update M1_biosciences_clustering

M1_biosciences_clustering/kmeans.tex

@@ -37,7 +37,7 @@ $$
 $$
 \item Step $[h+1]$: update labels $\zbf$ when centers $\overline{\xbf}_1^{[h+1]},\hdots,\overline{\xbf}_K^{[h+1]}$ are updated
 $$
-\widehat{z}_{i}^{[h]} = \arg \underset{1,\hdots,K}{\min} \left\{ d^2(\xbf_i,\overline{\xbf}_k^{[h+1]}) \right\}
+\widehat{z}_{i}^{[h+1]} = \arg \underset{1,\hdots,K}{\min} \left\{ d^2(\xbf_i,\overline{\xbf}_k^{[h+1]}) \right\}
 $$
 
 \end{itemize}
@@ -55,22 +55,22 @@ $$
 \frametitle{Decreasing inertia}
 \begin{itemize}
   \item Denoting by $[h]$ the step $h$ of the algorithm, $\zbf^{[h]},
-  \xbf^{[h]}$
-  \item The inertia depends on both quantities $\Ibf_W(\zbf^{[h]},\xbf^{[h]})$
+  \overline{\xbf}^{[h]}$
+  \item The inertia depends on both quantities $\Ibf_W(\zbf^{[h]},\overline{\xbf}^{[h]})$
   \item Updating centers
   $$
-  \Ibf_W(\zbf^{[h]},\xbf^{[h+1]}) = \sum_{i=1}^n \sum_{k=1}^K z_{ik}^{[h]}
+  \Ibf_W(\zbf^{[h]},\overline{\xbf}^{[h+1]}) = \sum_{i=1}^n \sum_{k=1}^K z_{ik}^{[h]}
   d^2(\xbf_i,\overline{\xbf}_k^{[h+1]}) 
   $$
   \item Updating labels
   $$
-  \Ibf_W(\zbf^{[h+1]},\xbf^{[h+1]}) = \sum_{i=1}^n \sum_{k=1}^K z_{ik}^{[h+1]}
+  \Ibf_W(\zbf^{[h+1]},\overline{\xbf}^{[h+1]}) = \sum_{i=1}^n \sum_{k=1}^K z_{ik}^{[h+1]}
   d^2(\xbf_i,\overline{\xbf}_k^{[h+1]}) 
   $$    
   \item The criterion decreases at each step
   $$
-  \Ibf_W(\zbf^{[h]},\xbf^{[h]}) \geq \Ibf_W(\zbf^{[h]},\xbf^{[h+1]}) \geq
-  \Ibf_W(\zbf^{[h+1]},\xbf^{[h+1]})$$
+  \Ibf_W(\zbf^{[h]},\overline{\xbf}^{[h]}) \geq \Ibf_W(\zbf^{[h]},\overline{\xbf}^{[h+1]}) \geq
+  \Ibf_W(\zbf^{[h+1]},\overline{\xbf}^{[h+1]})$$
 \end{itemize}
 \end{frame}