387

1 INTRODUCTION

Maritime Autonomous Surface Ships (MASS) are being

rapidly developed to address seafarer shortages and

prevent accidents. Objective safety evaluation is

essential for practical application, making simulation-

based testing crucial for cost-effectiveness and

comprehensiveness, as recommended by Class NK

guidelines [1]. While systematic frameworks for

evaluating collision avoidance algorithms exist [2], the

“lack of interaction with other vessels” remains a

challenge. In conventional simulations, target ships

typically follow “constant speed and course” or

“predefined trajectories,” failing to replicate the

probabilistic avoidance maneuvers seen in real-world

interactions. This discrepancy undermines the

reliability of MASS evaluations, particularly in head-on

situations. Although COLREGs mandate a starboard

turn, AIS data reveals diverse maneuvers. Realistic

evaluation thus requires a model capable of

reproducing these uncertain interactions. This study

develops a “Probabilistic Reactive Target Model” by

stochastically modeling existing ship behavior. We

focus on the temporal transition of the Points of

Potential Collision (PPC), derived from relative

motion. Instead of modeling the maneuvering action

itself, we represent the PPC’s passing direction and

distance as probability distributions to handle

decision-making uncertainty. The model combines

high-precision Artificial Neural Networks (ANN) with

interpretable linear models using AIS data, balancing

accuracy and interpretability for MASS safety

evaluation.

2 RELATED WORK

Safety evaluation frameworks integrating Operational

Design Domain (ODD) and Systems Theoretic Process

Incorporating Probabilistic Mutual Interactions

in Simulation-Based Safety Evaluation of Maritime

Autonomous Surface Ships

M. Ishii & H. Tamaru

Tokyo University of Marine Science and Technology, Tokyo, Japan

ABSTRACT: This paper proposes a “Probabilistic Reactive Target Model” that generates avoidance behaviors for

target ships to evaluate the collision avoidance algorithms of Maritime Autonomous Surface Ships (MASS) in a

realistic simulation environment. Focusing on the “shift of the Points of Potential Collision (PPC)” resulting from

collision avoidance maneuvers in one-on-one head-on situations, we conducted indirect probabilistic modeling

using AIS data. Specifically, we constructed a state transition probability model by estimating the directional

probability of the PPC shifting to either the starboard or port side using a linear binary classification model, and

by estimating the parameters of the passing distance distribution for each side using a neural network, assuming

a log-normal distribution. Furthermore, by iteratively sampling and evaluating transitions to target states that

follow this model, we demonstrated that it is possible to generate behaviors in a simulation environment where

target ships react to the movements of the MASS.

http://www.transnav.eu

the International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 20

Number 2

June 2026

DOI: 10.12716/1001.20.02.13

388

Analysis (STPA) provide a foundation for identifying

uncertainty and designing scenarios [3]. However,

these focus on high-level concepts rather than dynamic

behavioral models for interacting target ships.

Regarding risk assessment, studies have analyzed how

observation errors affect DCPA/TCPA [4] and

decision-making under uncertainty [5]. While these

clarify how noise propagates to evaluation metrics,

they do not model uncertainty as actual maneuvers.

Additionally, deep learning for trajectory prediction [6]

and Dynamic Bayesian Networks (DBN) for intention

estimation [7] primarily serve as “diagnostic” tools for

MASS situational awareness, which differ from

“generative” models required for simulation.

Understanding the obilistic nature of human decisions

is essential. Seafarers’ interpretation of COLREGs

suggests that avoidance is context-dependent rather

than deterministic [8]. This study builds on that insight,

proposing a model that generates target ship behavior

via PPC movement. The novelty lies in introducing

“behavior generation” with probabilistic interaction

into simulations, rather than focusing on “recognition”

technologies like intention estimation.

