## 3 Fusion Prediction Based on Data Augmentation
### Augmentation of Small-Scale Time Series  
In the framework proposed, the precondition for deep network prediction is the augmentation of the original data. Small samples should be expanded to meet the data size demand of the deep networks. Therefore, an augmentation algorithm based on random probability is proposed first.  
The original dataset is expressed as \(\{x_{1},x_{2},\cdots,x_{n}\}\), where \(n\) is the number of samples. The expansion of the small-scale time series is achieved through the following steps. First, the parameters related to resampling are set. The minimum data size of the network according to the needs of the augmented sample is set. The number of resamplings, \(m\), can be ascertained with the limitation of \(m\cdot n\geq\varepsilon\). The resampling is conducted following the Bootstrap method [34; 35]. If samples are unknown, multiple samplings in the existing partial samples construct the estimated confidence interval. In an abstract sense, the usual estimation from the samples does not extract all the information, while Bootstrap builds the confidence interval by exerting the residual value with the resampling. Third, for the original set, the sample is extracted with playback \(n\) times, and the new set is \(X_{j}(1\leq j\leq m)\).  
Meanwhile, the probability distribution parameter, \(\theta_{j}\), is calculated for the new set. After the resampling \(m\) times, the new sets are obtained and joined as \({Y_{k}}^{\prime}=\left\{X_{1},X_{2},\cdots,X_{m}\right\}^{k}\) and \(1\leq k\leq\alpha\), in which \(X_{j}\) consists of \(n\) samples. The corresponding parameter set is \(\{\theta_{1},\theta_{2},\cdots,\theta_{m}\}\) of which the \(a\)-th and \(b\)-th parameters are the lowest and highest points of the confidence interval:  
\[\left\{\begin{array}{l}a=\text{intfloor}\left(m\cdot\beta\right/\cdot\cdot 100 \%\right)\\ \quad 2\\ b=\text{intfloor}(m\cdot(1-\beta)\cdot 100\%)\end{array}\right. \tag{2}\]  
where \(\beta\) is the quantile parameter. Finally, the samples and the resampled new set are fused. \(b\) samples are selected from the original data and inserted evenly into the newly joined set. Small sample augmentation is performed through the process of resampling and union. First, the relevant resampling parameters are set, and the number of resampling times is determined. Second, confidence intervals for the estimates are constructed by taking multiple samples across the existing partial sample. Third, for the original set, samples are extracted to obtain a new set after n playbacks. At the same time, the probability distribution parameters for the new set are calculated. Finally, the samples and the resampled new set are fused. The augmentation can be conducted \(\alpha\) times considering the multiple models trained later. The augmentation Algorithm 1 is concluded as follows for a clear and logical presentation.  
```
Input: \(\{x_{1},x_{2},\cdots,x_{n}\}\), \(m\), \(\alpha\), \(\beta\) Output: \(Y_{k}\) Procedure For\(k=1\), \(k<=\alpha\), k++ For\(j=1\), \(j<=m\), j++ resampling of \(\{x_{1},x_{2},\cdots,x_{n}\}\to\ X_{j}\);  calculate probability distribution parameter \(\theta_{j}\); For end  join \(X_{j}\to\ Y_{k}^{\prime}=\{X_{1},X_{2},\cdots,X_{m}\}^{k}\);  join \(\theta_{j}\to\ \{\theta_{1},\theta_{2},\cdots,\theta_{m}\}\);  calculate confidence interval parameters a and b following Formula (2);  select \(b\) samples from \(\{x_{1},x_{2},\cdots,x_{n}\}\) evenly;  insert \(b\) samples selected above into \(Y_{k}^{\prime}\) evenly \(\to\ Y_{k}\); For end Procedure
```  
**Algorithm 1**Augmentation of a small sample.