Integrating Traffic Conflict Frequency and Severity Indicators to Estimate Pedestrian Safety Performance Functions for Signalized Intersections

Abstract

Assessing intersection safety for pedestrians based on crashes can be challenging because of the paucity of crashes. This is why surrogate measures of safety, typically traffic conflicts, have been used as an alternative approach for such assessments. Insufficient sample size has also presented a challenge in developing conventional safety performance functions (SPFs) for estimating pedestrian crash frequency in a variety of Highway Safety Manual applications. This challenge can also be resolved with the surrogate measures approach, in that these measures can logically capture the effects of geometric and operational variables that influence safety, thereby providing a useful complementary approach for developing SPFs, especially where sample sizes are small. This paper develops SPFs for pedestrian crashes at signalized intersections based on traffic conflicts, and in the process, it advances the methodology for this approach. Machine learning was used to develop a data-driven safety index that integrates frequency and severity indicators. This index was developed to classify conflicts into groups using a database of pedestrian conflicts derived from video observations at 44 intersections in five Canadian cities. The frequencies of conflicts in these groups, and crashes at these intersections, were then utilized to estimate SPFs. The results are promising in that they demonstrate the potential of using machine learning to estimate SPFs using surrogate measures. The approach is especially important given the focus on pedestrian crashes in Vision Zero plans and the reality that crash samples are typically too small for estimating pedestrian SPFs that directly capture the effects of multiple variables.

Keywords

surrogate measures of safety crash-conflict relationship machine learning-based safety index pedestrian safety safety performance functions

Traditionally, assessment of road safety relied on analyzing crash data reported by police. This is called direct assessment of safety based on crashes that present well-known challenges, including a slow and reactive approach because of the reliance on crash data, restricted availability of information for assessing crash risks, and a lack of real-time capabilities. Conflict-based safety analysis has been particularly valuable for analyzing rare crash types, such as those involving pedestrians. By using observed traffic conflicts, crashes can be estimated, so there is no need to wait several years for crash data to be collected for evaluating safety. That is why surrogate safety analysis (SSA) has been proposed as a complementary approach for safety assessments. This method relies on safety-critical events known as traffic conflicts instead of traffic crashes.

Traffic conflicts represent safety-critical events, and the assumption in this approach is that these events are most likely associated with the crash occurrence and, therefore, are strong indicators of safety. This assumption led to their use in safety assessments of road entities. Moreover, traffic conflicts can be useful for developing crash-based safety performance functions (SPFs) as an alternative or complementary approach to conventional SPFs used for a variety of Highway Safety Manual (HSM) applications. There are challenges with developing and calibrating robust, conventional SPFs for these HSM applications, as was evident in the most recent research aimed at meeting these challenges ( 1 ). In that research, the number of explanatory variables was limited, and several potentially influential variables could not be included because they were either correlated with included variables or were not observed. In addition, because of limitations in crash data availability, it was not possible to develop SPFs that separately capture interactions between turning vehicles and pedestrians, which are generally those of most interest in assessing and improving pedestrian safety at intersections. Developing such SPFs was specifically identified in a recent National Cooperative Highway Research Program (NCHRP) report ( 1 ) as an area for future research.

The application of conflicts to estimate crash-based SPFs for pedestrians is the focus of this paper. In the process, specific interactions between turning vehicles and pedestrians in these SPFs are considered. The appeal of using conflict measures as explanatory variables in SPFs is in their ability to logically capture the effects of multiple variables that relate to crashes, which, as noted, can be challenging to accomplish in conventional SPFs. Therefore, SPFs based on traffic conflicts can be used to proactively estimate crashes without requiring additional explanatory variables. This potential is especially important for quickly evaluating pedestrian safety improvements, in particular, new and innovative ones that are often the targets of Vision Zero programs.

Until relatively recently, conflict-based assessment has predominantly focused on one conflict indicator, that is, frequency based on either time to collision (TTC) or post encroachment time (PET). (TTC is the time required for two road users to collide if they continue at constant speeds and on the same path, while PET is the time difference between the moment a first road user leaves an area of potential collision and the moment of arrival of a second road user to this area.) Some early and notable efforts at incorporating another measure, that is, severity, include Ozbay et al. ( 2 ), who integrated conflict frequency and severity indicators using a crash index (CI). This index combined a modified TTC, relative speed, and relative acceleration of the vehicles involved. Later, Alhajyaseen ( 3 ) devised a “conflict index” that forecasts the kinetic energy released in a collision, with the mathematical formation of the conflict index equal to the change in total kinetic energy before and after collision divided by the PET value raised to an exponential power. Bagdadi ( 4 ) established a conflict severity index that relies on factors such as “delta-V” (i.e., the change in vehicle velocity), the mass of the individuals involved, and TTC. Wang and Stamatiadis ( 5 ) employed a multi-step approach, integrating various indicators to create an aggregated severe crash metric (ASCM). This method initially estimates crash probability using the TTC and maximum braking rate, followed by an estimation of severity based on power models of delta-V. More recently, Anarkooli et al. ( 6 ) devised a risk score (RS) for conflicts involving left-turn and opposing through vehicles by summing their speeds and dividing that sum by the PET value raised to an exponential power; the intent was to amplify the influence of PET at lower values while accounting for conflict severity. Despite the conceptual logic of these efforts to account for severity, these safety indexes nevertheless appear to be somewhat subjective and may not necessarily align with the most appropriate method for effectively integrating frequency and severity indicators in a real-world context.

The research for this paper aimed to further the earlier and more recent efforts in estimating pedestrian SPFs while complementing existing crash prediction methods. In so doing, machine learning (ML) was used to develop a data-driven safety index that integrates conflict frequency and severity indicators as a replacement for conventional integration methods. This index was then used to classify pedestrian conflicts derived from video observations at 44 intersections in five Canadian cities into groups. The frequencies of conflicts in these groups, and crashes at these intersections, were then utilized to estimate SPFs.

The rest of the paper is structured as follows. The next section presents some background on ML-based safety indexes, followed by a description of the data. The methodology is then outlined, and the results of integrating conflict frequency and severity indicators, along with the estimation of the SPF, are presented. The final section presents a summary and conclusions.

