Intrinsic Evaluation of Clusterings – Lecture Notes

\right\rVert ^2 } \\ \text{TSS} =& \hspace{1.6em}\text{WCSS}\hspace{2.1em} +\hspace{2.5em}\text{BCSS} \end{align*}\] \(\text{TSS}\) Total Sum of Squares \(=\frac{1}{2|X|}\sum _{x,y}\) \((x-y)^2\) \(\text{WCSS}\) [...] |x-\mu _X|^2 + |X| \cdot \sum\nolimits_{y\in Y} |y - \mu _X|^2 \\ =& |Y| \sum\nolimits_{x\in X} |x-\mu _X|^2 + |X| \cdot \sum\nolimits_{y\in Y} |y - \mu _Y|^2 + |X||Y| \cdot |\mu _Y - \mu _X|^2 \\ =& |Y|\cdot [...] _X - \mu _Y|^2 \\ 2|X| \cdot \text{TSS} =& \sum\nolimits_{x\in X} \sum\nolimits_{y \in X} |x-y|^2 = \sum\nolimits_{C_i,C_j} \sum\nolimits_{x\in C_i} \sum\nolimits_{y\in C_j} |x-y|^2 \\ =& 2|X|\sum\noli …

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+ Pi) + 2Pi+1wi+2wi − Piwi+2wi − 2wi+1Pi+1wi − Piwiwi+2 + 2Piwiwi+1 + O((t1 − t0) 3) = wi+2wi(Pi+2 − 2Pi+1 + Pi) + 2wi+2wi(Pi+1 − Pi) − 2wi+1wi(Pi+1 − Pi) + O((t1 − t0) 3) = wi+2wi(Pi+2 − 2Pi+1 + Pi) − [...] − t0) 2 n(n − 1) (a′′b − ab′′)(t0) = (wi+2Pi+2 − 2wi+1Pi+1 + wiPi)wi − Piwi(wi+2 − 2wi+1 + wi) + O((t1 − t0) 3) = (wi+2Pi+2 − 2wi+1Pi+1)wi − Piwi(wi+2 − 2wi+1) + O((t1 − t0) 3) = wi+2wi(Pi+2 − 2Pi+1 + [...] (v) ∑ k1+k2=i ∑ l1+l2=j〈ŵk1,l1(Pk1,l1 − P ), ŵk2,l2(Pk2,l2 − P )〉 ∑ i,j S2n i (u)S2m j (v) ∑ k1+k2=i ∑ l1+l2=j ŵk1,l1ŵk2,l2 . Die skalierte Bézierdarstellung des Abstandsquadrates hat also die Koeffizi- …

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