Container ship washer adoption factor analysis method based on time-varying survival analysis

Abstract

According to the container ship washer based on time-varying survival analysis, a factor analysis method is adopted, and influence factors influencing the installation decision of the container ship washer are identified from three dimensions of ship characteristics, market characteristics and policy environment; based on a statistical analysis method of correlation analysis, a variance expansion factor, chi-square test and a Cramer's V coefficient, selecting identified factors, performing conditional probability coding and PH test, and selecting a time function based on a Schoenfeld residual error, overall effect stability and a model fitting standard; fitting a time-varying Cox model; and summarizing the model and explaining the result. According to the method, the time-varying Cox proportional risk regression model is adopted to capture the dynamic characteristics of various factors, so that how the influence of different factors is evolved can be more accurately understood. An installation strategy of a specific industry can be more accurately determined by paying attention to the container ship. More influence factors are considered, so that the model is closer to reality, and the result is more reliable.

Description

Technical Field

[0001] The present invention relates to the technical field of container shipping systems, and in particular to a container ship scrubber adoption factor analysis method based on time-varying survival analysis. Background Art

[0002] Numerous studies have evaluated the economic feasibility of various compliance strategies, including installing scrubbers, fuel switching, and using alternative fuels such as liquefied natural gas. These studies typically use cost-benefit analyses and mathematical programming models to compare vessel costs under different scenarios. However, these studies rely on assumptions about key parameters such as fuel prices and market demand, which may not accurately reflect the complexity and dynamics of the market. Furthermore, most studies focus on specific routes or a small number of ships, potentially reducing the accuracy and applicability of their conclusions when the external environment or operating conditions change.

[0003] In contrast, some studies have adopted empirical approaches. Kim and Seo used the fuzzy analytic hierarchy process (FuzzyAHP) to analyze the factors influencing shipping companies' compliance with SOx emission reduction regulations based on 13 questionnaires from Korean shipping companies. Due to the small sample size, the representativeness of their conclusions for the broader shipping market remains to be verified. Subsequent studies have used statistical techniques to analyze scrubber installation decisions and other emission reduction strategies based on large-scale ship databases or Automatic Identification System (AIS) data. For example, Solakivi et al. focused on ships in Finnish ports and used binary logistic regression to examine scrubber installation decisions. Based on cross-sectional data on Finnish maritime trade in 2018, they analyzed the impact of ship characteristics such as age and deadweight tonnage. Subsequently, Bai et al. expanded their research to the global fleet, using AIS data from 2018 to 2019 and employing a logit model to analyze ships' choices between using low-sulfur fuel or installing scrubbers. Li et al. used multinomial logistic regression based on the Clarkson database to analyze the factors influencing the choice between low-sulfur fuel, scrubbers, and alternative fuels. The results showed that factors such as ship type, age, and flag had a significant impact. Zhang et al. specifically studied newbuildings, employing the same methodology but further incorporating market factors and company characteristics, expanding the analytical dimension of influencing factors. Given the significant influence of different ship types on strategy selection, some studies have begun focusing on specific ship types. Bao et al. employed a multivariate logistic regression model for cruise ships and found that the choice of emission reduction strategy was influenced by ship size, age, flag, and shipowner headquarters location. Kokosalakis et al. employed the generalized method of moments (GMM) method for container ships to explore the impact of ship characteristics, market conditions, and shipowner characteristics on emission reduction choices, finding that fuel efficiency, age, and flag were key factors influencing containership emission reduction options. Huang and Li used machine learning to analyze emission reduction options for newbuilding bulk carriers and found that deadweight tonnage and annual voyage distance were key influencing factors. These empirical studies typically rely on large-scale ship datasets such as Clarksons and AIS. However, existing studies primarily use logistic regression to analyze whether a ship adopts a scrubber. These models fail to capture the timing of adoption decisions, which is crucial for understanding the diffusion of scrubber technology. Furthermore, most studies analyze multiple ship types simultaneously, but previous research has shown that scrubber installation decisions vary significantly across different ship types. Therefore, aggregating different types of ships into a single analysis may mask important heterogeneity and lead to inaccurate results. Finally, many studies focus on the period before 2020. However, the implementation of the IMO 2020 regulations has led to significant changes in the regulatory and market environment. Ships sailing outside of Sulphur Emission Control Areas (SECAs) must now comply with a 0.5% sulphur limit, compared to 3.5% before 2020.Understanding scrubber adoption behavior after 2020 and comparing it with pre-2020 is crucial to understanding the diffusion dynamics of scrubber technology. Summary of the Invention

