1.3.1 An analysis of Net Present Value (NPV).

The NPV analysis is a crucial technique in evaluating the long-term profitability of a capital investment project. The analysis fundamentally considers the time value of money, emphasizing that the present value of a sum is not representative of its future worth. This concept, integral to investment decisions, acknowledges that money today is worth more than the same amount in the future due to its potential earning capacity. NPV calculates the difference between the present value of cash inflows and outflows over various time periods, a method essential for assessing the financial viability of any project. It helps in understanding the actual cost of a project when considering the time value of money, thereby allowing for a more accurate comparison with other potential investments. The NPV analysis is pivotal in ensuring that the selected project maximizes potential returns, considering both the scale of the investment and the time frame over which returns are expected [18]. where N = Total number of time periods (unit: year), R_n = Net cash flow at a single period, n = Time period (unit: year), i = interest rate at the time of calculation.

Following the highlighted significance of NPV analysis across various sectors, including transportation, many researchers contribute and extend the discourse into specialized methodologies and case studies, further validating the versatility and critical importance of NPV in railway project assessments. A study highlighted the application of NPV in evaluating the socio-economic benefits of the Heartland light rail project in Kansas City, USA, underlining the method’s ability to encompass broader societal impacts such as reduced automobile usage [19]. There is also a benefit-cost analysis model incorporating NPV to assess the safety and performance benefits against the costs of track upgrades in North America, aiming to minimize train derailments and enhance travel times [20]. In the context of High-Speed Rail (HSR) projects in Hong Kong and Mainland China, some existing research demonstrated NPV’s effectiveness in validating project viability pre-investment and facilitating comparisons with alternative investments [21]. Furthermore, Rahman et al. (2019) combined NPV with internal rate of return (IRR) analyses for evaluating light rail services in Jakarta, indicating substantial financial benefits for both the local government and passengers through minimized service costs [22].

Toporkova and Shylo (2018) delved into the analytical support of financial analysis at railway transport, emphasizing the necessity of tailored financial analysis methodologies that reflected the railway industry’s unique financial relationships and operational dynamics [23]. This study systematically explored the critical role of financial analysis in informing management decisions within the railway sector, thereby underpinning the foundational aspects of NPV calculations. Bai et al. (2011) discussed the financial analysis of railway construction projects from a network perspective, underscoring the comprehensive evaluation of project costs and benefits [24]. This analysis illuminated the nuanced considerations necessary for a thorough NPV assessment, including construction investments, operating costs, and the tangible benefits of improved transport quality and reduced costs. Their work asserted the indispensability of network-based financial analysis in ensuring the economic feasibility of railway projects. Another research presented a compelling case study on the electrification of the Cairo-Alexandria railway line, applying cost-benefit analysis to assess its economic and financial viability. Their findings, revealing a modest internal rate of return but substantial economic benefits when broader impacts were considered, illustrated the critical importance of comprehensive NPV analysis in capturing the full spectrum of a project’s potential value. However, the study revealed a significant pain point in obtaining the necessary data and information for comprehensive cost-benefit analysis, which was crucial for accurate NPV calculations. The difficulty in data collection could hinder the assessment of project viability, especially in regions with less developed information systems [25,26].

Also, Akhtar et al. (2017) on optimizing literature search systems underscored the importance of efficient information retrieval in supporting multi-domain research, including transportation economics [27]. While not directly related to NPV calculations, their research highlighted the broader academic infrastructure supporting rigorous financial analyses in the railway sector and beyond. These studies collectively underscored the complex interplay of financial, operational, and technological factors that had to be navigated in the NPV analysis of railway projects, reinforcing the methodology’s critical role in guiding strategic investment decisions.