Background on Machine Learning-Based Safety Indexes

Research on integrating conflict frequency and severity indicators is somewhat limited in the field of road safety; however, such research is quite prevalent in the domain of air traffic safety and has advanced ML techniques for this purpose. These techniques are so powerful that they can efficiently analyze and integrate high-dimensional safety-related indicators within a matter of seconds. The product of this integration is a unique index, known as an anomaly score (AS). This safety index quantifies the degree of separation of each sample within the population from the remainder of the data set ( 7 – 11 ).

Das et al. ( 12 ) were among the first researchers to use an unsupervised ML approach called Multiple Kernel Anomaly Detection (MKAD) to develop a safety index and detect safety-related anomalies in air traffic trajectories. They applied a density-based approach for integrating multiple factors and detecting potential safety anomalies in flight traffic data. To decrease the complexity and improve computational time, Janakiraman and Nielsen ( 13 ) considered another method called unsupervised extreme learning machines (ELMs) and replaced the kernel-based approach with ELMs. Using this method, they successfully identified crash risk in aviation data sets in overcoming the limitations of MKAD.

More advanced ML methods were introduced in the other studies to increase the accuracy of the results. For example, Puranik and Mavris ( 14 ) presented a novel combination of clustering and anomaly detection methods for identifying and visualizing flight records with high ASs. A year later, Fernández et al. ( 8 ) used a combination of clustering and anomaly detection methods to estimate flight crash frequency in almost all severity levels. These advances in flight monitoring went as far as to calculate the safety index and predict safety-related events a few seconds before their actual occurrence ( 9 ).

In the road safety field, as noted, there have been precious few attempts at developing a ML-based safety index. One such research effort is only marginally relevant to the current study in that it pertains to the safety of automated vehicles ( 15 ). That study presented a ML approach to distinguish regular (safe) from anomalous (unsafe) driving behavior for automated vehicles based on the belief that automated vehicles must be functionally safe, interact safely with other vehicles, and improve total traffic safety. It was claimed that autoencoders, categorized as reconstruction-based ML, are ideal for integrating high-dimensional variables and detecting anomalous data in time-series data, such as driving interactions.

Data

Miovision, a video analytics company, provided a data set from five Canadian cities (Toronto, Hamilton, York Region, Winnipeg, and Calgary) that contains information on conflict indicators, conflict speed (CS), and movements for all road users, including pedestrians, cyclists, and vehicles. Video data were recorded for over 24 h between 2020 and 2022 at 44 urban signalized intersections that share similar characteristics, for example, geometric layout (four-legged intersections) and operations (signal-controlled). This is a large database, according to the standards of previous research involving pedestrian crashes and video-derived conflicts ( 16 , 17 ).

The data were collected using four high-quality cameras placed at an elevated position mounted on 7 m high poles installed at each intersection. Figure 1 illustrates the layout, camera placements, and coverage (the shaded areas) of one of the studied intersections. At this intersection, one camera pole was positioned on the pedestrian refuge island at the northeast corner (b), while additional cameras were installed at the (a) Northwest (NW), (c) Southeast (SE), and (d) Southwest (SW) corners. These four cameras were necessary to capture an unobstructed view of all approaches and the pedestrian crosswalks at the intersection. These intersections are located within the same jurisdiction. Miovision indicated that the data were subject to a 17-step quality assurance and quality control (QAQC) process, which includes a manual review of every detected conflict using specialized tools by a team of software operators, as well as numerous spot checks and validation checks of results.

Figure 1.

Examples of camera positions and views from each overhead camera.

Conflict records were collected for potentially serious vehicle-to-pedestrian interactions. One such interaction in the database is shown in Figure 2. As rationalized later in the Methodology section, conflicts were identified on the basis of a modified TTC measure, denoted as $T_{2}$ . This measure was considered a better surrogate for crashes than TTC when pedestrians are involved in that it accounts for possible changes in speed and travel path leading up to an interaction between two road users heading toward a shared conflict area ( 18 ). Conflicts with $T_{2}$ values of less than 3 s were deemed to be the potentially serious ones of interest for this research. This assumption was based on Hydén ( 19 ), who noted that conflicts with TTC values exceeding 3 s are part of the regular interactions between road users at an intersection. It is also consistent with the highest threshold that is frequently mentioned in the literature ( 20 ). As seen on close examination, a $T_{2}$ value of 1.86 was recorded for the interaction between the vehicle and pedestrian in Figure 2, so this conflict was identified as a potentially serious one.

Figure 2.

Vehicle-to-pedestrian conflict captured by elevated high-quality cameras in Toronto.

Crash records for the 5 years from 2017 to 2021 were assembled for the 44 intersections. This relatively rich data set included injury severity levels (i.e., property damage only, non-fatal injury, and fatal injury), collision impact type, pavement condition, and vehicle movement directions. The pedestrian crashes were separately extracted for each approach and then added together to produce the total number of such crashes at the intersection.

Table 1 presents the descriptive statistics of the data for conflicts and crashes between pedestrians and vehicles for the studied intersections. Not surprisingly, the mean value of 2.53 s for $T_{2}$ implies that most recorded conflicts are not severe in a relative sense.

Table 1.

Summary of Conflict and Crash Data

City/province	No. of intersections	Variable	Mean	SD	Minimum	Maximum
Toronto, Ontario	10	$T_{2}$ (s)	2.56	0.36	1.24	3
		Conflict speed (km/h)	17.54	7.18	5.58	50.24
		Total pedestrian crashes for 5 years	2.8	2.94	0	7
Hamilton, Ontario	10	$T_{2}$ (s)	2.61	0.29	1.5	3
		Conflict speed (km/h)	16.18	5.84	5	38
		Total pedestrian crashes for 5 years	6.5	2.17	2	9
Winnipeg, Manitoba	12	$T_{2}$ (s)	2.43	0.46	1.07	3
		Conflict speed (km/h)	15.84	7.77	5.5	44.86
		Total pedestrian crashes for 5 years	2.42	2.47	0	8
Calgary, Alberta	9	$T_{2}$ (s)	2.51	0.39	1.46	3
		Conflict speed (km/h)	16.37	7.51	3.12	57.92
		Total pedestrian crashes for 5 years	2.33	2.12	0	7
York, Ontario	3	$T_{2}$ (s)	2.52	0.40	1.44	3
		Conflict speed (km/h)	18.48	11.00	7	69.43
		Total pedestrian crashes for 5 years	3.33	2.31	2	6
Summary	44	$T_{2}$ (s)	2.53	0.38	1.34	3.00
		Conflict speed (km/h)	16.88	7.86	5.24	52.09
		Total pedestrian crashes for 5 years	3.48	2.40	0.80	7.40

Note: SD = standard deviation.