[0004] This paper uses time-varying survival analysis methods (i.e., Kaplan-Meier models and time-varying Cox proportional hazards models) to investigate the determinants of scrubber installation decisions on global container ships. The aim is to answer two key questions: (1) What are the main factors that determine whether and when container ships install scrubbers? (2) How do the effects of these factors change over time, and what is their relative importance?

[0005] A factor analysis method for container ship scrubbers based on time-varying survival analysis is characterized by comprising the following steps:

[0006] Step S1: Data is collected from the Clarkson database to obtain the following information for each ship, including construction year, deadweight tonnage, horsepower, shipowner, operator, builder, flag state, fuel type, scrubber installation year, and scrubber type;

[0007] Step S2: Based on literature research, identify the factors that influence the decision to install scrubbers on container ships from three dimensions: ship characteristics, market characteristics, and policy environment;

[0008] Step S3: Based on the statistical analysis methods of correlation analysis, variance inflation factor, chi-square test, and Cramér's V coefficient, the factors identified in step S2 are selected to eliminate multicollinearity and correlation between variables and determine the main influencing factors;

[0009] Step S4: Conditional probability coding of the categorical variables in the main influencing factors;

[0010] Step S5: Perform PH test on the traditional Cox proportional hazard model using the Schoenfeld residual test method;

[0011] Step S6: When there is a PH test violation, a time-varying Cox model is used for analysis; that is, for variables that violate the PH test, the time function is selected based on the Schoenfeld residual, overall effect stability, and model fit criteria;

[0012] Step S7: fitting a time-varying Cox model;

[0013] Step S8: Model summary and result interpretation: Based on the fitting results of the time-varying Cox model, the decisive factors affecting the decision-making of container ship scrubber installation are analyzed.

[0014] First, this paper employs a time-varying Cox proportional hazards regression model to capture the dynamic nature of various factors, enabling a more accurate understanding of how their impact evolves. Second, focusing on container ships allows for more precise identification of installation strategies for specific industries. Third, considering more influencing factors makes the model more realistic and the results more reliable. Fourth, a larger dataset and a larger number of ships, covering both before and after the implementation of IMO 2020, provide a more comprehensive analysis. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Figure 1 It is a flow chart of the factor analysis method adopted in the present invention. DETAILED DESCRIPTION

[0016] The technical solution of the present invention is described in detail below with reference to the accompanying drawings:

[0017] like Figure 1 As shown in Figure 2, the factor analysis method for container ship scrubbers based on time-varying survival analysis includes the following steps:

[0018] Step S1: Data collection is performed in the Clarkson database; the data in this article covers 4,492 container ships built worldwide between 2000 and 2023. For each ship, various information is provided, including year of construction, deadweight tonnage (dwt), horsepower, owner, operator, builder, flag state, fuel type, scrubber installation year, and scrubber type.

[0019] Step S2: Based on literature research, identify the factors that influence the decision to install scrubbers on container ships from three dimensions: ship characteristics, market characteristics, and policy environment;

[0020] Among them, ship characteristics, including both physical and technical characteristics, are fundamental factors influencing the decision to install scrubbers. Ship age (defined as the number of years from construction to 2023) is a significant factor influencing scrubber installation. Numerous studies have shown a significant negative correlation between ship age and scrubber installation, due to the shorter payback period for older ships. Incorporating ship age into the model effectively measures the impact of a ship's lifecycle stage on the scrubber installation decision.

