Validation of Urban Freeway Models (2014)

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65 Overview This appendix presents the validation analysis of the L03 data- rich models. Pages 143–145 of the L03 report contain six model equations that predict the following travel time index (TTI) reliability statistics (http: //www.trb.org/Main/Blurbs/166935 .aspx): • mean TTI; and • 99th-, 95th-, 80th-, 50th-, and 10th-percentile TTI. In the L03 project, these models were termed “data-rich” because they predict a wide set of reliability measures based on different combinations of four data variables: • dccrit, the critical demand-to-capacity (D/C) ratio on the study section; • dcaverage, the average D/C ratio on the study section; • ILHL (incident-lane-hours-lost), the annual lane hours lost because of incidents on the study section, during the analysis time slice; and • Rain05Hrs, the annual hours of rainfall ≥0.05 in. on the study section, during the analysis time slice. The L03 project calibrated these data-rich models using data collected in a number of metropolitan areas, but only validated the models on roadway sections in the Seattle metropolitan area. The goal of this stage of the L33 project is to quantify the effectiveness of these models using new data sets collected from around the country. The rest of this appendix is organized as follows. Section 2 presents the validation procedure, including the data gather- ing and the techniques used to measure the effectiveness of the L03 data-rich models. Section 3 presents the validation results for each time slice, including results for the model overall and by region. Finally, Section 4 summarizes the con- clusions. There is also an attachment that contains detailed regional validation results. Validation Procedure Models There are six L03 data-rich models per analysis time slice (peak period, peak hour, weekday, and midday), resulting in a total of 24 data models to be validated. The models to be validated in this task are as follows: Peak Period Models 1. = ( )∗ + ∗ + ∗meanTTI 0.09677 dc 0.00862 ILHL 0.00904 05crite Rain Hrs 2. 99th-percentile TTI 0.33477 dc 0.012350 ILHL 0.025315 Rain05Hrscrite= ( )∗ + ∗ + ∗ 3. = ( )∗ + ∗ + ∗95th-percentile TTI 0.23233 dc 0.01222 ILHL 0.01777 Rain05Hrscrite 4. = ( )∗ + ∗ + ∗80th-percentile TTI 0.13992 dc 0.01118 ILHL 0.01271 Rain05Hrscrite 5. = ( )∗ + ∗50th-percentile TTI 0.09335 dc 0.00932 ILHLcrite 6. = ( )∗ + ∗10th-percentile TTI 0.01180 dc 0.00145 ILHLcrite Peak Hour Models 1. = ( )∗ + ∗ + ∗mean TTI 0.27886 dc 0.01089 ILHL 0.02935 Rain05Hrscrite 2. = ( )∗ + ∗99th-percentile TTI 1.13062 dc 0.01242 ILHLcrite 3. = ( )∗ + ∗ + ∗95th-percentile TTI 0.63071 dc 0.01219 ILHL 0.04744 Rain05Hrscrite 4. = ( )∗ + ∗80th-percentile TTI 0.52013 dc 0.01544 ILHLcrite 5. = ( )∗ + ∗50th-percentile TTI 0.29097 dc 0.0138 ILHLcrite 6. = ( )∗ + ∗10th-percentile TTI 0.07643 dc 0.00405 ILHLcrite A P P e n d i x C Data-Rich Validation

66 Midday Models 1. mean TTI 0.2599 dccrite= ( )∗ 2. 99th-percentile TTI 0.19167 dccrite= ( )∗ 3. 95th-percentile TTI 0.07812 dccrite= ( )∗ 4. 80th-percentile TTI 0.02612 dccrite= ( )∗ 5. 50th-percentile TTI 0.01134 dccrite= ( )∗ 6. 10th-percentile TTI 0.00389 dccrite= ( )∗ Weekday Models 1. mean TTI 0.00949 dc 0.00067average= ( )∗ + ∗e ILHL 2. 99th-percentile TTI 0.07028 dc 0.00222 ILHLaveragee= ( )∗ + ∗ 3. 95th-percentile TTI 0.03632 dc 0.00282 ILHLaveragee= ( )∗ + ∗ 4. 80th-percentile TTI 0.00842 dc 0.00117 ILHLaveragee= ( )∗ + ∗ 5. 50th-percentile TTI 0.0021 dcaveragee= ( )∗ 6. 10th-percentile TTI 0.00047 dcaveragee= ( )∗ Root mean square error (RMSE) and the alpha level of the model coefficients are the only model fit statistics presented in the L03 report for each of these models. Without the full model fit outputs, much of the L33 validation had to focus on evaluating the extent to which these models adhere to the assumptions required for generalized regression. Data The data used in the validation were collected from the Los Angeles, San Francisco Bay Area, Sacramento, and San Diego metropolitan regions (grouped together into a “California” data set); Minneapolis–St. Paul, Minnesota; Salt Lake City, Utah; and Spokane, Washington. Details about the study seg- ments, data sets, and data processing techniques are provided in the L33 Validation Plan report. The California, Salt Lake City, and Spokane data were collected from the 3-year period between January 1, 2010, and December 31, 2012. The Min- nesota data were collected from the 3-year period between June 1, 2009, and May 31, 2012. Validation was performed using data collected on week- days during the following analysis time slices: 1. Peak period: a continuous time period of at least 75 min during which the space mean speed is less than 45 mph; 2. Peak hour: a continuous 60-min period during which the space mean speed is less than 45 mph; 3. Midday period: 11:00 a.m.–2:00 p.m.; and 4. Weekday period: 12:00 a.m.–11:55 p.m. This is consistent with the time periods that L03 used to cali- brate and validate the data-rich models. Table C.1 summarizes the sample size of data by region and time period used in the validation. Each value represents the number of section-years for which the D/C ratios, ILHL, rain, and TTI reliability statistics were calculated from the col- lected data. Only the highlighted cells in Table C.1 were used in the data-rich validation analysis, as these were the loca- tions and time periods that had a sufficient sample size for analysis. In the validation, the input variables were plugged into the model equations to calculate the TTI reliability sta- tistics, which were then compared to the measured values. Table C.1 shows that far fewer section-year data points were generated for the peak period and peak hour time slices than for the midday and weekday time slices. This is because many segments did not meet the L03 definition of having a peak period or peak hour. In Spokane, none of the sections met these criteria. In Salt Lake City, only seven section-years met these criteria. This reduces the regional variation among the validation data sets, and suggests that the peak period definition needs to be re-evaluated in the model enhance- ment stage. In addition to the lack of a notable peak period in the Spokane and Salt Lake City data sets, in general, the travel times in these data sets exhibited much less variation and unreliability than in the California (CA) and Minnesota (MN) sites. This should be kept in mind when evaluating the validation results. Measures For each model, the goals of the validation were to quantify the model error and determine whether the model follows the key assumptions of generalized regression. This section first describes the method of determining the prediction error and then presents the performance measures that were evaluated. Table C.1. Final Data-Rich Validation Sample Sizes (Section-Years) Site Peak Period Peak Hour Midday Weekday CA 43 43 140 142 MN 19 25 60 60 Salt Lake City 3 4 32 30 City of Spokane 0 0 9 11 All Data 65 72 241 243