3 METHODS

3.1 Calculation of PPC

While DCPA and TCPA are the most common collision

risk indicators, they do not provide information about

the passing direction. Since passing direction is a

critical feature in this study, we adopt the PPC. The

PPC is defined as the coordinate where two ships will

collide, assuming the own ship’s speed and the target

ship’s course and speed remain constant. If the own

ship steers towdhe PPC, a collision occurs. For

example, if the PPC of an approaching vessel from

ahead shifts to the port side, it indicates a port-to-port

passing. Avoidance maneuvers can thus be interpreted

as steering to remove the PPC from the ship’s current

heading. Although the PPC assumes ships are point

masses, it is widely applied in collision area modeling

by incorporating hull sizes. Classic examples include

the Predicted Area of Danger (PAD), as well as modern

indicators like the Dangerous Area of Collision (DAC)

[9] and Obstacle Zone by Target (OZT) [10]. These are

used in both maneuvering support systems [11] and

traffic flow evaluations [12]. In this study, we treat hull

size as a separate feature, so assuming point masses for

the PPC calculation does not pose any issues. We

calculate the PPC using the collision course approach,

as illustrated in Figure 1. The own ship and the target

ship are separated by a distance d. The relative bearing

of the target ship from the own ship is



Az. Vown and Vtgt

represent the speeds of the own ship and target ship,

respectively. If the target ship’s course is Ctgt, the sine

rule based on the geometry in Figure 1 yields the

following relationship for the collision course:

( )

Az tgt

Az PPC

tgt own

sin





−

(1)

Solving for the collision course—the azimuth of the

PPC from the own ship



PPC:

( )

tgt

PPC Az Az tgt

own

sin sin

  

−



= − −





(2)

The collision course and PPC exist only if:

( )

tgt

Az tgt

own

sin C



−  − 

(3)

Once



PPC is determined, the distance to the PPC

(DPPC) and the time to collision (TPPC) are calculated

using the relative velocity Vrel:

tgt tgt own PPC

V sinC V sin



−

(4)

tgt tgt own PPCry

V V cosC V cos



=−

(5)

rel rx ry

V V V=+

(6)

PPC

rel

(7)

PPC own PPC

D V T=

(8)

Figure 1. Calculation of PPC

3.2 AIS Data Acquisition and Preprocessing

We analyzed one week of AIS data from the coastal

waters of Eastern Japan. To focus on medium-to-large

merchant ships, only vessels with a length overall (L) 

80 m were extracted, as smaller vessels like fishing

boats differ significantly in maneuverability. Data from

Tokyo Bay was excluded to focus on open-sea

maneuvers and eliminate the influence of vessel traffic

services. The resulting dataset primarily represents

along-coast traffic with few large crossing angles.

Therefore, this study focuses on head-on and near-

head-on situations, defined by the target’s relative

course range of 150 < Crel < 210, where both vessels are

expected to take reciprocal avoidance actions. The

asynchronous AIS data was synchronized via linear

interpolation and resampled at 30-second intervals.

Table 1 lists the statistics of the input features x. To

capture active collision risks, we selected data

satisfying 1.0 < XPPC,cur < 7.0 [nm] and -1.5 < YPPC,cur<1.5

[nm]. Other features include ship lengths (Ltgt, Lown),

own ship speed (Vown), and target’s relative heading

(Crel).

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Table 1. Mean and standard deviation of extracted features.

Feature

Unit

Mean ± SD

YPPC,cur

−0.12 ± 0.59

XPPC,cur

3.36 ± 1.44

Ltgt

102.9 ± 49.5

Lown

127.4 ± 49.1

Vown

knots

13.9 ± 2.9

Crel

deg

180.7 ± 15.1

3.3 Inverse Estimation Approach for Collision Avoidance

Maneuvers

3.3.1 Advantages of Focusing on State Transitions

Directly predicting changes in course and speed is

difficult because: (1) maneuvers are often triggered by

non-avoidance factors like waypoints; (2) the timing of

actions varies widely; and (3) frequently only one

vessel takes action. Instead, we focus on state

transitions. We propose a framework that

probabilistically models the time-series changes of

collision risk indicators and inversely estimates the

maneuvers required to achieve them. Specifically, we

use the PPC position, which integrates passing

direction and distance. As shown in Figure 2, PPCs

rarely stay directly ahead of the bow but distribute

laterally, suggesting that avoidance is geometrically

equivalent to shifting the PPC away from the ship’s

heading.