[0021] Vessel size (defined as the vessel's deadweight tonnage) directly impacts the daily fuel costs and potential fuel savings from scrubber installation. Incorporating size into the model allows for analysis of differences in scrubber installation decisions among vessels of different sizes. Power, the rated power of a vessel's main engine, reflects the vessel's energy consumption characteristics and is directly related to the scrubber's treatment capacity requirements. Research has shown a significant correlation between power and scrubber installation. This is primarily because greater power translates to greater potential fuel savings, thus increasing the economic attractiveness of scrubber installation. Fuel type, such as HSFO (3.5%), VLSFO (0.5%), MGO (0.1%), or LNG, is a key factor in balancing technical and economic considerations. Previous studies have typically considered fuel price as one of the core determinants of ship sulfur emission reduction strategy selection. However, fuel prices fluctuate significantly and are highly volatile due to factors such as international oil prices and geopolitical factors. Accurate, real-time price data is difficult to obtain when ships make scrubber installation decisions. Given that differences in fuel prices are inherently driven by differences in fuel type, this paper uses fuel type, rather than fuel price, as the research objective. More importantly, the choice of fuel type inherently reflects the economic trade-offs faced by shipowners when selecting a compliance strategy. Each fuel type has unique economic characteristics that balance operating expenses and capital investment. This approach preserves the core information about fuel differences while avoiding the difficulties of obtaining price data.

[0022] Market characteristics mainly include shipyards, shipowners and operators, which play different roles in the actual scrubber installation decision. Shipyards represent the capability level of technology suppliers. Shipyards in different regions and market positioning have different technical capabilities and development strategies, which have a significant impact on the installation of scrubbers. For example, some shipyards prioritize early compatibility in ship design and pre-allocate space and infrastructure to facilitate the future installation or modification of scrubbers, thereby reducing the complexity and cost of subsequent installation. Some shipyards and scrubber manufacturers have established strategic partnerships that enable ships to obtain discounts, thereby reducing costs. Including shipyards in the model helps to understand the potential impact of the technical capabilities and development strategies of technology suppliers on scrubber installation.

[0023] Shipowners reflect capital strength and investment preferences. Research shows that sulfur emission reduction decisions vary significantly among shipowners. This is primarily due to differences in capital structure, risk appetite, and long-term development strategies. For example, a shipowner's capital structure and financing capacity directly influence its likelihood of making large-scale investments in technology upgrades. Differences in risk appetite and investment return requirements among shipowners influence their acceptance of scrubbers. Incorporating shipowners into the model helps understand the mechanisms by which capital entity characteristics influence scrubber adoption.

[0024] Operators report the impact of multiple operational dimensions. Research indicates that operational factors such as speed, distance traveled, and time spent in ECAs all influence sulfur reduction strategies. However, these operational indicators fluctuate dynamically with seasonality and market demand, making accurate values ​​difficult to obtain when making scrubber installation decisions. Considering that ships from the same operator typically follow uniform route planning, speed strategies, and fuel policies, incorporating the operator into the model introduces operational factors, replacing more granular but difficult-to-quantify indicators such as speed, distance, and percentage of ECAs.

[0025] The policy environment includes two variables: International Maritime Organization (IMO) regulations and flag state, reflecting the impact of global and local environmental policies on scrubber technology adoption, respectively. IMO refers to whether the ship was built before or after 2020, reflecting the dynamics of global policy pressures. Research shows that as the implementation date of the IMO sulfur reduction policy approaches, the propensity for ships to adopt scrubbers increases significantly. Incorporating IMO regulations into the model helps understand the impact of global policies on scrubber adoption decisions.

[0026] The flag state refers to the country or region where a vessel is registered, reflecting the impact of differences in local regulations and policies. Differences in environmental policies and supporting measures implemented by different flag states may influence scrubber installation decisions. Related research also confirms that flag state significantly influences decisions about sulfur emissions reduction. Incorporating flag state into the model can capture variations in scrubber installation decisions caused by local policy differences.

