# Prediction of Arrival Time of Vessels Considering Future Weather Conditions
## 3 Proposed Method
### (B) Voyage Speed Calculation  
An overview of the voyage speed calculation is illustrated in Figure 7. The time required to travel between sites and the time of arrival at the destination port are determined by calculating the voyage speed for each transit site in steps B1 to B3 described below. In B1, past data are found for similar vessels at each transit site, i.e., the longitude and latitude of the transit site of interest and past data for vessels with similar GRTs and types. Here, there are only a few data with the exact longitudes and latitudes of the transit sites of interest, so data within \(\pm 0.1^{\circ}\) of the exact latitudes and longitudes are used. As an example, the data shown in Figure 6 are longitudes ranging from 120.05 to 120.25 and latitudes ranging from 22.58 to 22.78. In B2, Bayesian learning is used to derive the probability distribution of the voyage speeds by considering the bow direction and future weather conditions at each site, and the average speed is calculated. The probability distribution is updated on the basis of [PERSON] theorem given in Equation (2). \[\text{Posterior distribution}\ \propto\ \text{prior distribution}\times\text{likelihood function} \tag{2}\]  
Here, the prior and posterior distributions are the probability distributions of the estimation target before and after learning, respectively, and the likelihood is the probability distribution of the learning data. The symbol \(\propto\) denotes that the term on the right side is proportional to the terms on the left side. In this study, the estimation target is the voyage speed; the prior distribution is the probability distribution of the voyage speed generated from the data in B1; the likelihood function is the probability distribution of past data with similar bow directions, wave heights, wave periods, and wind speeds as the considered vessel; and the posterior distribution is the probability distribution of the voyage speed reflecting the likelihood function in the prior distribution. Note that both the prior distribution and likelihood function  
Figure 7: Overview of the procedure used to calculate the voyage speed. are histograms created using real data. Figure 8 illustrates how Bayesian learning is used to calculate the voyage speed. [PERSON] theorem is applied to each class of the histogram for the voyage speed. In Figure 8, the probability of distribution in the 10 knots class is 40%, which is calculated from the data in B1. Likelihood is calculated from the data of the ship whose heading is 60 degrees, HTSGW is 2.5 m, PERPW is 8 s, and wind speed is 10.5 m per second. Finally, the probability of posterior distribution is calculated by multiplying the prior distribution and likelihood, as shown in Equation (2), which is 11%. The same procedure is used for the other voyage speed classes, and the prior and posterior distributions are obtained by normalizing the calculated histogram to 1. As mentioned above, the likelihood function is calculated from data for vessels with similar bow directions, wave heights, wave periods, and wind speed parameters, but very little data with exactly matching parameters are available. Therefore, as shown below, the likelihood function is calculated by generating a probability distribution for each parameter and combining the probability distributions. The random variables used in this calculation are defined as follows. \(v\)  
Voyage speed  
_Heading_  
Bow direction  
_HTSGW_  
Wave height  
_PERPW_  
Wave period  
_Wind speed_  
Wind speed  
In the following equations, \(p\) represents the probability distribution, and the prior distribution of the voyage speed is represented by \(p(v)\). Substituting these variables into [PERSON] theorem given by Equation (2) yields the following expression. \[p(v|\text{Heading, HTSGW, PERPW, WindPower})\propto p(\text{Heading, HTSGW, PERPW, WindPower}|v)\times p(v) \tag{3}\]  
\(p(\text{Heading, HTSGW, PERPW, WindPower}|v)\) is the likelihood function for the conditional probability of the voyage speed \(v\).