## 2 Parameter Selection and Data Reduction
### Data Augmentation  
Limited by the upgrading of technology and the backwardness of modern equipment on the industrial site, there are not enough data for parameter optimization and calculation. The primary purpose of the method in this section is to generate a large amount of data so as to provide sufficient samples for data-driven methods to calculate. The GAN was proposed by [PERSON] et al. in 2014, and consists of two essential components, namely, the generator and the discriminator. Through the zero-sum game, GANs can be trained to generate more realistic data. GAN is widely used in image generation and recognition, speech generation and recognition, data generation, and other fields. The main thought behind GAN is using adversarial networks to improve the quality of generated data. The generator learns the probability distribution of the original data and generates artificial samples that mimic the pattern, using random noises. The discriminator discriminates the artificially generated data from the actual data and prompts the generator to produce better quality data in the next iteration. The general semantics flow chart of GAN is depicted in Figure 2.  
The GANs framework is equivalent to a minimax two-player game, which can be described by the following function [21]:  
\[\min_{G}\max_{D}V(D,G)=\min_{G}\max_{D}(E_{x\sim p_{x}(x)}[logD(x)]+E_{z\sim p_{ z}(z)}[log(1-D(G(z)))]) \tag{1}\]  
where \(x\) is a real datum with a distribution, and \(p_{x}\) and \(z\) is a random vector. \(z\) is made to be an input of \(G\) through which the generated datum \(G(z)\) can be obtained. \(p_{g}\) is denoted  
Figure 2: Typical architecture of GAN. as the distribution of \(G(z)\). Then, \(D\) represents the probability that \(x\) comes from \(p_{x}\), rather than from \(p_{g}\). Ideally, \(D(x)=1\) if \(x\sim p_{x}\) and \(D(x)=0\) if \(x\sim p_{g}\). \(D\) is trained to maximize the probability of assigning the correct label to both the real datum \(x\) and the generated datum \(G(z)\). In this paper, GAN is used to generate grinding industry data. The data include the crushing ore size, ore grade, magnetic tube \(mFe\), magnetic tube recovery rate, raw ore \(FeO\) content, raw ore \(mFe\) content, the water feed rate, and the fresh ore feed rate. After the actual field investigation records, 182 sets of data were used as the original sample. The data from the actual production process have some incorrect data. Some of the values are 0, while others are either unreasonably big or small. To get rid of the abnormal data, Formula (2) is used firstly for data screening. \[|X-\overline{X}|\geq\alpha\times\sqrt{\frac{1}{n}\sum_{i=1}^{n}(X_{i}- \overline{X})^{2}} \tag{2}\]  
where \(X_{i}\) is the \(i\)th parameter value; \(\overline{X}\) is the average of the values of the set of parameters. When one of the parameter values is greater than \(\alpha\) times, the standard deviation, the sample should be deleted. After data screening, 164 sets of data are left to generate samples. In the generator, the input is composed of the noise input, which contains 80-dimensional data, randomly generated by Gaussian distribution. There are two hidden layers, followed by the rectified linear unit (ReLU) as an activation function, and the kernel sizes are 120 and 240. The output of the generator is a one-dimensional data sample, using Sigmoid as the activation function. As for the discriminator, the input layer is with eight kernels, using LeakyReLU as an activation function, and the kernel size is 120. Another layer with 240 kernels using LeakyReLU is added and followed by dropout with the probability of 0.5. Finally, the model layer is flattened and linked to one fully connected layer with 0.5 dropouts. GAN generates 886 samples. To verify the accuracy of the generated samples, the back-propagation neural network (BP), support vector machines (SVM), and extreme learning machine (ELM) are used to compare the original data and the generated data. We used 30% of the original samples as test samples.