Methodology

In this section, the selection of the surrogate measures of safety and the ML-based safety index for integrating conflict frequency and severity indicators are discussed before presenting the study framework.

Rationale for the Selection of Safety Indicators

Generally speaking, traffic patterns are formed by aggregating the interactions among individual road users, including automobiles, pedestrians, and cars. An interaction can be an adaptation process or a response to the behavior of other road users. Traffic safety depends on these individual interactions. To be more precise, consider the frequency distribution function in Figure 3; increasing the conflict severity scale from left to right increases the risk of a collision, although the frequency of interactions decreases. These considerations are fundamental to the selection of the frequency and severity indicators.

Figure 3.

Conflict frequency distribution of traffic interactions as a function of severity.

Conflict Frequency Indicator

TTC is a commonly used frequency indicator that was first considered for this research. It was introduced back in 1972 ( 21 ) and defined as the time required for two road users to collide if they continue at constant speeds and on the same path. If the TTC value is less than a particular threshold value, the vehicles will likely be on a crash course. TTC is one of the most prevalent conflict measures of crash frequency ( 22 ) and has been used successfully for crash frequency prediction ( 23 ).

TTC, however, has some limitations as a vulnerable road user (VRU) conflict frequency indicator because of interactions when VRUs narrowly avoid impact and when the closeness of the interaction causes vehicles to stop for a few seconds before advancing; another issue with TTC arises when users are not on a precise collision course but were on a course to narrowly avoid a collision, a situation that is unsafe because of the low margin for error.

To overcome these issues with TTC, $T_{2}$ has been introduced as an alternative indicator of the temporal separation risk to classify VRU conflicts ( 16 , 23 ). Here, $T_{2}$ is a continuous indicator for which values are calculated as long as two road users are heading toward a shared conflict area. In calculating T₂, as shown in Figure 4, $d_{l}$ and $d_{2}$ represent the distance between the road user and the shared conflict location, while $v_{l}$ and $v_{2}$ represent the speeds of the road users. This approach accounts for possible changes in speed and travel path leading up to an interaction. When $\frac{d_{l}}{v_{l}}$ = $\frac{d_{2}}{v_{2}}$ , the road users are on a projected collision course, and $T_{2}$ equals TTC. The $T_{2}$ value is no longer calculated when the first road user passes the conflict zone; T₂ is, therefore, a generalized and flexible temporal proximity measure of particular cases, as illustrated in Figure 4. (Interested readers are referred to Arun et al. [ 16 ] for more details.) (Interested readers are referred to Laureshyn [ 18 ] for more details) The formula for the instantaneous value of T₂ is given below, and the T₂ value recorded for an interaction or conflict is the minimum of a time series of T₂ values:

T_{2} = Max (\frac{d_{l}}{v_{l}}; \frac{d_{2}}{v_{2}})

(1)

This paper employs T₂ because of its generalized and flexible nature, suitable for capturing diverse real-world scenarios, and its established status in the literature ( 18 ). It is worth noting in this context that although there is some debate in the literature about the best approach to measuring temporal proximity or nearness to a conflict, contributing to this debate by assessing the merits of different temporal proximity metrics is beyond the scope of the paper.

Figure 4.

Illustration of the T₂ concept.Source: Adapted from Laureshyn ( 18 ).

Conflict Severity Indicator

Delta-V or $Δ v,$ defined as the change in vehicle velocity resulting from a collision (assumed to be perfectly inelastic), was initially considered as the severity indicator by Shelby ( 25 ). A significant challenge in estimating delta-V is that it is only valid for completely inelastic collisions. As described by Laureshyn et al. ( 26 ), delta-V can be determined based on a rear-end collision in which the lead and trailing vehicles are perfectly inelastic. Since vehicle-to-pedestrian conflicts are not inelastic, other indicators, such as CS (the speed of the conflicting vehicle), can be more appropriate for evaluating severity, an understanding that is based on the notion that “speed kills” ( 23 , 27 ). Thus, the recorded speed of the vehicle conflicting with the pedestrian, CS, was selected as the severity indicator for this research.

Integration Method

This section presents the integration method, namely the autoencoder neural network (ANN). The aviation safety and similar literature suggest a reliance on autoencoders to integrate high-dimensional data by creating a single-dimension index called an AS ( 7 , 8 , 11 , 15 , 28 ). The ANN is categorized as a ML anomaly detection and a novel dimensionality reduction method. This method has a scoring technique that effectively integrates the input values and assigns a unique safety index to each observation. The latter capability makes it suitable for integrating traffic conflict indicators.

An autoencoder attempts to encode data by compressing them into lower dimensions, represented by a “bottleneck” layer or code, and subsequently decoding the data to reconstruct the original input. The bottleneck layer, which is the hidden layer where the encoding is produced, retains the compressed representation of the input data or, in other words, the AS. The mean reconstruction loss is minimized during the training process based on the frequent data points. Because of the relatively low frequency of anomalies in the observations, the autoencoder does not prioritize their reconstruction loss while training. As a result, the trained autoencoder reconstructs the safe interaction with the lowest reconstruction error, while anomalies have the highest AS ( 13 , 15 , 29 ).

