## 2 Methodology
### Quantitative Analyses of COLREGs
#### 2.2.1 Four-Stage Theory  
According to COLREGs, two power-driven vessels in a hazardous conflict approach one another in four stages if they maintain their speed and course [30]. The four-stage theory is established to expound on the timing to avoid the collision, which involves \"no risk of collision (No CR)\", \"risk of collision (CR)\", \"close-quarter situation (CS)\", and \"immediate danger stage (ID)\" as shown in Figure 2. For stand-on vessels and give-way vessels, the timing of taking measures to prevent collisions is distinct in the same encountering situation. Furthermore, actions taken by the vessel, which include altering courses and changing speed, are varying with the development of the stage and two vessels approaching each other [31].  
In stage 1: the vessel is too far to cause a collision, while both vessels are free to move during this time up until the first point of the CR. Instead of a series of minute changes in course and speed as required by Rule 8 and good seamanship in practice, any necessary course and speed changes must be large enough to be easily apparent to another vessel visually or by radar if the circumstances permit. Being able to observe another vessel's purpose clearly, ensures that the concerned vessel avoids the potential of misunderstandings. Furthermore, if the TS eventually invades the OS's ship domain on the basis that both ships keep their initial course and speed, the potential collision risk (PCR) exists between the ships. The first time-in-point of collision risk (FTCR) indicates the conclusion of this stage and also leads to the formation of stage 2.  
Figure 2: The diagram of the four-Stage theory.  
In stage 2: The conflict is increasing, and the giving-way vessel shall take action as soon as feasible to avoid the collision, as per COLREGs. Meanwhile, the stand-on vessel should maintain its speed and course in the first stages, but use all available means to monitor the success of the acts made as necessary by the other vessel. For the giving-way vessel, this is the \"golden time\" to take precautions. On the one hand, to act early is not only in the spirit of COLREGs, but also gives the ship more time to deal with the urgency. On the other hand, it provides the stand-on vessel more time to observe and check whether the conflict occurred or not after the collision-avoidance action. With the two vessels coming close together, the risk of a collision is high, and the stand-on vessel is permitted to take whatever actions necessary to avoid a collision after it is determined that the give-way ship is not acting correctly and effectively in accordance with COLREGs. This timing is determined by the situation and ship's manoeuvrability, and it is best defined by the ship captain's evaluation. CR that is the initial point exists at the confined sea only when \(TCPA\leq 20\) mi in this study [30].  
In stage 3: CS is created when both ships are unable to pass each other at a safe distance whichever collision avoidance action by each ship is taken.  
In stage 4: This is the most perilous stage when facing a situation where a disagreement has escalated to the point where it cannot be resolved just by the action of the give-way ship. Both vessels must execute collision-avoidance measures; otherwise, a collision will occur in a short period [31].  
In this paper, the model of CRI is utilised to quantify the encounter stage of ships, whose larger value indicates a higher risk, and the detail is shown in Section 3.1.