of $\mathbb{R}^4$ in $\mathbb{R}$ (immersion of $\mathbb{N}^4$ in $\mathbb{R}^4$).
+For the grad vector $q$ of components the partial derivatives of $q$
+with respect to variables $n$, $n_a$, $n_b$, $n_{a\wedge
+ \overline{b}}$ to define a gradient field - a particular vector
+field that we will also call implicit field - it must respect the
+Schwartz criterion of an exact total differential, i.e.:
+
+$$\frac{\partial}{\partial n_{a\wedge \overline{b}}}\left(
+\frac{\partial q}{\partial n_b} \right) =\frac{\partial}{\partial n_b}\left(
+\frac{\partial q}{\partial n_{a\wedge \overline{b}}} \right) $$
+and the same for the other variables taken in pairs. However, we have,
+through the formulas (\ref{eq2.3}) and (\ref{eq2.4})
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