# Evaluation and Prediction of Higher Education System Based on AHP-TOPSIS and LSTM Neural Network
## 5 Performance Evaluation
### Sensitivity Analysis  
To reflect the realities of prediction model errors and analyze the sensitivity of the model for subsequent optimization, we adopt a data prediction model with an optimization method based on root mean square error (\(RMSE\)), which is determined by the following formula:  
\[RMSE=\sqrt{\frac{\sum_{i=1}^{n}\left(p_{i}-a_{i}\right)^{2}}{n}} \tag{22}\]  
In Formula (22), \(a_{i}\) represents the real data and \(p_{i}\) represents the predicted data. We can find that \(RMSE\) is more sensitive to exceptional values. When there is a big difference between observed values and real ones, \(RMSE\) tends to be very large, so \(RMSE\) is used to verify the accuracy of network training.  
The detailed optimization proceedings are as follows:  
* Preprocessing: For variables to be predicted in the dataset, we calculate their periodic values and rank them from small to large. Next, we obtain the periodic values of the first N smallest variables to be predicted. Then, we calculate the correlation coefficients between the variables and those to be predicted and arrange the variables in order from large to small according to their corresponding correlation coefficients. Finally, we extract the variables with sums of correlation coefficients being greater than a coefficient threshold to form a training set.  
* LSTM modeling: We organize the training set into a time sequence and input the time sequence into the LSTM network composed of multiple connected LSTM units to obtain the current training model.
* Optimizing: We use the training model to calculate the values of the variables predicted at the set time and compare them with the actual values of the variables predicted to obtain the root mean square errors (RMSEs). Compared with the RMSEs of previous rounds of training, the smaller values are taken as the RMSEs of current rounds of training, and the corresponding training model is reserved as the optimal solution model.  
In the training of the LSTM neural network, we divide the dataset into the training dataset and the test dataset according to the ratio of 2:1. Then, we input the test dataset into the trained neural network to output the prediction dataset, and we calculate the RMSE values. The results are illustrated in Figure 12. We see that the predicted values of most indicators in our model are highly consistent with the observed ones, although the predicted curves of some indexes are quite different from the observed ones. For example, the predicted values of the student-teacher ratio are lower than the observed, and the predicted values of the proportions of higher education in GDP decrease and then rise sharply in the observed curve. In addition, the predicted values of the number of published papers and the proportions of employed persons receiving higher education are both obviously higher than their observed.  
Due to the small size of available data, the predictions of some indicators are not ideal, but overall trends are much more consistent with the true values, with errors within an acceptable range. If we increase the size of the original data, consistency becomes even better. Furthermore, it appears from Figure 12 that the RMSE of each indicator is relatively small, and it shows that the predicted values are close to their real ones, which verifies the rationality and robustness of the model.  
Figure 12: _Cont._