To express the mean reconstruction loss concept in mathematical terms, consider a training data set ${x_{1}, x_{2}, \dots, x_{N}}$ that has been normalized with an average value of 0 and a standard deviation of 1, where N is the size of the training data and $x \in R^{d} .$ An autoencoder training problem is then solved by optimizing Equation 2. Here, ${\hat{x}}_{i}$ represents a reconstruction corresponding to $x_{i}$ , ${\hat{x}}_{i} = g (f (x_{i}$ )), where f and g are the encoder and decoder functions, respectively. The encoder f transforms the input data $x_{i}$ into compressed code, and then the decoder g reconstructs the original data from this code with the reconstruction ${\hat{x}}_{i} \approx x_{i}$ . Assuming a normally distributed reconstruction error, the mean square error (MSE), as shown in Equation 3, is a reasonable choice of loss function or AS ( 8 ). It is unknown, a priori, whether $x_{i}$ is a normal or an anomalous conflict:

\min \sum_{i = 1}^{N} | | x_{i} - {\hat{x}}_{i} | |^{2}

(2)

Autoencoder AS = L_{MSE} (x_{i}, {\hat{x}}_{i}) = \frac{1}{N} \sum_{i = 1}^{N} | | x_{i} - {\hat{x}}_{i} | |^{2}

(3)

Methodological Framework

Figure 5 illustrates the study procedure and modeling framework. Using the ANN and the safety index developed by integrating $T_{2}$ and the CS, conflicts were first classified into different severity levels. The ML integrating process included (a) exporting and merging conflict frequency and severity indicators from the conflict data set (i.e., T₂ and the CS) for each city, (b) developing the safety index by running the ANN for each city, (c) labeling each conflict with the unique AS (i.e., safety index), and (d) classifying conflicts into different severity levels based on the AS thresholds. Then, a relationship between the frequency of the associated crashes and the categorized conflicts was developed. In the next step, SPFs were estimated using generalized linear models (GLMs) and linear regression models to relate crashes at the same intersections to the correspondingly classified conflicts. Lastly, comparisons were made based on the performance of each model.

Figure 5.

Methodological framework.Note: SPFs = safety performance functions.

Integration Results

The autoencoder model was selected to integrate conflict frequency and severity indicators, as described earlier. This model was run with Python using the pyod.models.auto_encoder, sklearn.preprocessing, and sklearn.model_selection library packages. Two encoders and decoders were used, indicating the number of layers responsible for encoding and decoding the input data. A single latent layer captured the compressed input data representation. The random state was set to “None,” meaning that no random seed is used for reproducibility. Shuffle was assumed to be “True,” signifying that training data are shuffled before each epoch. (This means that the order of the data samples is changed before each run, ensuring that the model does not learn patterns based on the order of the data. Shuffling the data helps prevent the model from being biased by the order of the samples and can lead to more robust and generalizable learning.) No early stopping was applied for the model during the training to prevent overfitting. Lastly, learning_rate_init was considered equal to 0.0001, representing the initial learning rate of the optimization algorithm used to update the model parameters during the training.

After training the data set based on the autoencoder with the described tuning parameters and using Equation 3, the reconstruction error for each observation was calculated. Figure 6 plots the autoencoder reconstruction error distribution for the combined data set of all intersections. This histogram provides a continuous reconstruction error distribution for each data point, which helps identify the anomaly decision threshold. A sharp decrease in the distribution ordinate suggests that the anomaly decision threshold for separating extreme conflicts from others should be close to that value. The conflicts that exceed the threshold have higher conflict severity levels. In Figure 6, the blue line represents the AS, and the red rectangle suggests the potential threshold range.

Figure 6.

Histogram of anomaly scores obtained from the autoencoder algorithm for all intersections.

Figure 7 shows a scatter plot of the estimated ASs for each traffic conflict with T₂ and CS on the x- and y-axes, respectively. To ensure that the classified conflicts based on the AS have the lowest T₂ and highest CS values, the center point of this scatter plot, that is, the densest area in this figure, was determined. Then, the conflicts with T₂ value of 2.7 s and more and speed values of 16 $km / h$ and less were labeled as “regular conflicts,” regardless of their safety index, as shown by the shaded area in Figure 7. In the next step, conflicts with T₂ of less than 2.7 s, CS of more than 16 $km / h$ , and ASs of more than 1, 1.5, 2, and 2.5 were labeled as “severe or extreme conflicts,” as can be seen by the grey, yellow, orange, and red dots, respectively, in Figure 7. These classified “extreme conflicts” based on different AS threshold values were employed as explanatory variables for estimating crashes. “Total conflicts” refers to all of the recorded conflicts: the sum of “extreme conflicts” and “regular conflicts.”

Figure 7.

Classifying traffic conflicts based on their autoencoder neural network-based safety index.

Estimation of Pedestrian Safety Performance Functions

After selecting the initial threshold range based on the AS histogram illustrated in Figure 6, different AS threshold values ranging from 1 to 2, with intervals of 0.5, were employed to determine which AS thresholds provided conflict frequencies that provided the best-fit SPFs for vehicle-to-pedestrian crashes. This preliminary modeling revealed that an AS threshold equal to 1 that classified conflicts into two groups, conflicts with AS < 1 and conflicts with 1 ≤ AS, was best for estimating SPFs.

Table 2 shows the number of total pedestrian crashes per year and the candidate independent variables for the SPFs—the classified vehicle-to-pedestrian conflicts per day in the two groups, the proportion of daily conflicts that occur at night, and the proportion of daily conflicts that involve a left-turning vehicle. As seen in Table 2, in Toronto, an average of 5.6 pedestrian crashes per year is associated with 121 severe conflicts daily. Of these conflicts, approximately 12% occur at night, and nearly half involve left-turning vehicles and pedestrian crossings. After classifying conflicts into different severity levels based on their ASs, the next step was establishing a relationship between categorized conflicts according to ASs and the reported crashes.

Table 2.

Number of Pedestrian Crashes and Conflicts for Each City

City	Pedestrian crashes/year	Daily conflicts with AS < 1	Daily conflicts with AS > 1	Proportion of conflicts at night	Proportion of left-turn conflicts
Toronto	5.60	238	121	0.12	0.44
Calgary	4.20	76	58	0.20	0.71
York	2.00	15	12	0.34	0.38
Winnipeg	5.80	28	12	0.07	0.79
Hamilton	13.00	85	46	0.40	0.52

Note: AS = anomaly score.

To extrapolate conflicts to crashes and develop a link between them, conventional (or fixed-effect) generalized linear and simple linear regression models were first estimated. Although Peesapati et al. ( 30 ) showed that a linear regression model structure outperforms a generalized linear in estimating crashes using traffic conflicts as an explanatory variable, both of these model structures are commonly used for establishing crash–conflict relationships ( 6 , 29 –33). Therefore, both of these model structures were investigated in his research.