67 Performance Measures Root Mean SquaRe eRRoR Because the data-rich models are in the exponential form, the prediction error is defined differently here than it was for the data poor validation. The research team assumes that the data-rich models are produced by first taking the logarithm transformation of the response data and then fitting it with a linear regression model. This implicitly assumes that the error term in the exponential form model is multiplicative and not additive. This is a typical way to develop a regression model for log-normally distributed data. Denote the predicted response values from the model as yˆ and the measured response values as y. The linear regression model form is ln yˆ X( ) = β The prediction error (residual) r is thus defined as: ln ˆ ln (C.1)r y y( ) ( )= − A positive mean r implies that the model systemically over- estimates values based on new data. Because the data-rich models are in exponential form and built by using the logarithm transformation of reliability measurements and linear regression method, the error (resid- ual) defined above is a multipliable term in the form of er. Taking the exponential function for both sides of Equation 1 leads to the following Equation 2: ˆ (C.2)y ye r= − or equally, ˆ e y y r = Then, subtracting 1 from both sides of the equation, the formula for the percent deviation of the predicted TTI reli- ability metric from its measured value is obtained. That is, 1 ˆ e y y y r ( ) − = − This definition states that the data-rich model overestimates or underestimates the response variable by a percentage of (er - 1) * 100% compared to the measured data. Note that when r is close to zero, er - 1 ≈ r. In the rest of this appendix, the RMSE values presented are the modified RMSE values, calculated according to the fol- lowing procedure: • The ordinary RMSE values are calculated using the residual defined as r = ln(yˆ) - ln(y); • The redefined RMSE is calculated by RMSE′ = eRMSE - 1; and • The presented values are RMSE′ * 100%, which is referred as RMSE in the rest of this data-rich model validation appendix. Note that the statistical analysis tables and the residual plots are produced using r, because it is r that needs to satisfy the linear regression assumptions. Student’S t-teSt The one sample Student’s t-test can be used to determine if the mean of the residuals is significantly different from zero in a statistical sense, which tests for systematic bias. With an unbiased model, the difference should be statistically insig- nificant. The t-value is calculated as 0 = − µ t r s n where r– is the residual mean, s is the standard deviation of residuals, n is the sample size, and µ0 is the specific mean value for comparison, set here to be zero. To draw a conclu- sion, if the calculated t value is larger than some threshold ta (e.g., a = 5%) using a two-tailed t distribution table, the null hypothesis that the residuals have a mean of zero can be rejected with (1 - a) level of confidence. Or, say that the residual mean is significantly different from zero at a level of probability. If the corresponding p value is used to draw a conclusion, it means that if the null hypothesis were correct, then we would expect to obtain such a large t value on at most p percentage of occasions. For the validation, we use a 90% level of confidence. ShapiRo-Wilk noRMality teSt The Shapiro-Wilk test can be used to determine whether the distribution of the residuals is significantly different from the normal distribution in a statistical sense. The null hypothesis in this test states that the residuals are normally distributed. To draw a conclusion, if the p-value is less than a threshold, the null hypothesis that the residuals are normally distributed can be rejected with (1 - a) level of confidence. The threshold used here is a = 10%. ReSidual plotS Ideally, residual r is a random variable that follows a normal distribution with zero mean. Plotting out the distribution of residuals allows for an assessment of the goodness of fit and the likelihood of the presence of bias and heteroscedasticity (unequal variance).

68 data-Rich Model Validation The data-rich models are categorized as peak period, peak hour, midday, and weekday models. For each category, there are six models for the different travel time reliability mea- sures: mean TTI, 99th-percentile TTI, 95th-percentile TTI, 80th-percentile TTI, 50th-percentile TTI, and 10th-percentile TTI. The following subsections are named as “time period— reliability measure—region of the data set.” Tables and fig- ures for all data are shown in these subsections, while tables and figures for specific regions are shown in the attachment (Tables C.52 to C.111 and Figures C.73 to C.252). Peak Period Peak Period—Mean TTI peak peRiod—Mean tti—all data Table C.2 presents the summary of the RMSE for this cate- gory. The largest RMSE appears in the CA data in the peak period mean TTI model validation. The RMSE for MN data is relatively small. Since the CA data account for the largest portion of the AllData set, the RMSE for AllData is also rela- tively large. These RMSE values only present a general impres- sion of the model performance; further investigation on the residual analysis results provides more details on the model performance. Table C.3 presents the statistical analysis of residuals result- ing from predicting the validation data with the data-rich peak period mean TTI model. Ideally, the residuals should follow a normal distribution with a mean of zero. As shown in Table C.3.c, the Student’s t-test for zero residual mean yields a p-value of 0.0119, meaning that we can reject the null hypothesis of zero residual mean with a confidence level of 90%. The Shapiro-Wilk normality test (Table C.3.d) rejects the hypothesis that the residuals follow a normal distribution as the p-value is less than 0.0001. The same conclusion can be drawn by observing the normality plot. From the plot of residuals versus the predicted mean TTI (Figures C.1 through C.3), we can see that the residuals have an increasing trend where the maximum residual is reached at the largest predicted response value, and when the predicted value is smaller than approximately 0.5 the model tends to underestimate the response variable. This nonrandom pattern indicates that the data-rich model cannot perfectly predict the validation data and may be improved. Table C.2. RMSE of Peak Period—Mean TTI RMSE All Data CA MN Mean TTI 96.94% 127.55% 21.59% Table C.3. Residual Analysis of Peak Period—Mean TTI—AllData Table C.3.a. Basic Statistical Measures Location Variability Mean 0.2086 Std deviation 0.6499 Median -0.007 Variance 0.4223 Minimum -0.348 Range 3.0176 Maximum 2.6696 Interquartile range 0.3150 Table C.3.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.5972 Pr < W <0.0001 Table C.3.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 2.5884 Pr > t 0.0119 Table C.3.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.2086 0.0476 0.3697 Std deviation 0.6499 0.5542 0.7858 Variance 0.4223 0.3071 0.6174 Figure C.1. Residual plot of peak period—mean TTI—AllData.

69 the models tend to underestimate the response variable as most residuals are under the zero reference line. These non- random patterns show that the model fits the validation data unsatisfactorily. All the associated plots and results for CA region are included in the attachment. peak peRiod—Mean tti—MinneSota In MN, the residuals increase with the predicted values. This nonrandom pattern shows that the model has the potential to be improved. The statistical tests show that we cannot reject the zero residual mean hypothesis but can reject the null hypoth- esis of normal distribution with a high confidence level. The histogram and the normality plot illustrate these conclusions. All the associated plots and results for MN region are included in the attachment. Peak Period—99th-Percentile TTI peak peRiod—99th-peRcentile tti—all data The summary results for this category provided in Table C.4 indicate that the estimation error for the CA data set is as large as 607.76%. As the CA data set constitutes the most part of the AllData set, the RMSE for the AllData set is also rela- tively large (403.44%). The RMSE for the MN validation data set is 63.67%. This is relatively small compared to the AllData and the CA data sets but still indicates nonsatisfactory esti- mation performance, since it means an estimation error as large as 63.67% of the measured value. The residual plot (Figure C.4) in validation analysis of the 99th-percentile TTI model using the AllData set shows a similar pattern to the validation of the mean TTI model using the same data set: residuals increase with the predicted value. Again, such nonrandom pattern shows that the model may have the potential for improvement. The normality plot shows that the residuals do not closely follow a normal distri- bution, which is also indicated in the normality test as a p-value less than 0.0001. The Student’s t-test shows that the research team can reject the null hypothesis of zero residual mean with a confidence level of 90%. Table C.5 and Figures C.4 to C.6 present these results. peak peRiod—99th-peRcentile tti—califoRnia The validation results for the peak period 99th-percentile TTI model using CA data set shows very similar pattern to that in the validation of percentile TTI model using AllData set. The Figure C.2. Residual histogram of peak period— mean TTI—AllData. Figure C.3. Residual normality plot of peak period—mean TTI—AllData. peak peRiod—Mean tti—califoRnia The residual plots in the peak period—mean TTI—CA looks similar to those in the AllData validation, since the CA vali- dation data set accounts for most of the samples in the AllData set. The normality test result rejects the null hypothesis that the residuals follow a normal distribution. The Stu- dent’s t-test for zero residual mean shows that we can reject the null hypothesis with a confidence level of 90%. The plot of residuals versus the predicted values shows an almost linear increasing relationship and tends to overestimate the mean TTI when the predicted value is larger than approximately 0.5. On the other hand, when the predicted value is smaller than 0.5, Table C.4. RMSE of Peak Period—99th-Percentile TTI RMSE All Data CA MN Mean TTI 403.44% 607.76% 63.67%

