2.2.1. Response Surface Method

The multivariate linear regression method [25–27] is adopted to convert the nonlinear system into a linear system. The partial least squares method is adopted to optimize and find the matching function. The method of minimum error squared is adopted to fit the model prediction. The principle of the method is as follows.


$$M\_{0k} = \alpha\_{k1} V\_{01} + \dots + \alpha\_{k\beta} V\_{0\beta} + M\_k \tag{1}$$

where *αk*1,... , *αk<sup>β</sup>* are the regression coefficients.

#### 2.2.2. Accuracy Test

After obtaining the data, the accuracy of the model needs to be tested. The multivariate RSM model was used in this work, so it is reasonable to use an *R*<sup>2</sup> test to judge the accuracy of the model. which is given as:

$$R^2 = 1 - \frac{\sum\_{j=1}^{m} \left[ f(\mathbf{x})\_j - \hat{f}(\mathbf{x})\_j \right]^2}{\sum\_{j=1}^{m} \left[ f(\mathbf{x})\_j - \overline{\hat{f}}(\mathbf{x}) \right]^2} \tag{2}$$

where *f*(*x*)*<sup>j</sup>* is the *j*th sample point response value, ˆ *f*(*x*)*<sup>j</sup>* is the calculated value of the response surface corresponding to the *<sup>j</sup>*th sample point, and <sup>−</sup> *f*(*x*) is the average value, which is obtained from <sup>−</sup> *f*(*x*) = <sup>1</sup> *<sup>m</sup>* <sup>∑</sup>*<sup>m</sup> <sup>j</sup>*=<sup>1</sup> *f*(*x*)*<sup>j</sup>* . The calculated *R*<sup>2</sup> is between 0 and 1, and the closer the value is to 1, the more accurate it is.

#### 2.2.3. Monte Carlo Sampling Analysis

In order to improve the accuracy of the predicted values represented by the response surface function, sample selection is significant. The MC method was selected for sampling in this study. To begin with, *x*<sup>i</sup> (i = 1, 2, ... , n) are the independent random parameters in the influence model function *f*(*x*), and each corresponding independent parameter *x*<sup>i</sup> (i = 1, 2, ... , n) is randomly sampled to obtain a random sample of *x*1, *x*2, ... , *x*n. Then, *x*<sup>i</sup> is substituted into a specific program for repeated random sampling and analysis. The response values of the M group structure are obtained, which are denoted as *f*(*x*)1, *f*(*x*)2, ... , *f*(*x*)*m*. Furthermore, the average value μ and standard deviation σ of *f*(*x*) are calculated, which can be expressed as:

$$\mu = \bar{f}(\boldsymbol{\pi}) = \frac{1}{m} \sum\_{j=1}^{m} f(\boldsymbol{\pi})\_j \tag{3}$$

$$\sigma = \sqrt{\frac{1}{m-1} \sum\_{j=1}^{m} \left[ f(\mathbf{x})\_j - \mu \right]^2} \tag{4}$$
