Algorithm 5 GMM

Input           X     input vectors, $\{{\overrightarrow{x}}_1,{\overrightarrow{x}}_2,\dots ,{\overrightarrow{x}}_n\}$           k     number of clusters     Output           C     centroids, $\{{\overrightarrow{c}}_1,{\overrightarrow{c}}_2,\dots ,{\overrightarrow{c}}_k\}$, $c_i=({\varphi }_i,{\mu }_i,{\sigma }_i^2)$     procedure GMM           for $k\leftarrow 1\dots K$ do                 ${\varphi }_k\leftarrow \frac{1}{K}$                 ${\mu }_k\leftarrow$ Random vector from X                 ${\sigma }_k^2\leftarrow$ Sample covariance           end for           while not converged do                 for $k\leftarrow 1\dots K,i\leftarrow 1\dots N$ do                       ${\gamma }_{ik}=\frac{{\varphi }_k\mathcal{N}\left(x_i\right|{\mu }_k{\sigma }_k^2)}{{\mathsf{\Sigma }}_{j=1}^K{\varphi }_j\mathcal{N}\left(x_i\right|{\mu }_k,{\sigma }_k^2)}$                 end for                 for $k\leftarrow 1\dots K$ do                       ${\varphi }_k={\mathsf{\Sigma }}_{i=1}^N\frac{{\gamma }_{ik}}{N}$                       ${\mu }_k=\frac{{\mathsf{\Sigma }}_{i=1}^N{\gamma }_{ik}{x}_i}{{\mathsf{\Sigma }}_{i=1}^N{\gamma }_{ik}}$                       ${\sigma }_k^2=\frac{{\mathsf{\Sigma }}_{i=1}^N{\gamma }_{ik}{(x_i-{\mu }_k)}^2}{{\mathsf{\Sigma }}_{i=1}^N{\gamma }_{ik}}$.                 end for           end while     end procedure