## 3 Probabilistic Neural Network Model Based on Symmetric Alpha Stable Distribution
### Improved Probabilistic Neural Network  
Improvement of the probabilistic neural network is based on the fact that the input dimension of the probabilistic work directly affects the accuracy, a large number of training samples are required, and the expression range of the basis function gaussian distribution is limited.  
(1) Optimize the probabilistic neural network structure This study uses a reduced concept lattice to reduce the input indexes of PNN, eliminate, as far as possible, those input indexes with a close relationship, reduce the dimension of the PNN input, optimize the structure of PNN, improve its efficiency of pattern recognition, compute correlation coefficients among different indexes (see Formula (8)) according to the related theory of [PERSON], and finally verify the reduction effect.  
\[r=\frac{\sum\limits_{i}^{N}\left(x_{p_{i}}-\overline{x_{p}}\right)\left(\left(x_{ q_{i}}-\overline{x_{q}}\right)\right)}{\sqrt{\sum\limits_{i}^{N}\left(x_{p_{i}}- \overline{x_{p}}\right)^{2}\frac{N}{i}\left(x_{q_{i}}-\overline{x_{q}}\right) ^{2}}},\overline{x_{p}}=\frac{1}{N}\sum\limits_{i}^{N}x_{p_{i}},\overline{x_ {q}}=\frac{1}{N}\sum\limits_{i}^{N}x_{q_{i}},p,q=1,2,\cdots,n \tag{8}\]  
Several questions arose after reduction; for instance, the number of training samples was small, and the number of training samples belonging to various categories was not the same. Such inequality of data may affect the accuracy of the output results. Thus, the SMOTE algorithm was used to insert new samples among a few samples with similar positions to ensure balance.  
Improve the radial basis function of the probabilistic neural network  
When \(\alpha=2\) and \(\beta=0\), the alpha stable distribution is consistent with the Gaussian distribution, indicating that the Gaussian distribution is a special form of the alpha stable distribution. To improve the radial basis function of the probabilistic neural network, we use the probability density function of the symmetric alpha stable distribution (\(S\alpha S\)) to replace the Gaussian distribution function of PNN and use it as the output of the model layer. This involves weighing and summing the vector of the input node and calculating the Euclidean distance between this vector and the vector of the sample input, before finally obtaining the similarity between this model and the standard model after operation of the \(S\alpha S\) activation function. Because the probability density function of an alpha stable distribution has closed expressions only in special cases, we utilize the MATLAB [38] toolbox function for the calculation, as follows:  
\[z=stblpdf(\|X-Y_{k}\|,\alpha,\beta,\gamma,\mu) \tag{9}\]  
This formula is used as the output of the model layer, in which \(\beta=0\), \(\mu=0\) and parameter \(\alpha\), \(\gamma\) must be estimated.  
Utilize the genetic algorithm to compute parameter \(\alpha\), \(\gamma\)  
Common methods to estimate the parameters of the \(S\alpha S\) distribution include the maximum likelihood estimation method, the sample quantiles method, the negative-order moments method, and the logarithm method. The genetic algorithm is a type of heuristic search algorithm based on swarm intelligence [39]. Because common methods used to estimate parameters of the \(S\alpha S\) distribution are always complex, the genetic algorithm can optimize the parameter \(\alpha\), \(\gamma\). The fitness function is defined as follows:  
\[f=1-\frac{K}{N}+W \tag{10}\]  
where \(N\) is the total number of training samples, \(K\) stands for the total number of training samples that have been predicted accurately, and \(W\) is a constant with a range from \(0<W<0.1\).  
To summarize, the detailed prediction model for roadway surrounding rock stability based on concept lattice reduction and a symmetric alpha stable distribution probabilistic neural network is shown in Figure 3.