Calibrating the Highway Capacity Manual Arterial Travel Time Reliability Model

In this step, two categories of input data are required. The first category, Step 1a in Fig. 3, is the base data set, alternative data set, and historical data set used in the HCM-6 TTR model. In this paper, the testbed weather, volume, and incident input data were obtained from local sources. A discussion of the input values is provided elsewhere (Tufuor and Rilett 2020).

The second category is shown as Step 1b in Fig. 3; it is the observed point-to-point travel time data. Recent advancements in intelligent transportation systems (ITSs), computer technology, and the Internet of things bring with it the potential for collecting more detailed and consistent real-time arterial point-to-point travel time data. Examples of widely used ITSs for data collection include crowdsourcing (e.g., INRIX 2020), connected and automated vehicles (e.g., Datta et al. 2016), and Bluetooth (BT)/wireless fidelity (Wi-Fi)/light fidelity (Li-Fi) detectors (e.g., Cotten et al. 2020).

In this paper, point-to-point ITS travel time data from BT detectors installed on the testbed were utilized. The BT data collection system, its validation, and analysis are discussed elsewhere (Tufuor and Rilett 2018, 2019). The BT data were aggregated, according to the HCM-6 protocol, to obtain average $15\text{-}\min$ weekday travel times $T_{ij}$ for the analysis period (e.g., 4:30–5:30 p.m.). The data were filtered to eliminate periods that occurred during special events (e.g., public holidays) in 2016. The resulting TTD will form the so-called ground truth, and the goal will be to adjust the TTR model parameters so that the TTD estimated by the HCM-6 TTR model replicates this empirical distribution.

In this step, the current HCM-6 parameter set is used to model the testbed and estimate the TTD.

In the first iteration, the current parameter set corresponds to the uncalibrated parameter values. The BT TTD and the HCM-6 TTD for the first iteration, as well as their corresponding cumulative distribution functions (CDFs), are shown in Fig. 4. It may be seen in Fig. 4 that the empirical BT TTD has considerably more spread compared to the corresponding HCM-6 TTD. It was these differences, described earlier, that motivated this paper.

In this step, the TTD estimated by HCM-6 (Step 2) is compared to the empirical BT TTD (from Step 1). Numerous statistical methods exist to test whether two samples from different populations are statistically similar (Spiegelman et al. 2010). Parametric tests, such as Student’s $t$-test and the $F$-test, are popular methods for statistical inference of mean and variance values, respectively. There are two disadvantages to these tests for the purposes of this paper. First, these parametric tests often require a prior assumption of the underlying distribution. This is problematic because often the form of the underlying TTD is unknown. More importantly, these tests do not indicate whether two distributions are statistically similar, and this can be problematic for urban arterial roadway analyses (Kim et al. 2005). Because the goal is to calibrate the HCM-6 TTR model to replicate the empirical TTD, it was critical to use nonparametric or distribution-free tests to statistically determine whether any differences between the distributions were statistically significant. Typical nonparametric tests for comparing two distributions are the Mann-Whitney-Wilcoxon tests and the Kolmogorov-Smirnov (KS) test.

In this paper, the KS test was used to test the hypothesis that the population of the HCM-6-estimated average travel times and the population of the empirical BT average travel times in Fig. 4 have the same distribution. The hypotheses of the KS test is as follows:

Null hypothesis, ${\mathrm{H}}_0$: $F_{HCM}(T_{ij})=F_{BT}(T_{ij}),\forall T_{ij}$.

Alternative hypothesis, ${\mathrm{H}}_{\mathrm{a}}$: $F_{HCM}(T_{ij})\ne F_{BT}(T_{ij}),\forall T_{ij}$.

Here, $F_{HCM}$ is the HCM-6 CDF, and $F_{BT}$ is the empirical BT CDF.

For this test example, a 5% significance level was chosen. Not surprisingly, for the HCM-6 TTD and the empirical BT TTD as shown in Fig. 4, the KS test showed that there were statistically significant differences between the two distributions at a 5% significance level (Tufuor and Rilett 2020).

In this step, the stopping criterion (criteria) is (are) checked to determine whether the calibration procedure should continue or not. Because the proposed methodology is an iterative process and there is no guarantee of convergence, at a minimum the analyst needs to set a stopping criterion that will stop the procedure after a maximum number of runs. There is a trade-off between the quality of the results of the iteration and the time spent (Agdas et al. 2018). Traditionally, a set number of iteration loops ($R$) is used or suggested for calibration (Spiegelman et al. 2010; Kramer 2017). For this paper, $R$ was used and set to 60 because preliminary study showed that 60 iterations (or generations) provided good results for this network. However, if there is no solution after 60 iterations, a new $R$ will be set. When the number of iterations equals $R$, the algorithm stops. If not, the algorithm proceeds to Step 5.