Figure 2. Relative plot of PPC positions.

3.3.2 Problem Formulation

Avoidance is modeled as a probabilistic event

transitioning from a “current situation” to a “PPC

passing state.” The input vector x consists of six items:

PPC,cur PPC,cur own tgt own rel

, , , , ,x Y X L L V C





(9)

The output y is the lateral position of the PPC when

it reaches the own ship’s beam (X = 0), denoted as

YPPC,abm (Figure 3). We approximate the distribution of

y by integrating three models: a binary classification

model for PPC passing direction (starboard/port) and

two distance distribution models for each side.

Observed PPC passing distances (Figure 4) show a

long-tailed distribution. To replicate this, we compared

log-normal and gamma distributions, selecting the one

with the highest log-likelihood. The parameters



for

these distributions are determined by an Artificial

Neural Network (ANN):

( ) ( )

( )

( ) ( )

ANN

,p y p y f



==x x x x∣

(10)

Figure 3. Conceptual image of PPC transitions toward the

abeam position.

Figure 4. Distribution of observed passing distances (y) for all

encounters.

3.3.3 Action Selection Process

In simulations, the target ship’s decision process

involves two steps: Step 1: Target State Generation A

target passing distance ytarget is sampled from the

predicted probability density:

( )

target

y p y x

(11)

Step 2: Inverse Action Estimation Let fmotion be the

motion model and be the set of possible actions.

The resulting passing distance for an action a  is

y=fmotion(x, a). The optimal action a



is found by

minimizing the deviation from ytarget:

( )

motion

argmin ,

target

a f a y



=−x

(12)

3.4 Passing Direction Classification Model

For the passing direction determination model, we

applied an ANN framework that performs binary

classification. This model outputs a value in the range

390

of 0 to 1, representing the probability of the direction in

which the passing will occur. For comparison, we

constructed an ANN with a hidden layer and a linear

model without a hidden layer. The two differ only in

the presence or absence of a hidden layer; the model

with a hidden layer used one layer with 32 nodes. The

loss function was binary cross-entropy, the activation

function for the hidden layer was ReLU, and the

activation function for the output layer was Sigmoid.

Other training conditions were identical. In the ANN

model, the input x undergoes a non-linear

transformation in the hidden layer, and the output pstb

is expressed by the following equation:

( )

stb ANN



= x

(13)

Here, fANN(x) represents the non-linear mapping in

the hidden layer, and () is the sigmoid function. This

output value p indicates the probability of a starboard

passing; pstb=1 means a starboard passing is certain, and

pstb=0 means a port passing is certain. Note that a

starboard passing is the result of a port turn, while a

port passing is the result of a starboard turn. On the

other hand, the linear model is a special form of the

ANN, corresponding to a structure where the hidden

layer is omitted. That is, the non-linear mapping fANN(x)

is replaced by a simple linear combination, expressed

by the following equation:

z x b=+w

(14)

This output is passed through the sigmoid function

to obtain the starboard passing probability pstb.

( )

stb

1 exp



+−

(15)

Here, since pstb=0.5 when z=0, it can be determined

that if z > 0, a starboard passing (i.e., port turn) is more

likely, and if z < 0, a port passing (i.e., starboard turn)

is more likely. The port passing probability pprt is

calculated as follows:

prt stb

1pp=−

(16)

3.5 Passing Distance Distribution Model

Next, the method for constructing the starboard and

port passing distance distribution models is described.

