Pedestrian safety is a topic of growing concern in the United States, especially in urban areas where 84% of pedestrian fatalities in traffic crashes occur (NHTSA, 2023). Widely considered to be the most vulnerable roadway users, pedestrians are disproportionately represented in fatality statistics, and their representation is growing at an alarming rate. Data from the Fatality Analysis Reporting System (FARS) indicates that 7 388 pedestrians were killed in traffic crashes in 2021, about 19% of all traffic fatalities in the United States. As the greatest pedestrian fatality total since 1981, that pedestrian fatality total also represents a 12.5% increase from the previous year, following a trend in yearly increases dating back to 2013 when pedestrian fatalities accounted for 3% less of all traffic fatalities (NHTSA, 2023). To address the growing concern for pedestrian safety, we must first understand the relationship between factors that contribute to pedestrian crashes and crash outcomes.

Often these relationships are quantified through statistical models that relate frequency of pedestrian crashes with observable factors that contribute to the crashes. Many different models have been used to investigate the relationships between pedestrian crashes and their contributing factors – most often including combinations of variables such as land use, socio-demographic information, and roadway characteristics. But perhaps the most important predictor of pedestrian safety outcomes is pedestrian exposure. Unfortunately, pedestrian exposure can be difficult to obtain because pedestrian volumes are not as widely available across a roadway network as vehicular volumes. This is because network-scale pedestrian counts are resource-intensive due to high costs associated with labor for manual counts and prices charged for automated counts. Some pedestrian volume data is more widely available from fitness-tracking sources such as Strava, but data from fitness-trackers is flawed in that they relay a self-reporting sample and the trackers are subject to error due to poor GPS reception, which is especially an issue in urban environments (Lee & Sener, 2020). Accurate volumes can also be difficult to obtain and even impractical to use due to highly variable daily volumes, short trips which may not be observed, and difficulties in detecting individuals (Lagerwey et al., 2015). Instead of directly using counts directly, pedestrian volumes are usually estimated through statistical models.

Several studies have developed pedestrian exposure models to predict the amount of pedestrian activity at a given level based on input variables that reflect the built environment, roadway features and other variables. These have traditionally been developed using log-linear ordinary least squares regression (OLS) or negative binomial (NB) regression (Behnam & Patel, 1977; Griswold et al., 2019; Hankey et al., 2012; Haynes et al., 2010; Lindsey et al., 2006; Lindsey et al., 2007; Liu & Griswold, 2009; Miranda-Moreno & Fernandes, 2011; Pulugurtha & Repaka, 2008; Schneider et al., 2009; Schneider et al., 2012). Pedestrian exposure has also been estimated using Tobit models and by modifying NB regression techniques to artificially inflate zero value pedestrian counts, in both cases relating the counts to demographics, land use, and traffic data (Lee et al., 2019). Still other studies suggest using stepwise linear regression to account for spatial variations in independent variables (Hankey & Lindsey, 2016; Hankey et al., 2017; Lu et al., 2018).

While these model types have their own merit, they also have various flaws when applied to estimating pedestrian exposure. OLS regression does not account for the count nature of volume data, and using a log-linear form assumes a logarithmic distribution that may not be observed. Similarly, Tobit models assume a normal distribution for the dependent variable which is not typically observed in pedestrian count data. Stepwise linear regression may result in atheoretical coefficient estimates, which makes interpretation complicated and may limit transferability to other datasets. NB regression is the most appropriate and the most common, because of its ability to account for overdispersion in fluctuating pedestrian volumes and the count nature of the data. Even so, all these models are flawed in that they are typically a ‘one-size-fits-all’ approach in which the same model is used to estimate pedestrian exposure at all locations within a transportation network. However, the relationships between pedestrian exposure and explanatory features—such as population, infrastructure design, and land use context—might differ significantly with respect to the context of an individual location, which may not be known a priori. Incorporating these differences could help improve the exposure model and provide more accurate predictions.

To help address this issue, this paper proposes a model-based recursive partitioning (MBRP) algorithm to develop pedestrian exposure models. The MBRP approach combines traditional statistical methods (e.g. NB regression) with the recursive data partitioning techniques commonly found in tree-based machine learning methods. The proposed method was tested on pedestrian exposure data obtained in North Carolina and shown to significantly improve predictions of pedestrian volumes by approximately 10%. Therefore, the MBRP algorithm presents a promising tool for advancing pedestrian safety analysis in practical applications.

The remainder of this paper is organized as follows. Section 2 provides a description of the methodology used for NB regression and the proposed MBRP approach. Section 3 describes the dataset used. Section 4 provides an analysis of the results from the MBRP model. Finally, Section 5 contains concluding remarks.

The goal of this research is to demonstrate the potential of the MBRP algorithm to estimate pedestrian exposure better than traditional regression methods. To do so, exposure models were developed over a training dataset using traditional NB regression methods and the MBRP algorithm. The performance of the MBRP model was judged relative to the NB regression model based on goodness of fit statistics over a separate test dataset and cumulative residual (CURE) plots.

