## 4 Experimental Process and Result Analysis
### Performance Analysis of the Improved MLP Model  
Next, the improved MLP model is trained and tested with the two datasets. To demonstrate the performance of the presented model more intuitively, the root mean square error (RMSE) is adopted in the experiments as the evaluation index of the estimation accuracy of the model. The smaller the RMSE value is, the closer the estimated value is to the true value, and the higher the accuracy of the model. The formula for calculating the RMSE [37] is  
\[\text{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)^{2}} \tag{18}\]  
where \(n\) denotes the number of samples, \(y_{i}\) represents the true value, and \(\hat{y}_{i}\) represents the estimated value. The \(x\)-axis and \(y\)-axis in Figure 7 are the training epochs and loss values of the model during the training process, respectively. A total of 500 epochs are iterated during the model training process. In accordance with Figure 7, the loss of the model is continuously reduced during the training process and is ultimately maintained at a low value, demonstrating that the model has converged normally and achieved better performance. The general idea of GNSS positioning is adopted, and the output results of the model are divided into three components: East, North, and Up. Figure 8 shows the experimental results of the proposed model on the three components of East, North, and Up. It can be seen from Figure 8 that the trained and converged model has a high estimation accuracy. After the two datasets are input into the model separately, the model gradually converges after several iterations of training. The presented method obtains RMSE values of 0.239 m and 0.029 m for the first and second datasets, respectively. The random forest (RF) algorithm is widely adopted in the field of GNSS positioning error estimation, and it has high estimation accuracy. Therefore, to demonstrate the performance of the presented method more clearly, the experimental results of the presented method and the RF algorithm will be displayed and comparatively analyzed next. The gray portion in Figure 9 indicates the confidence interval for the positioning error estimate. The confidence interval is denoted by the area of the estimated value minus two standard deviations and the estimated value plus two standard deviations. The confidence interval is a concept from statistics, which means that the true value has a certain probability of falling within a certain interval around the estimated value. The confidence interval is an interval range and is often used to indicate the credibility of the estimated value. Applying confidence intervals to the problem of positioning error estimation is  
Figure 7: Image of the loss function of the presented model during training. mainly used to describe the uncertainty range of positioning error estimation, which is conducive to evaluating the reliability and stability of the proposed model. As shown in Figure 9, the estimated values of the presented method have a large repetition range with respect to the true values, and they are basically within the confidence interval, which shows that the estimation results of the positioning errors of the presented method have high accuracy. However, the RF algorithm is sensitive to noise and outliers in the data, which affects the estimation accuracy of the algorithm. On the whole, the presented method has a larger repetition range of estimated values and true values compared to the RF algorithm. The estimation accuracy of the presented method is significantly better than the RF algorithm. It can be seen from Figure 10 that most of the estimation errors of the presented method are approximately distributed around 0, and the difference does not generally exceed 0.1. This shows that the estimated values obtained by the presented method are more accurate and have better performance. The RF algorithm is a model based on decision trees that cannot handle the interdependence in time series data well.