Ended algorithm description

Author: lamferzon

.gitignore

@@ -5,4 +5,5 @@
 *.out
 *.snm
 *.synctex.gz
-*.toc
+*.toc
+*.vrb

Presentation/Presentation.pdf

@@ -1,5 +1,13 @@
 \section{Fault detection and isolation on steel plates}
 
 \begin{frame}
-	TO DO
+	\frametitle{Dataset description}
+	The steel plates faults dataset comes from research by Semeion, Research Center of Sciences of Communication. The aim of the research was to correctly classify the type of surface defects in stainless steel plates. Below is some information about the dataset:
+	\begin{itemize}
+		\item number of fault classes: $6 + 1$ (no faults);
+		\item number of attributes: $27$;
+		\item number of instances: $1941$;
+		\item absence of missing values.
+	\end{itemize}
+	Unfortunately, no further details on the covariates are available.
 \end{frame}

Presentation/Presentation.vrb

@@ -10,6 +10,21 @@ \section{Description of the PLS  algorithm}
 	The first algorithm is more accurate than the other, however it requires more computational time than PLS2 to find the $\alpha$ eigenvectors into which project the \textit{m} covariates. 
 \end{frame}
 
+\begin{frame}
+	\frametitle{Data structures}
+	Before starting with the description of the algorithm, we recall that:
+	\begin{itemize}
+		\item the matrix $X \in \mathbb{R}^{n\times m}$ is decomposed into a \textbf{score matrix} $T \in \mathbb{R}^{n\times\alpha}$ and a \textbf{loading matrix} $P \in \mathbb{R}^{m\times\alpha}$ such that $X = \hat{X} + E = T\cdot P^\top + E$, where $E \in \mathbb{R}^{n\times m}$ is the (true) \textbf{residual} matrix for $X$;
+		\item the matrix $Y \in \mathbb{R}^{n\times p}$ is decomposed into a \textbf{score matrix} $U\in\mathbb{R}^{n\times\alpha}$ and a \textbf{loading matrix} $Q\in \mathbb{R}^{p\times\alpha}$ such that $Y = \hat{Y} + \widetilde{F} = U\cdot Q^\top + \widetilde{F}$, where $\widetilde{F}\in \mathbb{R}^{n\times p}$ is the (true) \textbf{residual matrix} for $Y$.
+		\item the matrix $B\in \mathbb{R}^{\alpha\times\alpha}$ is the \textbf{diagonal regression matrix} such that $\hat{U} = T\cdot B$.
+	\end{itemize}
+	Therefore:
+	\begin{center}
+		$Y = \hat{U}\cdot Q^\top + F =  T\cdot B\cdot Q^\top + F$
+	\end{center}
+	where $F$ is the \textbf{prediction error matrix}; $B$ is selected such that the induced $2$-norm of $F$ is minimized. 
+\end{frame}
+
 \begin{frame}[fragile]
 	\frametitle{MATLAB code}
 	The following MATLAB code implements the PLS2 algorithm:
@@ -47,8 +62,8 @@ \section{Description of the PLS  algorithm}
 	\textcolor{green}{% calculation of b and the error matrices}
 	b = (u'*t)/(t'*t); \textcolor{green}{% j-th column of the...}
 	    \textcolor{green}{% coefficient regression matrix}
-	E = E - t*p';
-	F = F - b*t*q';
+	E = E - t*p';  \textcolor{green}{% update of the residuals for matrix X}
+	F = F - b*t*q'; \textcolor{green}{% update of the residuals for matrix Y}
 	\end{Verbatim}
 \end{frame}
 
@@ -62,6 +77,6 @@ \section{Description of the PLS  algorithm}
 \textcolor{blue}{end}
 Y_hat = X*B2; \textcolor{green}{% computation of predictions}
 	\end{Verbatim}
-For each row of \verb|Y_hat| the fault class is chosen by assigning $1$ to the column whose value si greater than that of the others, $0$ otherwise. Moreover, to increase the performances of PLS it is necessary \textbf{normalize} both $X$ and $Y$ before running the algorithm.
+For each row of \verb|Y_hat| the fault class is chosen by assigning $1$ to the column whose value si greater than that of the others, $0$ otherwise. \\Moreover, to increase the performances of PLS it is necessary \textbf{normalize} both $X$ and $Y$ before running the algorithm.
 
 \end{frame}