Applying model-based recursive partitioning to improve pedestrian exposure models to support transportation safety analyses

NB regression

Pedestrian counts are always a non-negative integer and are therefore most appropriately modeled using count models. Though there are many count regression models, NB regression models are used most extensively in research due to their ability to account for overdispersion in the dataset, which is commonly observed in pedestrian count data. This paper’s NB regression procedure is adapted from Hankey et al. (2012) and Lee et al. (2019).

NB regression can be described through the following formulation: Let $i=1,2,3,\dots ,N$ represent the index of a given location where $N$ is the number of locations in the dataset. In the NB model, the pedestrian count at a location $i$ , takes an exponential form as shown in Equation 1:

where ${\lambda }_{i }$ is the predicted pedestrian count at location $i$ , $y_i$ is the observed pedestrian count at location $i$ , $\beta$ is the vector of estimated parameters, $X_{i }$ is the associated vector of explanatory variable values observed at location $i$ , and ${\epsilon }_i$ is an error term such that $exp\left({\epsilon }_i\right)$ has a gamma distribution. A maximum likelihood estimation (MLE) method was adopted using the probability distribution for NB regression as presented in Equation 2, and the likelihood function as presented in Equation 3. The MLE method allows for estimation of the coefficient parameter, $\beta$ , and the overdispersion parameter, $\alpha ,$ relying on the gamma function, denoted by $\Gamma (·)$ .

Selecting parameters which maximize the likelihood function presented in Equation 3 establishes a model expected to best fit the data. Typically, the likelihood value found by maximizing Equation 3 is very small, so frequently the natural log of Equation 3 is optimized in place and is reported as log-likelihood.

MBRP algorithm

The MBRP algorithm combines traditional statistical modeling, such as NB regression, with recursive data partitioning techniques commonly found in tree-based machine learning methods. In the MBRP algorithm, the root node is a parametric model fitted over the entire dataset. Child nodes are then formed through splitting based on a decision rule, which continues until the terminal node of the tree model is reached. Figure 1 depicts a general form of the tree model (Kashani & Mohaymany, 2011). The following formulation for the MBRP algorithm draws extensively from Seibold et al. (2016) and descriptions from Tang and Donnell (2019).

Developing a model using the MBRP algorithm is a process that occurs in three steps:

1. Fit a parametric model to all observations in the dataset;

2. Test coefficient stability over the splitting variable; and

3. Determine the optimal cut point of the splitting variable.

In Step 1, the parametric model with parameter vector, $\theta$ (represented by the root node in ) may be estimated through methods such as OLS or MLE which inform the objective function, $\Psi (·)$ as found in Equation 4. The coefficients are then estimated through a partial score function as shown in Equation 5, where $\psi (·)$ represents the score function, and $\beta$ is the estimated parameter (Seibold et al., 2016).

In Step 2, a generalized M-fluctuation test is used to test coefficient stability over splitting variables, with the null hypothesis stating that the partial score functions from Equation 5 are independent of partitioning variables. This hypothesis indicates that global estimates of an independent variable are appropriate. Equation 6 shows the functional form of the null hypothesis, where $Z$ is the splitting variable and $J$ is the number of splitting variables (Seibold et al., 2016).

The splitting variable is selected based on the greatest correlation with partial score functions, and the optimal cut point of the splitting variable is determined by evaluating segmented objective functions as shown in Equation 7 and choosing the minimum value. In Equation 7, $I_b$ is the set of observations belonging to $b$ under splitting rule ${\rm B}$ (Tang & Donnell, 2019).

The process outlined above results in child nodes after the splitting rule is followed. The process is then repeated on all child nodes until terminal nodes are reached, where there is no longer coefficient instability as determined in Step 2 of the process. Each of the terminal nodes represents a parametric model, and any observation in the root nodes that meets the splitting criteria to fall into the terminal node can be predicted using the corresponding parametric model. Additional information about the MBRP algorithm can be found in Zeileis and Hornik (2007) and Zeileis et al. (2008).

NB regression

All variables included in the final NB regression model were found to be statistically significant at the 95% confidence level or higher and were found to meet expectations consistent with engineering judgement:

• Count duration. In the dataset, the base condition is a 12-hour count. Therefore, it would be expected that a 13-hour count would yield a higher prediction due to a longer period for pedestrians to be observed. The same logic would indicate that a 16-hour count parameter should be greater than the 13-hour count parameter, however, 16-hour counts are typically conducted at two-way stop intersections being considered for signalization. Generally, these intersections have high traffic volumes which make pedestrian access uncomfortable, resulting in fewer pedestrians. Therefore, a smaller or even negative coefficient may be appropriate to describe the effects of a 16-hour count on pedestrians observed.

• Traffic conditions. As previously described, pedestrians are less comfortable walking near roadways with higher traffic volumes where they may be more vulnerable to crashes (Miranda-Moreno & Fernandes, 2011). That trend extends also to roadways with higher speed limits as crashes involving pedestrians and vehicles moving at higher speeds are known to result in more severe outcomes for pedestrians (Pulugurtha & Repaka, 2008). In the NB regression model, we find that the coefficients associated with the natural logarithm of AADT and speed limits greater than or equal to 40 miles per hour are negative, indicating a negative impact on pedestrian counts.

• Pedestrian-friendly infrastructure. Presence of features such as crosswalks and sidewalks are expected to contribute positively to pedestrian counts. Crosswalks provide priority to pedestrians crossing at intersections, and sidewalks provide a separate right of way for pedestrians to walk safely next to a road (Lee et al., 2019; Lu et al., 2018). These expectations are met by the positive coefficients associated with presence of a crosswalk or sidewalk in the NB regression model. Bus stops are also associated with greater pedestrian presence due to the access distance between the trip origin/destination and the transit stop (Hankey et al., 2012; Pulugurtha & Repaka, 2008).

