X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/book_chic.git/blobdiff_plain/9e9b8dc87f77206ba17bb278dc62b2bee644f487..c9f45698e9a535650b20a516d197dca86ed70d90:/chapter2.tex?ds=inline diff --git a/chapter2.tex b/chapter2.tex index 096f51f..ed3ec5b 100644 --- a/chapter2.tex +++ b/chapter2.tex @@ -29,7 +29,7 @@ gradually being supplemented by the processing of modal, frequency 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. @@ -298,7 +298,11 @@ It estimates a gap between the contingency $(card(A\cap \overline{B}))$ 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 @@ -435,3 +439,503 @@ and stability of the implication index as a function of small variations in the parameters involved in the study of its differential. +\section{Case of modal and frequency variables} +\subsection{Founding situation} + +Marc Bailleul's (1991-1994) research focuses in particular on the +representation that mathematics teachers have of their own teaching. +In order to highlight it, meaningful words are proposed to them that +they must prioritize. +Their choices are no longer binary, the words chosen by any teacher +are ordered at least at the most representative. +Mr. Bailleul's question then focuses on questions of the type: "if I +choose this word with this importance, then I choose this other word +with at least equal importance". +It was therefore necessary to extend the notion of statistical +implication to variables other than binary. +This is the case for modal variables that are associated with +phenomena where the values $a(x)$ are numbers in the interval $[0, 1]$ +and describe degrees of belonging or satisfaction as are fuzzy logic, +for example, linguistic modifiers "maybe", "a little", "sometimes", +etc. +This problem is also found in situations where the frequency of a +variable reflects a preorder on the values assigned by the subjects to +the variables presented to them. +These are frequency variables that are associated with phenomena where +the values of $a(x)$ are any positive real values. +This is the case when one considers a student's percentage of success +in a battery of tests in different areas. + +\subsection{Formalization} + +J.B. Lagrange~\cite{Lagrange} has demonstrated that, in the modal +case, +\begin{itemize} + \item if $a(x)$ and $\overline{b}(x)$ are the values taken at $x$ by + the modal variables $a$ and $\overline{b}$, with $(x)=1-b(x)$ + \item if $s^2_a$ and $s_{\overline{b}}^2$ are the empirical variances of variables $a$ and $\overline{b}$ +then the implication index, which he calls propensity index, becomes: + +\definition +$$q(a,\overline{b}) = \frac{\sum_{x\in E} a(x)\overline{b}(x) - + \frac{n_a n_{\overline{b}}}{n}} +{\sqrt{\frac{(n^2s_a^2+n_a^2)(n^2+s_{\overline{b}}^2 + n_{\overline{b}}^2)}{n^3}}}$$ +is the index of propensity of modal variables. +\end{itemize} + +J.B. Lagrange also proves that this index coincides with the index +defined previously in the binary case if the number of modalities of a +and b is precisely 2, because in this case :\\ +$n^2s_a^2+n_a^2=n n_a$,~ ~ $ n^2+s_{\overline{b}}^2 + n_{\overline{b}}=n + n_{\overline{b}}$~ ~ and ~ ~ $\sum_{x\in E} a(x)\overline{b}(x)=n_{a \wedge + \overline{b}}$. + + This solution provided in the modal case is also applicable to the + case of frequency variables, or even positive numerical variables, + provided that the values observed on the variables, such as a and b, + have been normalized, the normalization in $[0, 1]$ being made from the maximum of the value taken respectively by $a$ and $b$ on set $E$. + +\remark +In~\cite{Regniera}, we consider rank variables that reflect a +total order between choices presented to a population of judges. +Each of them must order their preferential choice among a set of +objects or proposals made to them. +An index measures the quality of the statement of the type: "if object +$a$ is ranked by judges then, generally, object $b$ is ranked higher +by the same judges". +Proximity to the previous issue leads to an index that is relatively +close to the Lagrange index, but better adapted to the rank variable +situation. + + +\section{Cases of variables-on-intervals and interval-variables} +\subsection{Variables-on-intervals} +\subsubsection{Founding situation} + +For example, the following rule is sought to be extracted from a +biometric data set, estimating its quality: "if an individual weighs +between $65$ and $70kg$ then in general he is between $1.70$ and +$1.76m$ tall". +A similar situation arises in the search for relationships between +intervals of student performance in two different subjects. +The more general situation is then expressed as follows: two real +variables $a$ and $b$ take a certain number of values over 2 finite +intervals $[a1,~ a2]$ and $[b1,~ b2]$. Let $A$ (resp. $B$) be all the +values of $a$ (resp. $b$) observed over $[a1,~ a2]$ (resp. $[b1,~ + b2]$). +For example, here, a represents the weights of a set of n subjects and b the sizes of these same subjects. + +Two problems arise: +\begin{enumerate} +\item Can adjacent sub-intervals of $[a1,~ a2]$ (resp. $[b1,~ b2]$) + be defined so that the finest partition obtained best respects the + distribution of the values observed in $[a1,~ a2]$ (resp. $[b1,~ b2]$)? +\item Can we find the respective partitions of $[a1,~ a2]$ and $[b1,~ + b2]$ made up of meetings of the previous adjacent sub-intervals, + partitions that maximize the average intensity of involvement of the + sub-intervals of one on sub-intervals on the other belonging to + these partitions? +\end{enumerate} + +We answer these two questions as part of our problem by choosing the +criteria to optimize in order to satisfy the optimality expected in +each case. +To the first question, many solutions have been provided in other +settings (for example, by~\cite{Lahaniera}). + +\subsubsection{First problem} + +We will look at the interval $[a1,~ a2]$ assuming it has a trivial +initial partition of sub-intervals of the same length, but not +necessarily of the same frequency distribution observed on these +sub-intervals. +Note $P_0 = \{A_{01},~ A_{02},~ ...,~ A_{0p}\}$, this partition in $p$ +sub-intervals. +We try to obtain a partition of $[a1,~ a2]$ into $p$ sub-intervals +$\{A_{q1},~ A_{q2},~ ...,~ A_{qp}\}$ in such a way that within each +sub-interval there is good statistical homogeneity (low intra-class +inertia) and that these sub-intervals have good mutual heterogeneity +(high inter-class inertia). +We know that if one of the criteria is verified, the other is +necessarily verified (Koenig-Huyghens theorem). +This will be done by adopting a method directly inspired by the +dynamic cloud method developed by Edwin Diday~\cite{Diday} (see also +\cite{Lebart} and adapted to the current situation. This results in +the optimal partition targeted. + +\subsubsection{Second problem} + +It is now assumed that the intervals $[a1,~ a2]$ and $[b1,~ b2]$ are +provided with optimal partitions $P$ and $Q$, respectively, in the +sense of the dynamic clouds. +Let $p$ and $q$ be the respective numbers of sub-intervals composing +$P$ and $Q$. +From these two partitions, it is possible to generate $2^{p-1}$ and +$2^{q-1}$ partitions obtained by iterated meetings of adjacent +sub-intervals of $P$ and $Q$ \footnote{It is enough to consider the tree structure of which $A_1$ is the root, then to join it or not to $A_2$ which itself will or will not be joined to $A_3$, etc. There are therefore $2^{p-1}$ branches in this tree structure.} respectively. +We calculate the respective intensities of implication of each +sub-interval, whether or not combined with another of the first +partition, on each sub-interval, whether or not combined with another +of the second, and then the values of the intensities of the +reciprocal implications. +There are therefore a total of $2.2^{p-1}.2^{q-1}$ families of +implication intensities, each of which requires the calculation of all +the elements of a partition of $[a1,~ a2]$ on all the elements of one +of the partitions of $[b1,~ b2]$ and vice versa. +The optimality criterion is chosen as the geometric mean of the +intensities of implication, the mean associated with each pair of +partitions of elements, combined or not, defined inductively. +We note the two maxima obtained (direct implication and its +reciprocal) and we retain the two associated partitions by declaring +that the implication of the variable-on-interval $a$ on the +variable-on-interval $b$ is optimal when the interval $[a1,~ a2]$ +admits the partition corresponding to the first maximum and that the +optimal reciprocal involvement is satisfied for the partition of +$[b1,~ b2]$ corresponding to the second maximum. + +\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, +physics,..., but also: weight, height, chest size,...). +The values taken by these variables for each individual are intervals +of positive real values. +For example, individual $x$ gives the value $[12,~ 15.50]$ to the math +score variable. +E. Diday would speak on this subject of symbolic variables $p$ at +intervals defined on the population. + + +We try to define an implication of intervals, relative to a variable +$a$, which are themselves observed intervals, towards other similarly +defined intervals and relative to another variable $b$. +This will make it possible to measure the implicit, and therefore +non-symmetric, association of certain interval(s) of the variable a +with certain interval(s) of the variable $b$, as well as the +reciprocal association from which the best one will be chosen for each +pair of sub-intervals involved, as just described in §4.1. + +For example, it will be said that the sub-interval $[2, 5.5]$ of +mathematical scores generally implies the sub-interval $[4.25, 7.5]$ +of physical scores, both of which belong to an optimal partition in +terms of the explained variance of the respective value ranges $[1, + 18]$ and $[3, 20]$ taken in the population. +Similarly, we will say that $[14.25, 17.80]$ in physics most often +implies $[16.40, 18]$ in mathematics. + + +\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 $b$ +are of a symbolic nature, in this case intervals of $\mathbb{R}^+$, it +is possible to extend the above algorithms\cite{Grasi}. +For example, variable $a$ has weight intervals associated with it and +variable $b$ has size intervals associated with variable $b$, due to +inaccurate measurements. +By combining the intervals $I_x$ and $J_x$ described by the