## 1 Introduction
Each PTDP data point has an ID for identifying mobile devices by day, and those linked to this ID are the associated attributes registered by the user of the device and the location data from the device. Although location data are acquired in one-minute units, most of the data for a single day are missing because the data acquisition timing is determined by an event, such as the user opening a specific application of the device. Thus, we use a simple method to impute the missing location data and regard these imputed data as the user trajectory to make the data length uniform. Furthermore, for this study, we select age groups as the specific user groups that are subjected to classification and assume that the age groups can be categorized into six classes: 0 to 19, 20 s, 30 s, 40 s, 50 s, and 60 s and over. Typically, mobile users' location data, which consist of longitude and latitude, are expressed in the geographic coordinate system. In other words, the location data that are part of a trajectory are represented as points in the region [180\({}^{\circ}\) W-180\({}^{\circ}\) E, 90\({}^{\circ}\) S-90\({}^{\circ}\) N]. Because using such trajectories as features is not realistic, we need to summarize a trajectory into feature vectors consisting of a finite number of dimensions. Hence, we generate \(n\)-dimensional feature vectors by assigning \(n\) representative points (RPs) to a target region and aggregating the user locations into RPs. A RP is generated using the Gaussian mixture model (GMM), which assumes a Gaussian mixture distribution (GMD) for the population distribution in a target region. In particular, the GMD is generated from the raw location data of all users in a target period, and we regard the centroids of each Gaussian distribution as the RPs for that distribution. Thus, the RPs are selected in locations where the population is concentrated. In this way, each user trajectory is aggregated into RPs, and \(n\)-dimensional feature vectors for classifying user age groups are generated. We employed ML models that use the \(n\)-dimensional feature vector as input and output an age group linked to the mobile users. For the ML models, we employed SVC, random forest (RF) [21], and deep neural network (DNN) [22] models, and for the hyperparameter tuning of these ML models, we used fivefold grid search cross-validation (GS+CV). With feature vectors generated using GMD in this manner, it is possible to construct a classification model that uses only spatial-temporal information without complicated preprocessing, which differs for each target region. In addition, this study aims to verify the effectiveness of features based on the GMM for classifying sparse trajectories without semantic information. To achieve this goal, a comparison is conducted with existing feature extraction methods for trajectory data. In particular, the comparison focuses on features extracted using the improved DNN (IDNN), developed by [PERSON] et al. [16], which is based on an SDA [15]. The IDNN does not assume that the trajectory is sparse, but part of this feature extraction method can be applied to our problem settings. This comparison demonstrates the superiority of the GMM-based features for sparse trajectories proposed in our past work, highlighting their effectiveness in this context. This allows for an evaluation of the relative performance and applicability of the GMM-based features. In our experiments, we demonstrate the performance of three ML models trained with GMM-based features via PTDP data acquired between January and June 2021 in Narashino, Chiba, Japan. Because the number of GMD components that determine the feature vector's dimensions can significantly influence the classification accuracies of the ML models, we conduct a sensitivity analysis on the number of components. In addition, we evaluate the accuracy and generalization performance of the combinations of three ML models and GMM- and IDNN-based features with respect to the level of missing value ratio, which quantifies the sparsity of trajectories.