Fixed-effect models assume that the intercept and/or parameter coefficients should remain constant across all observations in various cities. In reality, pedestrian conflicts might vary among cities because of variations in pedestrian and motorist behavior and intersection characteristics ( 34 ). Therefore, to address this possibility with respect to variations between cities, mixed-effect models were also estimated, allowing the intercept and/or parameter coefficients of the models to vary across cities.

Fixed- and mixed-effect generalized linear models (FE-GLM and ME-GLM) with a log link function can be written as follows:

μ_{ij} = \exp (β_{0} + ω_{j 0}) \times Extreme conflic {t_{ij}}^{(β_{1} + ω_{j 1})} \times \exp (\sum_{k = 2}^{n} (β_{k} + ω_{jk}) x_{ijk}) \times \exp (ϵ_{ij})

(4)

Fixed- and mixed-effect linear regression (FE-LR and ME-LR) models were then developed as follows:

μ_{ij} = (β_{0} + ω_{j 0}) + ((β_{1} + ω_{j 1}) Extreme conflic t_{ij}) (\sum_{k = 2}^{n} (β_{k} + ω_{jk}) x_{ijk}) + ϵ_{ij}

(5)

where $μ_{ij}$ is the expected number of crashes for intersection i in city j, $k = 2, \dots, n$ is a subscript for different explanatory variables, $β_{0}, β_{1}, \dots, β_{k}$ are systematic and fixed terms, and the coefficients associated with the independent variables, $ω_{jk}$ are normally distributed terms with mean zero and variance $σ_{k}^{2}, which is$ adapted to consider spatial heterogeneity, $x_{ij}$ are the explanatory variables that influence the probability of crashes, and $ϵ_{ij}$ is an error term.

In practical terms, when the standard deviation of a parameter is notably greater than zero, the mixed-effect model will be chosen. Conversely, the fixed-effect model may be appropriate if the standard deviation is small or close to zero. It should be noted that a fixed-effect model is equivalent to a mixed-effect model, as presented in Equations 4 and 5, with the $ω_{jk}$ term being zero and $j$ being removed from the equations. To estimate the FE-LR model, ME-LR model, FE-GLM, and ME-GLM, Python programming and Stata software were used.

One of the challenges encountered while pursuing this study was the identification of a suitable goodness-of-fit (GOF) measure for the negative binomial (NB) family of models that has the following properties: (a) it has a [0,1] bound; (b) it has a proportional increase concept where adding exploratory variables to the model one at a time will result in the same increase regardless of their order of selection; and (c) it is invariant with respect to the mean. Miaou ( 35 ) showed that the conventional $R^{2}$ for a perfect model (i.e., one with correct probability function, correct functional form, no omitted variable, and all parameters correctly estimated with no uncertainty) can be less than 1. In addition, Miaou demonstrated that by adding equally important and independent covariates to the model one at a time, the increase in the value of $R^{2}$ is not the same for each covariate. Lastly, $R^{2}$ is affected by increasing the mean of the response variable or excluding the intercept term of the model. That said, the objective was to identify a GOF measure encompassing all of these properties within the context of NB regression models. Specifically, the aim was to find a measure where a value of zero indicates that no covariates have been incorporated into the model, while a value of one indicates that all necessary covariates have been included. Therefore, the dispersion parameter-based $R^{2}$ introduced by Miaou ( 35 ), and shown in Equation 6, was used for this study to facilitate an intuitive evaluation of the model accuracy and promote informed judgment:

Dispersion - based R^{2} = 1 - \frac{\hat{α}}{{\hat{α}}_{Max}}

(6)

where $\hat{α}$ is the estimated dispersion parameter of the NB model and ${\hat{α}}_{Max}$ is the dispersion parameter estimated in a NB model with only the intercept term.

For the linear regression models, model performance was assessed by centered and uncentered $R^{2}$ . To elaborate on these concepts, the models without intercepts have a property called $uncentered R^{2}$ , defined in Equation 7, which is always greater than the conventional $centered R^{2}$ , defined in Equation 8. $Uncentered R^{2}$ describes how much variation in the dependent variable, y, has been explained, while $centered R^{2}$ assesses the improvement in accuracy of the linear model over just using the mean. In Equations 7 and 8, $y_{i}$ , ${\hat{y}}_{i}$ , and $\bar{y}$ are the vector of the observed dependent variable, the predicted value of the dependent variable, and mean of the dependent variable, respectively, and $SST$ , $SSR,$ and $SS T_{0}$ are sum of squares “total,” sum of squares “regression,” and sum of squares “total” without considering the mean:

Uncentered R^{2} = R_{0}^{2} = 1 - \frac{SSR}{SS T_{0}} = 1 - \frac{\sum_{1}^{n} {(y_{i} - {\hat{y}}_{i})}^{2}}{\sum_{1}^{n} y_{i}^{2}}

(7)

Centered R^{2} = R^{2} = 1 - \frac{SSR}{SST} = 1 - \frac{\sum_{1}^{n} {(y_{i} - {\hat{y}}_{i})}^{2}}{\sum_{1}^{n} {(y_{i} - \bar{y})}^{2}}

(8)

For the GLMs and the linear regression models, the Akaike information criterion (AIC), Bayesian information criterion (BIC), and P-values were also considered to complement the assessment and evaluation. In addition, cumulative residual (CURE) plots were used to determine whether the selected functional form fits an explanatory variable along the entire range of its values represented in the data ( 36 ). In cases where there can be multiple explanatory variables, the ranges can be consolidated by examining the CUREs based on the model-estimated values of the dependent variable, pedestrian crashes, in the case of this study ( 37 ).

Linear Regression Safety Performance Functions

The FE-LR models estimated for total pedestrian crashes per year are summarized in Table 3 with and without intercept. In developing the models, different model forms were tried, including multiple linear regression that incorporated, as additional independent variables, combinations of regular vehicle-to-pedestrian conflicts, left-turn extreme conflicts, right-turn and through movement extreme conflicts, the proportion of conflicts at night, and the proportion of left-turn conflicts. Left-turn, right-turn, and through movements extreme conflicts indicate the number of extreme conflicts that occur between pedestrians and vehicles turning left, right, and going straight through an intersection, respectively. The proportion of conflicts at night indicates the number of vehicle-to-pedestrian conflicts that occurred at night divided by the total vehicle-to-pedestrian conflicts. Left-turn extreme conflicts indicates the proportion of extreme conflicts that occur between left-turning vehicles and crossing pedestrians. Right turns and through extreme conflicts indicates the proportion of extreme conflicts that occurred between right turns or through movements and pedestrians at intersections.

