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Greetings! Could anyone enlighten me about the validity of this equation? I'm trying to prove it without success. $K$ is the number of clusters, $C_i$ is the $i$-th cluster, $m_i$ is the number of objects of $i$-th cluster and m is the total number of objects. Thanks in advance!

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  • $\begingroup$ It's the sum of all objects (x) that belong to i-th cluster! $\endgroup$
    – Ioannis Tsirovasilis
    Dec 25, 2019 at 10:22
  • $\begingroup$ @IoannisTsirovasilis $m_i$ should be means, not counts; otherwise the equality doesn't hold. Let $x_1=1,x_2=-1$, and let they belong to different clusters, i.e. $m_1=1,m_2=1$. Apparently, the equation doesn't hold. $\endgroup$
    – gunes
    Dec 25, 2019 at 10:43
  • $\begingroup$ Thank you! You are absolutely right! $\endgroup$
    – Ioannis Tsirovasilis
    Dec 25, 2019 at 11:51
  • $\begingroup$ Please choose a more descriptive title, e.g., by including the meaning of the equation, so that others can more easily find this. $\endgroup$
    – Has QUIT--Anony-Mousse
    Dec 26, 2019 at 8:43
  • $\begingroup$ @Anony-Mousse-ReinstateMonica Thanks for the tip I will keep that in mind! Happy new year's eve! $\endgroup$
    – Ioannis Tsirovasilis
    Dec 28, 2019 at 13:27

1 Answer 1

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Via a couple of algebraical tricks, you'll have your zero only if $\mu_i$ and $\mu$ are the means, not the counts: $$\begin{align}S&=\sum_{i=1}^K\sum_{x\in C_i}(x-\mu_i)\cdot(\mu-\mu_i)\\&=\sum_{i=1}^K\overbrace{\left(\sum_{x\in C_i} (x-\mu_i)\right)}^0\cdot (\mu-\mu_i)\\&=0\end{align}$$

If it were the counts, letting $x_1=1\in C_1,x_2=-1\in C_2$ will cause invalidity of the equation shown. The equation is also shift-variant if $m$ are counts.

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