70 Figure C.4. Residual plot of peak period— 99th-percentile TTI—AllData. Table C.5. Residual Analysis of Peak Period—99th-Percentile TTI—AllData Table C.5.a. Basic Statistical Measures Location Variability Mean 0.5030 Std deviation 1.5480 Median 0.0020 Variance 2.3963 Minimum -0.861 Range 7.4177 Maximum 6.5571 Interquartile range 0.9133 Table C.5.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.6312 Pr < W <0.0001 Table C.5.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 2.6197 Pr > t 0.0110 Table C.5.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.5030 0.1194 0.8866 Std deviation 1.5480 1.3201 1.8717 Variance 2.3963 1.7427 3.5033 Figure C.5. Residual histogram of peak period— 99th-percentile TTI—AllData. Figure C.6. Residual normality plot of peak period—99th-percentile TTI—AllData. zero residual mean hypothesis is rejected in the Student’s t-test, and the null hypothesis of normal distribution is rejected in the normality test. The histogram and normality plots also show that the residuals do not closely follow a normal distribution. The plot of residuals versus predicted response values shows an almost linear increasing pattern as the predicted value increases. All the associated results for CA region are included in the attachment. peak peRiod—99th-peRcentile tti—MinneSota In validating the 99th-percentile TTI model using the MN data set, the plot of residuals versus the predicted response

71 values does not indicate a good fit, evident from the increas- ing trend of the scattered points. The Student’s t-test shows that the null hypothesis of zero residual mean can be rejected with a confidence level of 90%, while the Shapiro-Wilk nor- mality test shows that null hypothesis of normal distribution cannot be rejected with 90% confidence level but can be rejected at a confidence level of 85%. All the associated results for MN region are included in the attachment. Peak Period—95th-Percentile TTI peak peRiod—95th-peRcentile tti—all data The RMSE values for the peak period 95th-percentile TTI model are smaller than the corresponding values in the vali- dation of the 99th-percentile TTI model. The largest value still comes from the CA data set, which is 359.19%, while the smallest is from the MN data, which is 45.85% as summarize in Table C.6. The residual analysis for the 95th-percentile TTI model using AllData set shows that the Student’s t-test rejects the null hypothesis of zero residual mean, as the p-value is only 0.0211. The Shapiro-Wilk rejects the null hypothesis of nor- mal distribution with a high level of confidence (Table C.7). The residual plot (Figure C.7) shows a similar pattern as that in the previous models: the residual keeps increasing when the predicted value increases. The histogram and the normal- ity plot (Figures C.8 and C.9) manifest the Shapiro-Wilk test result. peak peRiod—95th-peRcentile tti—califoRnia The residual analysis results produced in the CA data valida- tion shows that the null hypothesis of zero residual mean in the Student’s t-test can be rejected with a confidence level of 90%, and the null hypothesis of a normal distribution can also be rejected with a high confidence in the normality test, as demonstrated in the histogram and the normality plot. The residual keeps increasing as the predicted value increases. All the associated results and plots for the CA region are included in the attachment. peak peRiod—95th-peRcentile tti—MinneSota The residuals in the validation of the 95th-percentile model with the MN data set also present an increasing pattern in the residual plot. The Student’s test rejects the zero residual mean with an over 90% level of confidence, and the Shapiro-Wilk Table C.6. RMSE of Peak Period— 95th-Percentile TTI RMSE All Data CA MN Mean TTI 251.95% 359.19% 45.85% Table C.7. Residual Analysis of Peak Period—95th-Percentile TTI—AllData Table C.7.a. Basic Statistical Measures Location Variability Mean 0.3567 Std deviation 1.2161 Median -0.025 Variance 1.4789 Minimum -0.684 Range 5.7559 Maximum 5.0720 Interquartile range 0.5554 Table C.7.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.3567 0.0553 0.6580 Std deviation 1.2161 1.0371 1.4704 Variance 1.4789 1.0755 2.1622 Table C.7.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 2.3646 Pr > t 0.0211 Table C.7.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.6160 Pr < W <0.0001 Figure C.7. Residual plot of peak period— 95th-percentile TTI—AllData.

72 The residual analysis results (Table C.9) show that the zero residual mean hypothesis can be rejected in the Student’s t-test with a confidence level of 90%. The Shapiro-Wilk nor- mality test rejects the null hypothesis of normal distribution at a confidence level of 90%, as also demonstrated in the nor- mality plot. The residual plot (Figure C.10) shows an increas- ing trend in the residual value when the predicted response value increases. The residual histogram and normality plots are shown in Figures C.11 and C.12. peak peRiod—80th-peRcentile tti—califoRnia For the 80th-percentile TTI validation using CA data, the Stu- dent’s t-test rejects the null hypothesis of zero residual mean with a confidence level of 90%. The Shapiro-Wilk normality Figure C.8. Residual histogram of peak period— 95th-percentile TTI—AllData. Figure C.9. Residual normality plot of peak period— 95th-percentile TTI—AllData. normality test rejects the normal distribution hypothesis also with a high confidence level. All the associated results and plots for MN region are included in the attachment. Peak Period—80th-Percentile TTI peak peRiod—80th-peRcentile tti—all data For the 80th-percentile TTI (Table C.8), the largest RMSE is seen in the CA data set (206.54%) and the RMSE from the MN data set is much smaller, at 30.95%. From the 99th-percentile TTI validation to the 80th-percentile TTI validation, the RMSE for the AllData set has decreased. This is attributed to the fact that smaller percentiles result in smaller response variable range. However, one should note that the reduced RMSEs do not necessarily mean better model performance as it is not instructive to compare the models in this way. Table C.8. RMSE of Peak Period— 80th-Percentile TTI RMSE All Data CA MN Mean TTI 151.95% 206.54% 30.95% Table C.9.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.5929 Pr < W <0.0001 Table C.9. Residual Analysis of Peak Period—80th-Percentile TTI—AllData Table C.9.a. Basic Statistical Measures Location Variability Mean 0.2537 Std deviation 0.8954 Median -0.058 Variance 0.8018 Minimum -0.404 Range 4.0847 Maximum 3.6805 Interquartile range 0.4376 Table C.9.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.2537 0.0319 0.4756 Std deviation 0.8954 0.7636 1.0827 Variance 0.8018 0.5831 1.1722 Table C.9.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 2.2845 Pr > t 0.0257