Table 3.

Fixed effect linear regression parameter estimates (and standard errors)

Exploratory variable	Model 1		Model 2		Model 3		Model 4		Model 5		Model 6		Model 7		Model 8
	Est. (SE)	P value	Est. (SE)	P value	Est. (SE)	P value	Est. (SE)	P value	Est. (SE)	P value	Est. (SE)	P value	Est. (SE)	P value	Est. (SE)	P value
Intercept	na	na	na	na	na	na	na	na	na	na	na	na	0.57 (0.10)	<.0001	0.39 (0.12)	0.002
Extreme conflicts/day with AS>1	0.06 (0.01)	<.0001	na	na	0.03 (0.01)	<.0001	na	na	0.02 (0.02)	0.27	0.03 (0.01)	0.005	0.02 (0.01)	0.04	0.02 (0.01)	0.05
Regular conflicts/day with AS<1	na	na	na	na	na	na	na	na	0.002 (0.01)	0.80	na	na	na	na	na	na
LT extreme conflicts/day with AS>1	na	na	0.12 (0.03)	0.04	na	na	0.06 (0.03)	0.04	na	na	na	na	na	na	na	na
RT & Thru extreme conflicts/day with AS>1		na	0.03 (0.02)	0.001	na	na	0.02 (0.01)	0.08	na	na	na	na	na	na	na	na
Proportion of conflicts at night	na	na	na		1.63 (0.33)	0.001	1.53 (0.34)	<.0001	1.24 (0.32)	<.0001	1.24 (0.32)	0.001	na	na	0.95 (0.36)	0.01
Proportion of LT conflicts	na	na	na	na	na	na	na	na	0.43 (0.14)	0.003	0.42 (0.13)	0.003	na	na	na	na
Centered R²	-0.59		-0.43		0.02		0.04		0.22		0.21		0.10		0.23
Centered Adjusted R²	-0.55		-0.50		-0.03		-0.03		0.13		0.16		0.08		0.19
Uncentered R²	0.38		0.43		0.61		0.62		0.69		0.69		na		na
Uncentered adjusted R²	0.36		0.40		0.57		0.57		0.64		0.66		na		na
AIC	96.45		94.88		78.30		79.12		72.44		70.52		74.54		69.62
BIC	98.24		98.45		81.87		84.47		79.58		75.87		78.11		74.97
Observations	44		44		44		44		44		44		44		44

Note: Est. = estimate; SE: standard error; LT = left-turn; RT & Thru = right-turn and through movements; na = not applicable.

After comparing the centered $R^{2}$ , AIC, BIC, and P-values, Model 6, which has no intercept and contains, as independent variables, extreme conflicts/day with AS > 1, the proportion of conflicts at night, and the proportion of left-turn conflicts was selected as the best FE-LR model. In addition, the choice of a model without intercept is logical based on the reasoning that, without extreme conflicts, crashes are less likely to occur.

To capture heterogeneity between the cities, the ME-LR model was estimated, with the results shown in Table 4. Six models were developed by considering the different combinations of fixed and mixed parameters. Even though several exploratory variables were initially considered, comparing full log-likelihoods, centered $R^{2}$ , and P-values resulted in “classified extreme conflicts with AS > 1” being retained as the sole independent variable for the best ME-LR model (Model 4). In this model, the intercept is considered random with a variance of 0.35.

Table 4.

Mixed effect linear regression parameter estimates (and standard errors)

Exploratory Variable		Model 1		Model 2		Model 3		Model 4		Model 5		Model 6
		Est. (SE)	P value	Est. (SE)	P value	Est. (SE)	P value	Est. (SE)	P value	Est. (SE)	P value	Est. (SE)	P value
Intercept	Mean	na	na	na	na	0.00	na	0.00	na	na	na	na	na
	Variance	na	na	na	na	0.002 (0.006)	na	0.35 (0.59)	na	na	na	na	na
Extreme conflicts/day with AS>1	Mean	0.11 (0.04)	0.004	na	na	0.07 (0.03)	0.04	0.04 (0.01)	<.0001	0.03 (0.01)	<.0001	0.03 (0.01)	<.0001
	Variance	0.005 (0.01)	na	na	na	0.001 (0.006)	na	na	na	na	na	na	na
LT extreme conflicts/day with AS>1	Mean	na	na	0.07 (0.03)	0.02	na	na	na	na	na	na	na	na
RT & Thru extreme conflicts/day with AS>1	Mean	na	na	0.03 (0.01)	0.05	na	na	na	na	na	na	na	na
	Variance	na	na	0.03 (0.01)	0.05	na	na	na	na	na	na	na	na
Proportion of conflicts at night	Mean	na	na	1.24 (0.52)	0.02	1.19 (0.39)	0.002	na	na	1.01 (0.29)	<.0001	0.84 (0.40)	0.03
	Variance	na	na	0.29 (0.85)	na	na	na	na	na	na	na	0.07 (0.49)	na
Proportion of LT conflicts	Mean	na	na	na	na	na	na	na	na	0.41 (0.28)	0.14	0.43 (0.26)	0.10
	Variance	na	na	na	na	na	na	na	na	0.27 (0.57)	na	0.16 (0.49)	na
Full Log Likelihood		-44.87		-42.53		-39.71		-35.83		-33.46		-33.20
Centered R²		0.04		0.13		0.23		0.49		0.49		0.53
Observations		44		44		44		44		44		44

Note: Est. = estimate; SE = standard error; LT = left-turn; RT & Thru = right-turn and through movements; na = not applicable.

Comparing the best FE-LR and ME-LR models (Models 6 and 4, respectively), the former encompassed three independent variables, while the latter included one independent variable. In addition, centered $R^{2}$ showed a significant improvement from approximately 20% in the fixed-effect model to 50% in the mixed-effect model. Therefore, it can be concluded that ME-LR outperforms FE-LR.

