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\paragraph{} Apart from cross-validation, the optimal number of clusters is sometimes derived by Bayesian Information Criteria (BIC) or by Akaike's Information Criteria (AIC), which penalizes the model complexity through the relations,\\  \begin{eqnarray}  BIC &=& -2 \log L + p \log(N),\\  AIC &=& -2 \log L + \\ 2p,\\  \end{eqnarray}  where $p$ is the number of free parameters in the model. The lowest BIC and AIC values can both be used to find a trade off between modeling the data well and penalizing the complexity of the model. The BIC and AIC, as well as the cross validation results, can be seen in the following figure.