## 4 Statistical Models
### Comparisons of the Statistical Models  
This section compares the statistical models introduced in previous sections. The most reasonable and reliable approach to determine PA (probability of vessel aberrancy) should be based on long-term accidental statistics. However, when there is no adequate database, as stated by [PERSON] and [PERSON] [33], the AASHTO model is the most widely used model for  
Figure 5: An example for dangerous areas in the drift model (based on [14]).  
calculating the probabilities of collisions, because of its simplicity and practicality. The AASHTO predefined estimates of PA are derived from American data. The worldwide applicability could be the problem.  
Most of the models do not fully consider wind, visibility, navigational aids, or other factors of navigational circumstances. A comparison of the elements considered in each statistical model is given in Table 1. The elements in the table are derived from the models mentioned above. It is pointed out that these models are lacking in detailed descriptions of ship movements [12; 34]. The advantages and disadvantages of each model are discussed as follows.  
[PERSON]'s model, the Eurocode model, and the Three Random Variables model can mimic the critical steps for a ship collision. For a vessel that is completely out of control, the drift model can reflect the movement processes. However, the models show weaknesses in describing the dynamic behaviors of a nautical traffic system, such as describing ship behaviors such as collision avoidance, speed change, or course alteration.  
The ship distribution is considered in all three models. However, the formula of \(P_{c}(x,y)\) is not provided, which makes the model less practical than the AASHTO model. Further, only single ship routes are considered in the models. The problem with this is that these models may not be applicable in other areas. To overcome the deficiencies, the Three Random Variables model has the advantage of being able to consider more elements.  
[PERSON]'s model is applied in the design process of a large span bridge situated in the sea. However, there are disadvantages compared with other models. Firstly, the model does not consider overtaking encounters. Secondly, the effects of wind and currents are not considered. Further, the model assumes that the ship's drifting direction is evenly distributed over \(360^{\circ}\). These are simplifications compared with the Drift Model, in which the drifting ship moves under the force of wind and currents.  
The human factor is fully understood as an element in the probabilistic risk analysis. Human factor analysis should be studied and can be built into the model to calculate the probability of incidents.  
Figure 6: Three random variables collision model [11].  
This section has presented an overview of several models for calculating the probability of ship collisions with structures. The other kinds of probabilistic risks, such as collisions with wind farms, are based on the statistical models introduced in Section 4. After treatment of historical background, procedure, and the advantages and disadvantages of each model, comparisons are made for the risk elements that are considered in the existing statistical models. Consequently, the following discussions are made for further research: Firstly, more risk elements should be taken into consideration. Further research should consider all risk elements listed in Table 1. Moreover, it would be better to describe the ship movements in more detail. Secondly, the categories of ships based on ship types, ship tonnages, and ship dimensions, as well as ship maneuverability, need to be considered to make the model more accurate. Thirdly, different encountering situations must be included. Fourthly, the effects of wind and currents need to be considered. The wind and currents are not only the cause of position aberrance and a larger swept path, but can also drive the ships into collision (e.g., Drift Model). Therefore, the influence of the wind and currents needs to be examined and considered.