• Land development. Denser, more varied use of land is expected to result in greater pedestrian counts due to convenience of the walking mode. Increases in parcel count, a measure of urban density, should indicate shorter distances to potential destinations, while land use mix increases suggest a greater variety of residential, commercial, and industrial uses within a small radius (Gayah et al., 2022). Walking becomes more convenient over shorter distances, and therefore more pedestrians are likely to be observed with greater parcel count and land use mix. Additionally, alcohol sales locations are frequently accessed via walking due to the dangers associated with driving under the influence (Gayah et al., 2022).

• Demographics. People who reported that they commute by non-motorized means are vastly more likely to be observed in a pedestrian count, and therefore a greater proportion of non-motorized commuters is expected to increase pedestrian counts, as observed by the positive coefficient in the NB regression model (Griswold et al., 2019).

MBRP model

Under the framework discussed in the methodology section, a MBRP model splitting over the natural logarithm of parcel count was developed. Due to the relatively small dataset, the maximum depth of the tree structure was limited to ensure that each of the sub-models was trained on a sufficiently large dataset. This also allowed for greater control over variable inclusion based on statistical significance and consistency with theory. The optimal split value of the parcel counts was found to be 71, with 1 197 of the training dataset observations registering parcel counts less than or equal to the optimal split value. Figure 2 depicts the tree structure for the MBRP model developed.

In comparing the estimation results between the NB regression model and the MBRP model, land use mix was found to be statistically insignificant in the MBRP model and was therefore removed from the model. The indicator variables for crosswalks and bus stops were found to be insignificant in the higher urban density model of the MBRP model, though the coefficient for bus stops was consistent with theory, and the contribution from the crosswalk variable was minor enough to overlook given the magnitude of its contribution in the lower urban density model. Aside from these differences, the NB and MBRP model coefficients take the same signs and indicate the same relationships between explanatory variables and pedestrian count outcomes.

The MBRP model allows for additional interpretation of model coefficients between the sub-models, which captures the difference in relationships between the explanatory features and the pedestrian count outcome as they vary with urban density. For example, based on differences in the magnitudes of coefficients presented in the second and third columns of Table 4, we observe that pedestrian counts are more sensitive to the presence of alcohol sales locations in denser urban environments. On the other hand, pedestrians in denser urban environments are less sensitive to the presence of sidewalks, crosswalks, bus stops, and greater speed limits than their counterparts in less dense urban areas. Across all models, the coefficients estimated for the proportion of non-motorized commuters and the natural logarithm of AADT are approximately equal, which indicates that pedestrians everywhere are about as sensitive to both variables, regardless of urban density.

Model comparison

Following model development, both models' cumulative residuals were evaluated over the testing dataset. The results were plotted on a CURE plot (Figure 3) with the 95% confidence interval from the NB regression plotted to demonstrate how the MBRP model compares to the ‘baseline’ NB regression model. Overall, we can tell that both models are robust, plotting nearly all observations within the confidence interval. From the CURE plot, we see that for lower count values predicted, the cumulative residuals are about equal across both models. Both models appear to underpredict observations of approximately 200 pedestrians, though the residuals for the MBRP model are smaller. The trend of smaller residuals can be observed over most of the predicted range for the MBRP model, indicating a generally better fit for the data than the NB regression model.

The models were also tested for goodness of fit by standard error measurements to determine how well the models predict the pedestrian counts relative to the observed pedestrian counts in the test dataset. Table 5 presents the goodness of fit statistics for both models, where the better performance measure is bolded. To better depict the differences between the models, the goodness of fit statistics are presented for arbitrarily determined low-, mid-, and high-count ranges, as well as over the entire test dataset. Due to the nature of MAPE and the presence of zero-count observations, MAPE could not be applied to the low-count range, nor the full test dataset. In other studies, MAPE has been applied to zero-count locations through either log-transformation or by adding a negligibly small value to zero-counts. In this case, neither adjustment was applied to avoid artificially inflating the percentage error of zero-counts, for which any predicted value other than zero would incur immense errors.

Based on the AIC presented in Table 4, we find that the MBRP model is an overall improvement over the traditional NB regression model. But the statistics presented in Table 5 allow for more distinct quantitative results. Over the whole test dataset, the MAE and RMSE both indicate that the MBRP model predicts more accurately than the NB regression model by about 8.0% and 4.4% respectively. This trend is consistently observed across the low- and high-count ranges, where MAE improves by 14.0% and 7.2%, respectively, and the RMSE is improved by 22.2% and 6.3%, respectively. The exception to the trend is in the mid-count range, where the MAE improves by 3.9%, but the RMSE increases by 17.1%. This anomaly can be attributed to a single predicted value with a large residual, which is then highly weighted in the RMSE calculation due to the squared component of the error statistic. The evaluation that the MBRP model outperforms the NB regression model even in the mid-count range is further supported by the 9.0% improvement in the MAPE, which is also observed in the high-count range.

Accounting for evidence provided by the AIC as found in model development, the CURE plots generated for both models, and the goodness of fit statistics as calculated over the test dataset, the MBRP model improves upon the traditional NB regression model in predicting pedestrian count values. Though error evaluations differ in quantity, based on trends in the MAE, RMSE, and MAPE indicate that using MBRP to model pedestrian count data can improve prediction accuracy by approximately 10% compared to traditional NB regression models.