Note that the stopping criteria could be a combination of a maximum number of iterations and a specific convergence criterion, such as achieving a successful KS test. The algorithm would stop whenever either criterion was met. In addition, the stopping criteria will be application-specific and may require some experimentation on the part of the user. For example, larger road corridors may need more iterations.

In this step, a new set of input parameter values is identified. Several optimization algorithms, including the simplex method, genetic algorithm (GA), and simulated annealing, for example, may be used in this step (Kochenderfer and Wheeler 2019). The goal of the algorithm is to select a set of HCM-6 input parameter values for the $r$th run that will result, hopefully, in an estimated TTD that is “better” than the previous ($r-1$) estimated TTD. In this paper, a GA is used to perform this task. A detailed description of the GA process may be found elsewhere (Kramer 2017; Appiah et al. 2011). The GA parameters for this application were selected based on a literature search of previous engineering-related GA applications (e.g., Yao et al. 2012; Yang et al. 2016; Hassanat et al. 2019; Cimorelli et al. 2020). The literature review found that the GA operators, e.g., the crossover rate and mutation rates, ranged from 50% to 90% and 1.0% to 2.5%, respectively. For the testbed, the midrate of the crossover (e.g., 70 %) and mutation (e.g., 1.75%) rates were selected. Twenty parameter sets were analyzed in each generation, and the generation gap was 75%, which was also selected based on experience and previous studies (e.g., Angelova and Pencheva 2011; Roeva and Vassilev 2016). As previously, the best GA values to use will be a function of the application and will, in all likelihood, need to be identified through a combination of prior experience and experimentation.

Since $R$ was set to 60, it will be necessary to examine 1,200 parameter sets for the test example. Furthermore, because $M=4$, the process shown in Fig. 3 was run 4,800 times during the calibration process.

It should be noted that because a statistical test was used in the example problem, once the calibration is complete, there may be one solution, a set of acceptable solutions, or no solution (Kim et al. 2005). If a set of acceptable solutions is found, it will be necessary to develop criteria to select the “best” solution. Intuitively, there are many ways to select the “best” solution. Possible selection criteria include using the lowest error, choosing the parameter set that has the least amount of difference with the HCM-6 default parameter values, and engineering judgment of the “best” representation of local conditions.

As discussed previously, the calibration process for the example problem focused solely on 19 demand factors. Table 3 shows the HCM-6 uncalibrated and calibrated demand factors and the percentage change between the factors. Also in Table 3 is the calibrated demand factors using only 2 parameters (e.g., mean and variance of the 19 demand factors). The process will be explained in the next section.

It may be seen in Table 3 that the differences between the uncalibrated and calibrated month-of-year and time-of-day adjustment factors are relatively small. For example, the average absolute percentage difference is 4.8%, and the largest difference is 8.9%. In contrast, the average absolute percentage difference between the uncalibrated and calibrated factors for the day-of-week parameter factors is 22.0%. In addition, the largest difference for the Tuesday factor is 56.9%.

The demand factors for both the calibrated and uncalibrated conditions in Table 3 were used to determine their corresponding DMFs using Eq. (3). Fig. 7 shows the distribution of the DMFs for both the calibrated condition and the uncalibrated condition [from Fig. 2(a)].

From Fig. 7 it may be seen that the calibrated DMFs have considerably more spread compared to the uncalibrated DMFs. This is evidenced by the fact that the calibrated standard deviation is approximately 78% greater. The calibrated DMF distribution is bimodal. The appropriateness of the bimodality will be discussed subsequently.

To illustrate how the traffic demand volumes and the demand factors interact, consider the northbound through movement on the first segment of the testbed for a scenario where the base traffic demand volume is 1,100 vehicles per hour (vph). Figs. 8(a and b) respectively show the uncalibrated and calibrated traffic demand volume for only the northbound through movement for the first segment of the testbed. These volumes were obtained using the DMF and the baseline volume in Eq. (3).

It should be noted that similar histograms can be obtained for all intersections and segment movements. The through movement was selected because the HCM-6 TTD is focused on the performance of the major-street through movement (Bonneson 2014).