73 confidence level of 90%. All the associated results and plots for MN region are included in the attachment. Peak Period—50th-Percentile TTI peak peRiod—50th-peRcentile tti—all data For the 50th-percentile TTI validation, the RMSE values show a similar pattern to that in the previous models: the largest value comes from the CA data (Table C.10). The Student’s t-test shows that the null hypothesis of zero residual mean can be rejected with a confidence level of 90% (Table C.11). The Shapiro-Wilk normality test result rejects the null hypothesis of normal distribution at a confidence level of 90%, which is also illustrated in the histogram (Fig- ure C.14) and the normality plot (Figure C.15). The residual plot (Figure C.13) still shows an increasing trend similar to that in the previous models. peak peRiod—50th-peRcentile tti—califoRnia The Student’s t-test results show that the zero residual mean hypothesis can be rejected at a confidence level of 90%. The Shapiro-Wilk normality test result rejects the null hypothesis of normal distribution at the same threshold confidence level, which is also illustrated in the histogram and the normality Figure C.10. Residual plot peak period— 80th-percentile TTI—AllData. Figure C.11. Residual histogram of peak period— 80th-percentile TTI—AllData. Figure C.12. Residual normality plot of peak period—80th-percentile TTI—AllData. test rejects the null hypothesis that the residuals follow a nor- mal distribution at a confidence level of 90%. The residual plot shows a similar pattern to that in the AllData validation, with the residual increasing with the predicted value. This non- random pattern indicates that it may be possible to improve the data-rich model performance. All the associated results and plots for the CA region are included in the attachment. peak peRiod—80th-peRcentile tti—MinneSota In MN, the residuals also show an increasing trend as the pre- dicted response value increases. The Student’s t-test shows that the null hypothesis of zero residual mean cannot be rejected with a confidence level of 90% but can be rejected with a confidence level of 80%. The Shapiro-Wilk normality test shows that the null hypothesis that the residuals follow a normal distribution can be rejected with the preset threshold Table C.10. RMSE of Peak Period— 50th-Percentile TTI RMSE All Data CA MN Mean TTI 89.55% 116.63% 23.15%

74 Figure C.15. Residual normality plot of peak period—50th-percentile TTI—AllData. Figure C.14. Residual histogram of peak period— 50th-percentile TTI—AllData. Table C.11. Residual Analysis of Peak Period—50th-Percentile TTI—AllData Table C.11.a. Basic Statistical Measures Location Variability Mean 0.2044 Std deviation 0.6106 Median -0.017 Variance 0.3729 Minimum -0.368 Range 2.8998 Maximum 2.5315 Interquartile range 0.2944 Table C.11.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.2044 0.0531 0.3557 Std deviation 0.6106 0.5207 0.7383 Variance 0.3729 0.2712 0.5451 Table C.11.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 2.6990 Pr > t 0.0089 Table C.11.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.6172 Pr < W <0.0001 Figure C.13. Residual plot of peak period— 50th-percentile TTI—AllData. plot. The plot of residuals versus the predicted values shows an increasing trend similar to that in the previous model vali- dations. All the associated results and plots for the CA data are included in the attachment. peak peRiod—50th-peRcentile tti—MinneSota In MN, the Student’s t-test result shows that the null hypoth- esis of zero residual mean cannot be rejected with a confi- dence level of 90%, while the Shapiro-Wilk normality test result rejects the null hypothesis that the residuals follow a nor- mal distribution. The increasing trend in the plot of residuals

75 versus the predicted values still exists, and the histogram and the normality plot demonstrates the normality test results that the residuals are not closely following a normal distribution. All the associated results and plots for the MN data are included in the attachment. Peak Period—10th-Percentile TTI peak peRiod—10th-peRcentile tti—all data The RMSE values for the validation of peak period 10th- percentile TTI data-rich model are all within 15% (Table C.12). However, as the 10th-percentile TTI data are within a small range themselves, the smaller RMSE values do not necessarily mean good model performance. Further investigation on the validation results for each data set is required. The statistical validation using the AllData set shows that the zero residual mean hypothesis can be rejected with a con- fidence level of 90% in the Student’s t-test, and the null hypothesis of normal distribution can be rejected with a con- fidence level of 90% in the Shapiro-Wilk normality test (Table C.13). The plot of residual versus the predicted value (Figure C.16) shows the problem of an increasing trend of residuals as the predicted value increases, as well as the non- constant residual variance problem. These nonrandom pat- terns show that the model can potentially be improved further. The residual histogram and normality plots are shown in Fig- ures C.17 and C.18. peak peRiod—10th-peRcentile tti—califoRnia Validation of the CA data set generally produces similar results to the validation using the AllData set. The residual plot shows the increasing trend and the nonconstant residual variance problems. The Student’s t-test cannot reject the zero residual mean hypothesis at the threshold confidence level of 90% as the p-value is as large as 0.1820. The hypothesis that the residuals follow a normal distribution is rejected in the Shapiro-Wilk normality test with a confidence level of 90%, which is illustrated in the histogram and the normality plot. All the associated results and plots for the CA data are included in the attachment. peak peRiod—10th-peRcentile tti—MinneSota The MN data set validation results show that the assumption of zero residual mean is likely to be violated as the Student’s t-test yields a p-value less than 0.0001, while the null hypothesis Table C.12. RMSE of Peak Period—10th-Percentile TTI RMSE All Data CA MN Mean TTI 12.13% 14.43% 6.23% Table C.13. Residual Analysis of Peak Period—10th-Percentile TTI—AllData Table C.13.a. Basic Statistical Measures Location Variability Mean 0.0333 Std deviation 0.1104 Median 0.0303 Variance 0.0122 Minimum -0.344 Range 0.7344 Maximum 0.3906 Interquartile range 0.0600 Table C.13.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0333 0.0059 0.0606 Std deviation 0.1104 0.0941 0.1334 Variance 0.0122 0.0089 0.0178 Table C.13.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 2.4305 Pr > t 0.0179 Table C.13.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.8479 Pr < W <0.0001 Figure C.16. Residual plot of peak period— 10th-percentile TTI—AllData.

76 that the residuals follow a normal distribution cannot be rejected in the Shapiro-Wilk normality test as the p-value is 0.6593. The residual plot shows that the residuals are all posi- tive except for one. Meanwhile, a nonrandom increasing trend is obvious in the residual plot. All the associated results and plots for the MN data are included in the attachment. Conclusions of the Data-Rich Peak Period Model Validation The peak period data-rich model validation analyzed six models built in L03 using three data sets: the CA data set, the MN data set, and the AllData set, which included all the avail- able peak period data. The validation analysis indicates that the most significant problem is the increasing residual trend shown in the residual plots. A good regression model should present randomly scattered residuals without obvious trends. Additionally, the validation results rarely satisfy the zero residual mean or the normally distributed residual assumptions. Overall, the model form is not adequate and there is room to improve the models for better performance. Peak Hour Peak Hour—Mean TTI peak houR—Mean tti—all data The RMSE values of the three data sets used to validate the peak hour mean TTI model are all close to 25%, implying that on average the predicted mean TTI has an error in the magnitude of approximately one-fourth of the mean TTI itself (Table C.14). The corresponding RMSE in L03, at 26.4%, is of the same magnitude. Note that when comparing the L03 RMSE value with the validation RMSE values, the research team needs to keep in mind that the definition of RMSE in L33 may not be the same at that used in the L03, because of the limited knowledge of how the regression mod- els were developed in L03. The residual analysis presents relatively satisfying results (Table C.15). The Student’s t-test for the zero residual mean and the Shapiro-Wilk normality test both yield p-values larger than 0.1, indicating that the research team does not have strong evidence to reject the null hypotheses that the residuals satisfy the zero residual mean and the normal distri- bution assumptions. The residual versus the predicted value plot (Figure C.19) generally presents a random pattern, although it may have a slight tendency of overestimation when the predicted value is large. The histogram (Figure C.20) and the normality plot (Figure C.21) both show that the resid- uals approximately display a normal distribution pattern. Overall, this data-rich model generally performs well given the above analysis. peak houR—Mean tti—califoRnia The validation using the CA data set yields similar results to that using the AllData set. The Shapiro-Wilk normality test Figure C.18. Residual normality plot of peak period—10th-percentile TTI—AllData. Figure C.17. Residual histogram of peak period— 10th-percentile TTI—AllData. Table C.14. RMSE of Peak Hour— Mean TTI RMSE All Data CA MN Mean TTI 25.45% 26.97% 24.68%

