Validation of Urban Freeway Models (2014)

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23 Overview Enhancement was explored for the data-rich models, but no suitable enhancements resulting in performance improve- ments were found. For the data-poor enhancement, the original L03 models were recalibrated using data collected in the Los Angeles, San Diego, Sacramento, San Francisco, and Minneapolis regions. Additionally, the research team explored the perfor- mance of three new model forms to predict the 95th-, 90th-, and 80th-percentile TTIs: 1. A one-parameter power model (y = xb) 2. A two-parameter power model (y = a × xb) 3. A two-parameter polynomial model (y = a × x + b × x2) New models were explored for the 95th-, 90th-, and 80th- percentile models because they exhibited the worst perfor- mance in the validation assessment. The results of the recalibration were compared with the performance of the three new model forms. Overall, the team found that the error values (measured in mean square error) for the recalibration and the new models were similar 95th- and 90th-percentile TTI predictions. For the 80th-percentile TTI equation, the mean square error (MSE) of the new models was approximately half that of the recalibrated model. All of the new models exhibited a better adherence to the assumptions of regression than the original model form. In this chapter, the results section summarizes the results of the enhancement assessment and documents the equation outputs. The recommendations section discusses application guidelines for using the recommended models. Results Models The models shown in Table 4.1 were developed from the All- Data set (containing data from the regions in California and the Minneapolis region). Further discussion of these equa- tions, as well as the equations calibrated specifically to the California regions and the Minneapolis region, are included in Appendix E. Performance The performance of the recalibrated and new models was evaluated through consideration of the model statistics (mean square error and F-test results) and an assessment of how well the models meet the assumptions of regression. The details of the evaluation are presented in Appendix E in the form of model statistics, fit plots, observed versus predicted value plots, residuals versus predicted value plots, outlier and leverage plots, residual histograms, and normality plots. Table 4.2 shows the mean square error results for the recalibrated and new models. In terms of MSE, the new models show the most significant improvement over the recalibrated model for the 80th-percentile TTI. The MSEs between the recalibrated and new models for the 95th- and 90th-percentile TTI predictions are comparable. All of the recalibrated and new models satisfied the F-test, indicating overall validity. As seen in the residual versus pre- dicted value plots, all of the new models exhibited improved residual patterns over the recalibrated L03 models. The new models still exhibited some issues with non-constant variance and residuals that are not perfectly normally distributed. As with the error, the best improvements in the residual patterns were seen for the 80th-percentile TTI model. C h a p t e R 4 Enhanced Models and Application Guidelines

24 Table 4.1. Recalibrated and New Data-Poor Models Model Form 95th-Percentile TTI Recalibration 95th-percentile TTIAllData = 1 + 3.4201ln(meanTTI) 1-param power 95th-percentile TTIAllData = meanTTI1.9566 2-param power 95th-percentile TTIAllData = 1.0406 ∗ meanTTI1.8821 2-param polynomial 95th-percentile TTIAllData = 0.1494 ∗ meanTTI + 0.8902 ∗ meanTTI2 90th-Percentile TTI Recalibration 90th-percentile TTIAllData = 1 + 2.8189 ∗ ln(meanTTI) 1-param power 90th-percentile TTIAllData = meanTTI1.7324 2-param power 90th-percentile TTIAllData = 1.0099 ∗ meanTTI1.7137 2-param polynomial 90th-percentile TTIAllData = 0.3528 ∗ meanTTI + 0.6591 ∗ meanTTI2 80th-Percentile TTI Recalibration 80th-percentile TTIAllData = 1 + 2.1598 ∗ ln(meanTTI) 1-param power 80th-percentile TTIAllData = meanTTI1.4448 2-param power 80th-percentile TTIAllData = 0.9943 ∗ meanTTI1.4559 2-param polynomial 80th-percentile TTIAllData = 0.6166 ∗ meanTTI + 0.3809 ∗ meanTTI2 Standard Deviation TTI Recalibration StdDevTTIAllData = 0.7775 ∗ (meanTTI - 1)0.6810 PctTripsOnTime50mph Recalibration PctOnTimeTrip50mphAllData = e-2.0293∗[meanTTI-1] PctTripsOnTime45mph Recalibration PctOnTimeTrip45mphAllData = e-1.4874∗[meanTTI-1] PctTripsOnTime30mph Recalibration PctOnTimeTrip30mphAllData = 0.3401 + 0.6803 1 exp 4.5026 meanTTI 1.7890[ ]( )+ −p Table 4.2. Data-Poor Enhancement Mean Square Error Model Form MSE All Data CA MN 95th Percentile Recalibration y = 1 + a + ln(x) 0.0277 0.0255 0.0234 1-param power y = xb 0.0300 0.0273 0.0408 2-param power y = a × xb 0.0286 0.0264 0.0345 2-param polynomial y = a × x + b × x2 0.0289 0.0268 0.0352 90th Percentile Recalibration y = 1 + a + ln(x) 0.0137 0.0118 0.0110 1-param power y = xb 0.0122 0.0118 0.0151 2-param power y = a × xb 0.0121 0.0118 0.0144 2-param polynomial y = a × x + b × x2 0.0125 0.0121 0.0153 80th Percentile Recalibration y = 1 + a + ln(x) 0.00469 0.00410 0.00506 1-param power y = xb 0.00239 0.00178 0.00384 2-param power y = a × xb 0.00237 0.00176 0.00389 2-param polynomial y = a × x + b × x2 0.00245 0.00179 0.00436 Standard Deviation TTI Recalibrationa y = a × (x - 1)b 0.00668 0.00630 0.00364 PctTripsOnTime50mph Recalibrationa y = ea(x - 1) 0.00616 0.00765 0.00224 PctTripsOnTime45mph Recalibrationa y = ea(x - 1) 0.00363 0.00451 0.00123 PctTripsOnTime30mph Recalibrationa y = a + (b - a)/{1 + exp[c p (x - d )]} 0.00053 0.00050 0.00031 aThe recalibration for the 95th-percentile, 90th-percentile, and 80th-percentile models was performed following removal of two data points identified as outliers. The rest of the measures were recalibrated using all data points.

25 2. The new models allow for a consistent model form between different percentile TTI measures. 3. Since the variance of travel times tends to increase with the mean travel time, reliability model curves should show an increasing pattern at an increasing rate. The new models satisfy this characteristic in a way that the original L03 data-poor models do not. application Guidelines The research team recommends that the SHRP 2 program adopt the new L33 models. This recommendation is based on the following reasons: 1. The residual by predicted value plot shows improvement in the shape and in the balance of scatter around the origin, indicating that the new models better satisfy the assump- tions of regression.