M1_biosciences_dimension_reduction: update

M1_biosciences_dimension_reduction/PCA.tex

@@ -397,7 +397,7 @@ $$
 	$$
 	\item The correlation coefficient is a distance measure between variables:
 	$$
-	\operatorname{r}(\xbf^{j'},\xbf^{j}) = \frac{1}{2} \times  \left( 1 -  \frac{1}{n}\|\widetilde{\xbf}^j_c-\widetilde{\xbf}_c^{j'}\|^2 \right)
+	\operatorname{r}(\xbf^{j'},\xbf^{j}) = 1 - \frac{1}{2} \times  \frac{1}{n}\|\widetilde{\xbf}^j_c-\widetilde{\xbf}_c^{j'}\|^2 
 	$$
 	\end{itemize}
 \end{frame}

M1_biosciences_dimension_reduction/perspectives.tex

@@ -95,10 +95,10 @@ $\rightarrow$ Low-rank representation of $\mb X$
 \begin{columns}[c]
 	\begin{column}{.5\textwidth}
 		\begin{itemize}
-			\item In many situations only the distance $d_{ij}$ between individuals is available 
+			\item In many situations only the distance $d_{ii'}$ between individuals $(i,i')$ is available 
 			\item The objective of MDS is to find new coordinates $\ubf_1,\hdots,\ubf_n$ that minimize:
 			$$
-			\sum_{ij} \Big( d_{ij} - \|\ubf_i-\ubf_j \|^2 \Big)^2
+			\sum_{i,i'} \Big( d_{ii'} - \|\ubf_i-\ubf_{i'} \|^2 \Big)^2
 			$$
 			\item The information regarding the variables is not considered (not available)
 		\end{itemize}
@@ -120,7 +120,7 @@ $\rightarrow$ Low-rank representation of $\mb X$
 			\item These distances depend on a dot product
 			\item This dot product can be generalized by the so-called kernel
 			$$
-			K(\xbf_i,\xbf_j) =\langle \phi(\xbf_i), \phi(\xbf_j)\rangle
+			K(\xbf_i,\xbf_{i'}) =\langle \phi(\xbf_i), \phi(\xbf_{i'})\rangle
 			$$
 			\item $\phi$ is called the feature map and is unknown
 			\item Grounds most non linear methods (kernel-PCA, kernel MDS, etc)