## 2 Improved WITI Model Considering Gaussian Mixture Clustering  
### Gaussian Mixture Clustering  
The finite mixture model (FMM) is a method in which a complex density distribution is assessed by combining multiple simple distributions. Based on the FMM, the Gaussian mixture model is obtained by limiting all distributions to the Gaussian distribution [17].  
We first perform a cluster analysis of bad weather. It is assumed that sample set \(D=\{x_{1},x_{2},\ldots,x_{n}\}\) contains \(n\) unlabeled samples. Each sample \(x_{i}=\{x_{i1};x_{i2};\ldots;x_{im}\}\) is a m-dimension feature vector containing echo intensity and weather data information. The clustering algorithm divides sample D into \(k\) clusters \(\{C_{l}|l=1,2,\ldots,k\}\), where \(C_{l^{\prime}}\cap_{l^{\prime}\
eq l}C_{l}=\varnothing\) and \(D=\cup_{l=1}^{k}C_{l}\). Accordingly, \(a_{j}\in\{1,2,\ldots,k\}\) is used to represent the \"cluster mark\" of sample \(x_{j}\), \(x_{j}\in C_{aj}\). Therefore, the clustering result can be represented by a cluster label vector \(\overset{\rightarrow}{a}=\{a_{1};a_{2},\ldots;a_{n}\}\) containing n elements; this represents the final classification result of bad-weather types.  
The specific calculation formula is as follows:  
\[P(x)=\sum_{i=1}^{k}\lambda_{i}p\big{(}x\big{|}\mu_{i},\sum i\big{)},i=\{1,2, \ldots,k\} \tag{1}\]  
Figure 1: The architecture diagram of the overall system.  
where \(x\) is the vector in sample space D (i.e., echo intensity and weather phenomenon data), \(\lambda_{i}\) is the mixing coefficient, and \(P(x\,|\,u_{i},\sum i)\) is the \(i\)-th Gaussian distribution that both points to \(u_{i}\) and covariance \(\sum i\):  
\[p(x|\mu_{i},\sum i)=\frac{\exp[-1/2(x-\mu_{i})^{\frac{\sum_{i}}{\sum_{i}}(x- \mu_{i})}}{{(2\pi)}^{n/2}|\sum i|^{1/2}} \tag{2}\]  
If the values of parameters \(\lambda_{i}\), \(u_{i}\), \(\sum i\), and \(i\) are determined, Equation (2) can be used to obtain the probability \(P(x\,|\,u_{i},\sum i)\) of each Gaussian component generating sample X and density \(p(x)\) of the Gaussian mixture distribution.  
Let sample data set \(j\) be used to calculate the probability that the Gaussian component of generated sample \(x_{n}\) according to the Bayesian theorem, and record the largest Gaussian component of \(\gamma_{nj}\) as the category of \(x_{n}\). This is based on the principle of Gaussian mixture clustering.  
The parameter estimation takes the maximum-likelihood estimation as the optimization objective and uses the expectation maximization algorithm to solve the problem [18]. The initial parameters of Gaussian mixture clustering are set, and the posterior probability \(\gamma_{nj}\) that sample \(x_{n}\) a belongs to Gaussian component \(j\) is calculated using the current parameters:  
\[\begin{array}{l}\gamma_{nj}=p(c_{n}=j|x_{n})\\ \\ =\frac{p(c_{n}=j)p(x|\mu_{i},j)}{P(x_{n})}\\ \\ =\frac{\lambda_{i}p(x_{n}|\mu_{i},\sum j)}{\sum\limits_{i=1}^{k}\lambda_{i}p(x |\mu_{i},\sum i)}\end{array} \tag{3}\]  
Mean \(\mu_{j}\) and covariance \(\sum j\) are calculated as  
\[\lambda_{i}=\frac{1}{N}\sum\limits_{n=1}^{N}\gamma_{nj} \tag{4}\]  
\[\mu_{j}=\frac{\sum\limits_{n=1}^{N}\gamma_{nj}X_{n}}{\sum\limits_{n=1}^{N} \gamma_{nj}} \tag{5}\]  
\[\sum j=\frac{\sum\limits_{n=1}^{N}\gamma_{nj}(x_{n}-x_{j})(x_{n}-\mu_{j})^{T} }{\sum\limits_{n=1}^{N}\gamma_{nj}} \tag{6}\]  
The clustering results are then obtained by iterating to convergence.  
The Gaussian mixture clustering model can be used to classify bad weather in the sector according to the echo intensity and weather phenomenon. By inputting the sample data of the echo intensity and weather types in the sector, the bad weather is combined with the WITI model to measure the severity of the bad weather according to the echo intensity, so as to realize the classification of bad weather and calculate different weather types. The influence of bad weather on sector capacity can be quantified by using the weather-influence coefficient according to different categories.