## 2 Methods  
### Group Random Weight DBA Algorithm  
The GRW-DBA method proposed in this paper is based on the classic DBA method, which improves the grouping calculation and random weights and overcomes the shortcomings of the classic DBA method [34], which is easily affected by data outliers and cumbersome augmentation processes. The process of data augmentation by this algorithm will be introduced in detail below. First, divide calculation groups. Different from the classic DBA algorithm that uses all data to perform calculations, the GRW-DBA algorithm divides \(N\) sets of data into multiple computational groupings, such as any 2 as groups, any 3 as groups, any 4 as groups,..., and all \(N\) as groups. Each calculation group is independent of the other, and subsequent calculations are performed independently. Second, determine the sequence weight. In each calculation group, the initial sequence \(Q\) is first randomly selected and assigned a non-repeating random weight \(x\). Then, the DTW distance [33] between the initial sequence \(Q\) and any other sequence \(P\) in the group is calculated. Suppose the two sequences \(Q\) and \(P\) are \(Q=[q_{1},q_{2},q_{3},\cdots,q_{m}]\) and \(P=[p_{1},p_{2},p_{3},\cdots,p_{n}]\), respectively. The distance between elements \(q_{i}\) and \(p_{j}\) in the sequences is calculated by the formula:  
\[d(i,j)=(q_{i}-p_{j})^{2} \tag{1}\]  
The distances between all corresponding elements of two sequences form an \(m\times n\) distance matrix \(M\); \(M\) can be expressed as:  
\[M= \begin{bmatrix}d(m,1)&d(m,2)&d(m,3)&\cdots&d(m,n)\\ \vdots&\vdots&\vdots&\cdots&\vdots\\ d(3,1)&d(3,2)&d(3,3)&\cdots&d(3,n)\\ d(2,1)&d(2,2)&d(2,3)&\cdots&d(2,n)\\ d(1,1)&d(1,2)&d(1,3)&\cdots&d(1,n)\end{bmatrix}  \tag{2}\]  
Taking \(d(1,\,1)\) as the starting point, select one of the elements which is above, right, and top right of the starting point. Then, based on the element, repeat the same steps in this way until \(d(m,\,n)\) is reached. These selected elements form a path \(R\) between sequences \(Q\) and \(P\), which can be denoted as \(R=\{d(r_{1}),d(r_{2}),\cdots,d(r_{s}),\cdots,d(r_{N})\}\), where \(N\) denotes the total number of elements in the path, \(r\) is the coordinate of the point on the path, i.e., \(r_{s}=(i,j)\), and there are many of these paths \(R\). However, there must exist an optimal path in all path spaces \(R\) that minimizes \(\sum_{s=1}^{N}d(r_{s})\). Therefore, the DTW distance between sequences \(Q\) and \(P\) is:  
\[\text{DTW}(Q,P)=\text{min}(\sum_{s=1}^{N}d(r_{s})) \tag{3}\]  
where \(d(r_{s})\) is the path distance calculated between each corresponding value, and \(N\) represents the number of sequence data. To solve for the minimum \(\sum_{s=1}^{N}d(r_{s})\) value, the cumulative distance matrix \(D\) is calculated using the dynamic programming method and the corresponding distance \(d\) between two points as:  
\[D(i,j)=d(i,j)+\text{min}\{D(i,j-1),D(i-1,j-1),D(i-1,j)\} \tag{4}\]  
where \(isequences. Finally, the remaining (\(1-1.6x\)) weights are equally distributed to the other remaining sequences. Since \(x\), \(0.3x\), and \(1-1.6x\) represent the weight of each sequence, it is very important to ensure that all three values are between 0 and 1 according to the experience of the literature [35]. Therefore, it can be calculated that the value range of \(x\) in this case is between 0 and 0.625. The generation of random weights adopts the mature Mersenne Twister algorithm [36] in the computer, which generates pseudo-random numbers by continuously calling the algorithm. After the judgment process, non-repeating values that meet the requirements are used as random weights and parameters. We use this method in all calculation groups to calculate combinations of more than four sequences; due to the limitation on the number of sequences, combinations of two and three sequences do not satisfy the above weight distribution scheme, so these groups assign uniform weights to all sequences. Third, we find the weighted average by group.