77 shows that the null hypothesis of normal distribution cannot be rejected with the threshold confidence level of 90%. The Student’s t-test rejects the null hypothesis of zero residual mean with a high confidence level, which corresponds to the fact that the 95% confidence limits for the mean of residual both fall on the negative side. The residual versus the predicted value plot presents no strong unusual pattern except for a potential slight tendency of overestimation trend as the pre- dicted value ln(mean TTI) increases, and that there are more negative residuals than positive ones. Overall, this validation result is also relatively satisfying. The associated results and plots for this category are included in the attachment. Figure C.21. Residual normality plot of peak hour—mean TTI—AllData. Figure C.20. Residual histogram of peak hour— mean TTI—AllData. Table C.15. Residual Analysis of Peak Hour—Mean TTI—AllData Table C.15.a. Basic Statistical Measures Location Variability Mean -0.030 Std deviation 0.2262 Median -0.025 Variance 0.0512 Minimum -0.547 Range 0.9459 Maximum 0.3987 Interquartile range 0.3138 Table C.15.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.030 -0.084 0.0227 Std deviation 0.2262 0.1944 0.2707 Variance 0.0512 0.0378 0.0733 Table C.15.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t -1.144 Pr > t 0.2565 Table C.15.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9763 Pr < W 0.1916 Figure C.19. Residual plot of peak hour—mean TTI— AllData.

78 peak houR—Mean tti—MinneSota The validation of the peak hour mean TTI model using the MN data set shows that the null hypothesis of the zero resid- ual mean cannot be rejected with a confidence level of 90%. The normality test also shows that the null hypothesis can- not be rejected with the preset threshold confidence level but can be rejected with a confidence level of 85%. The residual plot displays a random pattern except for the potential increasing trend toward the upper right corner. Generally, this model performs satisfactorily. The associated results and plots for this category are included in the attachment. Peak Hour—99th-Percentile TTI peak houR—99th-peRcentile tti—all data The RMSE values are all close to 50% in the peak hour 99th- percentile TTI model validation (Table C.16). Because the 99th-percentile TTI usually represents some extreme or usual value in the TTI distribution, it is expected to have a relatively large prediction error. The RMSE value in the L03 report is 41.3%, smaller than all the RMSE values produced in this validation. This may indicate that the validation set presents different characteristics than the training data set. The residual analysis using the AllData set is summarized by Table C.17 and Figures C.22 to C.24. The Student’s t-test shows no strong confidence to reject the zero residual mean hypothesis, while the normality test rejects the null hypoth- esis of normal distribution with a confidence level of 90%. The residual plot presents a problematic pattern; the model tends to underestimate the response variable when the pre- dicted value is small, and it tends to overestimate the response variable when the predicted value is large. Such a pattern may indicate that the samples located in the bottom left and the upper right corners are potential outliers. A closer look into the details of the validation data set reveals that the coefficient of the D/C predictor is 1.13062, dominating the prediction results. Those potential outlier samples are likely to have a different relationship between the D/C ratio and the response variable when compared to that in the training data samples used in L03. The model performs better when the predicted ln(99th-percentile TTI) is within (1, 1.5) where the residuals are within (-0.5, 0.5), or that the predicted 99th-percentile TTI is within (2.72, 4.48). Table C.16. RMSE of Peak Hour— 99th-Percentile TTI RMSE All Data CA MN Mean TTI 50.74% 52.78% 47.46% Table C.17. Residual Analysis of Peak Hour—99th-Percentile TTI—AllData Table C.17.a. Basic Statistical Measures Location Variability Mean -0.057 Std deviation 0.4092 Median -0.080 Variance 0.1675 Minimum -0.911 Range 2.3174 Maximum 1.4057 Interquartile range 0.3618 Table C.17.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.057 -0.153 0.0391 Std deviation 0.4092 0.3516 0.4897 Variance 0.1675 0.1236 0.2398 Table C.17.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t -1.183 Pr > t 0.2407 Table C.17.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9464 Pr < W 0.0040 Figure C.22. Residual plot of peak hour— 99th-percentile TTI—AllData.

79 peak houR—99th-peRcentile tti—califoRnia Validation using the CA data presents similar results to that using the AllData set. Again, the model performs best when the predicted ln(99th-percentile TTI) is within (1, 1.5). The Student’s t-test shows that the zero residual mean hypothesis cannot be rejected, while the Shapiro-Wilk normality test rejects the null hypothesis of normal distribution with a con- fidence level of 90%. The histogram and the normality plot manifest these tests results. All associated results and plots are included in the attachment. peak houR—99th-peRcentile tti—MinneSota The residual plot using the MN data set do not presents any strong unusual pattern except for the three underestimated samples located in the bottom left corner of the plot, which may be potential outliers. The Student’s t-test rejects the zero resid- ual mean with a confidence level of 90%, while the Shapiro- Wilk test cannot reject the null hypothesis of normal distribution with the preset confidence level. All associated results and plots are included in the attachment. Peak Hour—95th-Percentile TTI peak houR—95th-peRcentile tti—all data The RMSE values from the validation of the 95th-percentile TTI are summarized in Table C.18. The largest value still comes from the CA data set, which is 40.19%. The RMSE for the AllData set is 38.38%, which is almost equal to the RMSE of 38.3% for the peak hour 95th-percentile TTI model in the L03 report. From this validation it is apparent that neither the zero residual mean hypothesis nor the normal distribution hypothesis can be rejected with the preset threshold confi- dence level, indicating satisfaction to these two model assumptions (Table C.19). The histogram and the normality plot also present good shape (Figures C.26 and C.27). However, the plot of residual versus the predicted reveals a problematic pattern: the increasing trend of residual as the predicted value increases (Figure C.25). It shows that the model tends to under- estimate ln(95th-percentile TTI) when this predicted value is small while overestimate it when this predicted value is large. This pattern shows that although the model satisfies the zero residual mean and the normal distribution of residual assump- tions, the model form may still be inadequate and might be improved. peak houR—95th-peRcentile tti—califoRnia The validation using the CA data set yields a similar pattern to that using the AllData set. The research team cannot reject the zero residual mean hypothesis or the normal distribution hypothesis with the preset threshold confidence, indicating these two model assumptions are satisfied. However, the residual plot shows a nonrandom increasing pattern, which indicates that the model form may not be adequate to predict the validation data set. All associated results and plots are included in the attachment. Figure C.23. Residual histogram of peak hour— 99th-percentile TTI—AllData.` Figure C.24. Residual normality plot of peak hour—99th-percentile TTI—AllData. Table C.18. RMSE of Peak Hour— 95th-Percentile TTI RMSE All Data CA MN Mean TTI 38.38% 40.19% 37.27%

