From 9e9b8dc87f77206ba17bb278dc62b2bee644f487 Mon Sep 17 00:00:00 2001 From: couturie Date: Sat, 9 Mar 2019 22:25:07 +0100 Subject: [PATCH] new --- book.tex | 1 + chapter2.tex | 140 ++++++++++++++++++++++++++++++++++++++++++++++++- references.tex | 9 ++++ 3 files changed, 149 insertions(+), 1 deletion(-) diff --git a/book.tex b/book.tex index e7afb00..0cdc1fd 100644 --- a/book.tex +++ b/book.tex @@ -30,6 +30,7 @@ \usepackage{newtxtext} % \usepackage{newtxmath} % selects Times Roman as basic font +\usepackage{diagbox} % see the list of further useful packages % in the Reference Guide diff --git a/chapter2.tex b/chapter2.tex index f492308..096f51f 100644 --- a/chapter2.tex +++ b/chapter2.tex @@ -215,7 +215,8 @@ between these two parts) and of the same respective cardinals as $A$ and $B$. Let $\overline{Y}$ and $\overline{B}$ be the respective complementary of $Y$ and $B$ in $E$ of the same cardinal $n_{\overline{b}}= n-n_b$. We will then say: -Definition 1: $a \Rightarrow b$ is acceptable at confidence level + +\definition $a \Rightarrow b$ is acceptable at confidence level $1-\alpha$ if and only if $$Pr[Card(X\cap \overline{Y})\leq card(A\cap \overline{B})]\leq \alpha$$ @@ -297,3 +298,140 @@ 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 $$q(a,\overline{b}) = \frac{n_{a \wedge \overline{b}}- \frac{n_a.n_{\overline{b}}}{n}}{\sqrt{\frac{n_a.n_{\overline{b}}}{n}}}$$ +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 +$\frac{n_a.n_{\overline{b}}}{n}\geq 4$), the variable +$Q(a,\overline{b})$ approximately follows the reduced centered normal +distribution. The intensity of implication, measuring the quality of +$a\Rightarrow b$, for $n_a\leq n_b$ and $nb \neq n$, is then defined +from the index $q(a,\overline{b})$ by: + +\definition +The implication intensity that measures the inductive quality of a +over b is: +$$\varphi(a,b)=1-Pr[Q(a,\overline{b})\leq q(a,\overline{b})] = +\frac{1}{\sqrt{2 \pi}} \int^{\infty}_{ q(a,\overline{b})} +e^{-\frac{t^2}{2}} dt,~ if~ n_b \neq n$$ +$$\varphi(a,b)=0,~ otherwise$$ +As a result, the definition of statistical implication becomes: +\definition +Implication $a\Rightarrow b$ is admissible at confidence level +$1-\alpha $ if and only if: +$$\varphi(a,b)\geq 1-\alpha$$ + + +It should be recalled that this modeling of quasi-implication measures +the astonishment to note the smallness of counter-examples compared to +the surprising number of instances of implication. +It is a measure of the inductive and informative quality of +implication. Therefore, if the rule is trivial, as in the case where +$B$ is very large or coincides with $E$, this astonishment becomes +small. +We also demonstrate~\cite{Grasf} that this triviality results in a +very low or even zero intensity of implication: If, $n_a$ being fixed +and $A$ being included in $B$, $n_b$ tends towards $n$ ($B$ "grows" +towards $E$), then $\varphi(a,b)$ tends towards $0$. We therefore +define, by "continuity":$\varphi(a,b) = 0$ if $n_b = n$. Similarly, if +$A\subset B$, $\varphi(a,b)$ may be less than $1$ in the case where +the inductive confidence, measured by statistical surprise, is +insufficient. + +{\bf \remark Total correlation, partial correlation} + + +We take here the notion of correlation in a more general sense than +that used in the domain that develops the linear correlation +coefficient (linear link measure) or the correlation ratio (functional +link measure). +In our perspective, there is a total (or partial) correlation between +two variables $a$ and $b$ when the respective events they determine +occur (or almost occur) at the same time, as well as their opposites. +However, we know from numerical counter-examples that correlation and +implication do not come down to each other, that there can be +correlation without implication and vice versa~\cite{Grasf} and below. +If we compare the implication coefficient and the linear correlation +coefficient algebraically, it is clear that the two concepts do not +coincide and therefore do not provide the same +information\footnote{"More serious is the logical error inferred from + a correlation found to the existence of a causality" writes Albert + Jacquard in~\cite{Jacquard}, p.159. }. + +The quasi-implication of non-symmetric index $q(a,\overline{b})$ does +not coincide with the correlation coefficient $\rho(a, b)$ which is +symmetric and which reflects the relationship between variables a and +b. Indeed, we show~\cite{Grasf} that if $q(a,\overline{b}) \neq 0$ +then +$$\frac{\rho(a,b)}{q(a,\overline{b})} = \sqrt{\frac{n}{n_b + n_{\overline{a}}}} q(a,\overline{b})$$ +With the correlation considered from the point of view of linear +correlation, even if correlation and implication are rather in the +same direction, the orientation of the relationship between two +variables is not transparent because it is symmetrical, which is not +the bias taken in the SIA. +From a statistical relationship given by the correlation, two opposing +empirical propositions can be deduced. + +The following dual numerical situation clearly illustrates this: + + +\begin{table}[htp] +\center +\begin{tabular}{|l|c|c|c|}\hline +\diagbox[width=4em]{$a_1$}{$b_1$}& + 1 & 0 & marge\\ \hline + 1 & 96 & 4& 100 \\ \hline + 0 & 50 & 50& 100 \\ \hline + marge & 146 & 54& 200 \\ \hline +\end{tabular} ~ ~ ~ ~ ~ ~ ~ \begin{tabular}{|l|c|c|c|}\hline +\diagbox[width=4em]{$a_2$}{$b_2$}& + 1 & 0 & marge\\ \hline + 1 & 94 & 6& 100 \\ \hline + 0 & 52 & 48& 100 \\ \hline + marge & 146 & 54& 200 \\ \hline +\end{tabular} + +\caption{Numeric example of difference between implication and + correlation} +\label{chap2tab1} +\end{table} + +In Table~\ref{chap2tab1}, the following correlation and implications +can be computed:\\ +Correlation $\rho(a_1,b_1)=0.468$, Implication +$q(a_1,\overline{b_1})=-4.082$\\ +Correlation $\rho(a_2,b_2)=0.473$, Implication $q(a_2,\overline{b_2})=-4.041$ + + +Thus, we observe that, on the one hand, $a_1$ and $b_1$ are less +correlated than $a_2$ and $b_2$ while, on the other hand, the +implication intensity of $a_1$ over $b_1$ is higher than that of $a_2$ +over $b_2$ since $q1