Here, the passing distance is denoted by the random

variable y, which corresponds to YPPC,abm defined in

Section 3.3.2. In this paper, the starboard side is defined

as positive distance and the port side as negative

distance; however, during model construction, the port

side is also treated as a positive value. Therefore, the

training procedure and settings for both models are

identical. Hereafter, unless a distinction is required,

these models are collectively referred to as the passing

distance distribution models. Since these distributions

exhibit asymmetric shapes with one-sided tails, the

log-normal and gamma distributions were selected as

candidates. Both were compared, and the one showing

the higher log-likelihood was adopted. The probability

density function of the log-normal distribution is

defined as follows:

( )

1 (ln )

, , 0

f y exp y





  



−

= − 





(17)

: Mean of the variable’s natural logarithm lny

: Standard deviation of the variable’s natural

logarithm lny

y: Passing distance as a random variable (positive

values only)

By using the Negative Log-Likelihood (NLL) of the

log-normal distribution as the loss function, it becomes

possible to learn the distribution parameters using an

ANN. The NLL for a sample

is expressed by the

following equation:

( )

(ln )

, ln , ln 2

L f y y



     





−

= − = +







(18)

Similarly, the probability density function of the

gamma distribution and its NLL are defined as follows:

( )

, , 0

f yk y e y



−−

=

(19)

k: Shape parameter



: Scale parameter

(k): Gamma function

y: Passing distance as a random variable (positive

values only)

( )

1 1 1

, ln , ln lnΓ 1 ln

N N N

i i i

L k f y k N k k k y y

  



= = =

= − = + − − +

  

(20)

3.6 Integration and Generation of Probabilistic Avoidance

Behavior Model

The integrated model is defined by combining the

classification model that determines the passing

direction with the passing distance distribution models

for the starboard and port sides. In this model, the

overall passing distance distribution is obtained by

weighting the respective passing distance distributions

with the probability of the passing direction. Let

fstb(y|stb,stb) and fprt(y|prt,prt) be the passing distance

distributions for the starboard and port sides,

respectively. The overall passing distance distribution

fint(y) by the integrated model is defined by the

following equation:

( )

int stb stb stb stb prt prt prt prt

,,f y p f y p f y

   

(21)

Here, fstb and fprt are passing distance distribution

models constructed based on the log-normal

distribution (or gamma distribution), and their

distribution parameters ,  (or k,



) are predicted by

the ANN. By integrating the probability of the passing

direction and the distance distributions in this manner,

the entire passing behavior can be expressed

probabilistically.

3.7 Training Conditions and Hyperparameters

TensorFlow was used for the training of each model,

and Adam [13] was adopted as the optimization

method. The total number of samples analyzed from

the AIS data was 821,972. Specifically, 296,007 samples

391

of starboard passing data were used for the starboard

passing distance distribution model, and 525,965

samples of port passing data were used for the port

passing distance distribution model. All data (821,972

samples) were used for the binary classification model.

In the training of all models, input features were

standardized to facilitate smooth ANN training. The

data was randomly split into training and testing sets

at a ratio of 8:2, and the training set was further

subdivided into training and validation sets at a ratio

of 8:2. Consequently, 64% of the total data was used for

training, 16% for validation, and 20% for testing.

Training was conducted under common conditions for

all models, with the number of epochs set to 20 and the

batch size set to 128.

4 RESULTS AND DISCUSSION

4.1 Evaluation and Analysis of Passing Direction

Classification Model

4.1.1 Performance Evaluation

A performance comparison was conducted

regarding the presence or absence of a hidden layer for

the model classifying the passing direction (starboard

or port). Classification performance was evaluated

using Accuracy, which indicates the proportion of

correctly classified instances. The results showed that

the model with a hidden layer achieved an accuracy of

0.9102, while the model without a hidden layer

achieved 0.9058. Although the model with a hidden

layer showed slightly higher performance, the

difference was not significant. Since the model without

a hidden layer corresponds to a linear model, it

possesses a simple structure and high interpretability.

Therefore, the linear model without a hidden layer is

adopted for the subsequent discussion.

