## 2 Materials and Methods
### Feature Selection and Data Augmentation  
When all samples are used for the pathological classification of GGNs, two problems remain, as follows:  
1. The number of samples is much less than that of the features, and some features are unnecessary.
2. The sample distribution is imbalanced; Table 1 shows that the number of samples in the majority class is 96, but there are only eight in the minority class.  
To overcome these problems, feature selection and sample augmentation are implemented to \(S_{FFDC}\) and \(S_{OFDC}\) in advance. Feature selection removes irrelevant and redundant features [26]. To identify the key features and reduce feature dimensionality, we applied the analysis of variance (ANOVA) method [27]. ANOVA is a single variable analysis method to test whether the effect of any independent feature is obvious for which we computed the three sums of squares in \(S\), \(SST\), \(SSW\), and \(SSB\)[28]. According to the four pathological subtypes of GGN and all samples, S consists of four groups of \(\{x_{ij}\}\) in which each contains \(n_{i}\) samples, \(i=1,2,3,4\); \(j=1,\ldots,n_{i}\).  
As a result, \(SST\) is computed by  
\[SST=\sum_{i=1}^{4}\sum_{j=1}^{n_{i}}\left(x_{ij}-\overline{X}\right) \tag{2}\]  
where \(\overline{X}\) is the mean of all samples in \(S\). \(SSB\) is computed as  
\[SSB=\sum_{i=1}^{4}n_{i}(\overline{X}_{i}-\overline{X}) \tag{3}\]  
Finally, \(SSW\) is calculated as  
\[SSW=SST-SSB \tag{4}\]  
To calculate the effect of each feature in \(S\), \(SSB\) is divided by its freedom degree of 3 to obtain an estimate of \(MSB\). \(SSW\) is divided by its freedom degree of 233 to obtain an estimate of \(MSW\). Finally, a statistical value of \(F\)-_ratio_ is computed as  
\[F=MSB/MSW \tag{5}\]  
We consulted the priori table of critical \(F\) values to obtain a significant \(p\) value. In this paper, we took a threshold of 0.05. If \(p>0.05\), the relative feature is rejected; if not, it is accepted for classification. In the following, the feature selection process is implemented in the IBM SPSS Statistics software. The feature number in \(S\) based on FFDC and OFDC was reduced to 142 dimensions and 680 dimensions, respectively.  
Figure 3: Type and number of extracted radiomics features.  
To overcome the problem of the imbalanced sample distribution in \(S\), the synthetic minority oversampling technique (SMOTE) [29; 30] is used to increase the balance radio between cases in four classes in \(S\). SMOTE randomly creates synthetic samples by adding a weighted difference between the \(j\)th sample and its \(k\) nearest neighbors. This enables oversampling of minority samples. These newly synthesized samples will enhance the generality of the classifiers, thereby avoiding overfitting to a certain extent [31]. Before data augmentation, all samples in the set on FFDC and the set on OFDC must be normalized according to the following form:  
\[F_{sta}=(F-\mu_{F})/\sigma_{F} \tag{6}\]  
where \(F_{sta}\) is the standardized feature; \(\mu_{F}\) is the mean value of the feature; and \(\sigma_{F}\) is the standard deviation of the feature.  
According to Table 1, SMOTE is configured with five nearest neighbors for oversampling to generate synthetic samples in \(S_{FFDC}\) and \(S_{\text{OFDC}}\). The SMOTE steps are as follows:  
1. For each sample \(a\) in the minority class, five nearest neighbors are found.
2. For each randomly selected nearest neighbor \(b\), a new sample \(c\) is constructed with the original sample \(a\) according to the following equation: \[c=a+rand(0,1)|a-b|\] (7)
3. The new sample set is thus obtained by the original and generated samples.