## 3 Experiments  
The experiment of this research starts by generating samples with multivariate data with the illustrated properties in Table 2. These samples are used to evaluate the performance of the proposed method. Each variable will be generated with specified parameters. However, the variables in each sample must have the given correlation coefficients. It is essential for the validation procedure to generate multivariate data using theoretical random distribution parameters and specific correlations. The inverse transform method is used for this purpose. The inverse transform method is a statistical technique used to generate random numbers from a given probability distribution. It uses the cumulative distribution function (CDF) of the desired distribution to transform a uniform random variable into a random variable with the desired distribution. For this purpose, a computer algorithm is used to generate the multivariate data with the desired correlation coefficients between variables (Figure 1). To begin, a set of random vectors is generated from the multivariate normal distribution, with a defined mean vector and covariance matrix. The number of vectors generated is equal to the desired sample size, and each vector has dimensions of 1 by \(p\), where p represents the number of variables or quality characteristics. The mean vector is set to zero, and the desired correlation matrix represents the variance-covariance matrix between all variables that need to be generated. Next, for each variable \(X_{i}\), where \(i\) ranges from 1 to \(p\), an \(n\)-by-1 vector is generated using the inverse of the desired cumulative distribution function (CDF). The inverse CDF uses the parameters of the corresponding \(X_{i}\) at the values of the cumulative distribution  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Distribution** & **Variable** & **USL** & \(\mathbf{\kappa}\) & \(\mathbf{\beta}\) & **Correlation** & **Actual PCI** \\ \hline gamma & X1 & 13 & 1 & 2 & 1 & 0.49 & & 0.89 \\  & X2 & 26 & 2 & 3 & 0.49 & 1 & & 0.89 \\ \hline gamma & X1 & 130 & 5 & 7 & 1 & \(-\)0.37 & 0.58 & & \\  & X2 & 58 & 6 & 3 & \(-\)0.37 & 1 & \(-\)0.28 & & 1.18 \\  & X3 & 150 & 2 & 8 & 0.58 & \(-\)0.28 & 1 & & \\ \hline Beta & X1 & 0 function of the normal distribution. Despite this, the output correlation between variables is almost equal to the desired correlation, and it varies in each run of the previous procedures. Therefore, the algorithm is run for several iterations until the difference between the desired and output correlation is less than suggested tolerance \(\varepsilon\). Furthermore, the algorithm for generating multivariate data with the desired correlation is used to generate data from the four samples' parameters in Table 2. To examine the performance of the suggested multivariate PCI with different sample sizes, four different sample sizes have been generated from each sample. The sample sizes are 50, 100, 500, and 1000. For example, one sample of two variables from the gamma distribution with a sample size of 50 has been generated. Then, the generated data \((X_{j})\) is transformed to normal distribution \((Y_{i})\). This is done by reducing the skewness of the data. Since a normal distribution has zero skewness, the skewness of the generated data samples must be reduced. There are some common methods for reducing skewness in data such as logarithmic transformation and square-root transformation. In this research, the root transformation is used to reduce the skewness of the data. Root transformation looks for an optimal root (_r_) such that if the data were raised to the power \(r\) (\(X^{r}\)), the skewness in the resulting data distribution would be almost zero. For this purpose, a software code was developed to search for the optimal root that would reduce the skewness of a given variable datapoint \((X_{j})\). This code is based on the iterative search either until the skewness is reduced to around zero or after performing a given number of iterations without improvement.