4.1.2 Contribution of Features to Maneuver Decisions

Table 2 shows the slopes (coefficients) and intercept

of the linear model. The sign of a coefficient represents

the direction of influence that each feature exerts on the

probability of a starboard passing. A positive

coefficient means that a larger value of the feature

increases the probability of a starboard passing (i.e., a

port turn), while a negative coefficient means it

increases the probability of a port passing (i.e., a

starboard turn). Specifically, YPPC,cur has a positive

value, indicating a tendency that the more the PPC is

located to the front-starboard of the own ship, the more

likely a starboard passing is selected. On the other

hand, Lown and Ltgt show negative values, indicating that

larger hull sizes tend to lead to the selection of a port

passing (starboard turn).

Comparing the absolute values of the coefficients

on the standardized scale shown in Figure 5, the

contribution of YPPC,cur is the largest, significantly

exceeding other features. Next, Lown and Vown, which

affect the own ship’s maneuverability, show relatively

large values.

Figure 5. Feature relative importance based on absolute

standardized coefficients. Red bars indicate positive

coefficients, while blue bars indicate negative coefficients.

4.1.3 Decision Boundaries for Starboard and Port Turns

Under COLREGs, a starboard turn is mandatory in

head-on situations. However, in practice, decision-

making becomes difficult when the PPC is slightly to

starboard of the bow, as navigators must choose

between a standard starboard turn or a port turn to

maintain a safe passing distance. To quantify this

“ambiguous zone,” we identified the YPPC,cur where the

probabilities of starboard and port turns are equal. By

inputting the mean values of all features except for the

YPPC,cur into the linear model, we found that the output

z=0 occurs at a YPPC,cur of 0.12 nm. While other factors

influence the final decision, this suggests that the

choice between starboard and port maneuvers is most

uncertain when the encounter involves a starboard-to-

starboard passing of approximately 0.12 nm. This

finding quantitatively defines the boundary of

decision-making uncertainty, providing critical insight

for probabilistic behavioral modeling.

4.1.4 Interpretability Analysis using Linear Model

Coefficients

The coefficients in the original scale represent the

extent to which each feature contributes to the linear

combination output z when the feature changes by one

physical unit. Since the transformation by the sigmoid

function is applied subsequently, the influence of a

change in each feature on the probability p is non-

linear, and its magnitude depends on the current state

(the value of z or p). Therefore, while these coefficients

do not represent a direct increase or decrease in

probability, they are general parameters that can be

published in a form that ensures the reusability of the

model. This format not only facilitates implementation

into navigation support systems and simulation

environments but also enables quantitative

comparison of maneuvering decision models in the

maritime field.

392

Table 2. Regression coefficients of the linear model in

standardized and original scales. Since the predicted

variable is a probability (dimensionless), the coefficients in

the original scale have the inverse units of each input

variable (e.g., [m⁻¹], [nm⁻¹]), whereas the standardized

coefficients are dimensionless.

Feature

Coefficient

(Standardized Scale)

Coefficient (Original

Scale)

Unit

YPPC,cur

3.09 × 10⁰

5.27 × 10⁰

[nm⁻¹]

XPPC,cur

−5.62 × 10⁻²

−3.91 × 10⁻²

[nm⁻¹]

Ltgt

−9.70 × 10⁻²

−1.96 × 10⁻³

[m⁻¹]

Lown

−3.56 × 10⁻¹

−7.25 × 10⁻³

[m⁻¹]

Vown

3.06 × 10⁻¹

1.04 × 10⁻¹

[knots⁻¹]

Crel

9.09 × 10⁻²

6.03 × 10⁻³

[deg⁻¹]

Bias

(Intercept)

−1.28 × 10⁰

−1.92 × 10⁰

—

4.2 Evaluation and Analysis of Passing Distance

Distribution Model

4.2.1 Comparison of Distribution Model Performance

In this study, the log-normal distribution and the

gamma distribution were compared as passing

distance distribution models. Table 3 shows the

Average Log-Likelihood. A larger value indicates that

the model explains the data better. For both the

starboard and port sides, the average log-likelihood of

the log-normal distribution exceeded that of the

gamma distribution, indicating a better goodness of fit.

