# Gaussian Process Model-Based Performance Uncertainty Quantification of a Typical Turboshaft Engine
## 3 Methodologies
### Gaussian Process Model-Based Latin Hypercube Sampling  
In this study, the LHS is employed to perform uncertainty quantification. In general, the LHS is a kind of stratified sampling, the key of which is to stratify the probability distributions of input variables [31; 32]. Considering an input variable that follows a certain distribution, the CDF of the variable is firstly stratified into a set of nonoverlapping intervals with the same length. The stratification can be projected to the variable through the inverse CDF. Thus, the variable can also be divided into the same number of equiprobable intervals. Note that these intervals are not necessarily with equal length. Next, one random value is needed to be selected from each equiprobable interval and a series of random values can then be obtained for the variable. For multiple input variables, each variable corresponds to a list of random values. After randomly combining the values from different variables with each other, a set of sample points can be obtained and each sample point contains a group of random values from different variables. Since the LHS spreads the random values more evenly across all possible values within the ranges of variables, it is more efficient than the MCS in most instances. Let \(X\in\left[0,1\right]^{d}\) denote a \(d\)-dimensional vector of normalized continuous random variables, and \(\mathbf{F}(\cdot)=\{F_{1}(\cdot),F_{2}(\cdot),\ldots,F_{d}(\cdot)\}\) denote a \(d\)-dimensional vector of CDFs of \(X\). Assuming that a sample size of \(n\) is specified, a general procedure of the LHS can be summarized as follows:  
1. Stratify the vertical axis on the plot of \(F_{i}(\cdot)\) of \(X_{i}\) into \(n\) nonoverlapping intervals with the same length: \([V_{i,0},V_{i,1})\), \([V_{i,1},V_{i,2}),\ldots,\)\([V_{i,n-1},V_{i,n}]\), where \(V_{i,j}=F_{i}(0)+j(F_{i}(1)-F_{i}(0))/n,i=1,2,\ldots,d,j=0,1,\ldots,n;\)
2. Map the boundary values \(V_{i,j}\) on the vertical axis to the values of \(X_{i}\) defined on the horizontal axis using the inverse CDF \(F_{i}^{-1}(\cdot)\);
3. Stratify the random variable \(X_{i}\) using the mapped values and obtain intervals: \(\left[0,F_{i}^{-1}(V_{i,1})\right),\)\(\left[F_{i}^{-1}(V_{i,1}),F_{i}^{-1}(V_{i,2})\right),\ldots,\)\(\left[F_{i}^{-1}(V_{i,n-1}),1\right];\)
4. Select a random value from each interval of \(X_{i}\) and generate a \(n\times d\) matrix where the \(i\)th column contains \(n\) random values of \(X_{i}\);
5. Order the values in every column of the matrix randomly and complete the LHS. When performing uncertainty quantification by coupling LHS with GPM, a set of samples are firstly generated according to the CDFs of inputs. Then, the response variable can be evaluated for every sample through the GPM. With the evaluated values, the statistics of response variable, such as mean and variance, can be obtained. Since posterior distribution of the response variable can be predicted by the GPM, the statistics can then be viewed as random variables and their distributions can also be derived. In this situation, the expectation can be regarded as the estimation of the statistic, while the variance can be used as an indicator of the estimation accuracy. Thus, it is possible to directly estimate the uncertainties of the two statistics (mean and variance) resulting from the surrogate model uncertainty. The estimation of model uncertainty is an advantage of the GPM over other models. Considering a \(n\)-dimensional response variables \(\mathbf{Y}_{n}=(y_{1},y_{2},\ldots,y_{n})^{T}\), a multivariate normal distribution of \(\mathbf{Y}_{n}\), where \(\mathbf{Y}_{n}\sim N(\mu,\Sigma)\), can be obtained based on the GPM.