1.3.2 Sensitivity analysis.

A sensitivity analysis is necessary to assess unusual circumstances that may arise during operations, such as natural disasters, vandalism, and unforeseen damages. One tangible benefit of conducting such an analysis is its applicability to financial projections for new projects, enabling the allocation of reserve capital [18,28]. The ISO 14040 standard recommends utilizing Monte Carlo Simulation (MCS) in life cycle analyzes to mitigate uncertainties. This approach has been employed in railway research, particularly in Life Cycle Assessment (LCA), where it minimizes uncertainties related to the recycling process. The probability density function for the triangular distribution is defined as follows: (1) where x is a lower limit value, y is an upper limit value, and z is a mode value; x ≤ y ≤ z

2.1.1 Railway infrastructure accidents.

The railway infrastructure accidents obtained from the US Federal Railroad Administration (FRA) include all incidents occurring on rail tracks and/or structures. The FRA is the US Department of Transportation’s agency that collects accident information annually per county. In this study, we obtained the accident data for three states (Washington, Oregon, and California) on the West Coast of the US, occurring from 2000 to 2022 [29] (Fig 2). Primarily, 15.2% of railway infrastructure accidents are of the T110 type, characterized by a wide gauge and defective or missing crossties. The T002 type (washout/rain/slide track) and the T207 type (detail fracture and shelling/head check) follow, representing 9.4% and 7.7% of accidents, respectively. Note that in some counties, no incident is recorded in the dataset, shown as transparent polygons in Fig 2.

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Fig 2. Cumulative railway infrastructure accidents in each county occurring from 2000 to 2022 from the US Federal Railroad Administration.

The map was created using QGIS [14].

https://doi.org/10.1371/journal.pone.0308255.g002

2.1.2 Seismic hazard variables.

We use three main datasets, which are earthquakes, active faults, and ground shaking (peak ground acceleration) as indicators of seismic hazard and refer to them as seismic hazard variables, noting that geological complexity and local site conditions are important aspects that are not considered in this study

1. (1) Earthquakes

We collected information on M≥3.5 earthquakes that occurred from 2000 to 2022 in the western U.S (see Supplementary Information for additional details on the dataset; Fig 1). This dataset is from the U.S. Geological Survey (USGS) website that lists earthquakes and includes information on hypocentral depths, magnitudes, and geographical locations of the earthquakes [9]. The total number of earthquakes in this dataset is 2,282 events. We selected earthquakes that had magnitudes greater than 3.5 because earthquakes starting from this magnitude can produce seismic waves that can be detected regionally by seismographs, offering more reliable data [30]. This facilitates the precise determination of earthquake parameters, including magnitude, location, and depth. Additionally, these earthquakes are more likely to cause noticeable surface effects, such as structural damage and felt ground shaking [31]. The maximum hypocentral depth among the events used in this study is 63 km occurring in Washington, and the highest magnitude among these earthquakes is M7.1 occurring in California. Since there is a wide range of both hypocentral depths and magnitudes of earthquakes (see Supplementary Information for additional details on the dataset), we divided the events based on these two factors. We then determined the number of earthquakes occurring annually in each county, using the following classification:

• EQ-MC1: Magnitude ≥M3.5 earthquakes, all depths,
• EQ-MC2: Magnitude ≥M4 earthquakes, all depths,
• EQ-MC3: Magnitude ≥M5 earthquakes, all depths,
• EQ-HD1: Magnitude≥M3.5 earthquakes and hypocentral depths ≤10 km,
• EQ-HD2: Magnitude ≥M3.5 earthquakes and hypocentral depths ≤20 km,
• EQ-HD3: Magnitude ≥M3.5 earthquakes and hypocentral depths ≤40 km.

1. (2) Active Faults

We acquired data on active faults in the western US from the US Geological Survey and California Geological Survey (2006) [11]. This dataset encompasses details such as fault names, types, lengths, and locations. Utilizing this information, we were able to calculate the fault density by determining the total length of faults per unit area within each county (Fig 3). This calculated metric was subsequently incorporated into our analysis to examine its significance as a predictor for railway infrastructure accidents. Areas in proximity to active fault lines are regarded as vulnerable zones, susceptible to damage from fault activity, whether through aseismic creep or earthquakes. By including fault density in our model, we aim to assess the long-term influence of fault activity on railway infrastructure, beyond just the immediate aftermath of seismic events. This approach allows us to gain a deeper understanding of the factors contributing to the resilience or vulnerability of railway systems in actively deforming regions.