Therefore, the log-normal distribution is adopted for

the subsequent analysis in this study.

Table 3. Comparison of Lognormal and Gamma Models for

Passing Distance Distribution. Avg. Log-Likelihood

represents the average log-likelihood per sample.

Side

Distribution

Avg. Log-Likelihood

Port

Lognormal

0.145

Port

Gamma

0.067

Starboard

Lognormal

0.338

Starboard

Gamma

0.255

4.2.2 Discussion on the Distribution Characteristics of

Passing Distance

The result that the passing distance resulting from

avoidance maneuvers fits the log-normal distribution

better than the gamma distribution suggests that the

formation process of the passing distance is governed

not by additive factors, but by the accumulation of

multiple proportional fluctuations—that is,

multiplicative uncertainty. Generally, when an

observed value is composed of the sum of multiple

additive errors, its probability distribution exhibits a

shape like a normal or gamma distribution. For

instance, letting D0 be the reference value and



i be the

additive errors, it can be expressed as:

0 i





(22)

If each



i > 0 follows an exponential distribution, D

follows a gamma distribution. On the other hand,

when errors accumulate as proportional fluctuations—

that is, when each factor has a multiplicative influence

representing “how much it increased or decreased

relative to the current value”—the observed value is

expressed as:

DD=



(23)

Here,



i is a random variable representing the

relative fluctuation due to each factor. Since (1+



i) is

positive, it satisfies



i > -1. Realistically,



i is considered

to be a small random fluctuation (e.g., around  tens of

percent). Taking the natural logarithm of both sides

gives:

( )

ln 1

lnD lnD



= + +



(24)

The right-hand side is the sum of independent

random variables. According to the Central Limit

Theorem, if each ln(1+



i) has a finite mean and

variance, their sum approaches a normal distribution

as the number of samples increases. Therefore,

( )

,lnD N





(25)

This implies that D follows a log-normal

distribution. The passing distance is a quantity

determined by the interaction of multiple factors, such

as observation errors of the target ship’s position,

individual differences regarding the desired passing

distance, and variance in hull control. It is reasonable

to consider that each of these factors influences the

outcome not as an absolute value change but as a

relative increase or decrease. Given this structure, it is

natural to assume that the passing distance tends to

exhibit multiplicative rather than additive uncertainty,

which is consistent with the property of showing log-

normal characteristics.

4.2.3 Analysis of Factors Determining Passing Distance

Permutation Feature Importance (PFI) [14] was

used to identify the primary features in the passing

distance model. Figures 6 and 7 show the relative

importance of features calculated for the port and

starboard passing distance models, respectively. Here,

the contribution of each feature was normalized so that

the total equals 100%. PFI is calculated based on the

average decrease in log-likelihood when feature values

are randomly permuted. The analysis results showed

that in both the port and starboard models, YPPC,cur

demonstrated the highest importance, showing a trend

similar to that of the passing direction classification

model. This was followed by a high contribution from

XPPC,cur, which is thought to be because the distance

remaining until collision strongly affects the

magnitude of uncertainty in subsequent manoeuvring

actions. It should be noted that while PFI indicates the

average importance across all test data, the dataset

includes cases where avoidance maneuvers have

already been completed (e.g., YPPC,cur = 1.0 nm, XPPC,cur =

1.0 nm). In such cases, the predicted YPPC,abm

distribution forms a sharp peak near the input YPPC,cur,

as the passing configuration is already established, and

consequently, there is a tendency for the contribution

of YPPC,cur and XPPC,cur to be evaluated highly.

393

Figure 6. PFI of starboard passing distance model (total =

100%)

Figure 7. PFI of port passing distance model (total = 100%)

4.3 Characteristic Analysis of the Integrated Probabilistic

Model

Individual Conditional Expectation (ICE) [15] is a

prominent method for visualizing and interpreting the

behavior of black-box machine learning models, such

as ANNs. It is characterized by the ability to

individually verify how the model output fluctuates

when the value of a specific input feature is changed.