GLM Safety Performance Functions

As can be seen in Table 5, SPFs were estimated using a FE-GLM form with a log link. The models were fitted with independent variables, including classified extreme conflicts, the proportion of conflicts at night, and the proportion of left-turn conflicts. It is worth noting that the GLMs only use 39 intersections instead of all 44 since the GLM framework includes the natural logarithm (Ln) of certain independent variables. For five intersections, the values of these independent variables (extreme conflicts) were zero, and the Ln of zero is undefined. This led to the exclusion of these intersections from the analysis to ensure the validity and accuracy of the model. It is seen that Model 1 has the lowest $R^{2}$ and that considering all three independent variables in a model (i.e., Model 3) has one variable with a high P-value of 0.52. On the other hand, the lowest AIC and BIC (184.33 and 191, respectively) and highest dispersion-based R² (0.34) are observed for the model containing two independent variables—“extreme conflicts with AS > 1” and “proportion of conflicts at night” (i.e., Model 2). As such, Model 2 is selected as the best FE-GLM to estimate crashes. Notably, models containing regular conflicts exhibited high P-values, leading to the decision to exclude them from the results in Table 5.

Table 5.

Fixed effect generalized linear model estimates (and standard errors)

Exploratory variable	Model 1		Model 2		Model 3
	Est. (SE)	P value	Est. (SE)	P value	Est. (SE)	P value
Family (Link)	NB (log)		NB (log)		NB (log)
Intercept	-0.76 (0.26)	0.003	-1.04 (0.26)	<.0001	-1.24 (0.41)	0.002
Ln (Extreme conflicts with AS>1/day)	0.29 (0.15)	0.05	0.25 (0.14)	0.06	0.27 (0.14)	0.05
Proportion of conflicts at night	na	na	1.30 (0.57)	0.02	1.40 (0.59)	0.02
Proportion of left-turn conflicts	na	na	na	na	0.27 (0.42)	0.52
Log-likelihood	52.00		54.48		54.69
Dispersion	0.48		0.35		0.35
AIC	187.31		184.33		185.92
BIC	192.29		190.98		194.24
Pearson chi²	22.9		23.6		22.5
Pseudo R²	0.11		0.25		0.26
$Dispersio n_{\max}$	0.53		0.53		0.53
Dispersion–based $R^{2}$	0.10		0.34		0.34
Observations	39		39		39

Note: na = not applicable.

As was done for the linear regression models, a ME-GLM was developed to account for variances across different cities. The results are presented in Table 6. Although several exploratory variables were initially considered, the generally high P-values led to the decision to retain “extreme conflicts with AS > 1” as the only independent variable. Model 2 outperformed Model 1 on the basis of the AIC, BIC, and dispersion-based $dispersion - based R^{2}$ . For this model (#2), the intercept is considered random, with a mean of −1.05 and a variance of 0.19. In addition, a comparison of the fixed-effect Model 2 and the mixed-effect Model 2 reveals that utilizing a mixed-effect model was a valid choice, as it improved the dispersion-based $R^{2}$ from 0.34 to 0.49.

Table 6.

Mixed effect generalized linear model estimates (and standard errors)

Exploratory variable		Model 1		Model 2
		Est. (SE)	P value	Est. (SE)	P value
Intercept	Mean	0.00	na	-1.05 (0.31)	0.001
	Variance	0.98 (0.84)	na	0.19 (0.15)	na
Ln (Extreme conflicts with 1≤AS)	Mean	0.33 (0.17)	0.05	0.42 (0.14)	0.003
Full Log-likelihood		-91.26		-87.07
AIC		188.51		182.14
BIC		193.51		188.79
Dispersion		0.21		0.18
$Dispersio n_{\max}$		0.35		0.35
Pseudo $R^{2}$		0.47		0.47
Dispersion–based $R^{2}$		0.39		0.49
Observations		39		39

Note: DIC = deviance information criterion; AS = anomaly score; na = not applicable.

Cumulative Residual Plots

As mentioned earlier, CURE plots are used to determine whether the selected functional form fits an explanatory variable along the entire range of its values represented in the data by offering a visually informative assessment of the GOF for the models. These plots were generated by first sorting intersections in ascending order based on the model-estimated pedestrian crashes. Then, the residuals (the difference between observed and predicted crashes) were calculated for each intersection, and cumulative values were plotted. The standard deviation boundaries, $\pm 2 σ$ , provide a visual reference for understanding the spread of residuals and are calculated using Equation 9:

σ^{* 2} = σ^{2} (n) [1 - \frac{σ^{2} (n)}{σ^{2} (N)}]

(9)

where N is the total number of data points (residuals) and $n$ is an integer between 1 and N ( 36 ). In general, the plot of CUREs should not exceed the $\pm 2 σ$ bounds.

In the interpretation of the plots, when the CUREs consistently drift upwards, crashes are underestimated by the model and, conversely, when the CUREs consistently drift down, the model overestimates crashes.

Figure 8 depicts the CURE plots for the four final models: the FE-LR model, ME-LR model, FE-GLM, and ME-GLM, shown as (a), (b), (c), and (d), respectively. As seen, the plotted lines oscillate around the abscissa and fall within the two standard deviation lines, indicating that the model fits the data well for a range of model-estimated pedestrian crashes. The ME-LR model seems best with respect to the smoothness of the oscillations along the x-axis and the values of the ordinates relative to the two standard deviations.

Figure 8.

Cumulative residual (CURE) plots for the crash model estimates based on the fixed-effect linear regression (FE-LR) model, mixed-effect linear regression (ME-LR) model, fixed-effect generalized linear model (FE-GLM), and mixed-effect generalized linear model (ME-GLM), shown as (a), (b), (c), and (d), respectively.

Development of Statistical Equations for Anomaly Score Boundaries

Classifying conflicts based on ASs enables a more precise estimation of crash frequency; however, this process necessitates training an ANN for each specific data set, which may discourage jurisdictions from estimating long-term crash frequency using short-term video observations. To foster the practical application of this method, straightforward equations that define the boundaries between consecutive AS levels can offer an alternative for classifying severe conflicts. This research adopted a simple quadratic form for the boundary equations, as a visual inspection of the data plotted in Figure 7 indicates that the threshold boundary is curved and could be effectively modeled by such a function.