subjects +$x$ of $E$ according to each of the variables $a$ and $b$ +respectively, we obtain two intervals $I$ and $J$ covering all +possible values of $a$ and $b$. +On each of them a partition can be defined in a certain number of +intervals respecting as above a certain optimality criterion. +For this purpose, the intersections of intervals such as $I_x$ and +$J_x$ with these partitions will be provided with a distribution +taking into account the areas of the common parts. +This distribution may be uniform or of another discrete or continuous +type. +But thus, we are back in search of rules between two sets of +variables-on-intervals that take, as previously in §4.1, their values +on $[0,~ 1]$ from which we can search for optimal implications. + + +\remark Whatever the type of variable considered, there is often a +problem of overabundance of variables and therefore difficulty of +representation. +For this reason, we have defined an equivalence relationship on all +variables that allows us to substitute a so-called leader variable for +an equivalence class~\cite{Grask}. + +\section{Variations in the implication index q according to the 4 occurrences} + +In this paragraph, we examine the sensitivity of the implication index +to disturbances in its parameters. + +\subsection{Stability of the implication index} +To study the stability of the implication index $q$ is to examine its +small variations in the vicinity of the $4$ observed integer values +($n$, $n_a$, $n_b$, $n_{a \wedge \overline{b}}$). +To do this, it is possible to perform different simulations by +crossing these 4 integer variables on which $q$ depends~\cite{Grasx}. +But let us consider these variables as variables with real values and +$q$ as a function that can be continuously differentiated from these +variables, which are themselves forced to respect inequalities: $0\leq +n_a \leq n_b$ and $n_{a \wedge \overline{b}} \leq inf\{n_a,~ n_b\}$ and +$sup\{n_a,~ n_b\} \leq n$. +The function $q$ then defines a scalar and vector field on +$\mathbb{R}^4$ as an affine and vector space on itself. +In the likely hypothesis of an evolution of a nonchaotic process of +data collection, it is then sufficient to examine the differential of +$q$ with respect to these variables and to keep its restriction to the +integer values of the parameters of the relationship $a \Rightarrow b$. +The differential of $q$, in the sense of Fréchet's +topology\footnote{Fréchet's topology allows $\mathbb{N}$ sections, + i.e. subsets of naturals of the form $\{n,~ n+1,~ n+2,~ ....\}$, to be + used as a filter base, while the usual topology on $\mathbb{R}$ + allows real intervals for filters. + Thus continuity and derivability are perfectly defined and + operational concepts according to Fréchet's topology in the same way + as they are with the usual topology.}, is expressed as follows by +the scalar product: + +\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).} +\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 +component vector the differential increases of these occurrence +variables, and $grad~ q$ the component vector the partial derivatives +of these occurrence variables. + +The differential of the function $q$ therefore appears as the scalar product of its gradient and the increase of $q$ on the surface representing the variations of the function $q(n,~ n_a,~ n_b,~ n_{a \wedge + \overline{b}})$. Thus, the gradient of $q$ represents its own +variations according to those of its components, the 4 cardinals of +the assemblies $E$, $A$, $B$ and $card(A\cap \overline{B})$. It +indicates the direction and direction of growth or decrease of $q$ in +the space of dimension 4. Remember that it is carried by the normal to +the surface of level $q~ =~ cte$. + +If we want to study how $q$ varies according to $ n_{\overline{b}}$, +we just have to replace $n_b$ by $n-n_b$ and therefore change the sign +of the derivative of $n_b$ in the partial derivative. In fact, the +interest of this differential lies in estimating the increase +(positive or negative) of $q$ that we note $\Delta q$ in relation to +the respective variations $\Delta n$, $\Delta n_a$, $\Delta n_b$ and +$\Delta n_{a \wedge + \overline{b}}$. So we have: + + +$$\Delta q= \frac{\partial q}{\partial n} \Delta n + \frac{\partial + q}{\partial n_a} \Delta n_a + \frac{\partial + q}{\partial n_b} \Delta n_b + \frac{\partial + q}{\partial n_{a \wedge + \overline{b}}} \Delta n_{a \wedge + \overline{b}} +o(\Delta q)$$ + +where $o(\Delta q)$ is an infinitely small first order. +Let us examine the partial derivatives of $n_b$ and $n_{a \wedge + \overline{b}}$ the number of counter-examples. We get: + +\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 + \label{eq2.3} +\end{equation} + + +\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 + \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 +also positive. This is interpreted as follows: if the number of +examples of $b$ and the number of counter-examples of implication +increase then the intensity of implication decreases for $n$ and $n_a$ +constant. In other words, this intensity of implication is maximum at +observed values $n_b$ and $ n_{a \wedge + \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})