It may be seen in Fig. 8 that the calibration had two effects on the demand distribution. First, the average through volume increased by 21%, from 1,133 to 1,371 vph. Second, the dispersion of the volumes in the calibrated condition is much greater, as evidenced by the fact that the standard deviation increased by 79%. The increase in both the mean volume and the variability in volume allowed the resulting HCM-6 TTD to match the empirical TTD.

It could be argued that the calibrated demand factors as shown in Table 2 no longer have a “physical” meaning. For example, why would a Tuesday (e.g., 0.411) have 66% less traffic than a Friday (e.g., 1.200), all else being equal? However, it is important to note that the HCM-6 TTD estimation methodology is a mechanistic approach where many solutions (e.g., combination of different values of the demand factors) will give the same answer.

Note that a user may wish for the calibrated DMF distribution to have a “physical” meaning. In this situation, the calibrated bimodal DMF distribution can be transformed into an alternative form that, first, fits the user’s prior knowledge of the demand factor distributions and, second, will still result in a statistically similar TTD when used in the HCM-6 TTR model. A Monte Carlo procedure can be used to randomly simulate and substitute the DMF values with the preferred DMF distribution. The values from the resultant DMF distribution can be used to derive the corresponding hour-of-day, day-of-week, and month-of-year demand factors using simple optimization techniques.

As an example, the user may wish for the DMF distribution to follow a Weibull distribution. The transformed DMF distribution is shown in Fig. 9(a). It may be seen that the distribution is approximately bell-shaped and has mean and variance values that are similar to the calibrated, bimodal DMF distribution. Fig. 9(b) shows the resulting traffic volume for the same through movements as shown in Fig. 8(b). Not surprisingly, this distribution is also bell-shaped. More importantly, it was found that the calibrated DMF distribution in Fig. 7 and the transformed DMF distribution in Fig. 9(a) resulted in similar TTDs when input into the HCM-6 TTR model. These TTDs were compared using the KS test, and there were no statistically significant differences at the 5% significance level. In other words, for this test network it was relatively straightforward to transform the calibrated DMF shown in Fig. 7 to those in Fig. 9(a). Both sets of demand factors will result in statistically valid TTDs. Note that the foregoing procedure was repeated for a lognormal and gamma distribution, and similar results were found.

Following from the previous discussions on the transformation of the DMFs, it is plausible to use an alternative procedure to calibrate the HCM-6 TTR model. In the alternative calibration process, the DMF distribution is defined by two parameters (e.g., mean and variance). For a given iteration, the current mean and variance values from the GA are used to back-calculate the 19 demand factors using an optimization code. In essence, the optimization identifies values for the 19 demand factors that result in the mean and variance and are subject to certain constraints (e.g., nonnegativity, no demand factor less than 0.8 or greater than 1.2). These 19 demand factors are then used in the HCM-6 procedure to calculate the TTD. This is in contrast to the original calibration where at each iteration 19 new demand parameters are identified in the GA.

The calibrated DMF distribution using the mean and variance are shown in Fig. 10. It may be seen that both calibrated DMF distributions are bimodal with similar means and variances. However, the values of the DMF distributions are considerably different. The associated demand factors for the DMF distribution shown in Fig. 10 are shown in Table 3. It may be seen that these demand factors are also considerably different than the demand factors for the original calibration that used 19 parameters. More importantly, the estimated TTDs from the modified calibration procedure were statistically the “same” as the empirical TTD using the KS test at the 5% level of significance. There was an average of approximately 5% error between the estimated TTDs from the modified calibration procedure and the empirical TTD. In summary, equally good results were achieved when 2 parameters, rather than all 19 of the demand factors, were used in the calibration.

There was no major difference in the time it took to run the original (e.g., 19 parameters) and the revised (e.g., 2 parameters) calibration procedure. This is not surprising because both used the same number of maximum iterations as the stopping criteria, and the extra step (e.g., identifying the 19 demand factors given a mean and variance of the DMF distribution) did not involve a significant amount of computing time. However, the revised calibration procedure was able to identify a statistically significant solution in the first iteration as compared to the fifth iteration in the original. In addition, the revised procedure identified 26, as opposed to 15, statistically valid solutions. It is easy to hypothesize that calibrating 2 parameters would be more efficient than calibrating 19 parameters, and this was found to be the case for this testbed.

As previously, the calibrated parameter set using the mean-and-variance process can be converted to a standardized bell-shaped histogram if the user desires. Alternatively, the calibration procedure could assume that there is an underlying PDF associated with the mean and variance identified from the GA. This could obviate the need to transform the demand factors, although this was not examined in this paper.