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Fig 3. Fault density map.

The density is calculated by dividing the length in km of the faults shown in Fig 1 by the size of each county in km². The map was created using QGIS [14].

https://doi.org/10.1371/journal.pone.0308255.g003

1. (3) Peak Ground Acceleration (%g)

Peak Ground Acceleration (PGA) is a crucial indicator in seismology and earthquake engineering, denoting the maximum ground acceleration during an earthquake and measured in percentage of gravitational acceleration units (%g). It is critical for developing earthquake-resistant designs and assessing seismic risks. PGA at any location depends on factors such as earthquake magnitude (with larger earthquakes generating stronger motions), proximity to the epicenter, local geological features (soft sediments amplify shaking more than bedrock), earthquake depth (shallower earthquakes commonly produce higher PGAs), earthquake source characteristics, and the direction of the earthquake’s rupture along with the site’s topography [32]. These aspects collectively influence the experienced ground shaking, positioning PGA as essential for effective earthquake response and structural planning. The map depicting ground shaking in the western US was sourced from the National Atlas of the United States [33,34] (Fig 4) and has the grid spacing of 0.05 degrees for the conterminous United States. The map illustrates the likelihood of increases in peak ground motion to various levels, with Fig 4 specifically detailing the anticipated percentage of acceleration due to the gravity (%g). A higher PGA value in a region signifies a stronger peak horizontal ground acceleration, characterized by a 10% probability of exceedance within a 50-year period, compared to regions with lower PGA values. It’s important to note that the hazard map we utilized from [33] is not the latest version available. However, we chose this map because it is more accessible and allows for easier extraction of the PGA values on a spatial basis. The most recent PGA map is available in the study conducted by [35].

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Fig 4. Peak ground acceleration (%g) map as a measure of ground shaking in the western US obtained from [33,34].

The warmer colors represent higher peak horizontal ground accelerations with a 10% probability of exceedance in 50 years. The map was created using QGIS [14].

https://doi.org/10.1371/journal.pone.0308255.g004

In our research, we calculated the average ground shaking for each county and incorporated these averages as one of the predictive variables in our model. This approach allows us to quantitatively assess the potential impact of ground shaking on regional railway infrastructure by integrating a key measure of seismic intensity—peak ground acceleration—as a determinant of the vulnerability of railway systems to seismic events.

2.2.1 Linear regression.

Linear regression is a fundamental statistical technique that models the relationship between a dependent (target) variable and one or more independent (feature) variables by fitting a linear equation to observed data. It assumes a straight-line relationship whereby it attempts to predict the dependent variable’s value based on the independent variables. The essence of linear regression is captured by the equation: (2) where y_i represents the dependent variable’s value (target), x_i denotes the independent variables (features), β is the slope indicating the change in y_i for a one-unit change in x_i, and α is the y-intercept, the value of y_i when x_i is zero.

This study explores four variations of linear regression models: simple linear (Least Squares), Lasso, Ridge, and Elastic Net regression. The latter three models incorporate regularization techniques to mitigate overfitting—a common issue where the model performs well on training data but poorly on unseen (test) data. Regularization works by imposing penalties on the size of the coefficients, effectively simplifying the model to improve its generalization capabilities. Additionally, these regularization methods can address multicollinearity, a situation where independent variables are highly correlated, which can destabilize the standard least squares estimation. Unlike traditional least squares or simple linear regression, which solely focuses on minimizing the sum of squared differences between observed and predicted values, regularization adds a complexity penalty to the model selection criteria, balancing the trade-off between fit and complexity.

