## 6 Numerical Experiment
### Results and Discussions: Performance by the Missing Value Rate of the User Trajectory  
On the basis of Section 6.4, we analyze the trend of the ML models using the feature vectors \(\mathbf{f}_{k}^{\text{GMM}}\) and \(\mathbf{f}_{k}^{\text{IDNN}}\) performance with respect to trajectory sparsity. First, to consider the effect of trajectory sparsity on classification performance, we visualize the trends of the accuracy values for the test and training data against the missing value ratio \(\theta_{\text{th}}\) in Figure 11 and Figure 12, respectively. For both figures, the vertical and horizontal axes represent the accuracy \(A_{\text{M}}\) and missing value ratio \(\theta_{\text{th}}\), respectively. In addition, the red, blue and green symbols represent the SVC, RF, and DNN, respectively, and the circle and cross symbols indicate the GMM- and IDNN-based features, respectively. Fewer differences between the accuracy of training and test data indicate higher generalization performance. In Figure 11, the horizontal and vertical dashed lines represent the best accuracy and the corresponding \(\theta_{\text{th}}\) among the combinations of feature extraction methods and ML models, respectively. In the SVC and RF methods that use GMM and IDNN features, given that the trends in which a lower \(\theta_{\text{th}}\) leads to higher accuracy were confirmed between the threshold of the best accuracy, i.e., the dashed line in Figure 11 and \(\theta_{\text{th}}=1.00\), the sparsity of the trajectory would decrease the model performance. Conversely, if \(\theta_{\text{th}}\) is too small, low performance can also occur. This is because an extremely small \(\theta_{\text{th}}\) yields low numbers of samples relative to the feature space. Conversely, for both the GMM and IDNN features, the DNN tends to  
Figure 11: This figure illustrates the trends in classification accuracy for the missing value ratio \(\theta_{\text{th}}\) based on the combinations of feature extraction methods and ML models for each month. Averages of 20 accuracies on test data. The dashed horizontal line indicates the best accuracy among the combinations. The title shows the month with the best average accuracy (standard deviation), the combinations of the feature extraction method and the ML model provide the best accuracy, and the results of the Shapiro–Wilk test for the accuracy distribution of 20 seeds, where n.s. is \(p\geq 0.050\). The best average accuracy values are higher than those of random classification (accuracy \(\approx 0.170\)) in the range of 3 SD in any month. decrease in accuracy as \(\theta_{\text{th}}\) decreases in the range of \(\theta_{\text{th}}=0.95\)-\(0.98\), which contrasts with the trend observed in SVC and RF, where accuracy increases as \(\theta_{\text{th}}\) decreases. This suggests that, in this range of missing value ratios, the amount of training data may be insufficient to adequately train the DNN models. This result is supported by the findings shown in Figure 12 for the DNN on the training data. Because the trends in accuracy for the training and test data are similar, it can be inferred that insufficient data availability prevents the DNN from learning effectively, thereby hindering its performance improvement. Focusing on the best accuracy values in Figure 11, GMM-RF achieves the highest accuracy for all months. In particular, for April to June, the test accuracy of GMM-RF is higher than that of the other combinations for all the missing value ratios. In addition, the test accuracy of GMM-SVC is not as high as that of GMM-RF, but it is confirmed that GMM-SVC has high generalization performance because of the smaller differences between the accuracies of the training and test data. This is implied from Figures 11 and 12. On the other hand, while the test accuracies of the GMM and IDNN features are close, the training accuracy of the IDNN features tends to be greater than that of the GMM features, particularly for \(\theta_{\text{th}}=0.98\)-\(1.00\).