[0027] Step S3: Based on statistical analysis methods such as correlation analysis, variance inflation factor (VIF), chi-square test, and Cramér's V coefficient, the factors identified in step S2 are selected to eliminate multicollinearity and correlation between variables and determine the main influencing factors;

[0028] Specifically, given the potential for intercorrelations between factors, the presence of multiple categorical variables, and the imbalanced subcategories, careful data preprocessing and factor selection are crucial for ensuring model robustness. For numerical variables (age, body size, and power), correlation analysis was performed to assess their linear relationships. The correlation coefficient threshold was set at 0.7. When the absolute value of the correlation coefficient between two variables exceeds this threshold, it indicates the possibility of a strong linear correlation. Furthermore, the variance inflation factor (VIF) was used to quantitatively assess multicollinearity between variables. According to Hair et al. (2014), a VIF value greater than 5 indicates the presence of severe multicollinearity, while a value greater than 10 indicates that multicollinearity has severely impacted the reliability of the estimates. In this paper, a stepwise elimination method was used to screen factors, using a threshold of 5. Specifically, the variable with the largest VIF value was eliminated each time until all variables had a VIF value less than 5. Preliminary results indicate that power and scale are highly correlated (r = 0.81), with high VIF values ​​(15.44 and 10.03, respectively). Given the importance of scale in existing research, this paper eliminated power while retaining scale.

[0029] For categorical variables (fuel type, builder, shipowner, operator, IMO, and flag state), the chi-square test was used to assess their independence. Given that the chi-square test is sensitive to sample size and cannot directly reflect the strength of association, the Cramér V coefficient was further used to quantify the degree of association between categorical variables. Based on Cohen's effect size criterion, a threshold of 0.5 was used. A strong association was found between shipowners and operators (Cramér V = 0.62). In practice, many ships are owned and operated by the same person, and since operators are more directly involved in day-to-day decision-making, shipowners were excluded from the analysis.

[0030] Step S4: Conditional probability coding of the categorical variables in the main influencing factors;

[0031] The dataset contains multiple categorical variables (e.g., flag state, operator, builder), each with many subcategories, some of which have very few observations. Directly encoding these categorical variables with dummy variables would produce too many indicator columns, leading to sparse observations and the risk of overfitting. To address this issue, this study employed a conditional probability encoding approach. This approach converts categorical variables into continuous risk scores based on observed adoption patterns. For each categorical variable, the conditional probability of scrubber adoption is calculated:

[0032]

[0033] Among them, m c Indicates the number of container ships in category C equipped with scrubbers. M c Indicates the total number of container ships in category C.

[0034] These probabilities are then normalized to the interval [0,1] using the following formula:

[0035]

[0036] Among them, q min and q max Represent the minimum and maximum probabilities between different categories. The final risk score is a continuous coded value for the category, reflecting its relative tendency to adopt a scrubber. This approach preserves the ordinal relationship between categories while reducing the model dimension and maintaining interpretability.

[0037] Step S5: Perform PH test on the traditional Cox proportional hazard model using the Schoenfeld residual test method;

[0038] The traditional Cox model relies on the proportional hazards (PH) assumption, which states that the effect of each factor remains constant over time:

[0039]

[0040] where h(t|x) represents the risk function given a set of factors x, where x represents the factors selected in step 3. i Indicates that the factor x i The corresponding regression coefficient

[0041] When the pH assumption is violated, traditional Cox models may produce biased or inaccurate results. To test the validity of this assumption, the Schoenfeld residual test can be used. This test assesses the correlation between the scaled residuals and the transformed time factor, providing statistical and graphical diagnostics for detecting time-varying effects.