Lasso Regression, also known as Least Absolute Shrinkage and Selection Operator (L1 regularization), modifies traditional linear regression by introducing a penalty on the absolute size of regression coefficients. This technique encourages the reduction of less important feature weights to zero, thereby simplifying the model and aiding in feature selection. The goal of Lasso Regression is to minimize the loss function: (3) where the first term, , represents the sum of squared residuals, and the second term, , is the penalty applied to the size of the coefficients, with λ controlling the degree of weight shrinkage. As λ increases, more coefficients are driven to zero, effectively eliminating less significant features from the model. This process of penalizing the sum of absolute values of the coefficients helps in dealing with multicollinearity and in automatically selecting the most relevant features for the model [36,37].

On the other hand, Ridge Regression, also known as L2 regularization, modifies linear regression by introducing a penalty on the square of the magnitude of coefficients. This technique aims to minimize the function [36,38,39]: (4) where the first term, , represents the sum of squared residuals, and the second term, , applies a penalty to the size of the coefficients squared, with λ as the regularization parameter controlling the extent of the penalty applied. Unlike Lasso regression, which can completely eliminate coefficients, Ridge regression reduces the magnitude of coefficients but does not set them to zero. This method is particularly useful in preventing multicollinearity by ensuring that the model coefficients are not overly sensitive to changes in the model’s input data, thereby reducing the model’s complexity and improving its generalization capability [36,38,39].

Moreover, Elastic Net Regression effectively merges the attributes of L1 and L2 regularization techniques, integrating both absolute and squared value penalties to optimize the following objective function [40]: (5) where α is a parameter for the combination of Ridge (α = 0) and Lasso (α = 1). The Elastic Net uniquely blends these regularization methods, offering a tunable approach (λ parameter) that linearly combines L1 regularization, associated with Lasso, and L2 regularization, identified with Ridge regression. This dual approach not only facilitates coefficient shrinkage but also enables the model to perform automatic feature selection by potentially reducing the coefficients of less significant variables to zero. Particularly advantageous in scenarios dealing with multicollinearity and overfitting, especially in high-dimensional datasets, Elastic Net stands out by leveraging the strengths of both Lasso and Ridge regression to produce a more robust and versatile model [40]. This balanced mechanism makes Elastic Net highly effective for complex datasets where both the selection of relevant features and the mitigation of overfitting are crucial.

2.2.2 Non-linear regression.

In this study, we employed Decision Tree and Random Forest regressions, both of which are non-parametric, supervised learning models designed for predicting target values. Each model employs a tree structure to facilitate regression analysis.

The Decision Tree Regression model constructs a tree from the top down, where each leaf represents a numerical prediction of the target value [41,42]. The root of the tree is selected based on the predictor (feature and threshold, e.g., Feature A < threshold) that minimizes the sum of squared residuals (SSR), where residuals are the differences between actual and predicted values. The tree grows by branching out from the root node, with each subsequent division designed to further reduce the SSR. The branching continues until terminal leaves, which represent the final predictions, are reached. While Decision Trees can adeptly handle complex datasets, they are prone to overfitting, particularly if the tree is allowed to grow too deep without restrictions. To mitigate this, we limited the maximum depth of the tree in our analysis, enhancing the model’s generalizability to new, unseen data.

The Random Forest Regression model extends the concept of Decision Trees by constructing an ensemble of multiple trees [43,44]. It operates by randomly sampling observations and building several decision trees from these subsets. The final prediction is then derived by averaging the predictions from all trees. This methodology significantly reduces the risk of overfitting, a common drawback of single Decision Trees, by aggregating the diverse outcomes of multiple trees. The Random Forest model, by virtue of its ensemble approach, typically delivers more robust and accurate predictions than a single Decision Tree, making it a powerful tool for handling complex predictive tasks.

2.2.3 Modeling.

In our study, we employed the scikit-learn library [45], a comprehensive tool within the Python programming language, for our modeling efforts. The modeling process was structured into three main stages: data preparation, hyperparameter tuning, and model execution.

We assumed that the residuals of the models follow a normal distribution, validated by examining the residual plots and conducting normality tests. It was also assumed that the observations are independent of each other, a crucial assumption for the validity of the statistical tests used. All data were standardized to have a mean of zero and a standard deviation of one, and missing values were imputed using the mean imputation method.

