# A Robust Wind Turbine Component Health Status Indicator
## 2 Materials and Methods
### The Gaussian Mixture Copula Model
#### 2.5.2 Model Fitting  
The procedure for fitting the GMCM, including the optimization of the parameters is applied as described in [32; 52]. The main steps considered to model the multivariate copula are as follows:  
* Compute the scaled ranks for each column: A range scaling process is applied to the input data of the model. This helps to reduce the impact of outliers, ensures that the scales between variables are comparable, and maintains the relationship of dependency between the variables.
* Parameter initialization: suitable starting parameters for the GMCM model have to be selected. The K-means algorithm is used to choose the starting values heuristically. The parameters defined within the GMCM model are shown in Table 7. The model fitting heavily relies on the selected initial parameters. Therefore, a seed is defined at the beginning of this process, to ensure result replicability and to obtain model-fitting results for different initial parameters.
* Fitting the model: in this step, the final GMCM model parameters are estimated. Here, the optimization methods considered for model coefficients calculation are the Nelder-Mead (NM) and Pseudo-Expectation-Maximization algorithm (PEM) as described in Section 2.5.1.
* Joint probability calculation: Once the models are fitted, the joint probability calculation is conducted. This step generates the log-likelihood for each multivariate observation on the analyzed period.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**NM Algorithm**} & \multicolumn{2}{c}{**PEM Algorithm**} \\ \hline
**WTG** & **Converge.** & **elaps. Time (s)** & **Converge.** & **elaps. Time (s)** \\ \hline S2\_GBX\_F1 & YES & 66.30 & NO & 480.03 \\ \hline S2\_GBX\_F2 & YES & 65.06 & NO & 91.27 \\ \hline S2\_GBX\_H1 & YES & 231.97 & NO & 159.12 \\ \hline S2\_GBX\_H2 & YES & 77.97 & NO & 216.11 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Performance of optimization methods (convergence = converge., elapsed = elaps.).  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Parameter** & **Description** \\ \hline k & the number of components in the mixture \\ \hline p & the dimension of the mixture distribution (input variables in the model) \\ \hline \(\pi\) & vector of length k related to mixture proportions, where the sum of them is 1 \\ \hline \(\mu\) & list of length k of numeric vectors of length p for each component \\ \hline \(\Sigma\) & list of length k of variance-covariance matrices for each component \\ \hline \hline \end{tabular}
\end{table}
Table 7: Parameters in GMCM models.  
As an example of the adjusted parameters in each case presented here, Table 8 shows all the adjusted parameters for the S2_GBX_F1 case.  
In this study, 20 real cases have been analyzed. As a result, once the variable selection is made (Section 2.4), 20 GMCM models have been generated.  
The GMCM model estimates the joint probability density function from the multivariate data, obtaining a probability density map from the multivariate environment. In this way, it is possible to define a boundary based on the joint probability (joint log-likelihood, in this case) [52]. Based on this joint probability, it is possible to define condition monitoring parameters such as the proposed in this work (see Section 2.5.3). It allows for identifying data whose probability of fulfilling the model is lower, or in other words, to detect periods where a component is working in a failure mode.  
To perform the described steps regarding the use of GMCM models, in this study, we have relied on the tools integrated within the R library \"GMCM\" [52].