80 peak houR—95th-peRcentile tti—MinneSota The validation results using the MN data set show that the research team cannot reject the zero residual mean assump- tion or the normal distribution of residual assumption with a confidence level of 90%, as the p-values in the Student’s t-test and the Shapiro-Wilk test are both larger than 0.1. The histogram and the normality plot manifest these hypothesis- testing results. However, the plot of residual versus the pre- dicted value displays a possible increasing trend, implying that the model may be biased. All associated results and plots are included in the attachment. Figure C.27. Residual normality plot of peak hour—95th-percentile TTI—AllData. Figure C.26. Residual histogram of peak hour— 95th-percentile TTI—AllData. Table C.19. Residual Analysis of Peak Hour—95th-Percentile TTI—AllData Table C.19.a. Basic Statistical Measures Location Variability Mean -0.037 Std deviation 0.3250 Median 0.0129 Variance 0.1056 Minimum -0.757 Range 1.7162 Maximum 0.9590 Interquartile range 0.4056 Table C.19.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.037 -0.113 0.0397 Std deviation 0.3250 0.2792 0.3889 Variance 0.1056 0.0780 0.1512 Table C.19.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t -0.957 Pr > t 0.3419 Table C.19.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9850 Pr < W 0.5515 Figure C.25. Residual plot of peak hour— 95th-percentile TTI—AllData.

81 Peak Hour—80th-Percentile TTI peak houR—80th-peRcentile tti—all data The RMSE values for the validation of the 80th-percentile TTI model using the AllData, CA, and MN data sets are close to one another, at around 35% (Table C.20). The RMSE for the corresponding model in the L03 report is 34.1%, which is close to the validation RMSE values. Table C.21 and Figures C.28 to C.30 summarize the vali- dation results for the 80th-percentile TTI model using the AllData set. The Student’s t-test rejects the null hypothesis of zero residual mean with a confidence level of 90%, while the Shapiro-Wilk normality test yields a p-value of 0.2959, indi- cating that the null hypothesis of normal distribution cannot be rejected with the preset threshold confidence level. The plot of residual versus the predicted ln(80th-percentile TTI) implies that bias might exist shown as the tendency of under- estimating the response variable when the predicted value is small and overestimating it when the predicted value is large. peak houR—80th-peRcentile tti—califoRnia The validation of the 80th-percentile TTI model using the CA data set shows that the model does not perform well enough. The Student’s t-test and the Shapiro-Wilk normality test reject their null hypotheses with a confidence level of 90%, implying violation of the zero residual mean assumption and Table C.20. RMSE of Peak Hour— 80th-Percentile TTI RMSE All Data CA MN Mean TTI 35.13% 36.89% 34.06% Table C.21. Residual Analysis of Peak Hour—80th-Percentile TTI—AllData Table C.21.a. Basic Statistical Measures Location Variability Mean -0.116 Std deviation 0.2798 Median -0.143 Variance 0.0783 Minimum -0.738 Range 1.4482 Maximum 0.7099 Interquartile range 0.3335 Table C.21.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.116 -0.182 -0.050 Std deviation 0.2798 0.2403 0.3347 Variance 0.0783 0.0578 0.1120 Table C.21.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t -3.517 Pr > t 0.0008 Table C.21.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9797 Pr < W 0.2959 Figure C.29. Residual histogram of peak hour— 80th-percentile TTI—AllData. Figure C.28. Residual plot of peak hour— 80th-percentile TTI—AllData.

82 the normal distribution of residual assumption. The residual versus predicted value plot shows that the model tends to overestimate when the predicted value is large, and that there are more negative residuals than positive ones. All these obser- vations suggest that the model can be improved. All associated results and plots are included in the attachment. peak houR—80th-peRcentile tti—MinneSota The statistical analysis results summarized in the attachment show that the Student’s t-test rejects the null hypo thesis of zero residual mean with a confidence level of 90%, and the Shapiro-Wilk normality test rejects the null hypothesis of normal distribution with the same confidence level. However, the plot of residual versus the predicted value presents no strong nonrandom pattern, except for the three potential outliers located at the bottom left corner. Because the basic model assumptions are violated, the model performance may be improved. Note that the failure to pass the statistical tests may result from the existence of those potential outliers. Peak Hour—50th-Percentile TTI peak houR—50th-peRcentile tti—all data The RMSE values for the peak hour 50th-percentile TTI model validation are summarized in Table C.22. The largest RMSE value comes from the CA data set. This is the same as all previ- ous validation exercises presented in this appendix. The RMSE for the corresponding model in L03 is 28.3%, which is close to the RMSE for AllData set. The validation results of the peak hour 50th-percentile TTI model using the AllData set are summarized in Table C.23 and Figures C.31 to C.33. The Student’s t-test yields a p-value of 0.0115, showing strong evidence to reject the null hypoth- esis of zero residual mean. The Shapiro-Wilk test yields a p-value of 0.6028, indicating that the null hypothesis of nor- mal distribution of residuals cannot be rejected. The plot of residuals versus predicted values shows that the residual vari- ance is much larger when the ln(50th-percentile TTI) is around 0.4 than it is otherwise. The histogram also shows that the mean of the residual distribution is shifted to the left Figure C.30. Residual normality plot of peak hour—80th-percentile TTI—AllData. Table C.22. RMSE of Peak Hour— 50th-Percentile TTI RMSE All Data CA MN Mean TTI 28.85% 32.41% 24.22% Table C.23. Residual Analysis of Peak Hour—50th-Percentile TTI—AllData Table C.23.a. Basic Statistical Measures Location Variability Mean -0.075 Std deviation 0.2440 Median -0.066 Variance 0.0595 Minimum -0.624 Range 1.0277 Maximum 0.4035 Interquartile range 0.3580 Table C.23.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.075 -0.132 -0.017 Std deviation 0.2440 0.2096 0.2919 Variance 0.0595 0.0439 0.0852 Table C.23.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t -2.596 Pr > t 0.0115 Table C.23.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9859 Pr < W 0.6028

83 side of the zero reference line, which corresponds to the 95% limits for residual mean shown in Table C.23. peak houR—50th-peRcentile tti—califoRnia The validation using the CA data set yields similar results to that using the AllData set: the Student’s t-test rejects the null hypo- thesis of zero residual mean while the Shapiro-Wilk normality test cannot reject the null hypothesis of normal distribution. The histogram and the normality plot demonstrate these conclu- sions. The residual plot shows that there are many more nega- tive residual samples than positive residual samples, and the prediction variance is much larger when the predicted ln(50th- percentile TTI) is around 0.4 than otherwise. The 95% confi- dence limits of residual mean are both negative, manifesting the same fact that there are more negative residuals than posi- tive ones. All associated results and plots are included in the attachment. peak houR—50th-peRcentile tti—MinneSota The validation analysis using the MN data set presents satis- fying results. The Student’s t-test and the Shapiro-Wilk test show evidence that we cannot reject the zero residual mean hypothesis and the normal distribution hypothesis. The his- togram and normality plot demonstrate these conclusions. The plot of residuals versus predicted values does not display any strong unusual pattern, except for two potential outliers located in the upper right corner. Given the RMSE of 24.22%, which is smaller than the corresponding RMSE value in L03, the research team concludes that this model performs satis- factorily for the MN validation data set. All associated results and plots are included in the attachment. Peak Hour—10th-Percentile TTI peak houR—10th-peRcentile tti—all data The RMSE values in the validation of the 10th-percentile TTI model are summarized in Table C.24. The largest one comes Figure C.33. Residual normality plot of peak hour—50th-percentile TTI—AllData. Figure C.32. Residual histogram of peak hour— 50th-percentile TTI—AllData. Figure C.31. Residual plot of peak hour— 50th-percentile TTI—AllData. Table C.24. RMSE of Peak Hour— 10th-Percentile TTI RMSE All Data CA MN Mean TTI 18.50% 22.24% 12.14%