In the integrated model of this study, for example, it is

possible to analyze how the output passing distance

distribution and passing direction probability change

when one of the input features is varied. In this study,

we selected one input feature of interest, varied its

value across three levels within the range of the

training data distribution, and compared the changes

in the probability distributions output by the

integrated model. Other input features were fixed at

the mean values of the training data. Overall, these ICE

analyses confirm that the integrated probabilistic

model captures physically reasonable and

interpretable trends in human collision avoidance

behavior.

1. Variation in YPPC,cur When the input feature YPPC,cur

was set to three values: −0.2 nm, 0 nm, and +0.2 nm,

no significant change was observed in the shape of

the passing distance distribution itself; however,

the probability of the passing direction

(starboard/port turn) changed significantly as

shown in Figure 8 and Table 4.

2. Variation in XPPC,cur When XPPC,cur was set to 6 nm, 3

nm, and 1 nm, a tendency was confirmed where the

peak of the log-normal distribution became steeper

as XPPC,cur decreased as shown in Figure 9 and Table

5. This is considered to reflect the fact that in

situations where the longitudinal distance to the

PPC is short, it is difficult to secure a sufficient

passing distance, resulting in a narrower spread of

the distribution.

3. Variation in Lown and Ltgt When Lown and Ltgt were set

to three levels: 100 m, 200 m, and 300 m, the results

showed that for both features, as the ship length

increased, the peak of the log-normal distribution

shifted outward, indicating a tendency to take a

larger passing distance as shown in Figures 10 and

11 and Tables 6 and 7. This is thought to reflect

maneuvering behavior that ensures a wider safety

margin for larger vessels.

Table 4. Log-space parameters (, ) of port and starboard

passing distance models, and the probability of starboard

passing at different YPPC,cur values.

YPPC,cur

[nm]



prt



prt



stb



stb

pstb

0.0

-1.0338

0.4166

-1.1248

0.3782

0.3434

0.2

-1.0639

0.5047

-1.0651

0.3268

0.6000

-0.2

-0.9752

0.3558

-1.0900

0.4368

0.1542

Figure 8. Predicted distribution of the integrated model when

the Y_{\mathrm{PPC,cur}} value is varied (−0.2 nm, 0 nm,

+0.2 nm).

Table 5. Log-space parameters (, ) of port and starboard

passing distance models, and the probability of starboard

passing at different XPPC,cur values.

XPPC,cur [nm]



prt



prt



stb



stb

pstb

6.0

-0.8391

0.4699

-0.7919

0.4517

0.2009

3.0

-1.0610

0.3622

-1.1841

0.4052

0.2204

1.0

-1.5779

0.2388

-1.6372

0.2580

0.2341

Figure 9. Predicted distribution of the integrated model when

the XPPC,cur value is varied (6 nm, 3 nm, 1 nm).

394

Table 6. Log-space parameters (, ) of port and starboard

passing distance models, and the probability of starboard

passing for different target ship lengths.

Ltgt [m]



prt



prt



stb



stb

pstb

100

-1.0137

0.3790

-1.1196

0.4150

0.2190

200

-0.7189

0.3912

-0.5963

0.3549

0.1873

300

-0.4307

0.3411

-0.1613

0.2924

0.1593

Figure 10. Predicted distribution of the integrated model

when the target ship length is varied (100 m, 200 m, 300 m).

Table 7. Log-space parameters (, ) of port and starboard

passing distance models, and the probability of starboard

passing for different own ship lengths.

Lown [m]



prt



prt



stb



stb

pstb

100

-1.2353

0.4018

-1.1744

0.4071

0.2537

200

-0.6046

0.3463

-0.8472

0.4731

0.1414

300

-0.4140

0.2783

-0.5646

0.4831

0.0739

Figure 11. Predicted distribution of the integrated model

when the own ship length is varied (100 m, 200 m, 300 m).