[0042] Step S6: When the PH test is violated, a time-varying Cox model is used for analysis. That is, for variables that violate the PH test, the time function is selected based on the Schoenfeld residual, overall effect stability, and model fit criteria;

[0043] The total effect of the factor at time t is E i (t) = β i +γ i f i (t)(4)

[0044] The choice of time function is crucial, and several candidate forms are typically available, such as linear, logarithmic, polynomial, or spline functions. In this study, we selected the time function based on Schoenfeld residuals, overall effect stability, and model fit criteria. Schoenfeld residual plots help infer the shape of the time function. The overall effect at multiple time points is calculated and evaluated to determine whether it remains stable. The Akaike Information Criterion (AIC) and cross-validation can compare the fit results of different time function models. Lower AIC values ​​indicate a better balance between goodness of fit and complexity.

[0045] Step S7: Perform time-varying Cox model fitting.

[0046] The time-varying Cox model is

[0047] h(t|x)=h0(t)exp((β1+γ1f1(t))x1+(β2+γ2f2(t))x2+…+(β k +γ k f k (t))x k ) (5)

[0048] Among them, f i (t) represents the factor x i The time function of γ i Representation factor x i The coefficient of the time-varying component of .

[0049] Step S8: Model summary and result interpretation: Based on the fitting results of the time-varying Cox model, the decisive factors affecting the decision-making of container ship scrubber installation are analyzed.

[0050] Table 1: Time-varying Cox model results

[0051] factor coefficient Hazard ratio Standard error P-value size 0.66 1.94 0.11 <0.005 Ship age -3.19 0.04 0.14 <0.005 IMO Policy 0.67 1.96 0.11 <0.005 flag state 0.32 1.38 0.12 0.01 Fuel type 2.82 16.86 0.08 <0.005 Operator 0.4 1.49 0.1 <0.005 Builder 0.32 1.37 0.11 <0.005 IMO Policy_Time Change 1.09 2.97 0.12 <0.005 Fuel Type_Time Varying -0.01 0.99 0.01 0.02

[0052] The results show that among the influencing factors, the order of influence from large to small is: fuel type > ship age > IMO policy > size > operator > flag state > builder.

Claims

1. A factor analysis method for container ship scrubbers based on time-varying survival analysis is characterized by: The steps include: Step S1: Data is collected from the Clarkson database to obtain the following information for each ship, including construction year, deadweight tonnage, horsepower, shipowner, operator, builder, flag state, fuel type, scrubber installation year, and scrubber type; Step S2: Based on literature research, identify the factors that influence the decision to install scrubbers on container ships from three dimensions: ship characteristics, market characteristics, and policy environment; Step S3: Based on the statistical analysis methods of correlation analysis, variance inflation factor, chi-square test, and Cramér's V coefficient, the factors identified in step S2 are selected to eliminate multicollinearity and correlation between variables and determine the main influencing factors; Step S4: Conditional probability coding of the categorical variables in the main influencing factors; Step S5: Perform PH test on the traditional Cox proportional hazard model using the Schoenfeld residual test method; Step S6: When there is a PH test violation, a time-varying Cox model is used for analysis; that is, for variables that violate the PH test, the time function is selected based on the Schoenfeld residual, overall effect stability, and model fit criteria; Step S7: fitting a time-varying Cox model; Step S8: Model summary and result interpretation: Based on the fitting results of the time-varying Cox model, the decisive factors affecting the decision-making of container ship scrubber installation are analyzed.

2. The container ship scrubber based on time-varying survival analysis adopts factor analysis method according to claim 1, characterized in that: In step S2 above, based on literature research, factors influencing the decision to install scrubbers on container ships were identified from three dimensions: ship characteristics, market characteristics, and policy environment. Specifically: Ship characteristics are fundamental factors influencing the decision on whether to install a scrubber, including the ship's physical and technical characteristics; Ship age is an important factor affecting whether to install scrubbers; Vessel size directly impacts daily fuel costs and potential fuel cost savings from installing a scrubber; Power is the rated power of the ship's main engine, which reflects the energy consumption characteristics of the ship and is directly related to the treatment capacity requirements of the scrubber; Fuel type is a key factor in balancing technical and economic considerations; fuel price is considered one of the core determinants affecting the selection of ship sulfur emission reduction strategies; Market characteristics mainly include shipyards, shipowners and operators; The policy environment includes two variables: IMO regulations and flag state, which reflect the impact of global and local environmental policies on the adoption of scrubber technology, respectively; The flag state refers to the country or region where the ship is registered, reflecting the impact of differences in local laws and policies.