84 from the CA data set. The RMSE for this model in the L03 report is 15.2%, which is larger than the MN RMSE but smaller than the other two. Judging by the single criteria of RMSE this model performs the best for the MN data set and performs the poorest for the CA data set. The validation of the 10th-percentile TTI model using the AllData set shows that the zero residual mean hypothesis in the Student’s t-test cannot be rejected with a confidence level of 90%, while the normal distribution of residual hypothesis can be rejected at the preset threshold confidence level (Table C.25). The plot of residuals versus predicted values (Figure C.34) shows that the prediction variance is much larger when the predicted ln(10th-percentile TTI) falls within (0.1, 0.15) than otherwise. Those seven samples located below the -0.2 refer- ence line may be outliers that this model cannot predict well, and they may represent different relationships between the three independent variables and the dependent variable other than those represented in the data-rich model. The histogram (Figure C.35) shows that the distribution of residuals has a long left tail and that there are more positive than negative samples. The residual normality plot is shown in Figure C.36. peak houR—10th-peRcentile tti—califoRnia The validation using the CA data set shows that the zero residual mean hypothesis and the normal distribution hypothesis can be rejected with a confidence level of 90% in their respective statistical tests. The histogram and the nor- mality plot demonstrate these conclusions. The plot of resid- uals versus the predicted ln(10th-percentile TTI) has the same nonconstant variance and potential outlier problems as seen in the validation using the AllData set. All associated results and plots are included in the attachment. Table C.25. Residual Analysis of Peak Hour—10th-Percentile TTI—AllData Table C.25.a. Basic Statistical Measures Location Variability Mean -0.016 Std deviation 0.1702 Median 0.0392 Variance 0.0290 Minimum -0.640 Range 0.8136 Maximum 0.1739 Interquartile range 0.1643 Table C.25.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.016 -0.056 0.0241 Std deviation 0.1702 0.1462 0.2037 Variance 0.0290 0.0214 0.0415 Table C.25.c. Tests for Location: Mu0 Test Statistic p-Value Student’s t t -0.794 Pr > t 0.4297 Table C.25.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.7949 Pr < W <0.0001 Figure C.34. Residual plot of peak hour— 10th-percentile TTI—AllData. Figure C.35. Residual histogram of peak hour— 10th-percentile TTI—AllData.

85 peak houR—10th-peRcentile tti—MinneSota The validation using the MN data set shows that the zero resid- ual mean and the normal distribution of residuals assumptions are likely violated as the Student’s t-test and the Shapiro-Wilk normality test both reject their null hypotheses. The plot of residuals versus predicted values shows that most of the residu- als are positive, and there is one potential outlier located at the bottom right of the figure. The violation of assumptions and the unusual patterns shown in the residual plots indicate that the peak hour 10th-percentile TTI does not perform satisfacto- rily using the MN data set. All associated results and plots are included in the attachment. Conclusions of the Data-Rich Peak Hour Model Validation Based on the RMSE values, the data-rich peak hour models generally perform the best with the MN data set and the poor- est with the CA data set. The most satisfying validation results were seen in the validation of the 50th-percentile TTI model using the MN data set, where both the zero residual mean assumption and the normal distribution assumption passed their respective statistical tests, and the residual plot did not show any unusual patterns. It is apparent that violations of basic regression assump- tions were common in these validation results. The models may not be able to sufficiently describe the relationship between the independent variables and the dependent vari- ables as there are some nonrandom patterns in the residual plots. All of these findings indicate that the models can poten- tially be improved for better performance. Midday Models The midday models depend only on the critical D/C ratio, in the form of y e x= β Thus, the prediction accuracy relies on the correction of the exponential relationship between the D/C ratio and the mean TTI. Midday—Mean TTI Midday—Mean tti—all data The RMSE for the training data set is 7.5% in the L03 report (Table C.26). In the validation, the largest RMSE comes from the CA data set, which is almost the same as the L03 report. The smallest is 3.52%, which is almost half of the L03 report. Based on the single criteria of RMSE, this data-rich model performs satisfactorily for the validation data sets, and even better than its performance for the training data set. However, the small RMSE value may result from the characteristics of the data sets; it does not necessarily indicate satisfying performance. The validation of the midday mean TTI model using the AllData set shows that the Student’s t-test rejects the zero resid- ual mean hypothesis with a confidence level of 90%, and the Shapiro-Wilk normality test also rejects the null hypothesis of normal distribution (Table C.27). Thus, it is likely that the zero residual mean assumption and the assumption that the residuals follow a normal distribution are violated. The distorted shape in the normality plot (Figure C.37) and the histogram (Fig- ure C.38) demonstrate these hypothesis-testing conclusions. The plot of residuals versus predicted values (Figure C.39) pre- sents some problematic patterns: there are more positive residu- als than negative ones, and there are some potential outliers with large absolute residual values. These observations indicate that the model may not perform satisfactorily. Midday—Mean tti—califoRnia The validation of the midday mean TTI model using the CA data set presents similar results to that using the AllData set. The null hypothesis of zero residual mean can be rejected in Figure C.36. Residual normality plot of peak hour—10th-percentile TTI—AllData. Table C.26. RMSE of Midday—Mean TTI RMSE All Data CA MN Salt Lake City Mean TTI 6.24% 7.57% 4.07% 3.52%

86 the Student’s t-test with a confidence level of 90%. The null hypothesis of normal distribution can also be rejected with the threshold confidence level in the normality test. The plot of residuals versus predicted values shows that most samples have positive residuals, while there are also potential outliers with large negative residuals. All associated results and plots are included in the attachment. Midday—Mean tti—MinneSota The validation of the midday mean TTI model using the MN data set also shows similar pattern to that using the AllData set. The zero residual mean and the normal distribution of residual hypotheses can be rejected in the statistical tests with Figure C.39. Residual plot of midday—mean TTI— AllData. Figure C.38. Residual histogram of midday— mean TTI—AllData. Table C.27. Residual Analysis of Midday— Mean TTI—AllData Table C.27.a. Basic Statistical Measures Location Variability Mean 0.0199 Std deviation 0.0573 Median 0.0349 Variance 0.0033 Minimum -0.443 Range 0.5167 Maximum 0.0741 Interquartile range 0.0329 Table C.27.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0199 0.0126 0.0272 Std deviation 0.0573 0.0526 0.0630 Variance 0.0033 0.0028 0.0040 Table C.27.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 5.3837 Pr > t <0.0001 Table C.27.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.5446 Pr < W <0.0001 Figure C.37. Residual normality plot of midday— mean TTI—AllData.