The proposed function form that aligns with the data is as follows:

(CS) + a {(T_{2} - b)}^{2} > c

(10)

where a, b, and c are estimated parameters and $T_{2}$ and $CS$ are the vectors of the safety indicators.

To estimate the parameters a, b, and c, an initial value and upper and lower boundary values were chosen for each parameter based on a visual inspection of the scatter plot. Python programming and essential data manipulation and visualization libraries were utilized, including “NumPy” for numerical computations, “pandas” for data handling, and “Matplotlib” for creating plots. SciPy’s curve_fit function was then used for curve fitting using the least-squares optimization method. The optimization method was defined to minimize the total number of misclassified observations in each iteration. One thousand iterations were run, and the estimated values of a, b, and c that led to the least misclassified observations were selected as the final values

Equation 11 shows the parameters of the estimated boundary equations for extreme conflicts defined by an AS greater than 1, which can be used for estimating total pedestrian crashes. This means that when the results of the expressions on the left-hand side exceed 25, the corresponding extreme conflict is associated with an approximate AS of 1:

(CS) + 60 {(T_{2} - 2.6)}^{2} > 25

(11)

This equation is plotted in Figure 9.

Figure 9.

Plot of the statistical equation for the anomaly score boundary.

Summary and Conclusions

A primary objective of the research was to use traffic conflicts for estimating crash-based pedestrian SPFs as a complementary approach to conventional SPFs used for HSM applications. The idea was to overcome the challenges of developing and calibrating robust, conventional SPFs for HSM applications, which were evident in recent research aimed at estimating those SPFs.

Using conflict measures as explanatory variables in SPFs can logically capture the effects of multiple variables, including specific interactions that relate to crashes that, as was evident in the recent HSM research ( 1 ), can be challenging to accomplish in conventional SPFs. This potential is especially important for quickly evaluating pedestrian safety improvements, particularly new and innovative ones that are often the targets of Vision Zero programs. For such evaluations, it is desirable to account for interactions between left-turning vehicles and pedestrians, which was possible for the approach used in this study.

Another key objective of this study was to advance the methodology for defining conflicts that are most closely related to crashes. To this end, the study examined an ANN model as a replacement for conventional approaches for integrating conflict frequency and severity indicators. For this exploration, a relatively rich pedestrian conflict database of 44 intersections in five Canadian cities covering 5 years of crash data and 24-h video-derived traffic conflicts with more than 1000 vehicle-to-pedestrian interactions was used. Then, conflicts were labeled and classified based on the unique data-driven safety index (i.e., AS), which was determined by integrating conflict frequency and severity indicators.

Once extreme conflicts were classified, SPFs were estimated using the ME-LR model, FE-LR model, ME-GLM, and FE-GLM to relate crashes to extreme conflicts and to combinations with other variables, including left-turn extreme conflicts, right-turn and through-movement extreme conflicts, the proportion of conflicts at night, and the proportion of left-turn conflicts. In general, the linear regression models outperformed the GLMs, a result that is consistent with previous research ( 30 ). In addition, mixed-effect modeling resulted in a considerable improvement in both LR models and GLMs, supporting the logic of estimating the crash–conflict relationship in such a way as to account for heterogeneity in the conflict distribution among different cities, which is also consistent with the previous research ( 34 ).

This study is unique in the sense that it attempts to demonstrate the potential of using ML methods for categorizing conflicts based on a data-driven safety index to reliably predict annual pedestrian crash frequency at an intersection in a short-term study. The estimated models revealed that the investigated approach is viable in that the best models were those that related crashes only to those conflicts with ASs larger than a threshold value. This emphasizes the need for proper classification of conflicts based on the AS threshold. The approach is especially important given the focus on pedestrian crashes in Vision Zero plans and the reality that observed frequencies of these crashes are typically too low for assessing the safety of intersections and for estimating models to capture the effects of multiple variables on crashes.

The practical application of these findings for other regions could be considered, albeit with due caution, as the safety index is derived from specific data and may vary with changes in the data set. To facilitate this practical application, simple statistical equations were developed based on Figure 7 that approximate the boundaries between consecutive AS levels. This approach would be particularly useful for jurisdictions that lack the resources to train an autoencoder for classifying conflicts within their own data sets. To be more precise, the conflict data set can be classified first using the developed equations, not the autoencoder model, and then the classified conflicts can be utilized to estimate total and/or severe crashes using the developed models.

With respect to data collection and safety indicators, it is important to note that analyzing vehicle-to-pedestrian conflicts poses more challenges than vehicle-to-vehicle conflicts, primarily because of the difficulty in detecting pedestrians—who are smaller in size—using image processing techniques. In addition, common safety indicators such as TTC may not fully capture the complex interactions that influence road user safety, particularly the immediate reactions between VRUs and drivers. Thus, future research could focus on developing safety indicators that better quantify these immediate reactions for vehicle-to-pedestrian conflicts.

Further work could also evaluate a larger sample and perhaps a wider variety of intersections to make the results more generalizable and facilitate the further development of the statistical models for the crash–conflict relationship and for establishing the AS boundaries. Such research can be enhanced by investigating other ML integration methods.

Footnotes

Author Contributions

The authors confirm contribution to the paper as follows: study conception and design: M. Hasanpour, B. Persaud; data collection: M. Hasanpour, C. Milligan: analysis and interpretation of results: M. Hasanpour, B. Persaud, C. Milligan; draft and final manuscript preparation: M. Hasanpour, B. Persaud. All authors reviewed the results and approved the final version of the manuscript.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by Discovery and Alliance Grants (Appl. ID RGPIN-2017-04457, RGPIN-2023-03787, and ALLRP 566916–21) from the Natural Sciences and Engineering Research Council of Canada (NSERC). The Alliance grant partners included the City of Toronto, which provided a financial grant, and MicroTraffic (now part of Miovision), which provided data, ideas, and in-kind financial contributions. This support is gratefully acknowledged. During the completion of the research for the paper, two authors (C. Milligan and M. Hasanpour) were employed by Miovision.

ORCID iDs

Maryam Hasanpour

Bhagwant Persaud

Craig Milligan

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