## 6 Numerical Experiment
### Methods
#### 6.1.2 Comparison of GMM and IDNN  
The performances of the ML models that use the feature vectors \(f_{k}^{\text{GMM}}\) and \(f_{k}^{\text{IDNN}}\) are compared. The IDNN is trained via the same training and test datasets with \(\theta_{\text{th}}=1.00\) as the ML models in Section 6.1.1. In [16], a clipping process is applied to center the trajectory centroid within the image, as their objective is to classify transportation modes, focusing primarily on the shape of the trajectory. By contrast, this study aims at attribute classification, where the relative position of the trajectory within the region, in addition to its shape, is crucial. Therefore, no clipping process is applied. The IDNN has several hyperparameters that need to be tuned. It is not realistic to tune all the hyperparameters in terms of the computational cost; thus, we focus on the number of hidden layers \(\lambda\) and the number of neurons in the \(l\)-th layer \(n^{(l)}\) to tune these parameters via the GS+CV. To simplify the process, we assume that all layers have the same number of neurons, i.e., \(n^{(1)}=n^{(2)}=\cdots=n^{(\lambda)}\). We suppose the search ranges for these hyperparameters with \(\lambda\in\{2,3,4\}\) and \(n^{(l)}\in\{100,150,200\}\), \(\forall l\in\{1,2,\ldots,\lambda\}\). The other parameters and settings for the IDNN are as follows:  
* Image size: \(W_{m}=H_{m}=32\)
* Noise distribution \(q_{D}\): masking noise (rate = 0.2)
* Activation functions:
* \(\varphi(\cdot)\), \(\varphi^{\prime}(\cdot)\): ReLU
* \(\varphi_{\text{supv}}(\cdot)\): Softmax
* Loss functions:
* \(\mathcal{L}\), \(\mathcal{L}_{\text{supv}}\): mean squared error
* Optimizer for each DEA and fine tuning of the DNN: Adam  
The image size determines the expressiveness of the trajectory representation. A larger image size is anticipated to enhance the representational capacity of the trajectory; however, it may also increase susceptibility to noise. Conversely, a smaller image size is more robust to noise but may compromise the expressive power of the trajectory representation. In [16], an image size of \(W_{m}=H_{m}=40\) was reported to achieve the best accuracy and demonstrate good generalization performance. In this study, the task involves a higher degree of trajectory sparsity and noise compared to [16]. To address these challenges while avoiding significant loss in representational capability due to an overly small image size, the image size was set to \(W_{m}=H_{m}=32\). Masking noise is used during the pre-training phase of SDA to deliberately introduce noise, with the goal of improving the robustness of extracted features. In [16], it was reported that a masking noise rate of 0.4 achieved the highest accuracy and improved generalization performance. However, because the trajectories in this study inherently contain noise, using a high masking noise rate would significantly reduce accuracy. Consequently, the masking noise rate was set to 0.2 in this study. In [16], sigmoid functions were used as activation functions. However, sigmoid functions are known to cause the vanishing gradient problem. In recent years, ReLU functions have gained popularity in many tasks [51]. Furthermore, since the output layer during fine-tuning is compared with binary vectors representing the classes, the Softmax function is more suitable than the sigmoid function. For these reasons, this study adopted ReLU for \(\varphi(\cdot)\) and \(\varphi^{\prime}(\cdot)\), and the Softmax function for \(\varphi_{\mathrm{sup}}(\cdot)\). For the loss function, although Cross-Entropy is commonly used for classification tasks, some studies report that mean squared error (MSE) performs better for certain problems [52]. Since there is no definitive conclusion about the superiority of one over the other, this study follows [15; 16] and adopts MSE as the loss function. Regarding the optimizer, SGD was utilized in [15; 16]. However, SGD requires careful tuning of the learning rate and is prone to becoming stuck in local optima if not configured correctly.