# A Robust Wind Turbine Component Health Status Indicator
## 2 Materials and Methods
### The Gaussian Mixture Copula Model
Selected variables are color-coded by component. Numbers 1 and 2 refer to the two studied sites. the \(j\)-th variable. A single superscript indicates the index along the variables while a single subscript is used for the index along the observations. Random variables are denoted by uppercase letters and their realizations are represented by lowercase letters. Bold font is used for vectors of random variables, observations, or parameters. Plain font is used for scalars. The Gaussian Mixture Model (GMM) is a probabilistic model used for representing complex data distributions. It assumes that the data are generated from a mixture of several Gaussian (Normal) distributions. The GMM represents the probability density function as a weighted sum of Gaussian distributions. Each Gaussian component represents a cluster or mode in the data. The probability density function (PDF) of a \(K\)-component GMM is given by:  
\[f_{GMM}(\mathbf{x};\mathbf{\theta})=\sum_{k=1}^{K}\pi_{k}\mathbf{\phi}(\mathbf{x};\mathbf{\mu}_{k},\mathbf{\Sigma}_{k}) \tag{1}\]  
where \(\pi_{k}\) are the mixing proportions, with \(\pi_{k}>0\) and \(\sum_{k=1}^{K}\pi_{k}=1\), while \(\mathbf{\mu}_{k}\) and \(\mathbf{\Sigma}_{k}\) are the component-specific vectors of means and covariance matrices, respectively. For simplicity, we use the parameter \(\mathbf{\theta}=(\pi_{1},\ldots,\pi_{k},\mathbf{\mu}_{1},\ldots,\mathbf{\mu}_{k},\mathbf{ \Sigma}_{1},\ldots,\mathbf{\Sigma}_{k})\) to jointly represent all parameters. A copula is a mathematical function that characterizes the joint distribution of multiple random variables by representing their dependence structure separately from their marginal distributions. Mathematically, a copula can be described as follows. Let \(F_{j}\) be the marginal cumulative distribution (CDF) of \(\mathbf{X}^{j}\). The copula function \(C(\mathbf{U}^{j})\) is a multivariate function defined on the unit hypercube \([0,1]^{n}\) where \(\mathbf{U}^{j}\) are the CDFs of \(\mathbf{X}^{j}\), \(\mathbf{U}^{j}=F_{j}(\mathbf{X}^{j})\). The copula function \(C(\mathbf{U}^{1},\ldots,\mathbf{U}^{p})\) maps the marginals to the joint CDF, such that for every joint distribution with continuous marginals, \(F(\mathbf{X}^{1},\ldots,\mathbf{X}^{p})\), there exists a unique copula function such that \(F(\mathbf{X}^{1},\ldots,\mathbf{X}^{p})=C(F_{1}(\mathbf{X}^{1}),\ldots,F_{p}(\mathbf{X}^{p}))\). It can also be shown that the corresponding joint density can be written as the copula density, \(c\), multiplied by the individual marginal densities \(f_{j}\):  
\[f(\mathbf{x})=c(F_{1}(\mathbf{x}^{1}),\ldots,F_{p}(\mathbf{x}^{p}))\prod_{j=1}^{p}f_{j}( \mathbf{x}^{j}) \tag{2}\]  
The Equation (2) illustrates how copulas provide a flexible way to construct multivariate density functions. Copulas achieve this by separating the specification of marginal distributions, \(f_{j}\) from the characterization of the dependence structure, \(c\). This decoupling allows independently choosing the parametric family for each aspect, enabling the customization of the marginals and the dependency modeling according to the specific needs. From Equation (2) the copula family can be obtained as:  
\[c(F_{1}(\mathbf{x}^{1}),\ldots,F_{p}(\mathbf{x}^{p}))=\frac{f(\mathbf{x})}{\prod_{j=1}^{p }f_{j}(\mathbf{x}^{j})} \tag{3}\]  
If the joint density function is chosen to be equal to the multivariate normal density \(\mathbf{\phi}\), with normal densities \(\mathbf{\phi}_{j}\), the Gaussian copula density is given as \(c_{\mathbf{\phi}}=\frac{\mathbf{\phi}(\mathbf{x})}{\prod_{j=1}^{p}\mathbf{\phi}_{j}(\mathbf{x}^{j})}\). Copula-based models can be viewed as generative models where the goal is to learn the underlying distribution of the observed data and then use that distribution to generate new samples that resemble the original data. These models are defined on the CDF-transformed data, \(\mathbf{U}^{j}=F_{j}(\mathbf{U}^{j})\), also called pseudo or latent observations.