From d2603b08d49ab1470a226e60976d1af940e4301b Mon Sep 17 00:00:00 2001
From: Laurent Modolo <laurent.modolo@ens-lyon.fr>
Date: Mon, 16 Oct 2023 09:50:49 +0200
Subject: [PATCH] update M1_biosciences_clustering
---
M1_biosciences_clustering/kmeans.tex | 14 +++++++-------
1 file changed, 7 insertions(+), 7 deletions(-)
diff --git a/M1_biosciences_clustering/kmeans.tex b/M1_biosciences_clustering/kmeans.tex
index 455d9fc..e1836ce 100644
--- a/M1_biosciences_clustering/kmeans.tex
+++ b/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}
--
GitLab