## 2 Materials and Methods
### SMOTE Algorithm Based on IOWA Operator  
The synthetic minority oversampling technique (SMOTE) [28], which uses the interpolation method, is used to balance the unbalanced dataset by inserting the new minority samples over the line between each minority sample and its \(K\) neighbor sample. The classic SMOTE method uses the Euclidean distance (Equation (2)) to calculate the nearest neighbor set of the minority sample set. Then, the new synthesizing samples are inserted between the minority sample set and the \(k\)-nearest neighbor sample set until an appropriate number of the new sample set is obtained:  
\[D=\sqrt{\sum_{i=1}^{n}(X_{i}-X_{k})^{2}} \tag{2}\]  
where there are \(n\) samples in the minority sample set \(X\), \(X_{i}\) is the \(i\)-th sample of \(X\), \(X_{k}\) is the \(k\)-th nearest neighbor sample of \(X_{i}\) and \(D\) is the distance between the calculated minority sample \(X_{i}\) and its nearest neighbor \(X_{k}\). The classical OWA operator assumes that the values of the elements to be aggregated are related to the defined weights, but the weights for each item of high-dimensional data are not balanced. So, the IOWA (induced ordered weighted average) algorithm was introduced [29]. The basic principle of the IOWA operator is to calculate the accuracy of  
Figure 1: Data synthesis based on weight and extension algorithms flow charts. each moment through the predicted value and assign weight coefficients to every single model in the order of accuracy from high to low. The predicted value and the actual value are from the sum of the square's error function. As follows, it establishes the combined prediction model with the minimum value as the target. Let \((\langle a_{1},x_{1}\rangle,\langle a_{2},x_{2}\rangle,\ldots,\langle a_{n},x_{n}\rangle)\) be \(n\) two-dimensional arrays, then the IOWA operator can be obtained from Equation (3):  
\[\text{IOWA}_{W}(\langle a_{1},x_{1}\rangle,\langle a_{2},x_{2}\rangle,\ldots, \langle a_{n},x_{n}\rangle)=\sum\limits_{i=1}^{n}w_{i}b_{i} \tag{3}\]  
where, IOWA\({}_{W}\) is the n-dimensional-induced ordered weighted average operator generated by \(a_{1},a_{2},\ \ldots\ a_{n}\), \(b_{i}\) is the \(i\)-th largest input vector in \(a_{1},a_{2},\ \ldots,a_{n}\) in order from the largest to the smallest, \(W=(w_{1},w_{2},\ldots,w_{n})^{\text{T}}\) is the weighted vector of OWA which satisfies \(\sum\limits_{i=1}^{n}w_{i}=1,w_{i}\geq 0\). The IOWA operator is an orderly weighted average of the corresponding values of induction value \(x_{1},x_{2},\ \ldots,x_{n}\) sorted from large to small. The sizes of \(w_{i}\) and \(x_{i}\) are independent of their positions but are related to their positions. The SMOTE process based on the IOWA operator is as follows:  
* Fisher scores of each feature variable in dataset X were calculated to obtain the weight matrix w for all features by Equation (4): \[w=\frac{\sum\limits_{i=1}^{C}\frac{n_{i}}{n}\left(\mu_{i}^{(m)}-\overline{\mu ^{(m)}}\right)^{2}}{\frac{1}{n}\sum\limits_{i=1}^{C}\sum\limits_{x\in w_{i}} \left(x^{(m)}-\mu_{i}^{(m)}\right)^{2}}\] (4) where the total of \(N\) samples in the dataset belongs to \(C\) categories, and each category contains \(n_{i}\) samples. \(x^{(m)}\) represents the value of sample x on the m-th feature, \(\mu_{i}^{(m)}\) represents the mean value of class \(i\) samples on the \(m\)-th feature and \(\overline{\mu^{(m)}}\) represents the mean value of all classes of samples on the \(m\)-th feature. * The minority samples in the training set were taken out. The variant form of Minkowski distance weighted by the IOWA was considered to define the neighborhood using Equation (5): \[D_{IOWA}=\left(\sum\limits_{i=1}^{n}w_{i}b_{i}^{p}\right)^{1/p}\] (5) where the well-known options for this parameter are \(p\in\{1,2,\infty,\ p\geq 1,\}\), it is the traditional Euclidean norm when \(p=2\). Similar to the IOWA operator, \(b_{i}\) is the value of the \(i\)-th larger input vector sampled from X. In addition, it satisfies \(\sum\limits_{i=1}^{n}w_{i}=1\) and \(w_{j}\in[0,1]\).