5 MODEL APPLICATION IN SIMULATION-

BASED SAFETY EVALUATION

This section evaluates the proposed models by

implementing them as a “Probabilistic Reactive Target

Model” within a Model-in-the-Loop Simulation (MILS)

environment. Conventional simulations often rely on

deterministic target ship behaviors, such as Constant

or Predefined models. While useful for basic functional

testing, these models fail to capture the uncertainties

and complex interactions encountered in real-world

maritime environments. In contrast, our model enables

the generation of diverse and realistic scenarios where

target ships make probabilistic decisions

(starboard/port maneuvers) and dynamic distance

adjustments based on the behavior of the MASS. This

allows for the robustness testing of MASS against

“boundary-case decisions” and “accidental risks” that

deterministic scenarios cannot cover. To demonstrate

this, we simulated a head-on situation between a

MASS (equipped with a basic avoidance algorithm)

and a target ship using the proposed model. The initial

conditions were set with a 5 nm range and a 180-degree

relative heading, with both ships offset by 2.5 degrees

to their respective starboard sides (PPC offset by ~0.2

nm). This represents an “ambiguous zone” where a

starboard maneuver is standard but a port maneuver

remains a possibility. Turn rates were capped at 15

deg/min at a constant speed. Figure 12 shows the

trajectories of 100 Monte Carlo simulation trials. In the

figures, the red trajectories correspond to the MASS,

whereas the green trajectories correspond to the

proposed model. Despite identical initial conditions,

the proposed model generated varying avoidance

decisions and passing distances. In addition, 100,000

Monte Carlo trials were conducted under the same

initial conditions, and high-risk cases were extracted

from these large-scale simulations and analyzed

(Figure 13). A “hazardous event” was defined as a case

where the Distance at Closest Point of Approach

(DCPA) falls below 0.3 nm while the range is within 3

nm. The results revealed “conflicting maneuvers,”

where the MASS and the target ship turned in opposite

directions (e.g., one to starboard and the other to port),

maintaining a high collision risk. Notably, such

dangerous interactions are difficult to reproduce with

conventional deterministic models. By introducing this

probabilistic approach, these latent risks can be

manifested and evaluated in a simulation

environment. This demonstrates that the proposed

method is a useful tool for identifying hazardous

encounter scenarios for MASS.

Figure 12. Simulated trajectories of 100 Monte Carlo trials

under identical initial conditions using the proposed

probabilistic reactive target model.

Figure 13. Extracted hazardous trajectories from the Monte

Carlo trials, where conflicting maneuvers result in DCPA <

0.3 nm.

395

6 CONCLUSION

Direct modeling of avoidance maneuvers using time-

series data is challenging because maneuvers in AIS

data contain factors other than avoidance. This study

instead focuses on collision risk indicators with clear

distribution characteristics, modeling their time-series

changes as probabilistic state transitions to inversely

estimate maneuvers. Specifically, we use the positional

change of the PPC to represent passing direction and

distance. The model comprises three sub-models: a

Passing Direction Classification Model and two

Passing Distance Distribution Models (starboard/port).

For passing direction, a linear model was adopted over

an ANN because it offered comparable accuracy

(0.9058 vs. 0.9102) with superior interpretability. For

passing distance, maximum likelihood estimation via

ANNs showed that a log-normal distribution fit better

than a gamma distribution. This suggests that passing

distance is governed by multiplicative uncertainty,

where proportional fluctuations in factors like

observation errors, individual preference, and control

variance accumulate. Parameter analysis (coefficients

and PFI) clarified feature contributions. Model

responses—such as port-to-port passing being

preferred when the initial PPC is port-side, and larger

ships maintaining greater passing distances—align

with maritime expertise, validating the model.

Integrated into a simulation, the model successfully

generated target ship behavior based on probability

distributions. It reproduced conflicting avoidance

maneuvers, a risk difficult to capture with

deterministic models, enabling more realistic

evaluation of MASS algorithms. Future work will

extend the model to crossing, overtaking, and multi-

vessel scenarios under land constraints.

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