Initially, we divided our dataset, which includes all seismic variables and instances of railway infrastructure accidents as detailed in the Methods 2.1 section, into two distinct sets: training and test. The dataset was randomly split into training (80%) and testing (20%) sets to ensure that model performance was evaluated on unseen data. This randomization was performed using a fixed random seed to ensure reproducibility. We used 5-fold cross-validation during the training phase to optimize hyperparameters and prevent overfitting. The choice of 5-fold cross-validation was based on balancing computational efficiency with robust performance evaluation.

Following the data segmentation, we normalized our features to ensure uniformity in scale for both the training and test sets. Given that all our variables are numerical, we employed the MinMaxScaler function from scikit-learn, which adjusts the data to a range between 0 and 1 [45].

Subsequent to the scaling process, we embarked on hyperparameter tuning for our chosen models—Ridge, Lasso, Elastic Net Regression, and Ensemble Models—utilizing the GridSearch with 5-fold cross-validation from the scikit-learn library [45]. This step is crucial for identifying the optimal hyperparameters for each model, thereby enhancing their performance [46]. For linear regression models, we focused on adjusting the alpha parameter (λ), as outlined in the Methods 2.2 section. For ensemble models, the hyperparameters tuned included splitter, max_depth, min_samples_leaf, and max_features for Decision Trees, and max_depth, max_features, min_samples_leaf, and n_estimators for Random Forests (Table 1).

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Table 1. Hyperparameters used in the GridSearchCV tuning techniques to find optimal values for each parameter of each model.

https://doi.org/10.1371/journal.pone.0308255.t001

Upon fine-tuning the hyperparameters, we proceeded with the model execution phase, resulting in the derivation of coefficient values for the linear regression models and feature importance for the ensemble models. To assess the accuracy of our models, we utilized the root mean square error (RMSE) metric, which will be elaborated upon in the subsequent section of our analysis.

2.2.4 Modeling result validation metric.

As previously outlined, we employ the Root Mean Square Error (RMSE) metric to evaluate the performance of our models and to determine their efficacy in predicting the target variable—the number of railway infrastructure accidents. The RMSE serves as a measure for quantifying the prediction errors, essentially representing the average discrepancy between the observed actual values and the values predicted by our models. This metric is particularly useful for assessing the accuracy of predictions, as it provides insight into the average magnitude of prediction errors across all observations. The RMSE is calculated with the following formula [47,48]: (6) where y_i represent actual values or observations, are predicted values, n is a number of samples.

We chose RMSE as our model validation metric because it is particularly sensitive to large errors, squaring the deviations before averaging them. This sensitivity makes RMSE a stringent metric, ensuring that models with significant errors are appropriately penalized [47]. Given our study’s focus on accurately predicting extreme values, such as large-scale railroad accidents due to major earthquakes, RMSE is essential for highlighting these deviations.

Additionally, RMSE provides a comprehensive measure of prediction accuracy by considering the magnitude of errors, offering a clear indication of model performance across all data points [47]. Expressed in the same units as the dependent variable, RMSE simplifies interpretation, helping us understand the scale of prediction errors in the context of railway accidents and infrastructure costs.

While other metrics, such as Mean Absolute Error (MAE) and R-squared are valuable, we opted for RMSE due to its robust evaluation capabilities and its ability to handle the complexity of our models. RMSE’s sensitivity to larger errors provides a nuanced view of model performance, especially in scenarios where significant errors can have substantial real-world implications.

To compare the performance of various models, we calculated the percentage differences in RMSE values between each model and the baseline Least-Squares (linear regression) model. This allowed us to determine whether alternative models offer significantly improved predictive accuracy for the target values. Additionally, we compared the average RMSE of linear regression models with regularization techniques (Ridge, Lasso, and Elastic Net) against the RMSE of non-linear models (decision trees and random forests). This comparison aimed to investigate whether more sophisticated modeling approaches can more accurately capture the relationship between seismic hazard variables and railway infrastructure accidents, thereby offering a deeper understanding of the underlying patterns and factors at play.