# A Robust Wind Turbine Component Health Status Indicator
## 2 Materials and Methods
### The Gaussian Mixture Copula Model
#### 2.5.1 Estimation of the GMCM Parameters  
One of the major challenges in using the GMCM is the determination of the parameters and it may require a high computational cost. This is conducted by finding the parameters that maximize the likelihood function given in Equation (6). The optimization of the likelihood function is not trivial and there are intrinsic problems such as convergence and identifiability of the GMCM parameters which may affect any estimation procedure [52]. In this study two optimization algorithms have been chosen, the Pseudo  
Figure 6: Multivariate copula. Expectation Maximization algorithm (PEM) and the Nelder-Mead (NM) algorithm, which are described below. The EM algorithm is a method for performing maximum likelihood estimation in the presence of latent variables. It does this by first estimating the values of the latent variables (Expectation step), then optimizing the model (Maximization step), and finally repeating these two steps until convergence is achieved. This is an efficient and general approach and is mostly used for density estimation with missing data, such as clustering algorithms like the Gaussian Mixture Model [53]. When the EM algorithm is applied to GMM, the inputs of the model, \((\mathbf{X}^{1},\ldots,\mathbf{X}^{p})\), remain fixed, as the parameters are iteratively updated through a sequence of alternating Expectation and Maximization steps. On the other hand, the inputs to a GMCM are the marginal values of the CDF, \((\mathbf{U}^{1},\ldots,\mathbf{U}^{p})\), which are used to obtain the values of the inverse distribution, \((\mathbf{Y}^{1},\ldots,\mathbf{Y}^{p})\). Since the inverse distribution functions along the margins change with each update of the parameters, so do the values of the inverse distribution. As a result, the assumption of fixed observations, made by the EM algorithm for GMM, is violated for GMCM and can not be applied directly here. In order to overcome this problem, modifications to the EM algorithm have been developed and are typically called the Pseudo Expectation Maximization (PEM) algorithm. The approach used here uses the initialization parameters to compute the pseudo-data. Then it iterates between the two EM stages, maximizing the pseudo-likelihood of obtaining new model parameters based on the pseudo-data and updating pseudo-data in the expectation stage [54]. The use of nonconstant pseudo-data in the likelihood does not guarantee the convergence of the process and can even achieve convergence to incorrect parameters. The Nelder-Mead optimization algorithm is a popular approach for optimizing non-differentiable objective functions [55]. It is employed for global or local searches in challenging problems involving noisy, nonlinear, and multi-modal functions. A notable advantage of the Nelder-Mead algorithm is that it does not rely on function gradient information, making it applicable to situations where the gradient is unknown or difficult to compute accurately. The NM algorithm is a simplex-based method developed to solve the optimization problem of minimizing (maximizing) a given nonlinear function \(f:\mathbb{R}^{n}\rightarrow\mathbb{R}\) without constraints. The method uses only function values calculated at some selected points in the real multidimensional space \(\mathbb{R}^{n}\) and does not need any gradient estimate at those points. The NM algorithm begins with a simplex \(S\) in \(\mathbb{R}^{n}\), defined as the convex hull of \(n+1\) vertices, \((x_{0},\ldots,x_{n})\in\mathbb{R}^{n}\) (a simplex is a geometric shape defined in the \(\mathbb{R}^{n}\) domain and used as a'vehicle' to perform the search for the optimum solution in that domain). The algorithm then starts iterating and proceeds to reshape/move this simplex, vertex by vertex, toward an optimal region in the search space. At each step, it tries one or more modifications of the current simplex and chooses one that moves it towards a 'better' region of the domain. Finally, the simplex vertex that yields the most optimal target value is returned.