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### Fuzzy Clustering Using the EM Algorithm

- Initially, let c1 = a and c2 = b
- 1st E-step: assign o to c1,

\[w. wt =\frac{\frac{1}{dist(o,c_{1})^2}}{\frac{1}{dist(o,c_{1})^2}+\frac{1}{dist(o,c_{2})^2}}=\frac{dist(o,c_{2})^2}{dist(o,c_{1})^2+dist(o,c_{2})^2}\]

\[w_{c,c_{1}}=\frac{41}{45+41}=0.48\] - 1st M-step: recalculate the centroids according to the partition matrix, minimizing the sum of squared error (SSE)

\[ c_{j}=\frac{\sum_{eachpoint_{o}}w_{o,c_{j}}^{2}o }{\sum_{eachpoint_{o}}w_{o,c_{j}}^{2}} \]

\[c_{1}=(\frac{1^{2}\times 3+ 0^{2}\times4+ 0.48^{2}\times9+ 0.42^{2}\times14+ 0.41^{2}\times18+ 0.47^{2}\times21}{1^{2}+0^{2}+0.48^{2}+0.42^{2}+0.41^{2}+0.47^{2}},\frac{1^{2}\times3+ 0^{2}\times10+ 0.48^{2}\times6+ 0.42^{2}\times8+ 0.41^{2}\times11+ 0.47^{2}\times7}{1^{2}+0^{2}+0.48^{2}+0.42^{2}+0.41^{2}+0.47^{2}}) =(8.47,5.12) \]

- Iteratively calculate this until the cluster centers converge or the change is small enough

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