## 4 Simulation and Experiment
### Test on Typical Simulated Time Series  
#### 4.1.1 Simulation Data  
For time series predictions, typical data trends are usually studied. The following part selects two time series benchmarks, the Mackey-Glass (MG) system and the multiple superimposed oscillator (MSO) problems. For each dataset, 1350 sets of data are selected for experiments, and the first 1000 sets are the training set of the model, which is used to train the proposed model. The following 350 sets of data are divided into the first 150 sets to verify the model's accuracy, and the last 200 sets of data are used to test the model's predictive performance. The MG system is a typical series model with a chaotic attractor. It is derived from a time-delay differential system:  
\[\frac{dy(t)}{dt}=\frac{ay(t-\tau)}{1+y^{n}(t-\tau)}+by(t) \tag{9}\]  
where \(n\) is the order of the time series variable, and \(a\) and \(b\) are the adjusting parameters. The MG system will be chaotic when \(\tau<16.8\). In the experiment, the parameters are set as \(\tau=16\), \(n=4\), \(a=0.1\), and \(b=0.1\). Then, 1350 sets of data are generated. The original data of the MG system is shown in Figure 4a. Data in the MSO system are generated by overlaying several simple sine wave functions, expressed as follows:  
\[y(n)=\sum_{i=1}^{k}\sin(\alpha_{i}n) \tag{10}\]  
where \(n\) is the time step order, \(k\) is the number of sine waves, and \(\alpha_{i}\) is the frequency of the sine waves. In the experiment, \(k=6\), and \(\alpha_{i}\) is set as a random value in \([0,1]\). The data size is the same as that of the MG system. The original data are presented in Figure 4b. The simulation data in MG and MSO systems are relatively periodic. Parts of the data are shown in Figure 4 for an intuitional observation. Based on the state trend characteristics of the time series data, the STL algorithm can decompose the data into three parts: seasonal, trend, and random items. The simulation data in the MG and MSO systems are decomposed into time series using the STL algorithm, and the seasonal subsequences are obtained, as shown in Figure 5. In the proposed time series prediction framework, input data are subjected to a sample enhancement algorithm to obtain multiple enhancement sets. Similarly, the enhancement set obtained through the sample enhancement algorithm is decomposed using STL to obtain seasonal subsequences, as shown in Figure 6. The overall trend of the data fluctuates significantly, with a certain  
Figure 4: Original simulation data used in the experiments. (**a**) Data from MG system. (**b**) Data from MSO system. degree of seasonality. The seasonal trend of the enhancement set after the enhancement algorithm is similar to the seasonal trend of the original simulation data in Figure 6. In addition, each enhancement set will also be used as input for the corresponding multi-model prediction in the next step of training. Concrete model building is conducted following the method in Section 3, including training data augmentation, data trend decomposition, parallel prediction with multiple GRUs, and a result fusion based on the error covariance intersection. Because the proposed method mainly consists of the sample augmentation and covariance intersection fusion of multiple GRUs, it is abbreviated as A-CI-GRUs. Other typical prediction methods are set as the contrast, as shown in Table 3. In order to exclude other interferences, the same model parameters in the experiments are set consistently. Figure 5: Seasonal subsequences of original simulation data. (**a**) Seasonal subsequences of the MG system. (**b**) Seasonal subsequences of the MSO system. Figure 6: Seasonal subsequences of the augmentation set from the original simulation data. (**a**) Seasonal subsequences of the augmentation set from the MG system. (**b**) Seasonal subsequences of the augmentation set from the MSO system.