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note_challenge.lyx
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#LyX 2.0 created this file. For more info see http://www.lyx.org/
\lyxformat 413
\begin_document
\begin_header
\textclass article
\begin_preamble
\usepackage{lmodern}
\usepackage{dsfont}
\end_preamble
\use_default_options true
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\end_header
\begin_body
\begin_layout Title
Challenge MIMSE 2014
\end_layout
\begin_layout Author
Adrien Todeschini
\end_layout
\begin_layout Standard
\begin_inset CommandInset toc
LatexCommand tableofcontents
\end_inset
\end_layout
\begin_layout Section
Fonction de coût
\end_layout
\begin_layout Standard
Soit
\begin_inset Formula $\hat{Y}\in\mathcal{Y}$
\end_inset
la sortie prédite d'un individu et
\begin_inset Formula $Y\in\mathcal{Y}$
\end_inset
sa vraie valeur.
Soit
\begin_inset Formula $L\,:\,\mathcal{Y}\times\mathcal{Y}\rightarrow\mathbb{R}$
\end_inset
une fonction de coût telle que
\begin_inset Formula $L(Y,\hat{Y})$
\end_inset
est le coût associé à la décision :
\begin_inset Quotes eld
\end_inset
prédire
\begin_inset Formula $\hat{Y}$
\end_inset
alors que la vraie valeur est
\begin_inset Formula $Y$
\end_inset
\begin_inset Quotes erd
\end_inset
.
\end_layout
\begin_layout Standard
En classification, on a
\begin_inset Formula $\mathcal{Y}=\{1,...,K\}$
\end_inset
et la fonction de coût peut être résumée par une matrice de coûts
\begin_inset Formula $C$
\end_inset
de taille
\begin_inset Formula $K\times K$
\end_inset
telle que
\begin_inset Formula $C_{kl}=L(k,l)$
\end_inset
(voir table
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Matrice-de-coûts"
\end_inset
).
\end_layout
\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
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<lyxtabular version="3" rows="6" columns="6">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="2cm">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
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<column alignment="center" valignment="top" width="0">
<row>
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\begin_inset Text
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<cell alignment="center" valignment="top" usebox="none">
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\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell multicolumn="1" alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
classe prédite
\end_layout
\end_inset
</cell>
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
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\begin_inset Text
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\end_layout
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $l=1$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $l=2$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $...$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $l=K$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell multirow="3" alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
vraie classe
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $k=1$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $C_{11}\leq0$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $C_{12}\geq0$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $...$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $C_{1K}\geq0$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell multirow="4" alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $k=2$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $C_{21}\geq0$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $C_{22}\leq0$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $...$
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</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $C_{2K}\geq0$
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<row>
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\begin_inset Text
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<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
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\begin_inset Formula $...$
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\vdots$
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\vdots$
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</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\ddots$
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</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
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<row>
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\begin_layout Plain Layout
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</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $k=K$
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\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $C_{K1}\geq0$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $C_{K2}\geq0$
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\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $...$
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\end_layout
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</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $C_{KK}\leq0$
\end_inset
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption
\begin_layout Plain Layout
Matrice de coûts
\begin_inset CommandInset label
LatexCommand label
name "tab:Matrice-de-coûts"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
La fonction de coût s'écrit :
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
L(Y,\hat{Y})=C_{Y\hat{Y}}=\sum_{k=1}^{K}\sum_{l=1}^{K}C_{kl}\mathds1(Y=k\mbox{ et }\hat{Y}=l)
\]
\end_inset
\end_layout
\begin_layout Section
Matrice de confusion
\end_layout
\begin_layout Standard
Soit un échantillon
\begin_inset Formula $(Y_{i},\hat{Y}_{i}),$
\end_inset
\begin_inset Formula $i=1,...,n$
\end_inset
de couples classe réelle, classe prédite.
La matrice de confusion comptabilise les occurences des prédictions en
fonction des vraies valeurs dans un tableau de contingence (voir table
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Matrice-de-confusion"
\end_inset
).
\end_layout
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wide false
sideways false
status open
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classe prédite
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\begin_layout Plain Layout
\end_layout
\end_inset
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<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
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\begin_inset Formula $l=1$
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<cell alignment="center" valignment="top" bottomline="true" usebox="none">
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\begin_layout Plain Layout
\begin_inset Formula $l=2$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $...$
\end_inset
\end_layout
\end_inset
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<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $l=K$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell multirow="3" alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
vraie classe
\end_layout
\end_inset
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<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $k=1$
\end_inset
\end_layout
\end_inset
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $p_{11}\times n$
\end_inset
\end_layout
\end_inset
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $p_{12}\times n$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $...$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $p_{1K}\times n$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell multirow="4" alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $k=2$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $p_{21}\times n$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $p_{22}\times n$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $...$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $p_{2K}\times n$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell multirow="4" alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $...$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\vdots$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\vdots$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\ddots$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\vdots$
\end_inset