Pedestrian count data analyzed in this study comes from the North Carolina Department of Transportation (NCDOT) and was collected and analyzed for a previous study investigating factors contributing to pedestrian crash outcomes (Gayah et al., 2022). The counts are not a random sample, arising from a variety of convenience samples such as turning movement counts (TMCs) and pedestrian safety studies at particularly dangerous intersections or intersections with high-volumes relative to other intersections in the city. The non-random sample is still considered to be a valid sample because the counts are known to demonstrate a wide range of pedestrian count values, as well as to accurately represent the expected levels of pedestrian activity near the count location. The counts come from the following sources:

Each observation in the dataset included vehicular volume data, roadway features at the count location, census-level demographic statistics, and land-use statistics for a 0.5-mile radius surrounding the count location. A summary of the variables that were included in final models is provided in Table 1 and Table 2. The land use mix variable is an aggregate of land uses in the surrounding area adapted from the methodology outlined in Frank et al. (2004) and Gayah et al. (2022). The land use mix value is based on four land use types: high intensity developed, medium intensity developed, low intensity developed, and all other land use classifications combined, where a value of 1.00 represents a perfect balance of all four land uses, and 0.00 represents only a single land use present. Alcohol sales locations are not commonly considered as an indicator variable; however, including this variable in indicator form can be justified through contextualization—the two categories represent the difference between an area with single liquor store and a couple of restaurants, and an area with densely clustered bars expecting high foot traffic and fewer vehicles. Including this variable in this form was found to improve model performance relative to a continuous form of alcohol sales locations.

Of the 3 618 counts provided, the majority were performed at intersections, and therefore intersections were selected as the unit of analysis. Count locations that were not intersections, in non-urban environments, and count durations less than or equal to 2.5 hours or equal to 24 hours were removed from the analysis to limit the analysis to characteristically similar count locations. The remaining 2 430 pedestrian counts ranged from 0 to 14 854 pedestrians over the respective durations, though 75% of counts registered 67 or fewer pedestrians. An 80/20 split was performed over the full 2 430 counts, creating a training dataset (1 942 observations) and a testing dataset (488 observations) such that both the NB and MBRP models would be estimated over the training dataset, and the models’ fitness would be assessed using the goodness of fit statistics based on their predictions over the test dataset. The training and test datasets were selected to be representative samples of the full dataset with approximately equal summary statistics, but each dataset was also vetted to ensure representation of outlier counts. Due to the nature of the outlier counts, exact matches for maximum count values could not be achieved, and the training dataset is known to contain a greater proportion of lower count values because of the higher maximum count but otherwise similar summary statistics. Table 3 shows the summary statistics of the final split datasets.

This section describes the estimation results from the NB regression model and the MBRP model, including the assessments of goodness of fit through mean absolute error (MAE), mean absolute percentage error (MAPE), and root mean square error (RMSE). Additionally, a cumulative residual (CURE) plot was generated with both models plotted against the 95-percent confidence interval as determined for the NB regression model, according to the methodology outlined in Hauer (2015).

The results of the MLE process for the NB regression model and the model generated from the MBRP algorithm are presented in Table 4. In the modeling process, only the variables found to be statistically significant at the 5% significance level were retained. Exceptions were made for some variables found to be insignificant at the 5% level but were deemed theoretically important predictors of pedestrian counts. Additionally, any variables that were found to be inconsistent with theory regarding the sign of the estimated parameter were vetted for their contribution to the model’s predictive performance before being considered for removal.

This study investigates a possible improvement in pedestrian exposure modeling, by augmenting traditional NB regression models with a tree-based machine learning component, the MBRP algorithm. It was hypothesized that the inclusion of the MBRP algorithm would not only improve predictive accuracy but would allow for additional interpretation of the relationships between explanatory variables and pedestrian exposure estimates which may differ contextually in ways that are not known a priori. Data from North Carolina is subdivided into training and test dataset. Training data is applied to develop a traditional NB regression model and an MBRP model, and the two models’ predictive performance is assessed based on the predictions for the test dataset.

The use of the MBRP algorithm resulted a model indicating that a pedestrian count’s relationship with explanatory variables varies with the natural logarithm of parcel count, which suggests that pedestrian counts in denser urban environments are more sensitive to presence of alcohol sales locations, and less sensitive to presence of sidewalks, crosswalks, bus stops, and roadways with higher speed limits. These trends are not observable through traditional NB models, which are typically more of a ‘one-size-fits-all approach which does not consider how the relationships between explanatory variables may vary with context.

The model developed using the MBRP algorithm improved predictions of pedestrian counts over the test dataset by approximately 10% as measured by MAE, MAPE, and RMSE. The model’s fitness was also shown to be an improvement over the NB regression model through lower AIC and the MBRP plotting more points within the confidence interval and nearer to zero on a CURE plot.

The proposed use of the MBRP algorithm to improve pedestrian exposure estimates overcomes the shortcomings of traditional NB regression methods by capturing relationships between explanatory variables in context. Fortunately, the methodology is quite transferable to other regions outside of North Carolina, given that the MBRP algorithm inherently contextualizes coefficient estimates and naturally defines its own optimal splitting points. However, given the difficulty of retaining statistically significant variables across sub-models in the MBRP model, it should be noted that the variables selected for the MBRP model presented in this paper may not be statistically significant in another region. Further optimization may be required for application elsewhere. The MBRP algorithm and other machine learning methods are also known to be ‘data hungry’, and therefore the predictive power of this model may be limited by the size of the available dataset. In practice, a larger dataset would yield better results and could allow for a deeper understanding of the relationships between explanatory variables, which may be more complex than the restricted model presented in this paper suggests.

No external funding was used in this research.

During the preparation of this work the authors used GPT-4 to check writing for grammatical errors and improve clarity of language in the final manuscript. The outputs were reviewed and revised by the authors who take full responsibility for the content of the publication.

The authors declare no competing interests.

The research performed in this paper does not qualify as human subjects research and thus does not require any IRB protocol.