3. The container ship scrubber based on time-varying survival analysis adopts factor analysis method according to claim 2, characterized in that: In step S3, the factors identified in step S2 are selected based on statistical analysis methods such as correlation analysis, variance inflation factor, chi-square test, and Cramér's V coefficient to eliminate multicollinearity and correlation between variables and determine the main influencing factors. Specifically: For numerical variables, namely, construction year, deadweight tonnage, and horsepower, correlation analysis was performed to assess their linear relationship. The correlation coefficient threshold was set at 0.

7. When the absolute value of the correlation coefficient between two variables exceeded this threshold, it indicated that there was a strong linear correlation. Variance inflation factor is used to quantitatively assess multicollinearity between variables. If the variance inflation factor value is greater than 5, it indicates that there is a serious multicollinearity problem; if the variance inflation factor value is greater than 10, it means that the multicollinearity problem has seriously affected the reliability of the estimate. For categorical variables, namely fuel type, builder, owner, operator, IMO, and flag state, the chi-square test was used to assess their independence, and the Cramér V coefficient was used to quantify the degree of association between categorical variables; according to Cohen's effect size criterion, a threshold of 0.5 was used.

4. The container ship scrubber based on time-varying survival analysis adopts factor analysis method according to claim 3, characterized in that: In the above step S4, conditional probability coding is performed on the categorical variables in the main influencing factors; specifically: The dataset contains multiple categorical variables, including flag state, operator, and builder, each of which contains many subcategories. A conditional probability coding method is used to convert categorical variables into continuous risk scores based on the observed adoption patterns. For each categorical variable, the conditional probability of scrubber adoption is calculated: Among them, m c represents the number of container ships in category c equipped with scrubbers, M c represents the total number of container ships in category C; These probabilities are then normalized to the interval [0,1]: Among them, q min and q max They represent the minimum and maximum probabilities between different categories, respectively. The final risk score is the continuous coded value of the category, reflecting its relative tendency to adopt a scrubber.

5. The container ship scrubber based on time-varying survival analysis adopts factor analysis method according to claim 4, characterized in that: In the above step S5, the Cox proportional hazard model is subjected to a PH test using the Schoenfeld residual test method; specifically: where h(t|x) represents the risk function for a given set of factors x, where x represents the factors selected in step S3; β i Indicates that the factor x i The corresponding regression coefficients; When the PH assumption is violated, the Schoenfeld residual is used to test the validity of the assumption; Evaluate the correlation between scaled residuals and transformed time factors, providing statistical and graphical diagnostics for detecting time-varying effects.

6. The container ship scrubber based on time-varying survival analysis adopts factor analysis method according to claim 5, characterized in that: In step S6 above, when the PH test is violated, a time-varying Cox model is used for analysis; that is, for variables that violate the PH test, a time function is selected based on the Schoenfeld residual, overall effect stability, and model fit criteria; specifically: The total effect of the factor at time t is E i (t)=β i +g i f i (t)(4) The time function was selected based on the Schoenfeld residual, overall effect stability, and model fit criteria; the Schoenfeld residual plot helped to infer the shape of the time function and calculate and evaluate the overall effect at multiple time points to determine whether it remained stable; the Akaike Information Criterion and cross-validation were used to compare the fitting results of different time function models, and a lower Akaike Information Criterion value indicated a better balance between fit and complexity.

7. The container ship scrubber based on time-varying survival analysis adopts factor analysis method according to claim 6, characterized in that: In step S7 above, a time-varying Cox model is fitted; specifically: The time-varying Cox model is h(t|x)=h0(t)exp((β1+γ1f1(t))x1+(β2+γ2f2(t))x2+…+(β k +g k f k (t))x k (5) Among them, f i (t) represents the factor x i The time function of γ i Representation factor x i The coefficient of the time-varying component of .

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