# Optimal Prediction of Individual Vessel Trajectories Based on Sparse Gaussian Processes
## 1 Introduction  
The advancement of autonomous surface vessels is rapidly increasing due to the rapid growth of information and communication technology in the era of the 4 th industrial revolution. The European Maritime Safety Agency (EMSA) has compiled statistics showing that 43% of all ship accidents that occurred between 2014 and 2020 were related to navigation, specifically involving collisions, contacts, and groundings/strandings [1]. Furthermore, human factors were responsible for 53.5% of all individual accidents documented in safety investigations during this time limit [1]. Although the vessels' navigational equipment, including RADAR, ECDIS, and AIS, provides ample information, it is evident that a thorough assessment of the collision risk was not conducted appropriately, resulting in the mariners' failure to prevent a collision. To enhance the ability of a navigator to avoid collisions, it is possible to incorporate an intelligent navigation system that includes a predictive trajectory for the ship. This allows encountering ships to take avoidance action in advance [2]. The errors can be minimized by utilizing sophisticated predictions when making decisions [3]. This study presents a methodology for forecasting the path of a vessel, which can be employed in the collision avoidance system of an autonomous surface vessel. [PERSON] and [PERSON] demonstrated that course prediction is beneficial for avoiding collisions [4]. Anticipating the future paths of vessels is crucial for ensuring the safety and optimizing the efficiency of navigation. Extensive research has been conducted not only in the domain of vessel traffic but also in the domains of air traffic and road traffic. Additionally, it has been utilized to create algorithms for traffic monitoring, anomaly detection, collision avoidance, and traffic management. The Gaussian process (GP) is a widely recognized Bayesian nonparametric model that generates posterior probabilistic outcomes. It is well-suited for analyzing continuous-domain data [5] and highly resilient when dealing with noisy observational data [6]. Despite the advantages of GP, it is inefficient to apply it to large datasets with more than 3000 training points due to its time complexity of \(\mathcal{O}\left(n^{3}\right)\) and storage cost of \(\mathcal{O}\left(n^{2}\right)\)[7, 8]. To address these constraints, the sparse Gaussian process (SGP) was introduced. Thus,this study proposes a probabilistic approach for predicting vessel trajectories by utilizing SGP, which enhances the performance of GP. Vessels are expected to follow established traffic patterns when navigating in restricted coastal waters. The AIS data were used to model the vessel trajectory patterns using an efficient Bayesian nonparametric method. The optimal approximation method for the performance of the SGP was selected using a genetic algorithm (GA) from three significant methods: subset of data (SoD) [9], subset of regressors (SoR) [10], and fully independent conditional (FIC) [11]. The predictive capabilities of the proposed model were validated by comparing its performance against conventional models, including the GP and support vector machine (SVM) [12]. SVM, like SGP, employs a sparse kernel method. However, unlike SGP, SVM provides deterministic rather than probabilistic results. This study aims to enhance the development of trajectory prediction systems for autonomous surface vessels to mitigate collisions and promote sustainable practices. The contributions of this paper are summarized as follows:  
* By applying the SGP, which produces posterior probabilistic results using the Bayesian sparse kernel technique, to ship trajectory prediction, both prediction accuracy and computational efficiency are significantly enhanced compared to the full Gaussian process (FGP) [7]. Furthermore, SGP demonstrates superior accuracy over SVM, which utilizes the same sparse kernel technique but yields deterministic results.