87 a confidence level of 90%. The histogram and the normality plot show the deviation of the residual distribution from a normal distribution. The residual versus the predicted value plot displays that there are more positive residuals than nega- tive ones and that there is a potential outlier located at the bottom right of the figure. This indicates that the model may be problematic in predicting the mean TTI for MN data. All associated results and plots are included in the attachment. Midday—Mean tti—Salt lake city The validation results for the midday mean TTI model using the Salt Lake City data are summarized in the attachment. It is apparent that the zero residual mean hypothesis can be rejected with a confidence level of 90%, and the normal dis- tribution of residual hypothesis can also be rejected with the same threshold confidence level. The histogram and the nor- mality plot demonstrate that the residual distribution devi- ates from the normal distribution. The zero reference line in the plot of residual versus the predicted value acts like a sepa- rating line. It separates the residuals into two sets, indicating that the model may not perform satisfactorily. Midday—99th-Percentile TTI Midday—99th-peRcentile tti—all data The RMSE values for each validation data set are summarized in Table C.28. The corresponding RMSE value in the data- rich appendix is 33.4%, which is close to the RMSE values for the AllData set, the CA data set, and the Salt Lake City data set. The RMSE for MN data set is smaller. A closer look at the validation details provides further details on the model per- formance for each validation data set. The validation of the midday 99th-percentile TTI model using the AllData set shows that the zero residual mean assumption, the assumption that the residuals follow a nor- mal distribution and the constant residual variance assump- tions may be violated (Table C.29). The Student’s t-test rejects the null hypothesis of zero residual mean with a confidence level of 90%, and the Shapiro-Wilk normality test rejects the null hypothesis of normal distribution with the same thresh- old confidence level. The plot of residual versus the predicted value (Figure C.40) shows that the variance of residual tends Table C.28. RMSE of Midday— 99th-Percentile TTI RMSE All Data CA MN Salt Lake City Mean TTI 32.32% 34.95% 25.86% 34.01% Table C.29. Residual Analysis of Midday— 99th-Percentile TTI—AllData Table C.29.a. Basic Statistical Measures Location Variability Mean 0.1946 Std deviation 0.2019 Median 0.2451 Variance 0.0407 Minimum -0.535 Range 1.2456 Maximum 0.7110 Interquartile range 0.1892 Table C.29.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.1946 0.1689 0.2202 Std deviation 0.2019 0.1853 0.2217 Variance 0.0407 0.0343 0.0491 Table C.29.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 14.962 Pr > t <0.0001 Table C.29.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.8792 Pr < W <0.0001 Figure C.40. Residual plot of midday— 99th-percentile TTI—AllData.

88 to increase with the increase in predicted values. All these observations imply that the predictive capability of the model may be insufficient. The residual histogram and normality plot are shown in Figure C.41 and Figure C.42. Midday—99th-peRcentile tti—califoRnia Since the CA data set constitutes the largest portion of the AllData set, the validation of the midday 99th-percentile TTI model using the CA data set shows similar results to that using the AllData set. The zero residual mean hypothesis test and the normality hypothesis test both reject the null hypo theses, respectively, with a confidence level of 90%. The residual ver- sus the predicted value plot shows that a nonconstant resid- ual variance problem may exist. All associated results and plots are included in the attachment. Midday—99th-peRcentile tti—MinneSota In this validation the zero residual mean hypothesis and the normal distribution of residual hypothesis again fail to pass the statistical tests, respectively. The histogram and the nor- mality plot demonstrate these conclusions. The plot of resid- uals versus the predicted values shows two unusual patterns: more positive residuals and the potential outliers when the predicted value is around 0.45. Thus, the model may not per- form satisfactorily. All associated results and plots are included in the attachment. Midday—99th-peRcentile tti—Salt lake city The Salt Lake City data validation of the midday 99th-percentile TTI model shows that the zero residual mean hypothesis and the normal distribution of residual hypothesis can be rejected again with a confidence level of 90% in the respective statisti- cal tests. The plot of residuals versus the predicted values show that the model constantly overestimates ln(99th-percentile TTI), excluding one sample with a large negative residual located at the bottom of the figure. Midday—95th-Percentile TTI Midday—95th-peRcentile tti—all data The RMSE for this midday 95th-percentile TTI model for the training data set in the L03 report is 21.8%. The research team can see that the RMSE values for all the validation data sets are smaller than 21.8% (Table C.30). This could be a result of the varying characteristics of the data sets instead of improved performance. The research team needs to investi- gate the validation details to see if the model satisfies the basic regression model assumptions. The validation results for the midday 95th-percentile TTI model using the AllData set shows that the zero residual mean hypothesis can be rejected with a confidence level of 90% in the Student’s t-test, and that the normal distribution hypothesis can be rejected with the same threshold confi- dence level in the normality test (Table C.31). The histogram (Figure C.44) and the normality plot (Figure C.45) demon- strate the deviation of the residual distribution from a Figure C.41. Residual histogram of midday— 99th-percentile TTI—AllData. Figure C.42. Residual normality plot of midday— 99th-percentile TTI—AllData. Table C.30. RMSE of Midday— 95th-Percentile TTI RMSE All Data CA MN Salt Lake City Mean TTI 15.62% 17.29% 14.01% 12.55%

89 normal distribution. The plot of residual versus the predicted value (Figure C.44) presents an overestimation problem and the nonconstant residual variance problem. Generally this model does not perform satisfactorily. The plot of residual versus the predicted value is shown in Figure C.43. Midday—95th-Percentile tti—california Validation of the CA data set presents similar results as that of the validation using the AllData set. The zero residual mean assumption and the normal distribution assumption are likely to be violated since they fail to pass the Student’s Table C.31. Residual Analysis of Midday— 95th-Percentile TTI—AllData Table C.31.a. Basic Statistical Measures Location Variability Mean 0.0752 Std deviation 0.1244 Median 0.1084 Variance 0.0155 Minimum -0.662 Range 0.9157 Maximum 0.2537 Interquartile range 0.0821 Table C.31.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0752 0.0594 0.0910 Std deviation 0.1244 0.1142 0.1366 Variance 0.0155 0.0130 0.0187 Table C.31.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 9.3841 Pr > t <0.0001 Table C.31.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.6845 Pr < W <0.0001 Figure C.43. Residual plot of midday—95th-percentile TTI—AllData. Figure C.44. Residual histogram of midday— 95th-percentile TTI—AllData. Figure C.45. Residual normality plot of midday— 95th-percentile TTI—AllData.

90 t-test and the normality test. The residual plot shows that the model tends to overestimate the response variable. There are also some potential outliers located on the bottom side to the zero reference line when the ln(95th percentile TTI) is around 0.2. All associated results and plots are included in the attachment. Midday—95th-peRcentile tti—MinneSota The validation of the MN data set presents similar problems as that of the validation using the CA data set. The zero resid- ual mean assumption and the normal distribution assump- tion are likely to be violated as the Student’s t-test and the Shapiro-Wilk normality test reject the null hypotheses with a confidence level of 90%. The plot of residuals versus the pre- dicted values shows that the model tends to overestimate the response variable while some potential outliers with large negative residuals exist when the ln(95th-percentile TTI) is around 0.18. All associated results and plots are included in the attachment. Midday—95th-peRcentile tti—Salt lake city The validation of the midday 95th-percentile TTI model using the Salt Lake City data set shows that the zero residual mean assumption and the normal distribution assumption are unlikely to be satisfied. The plot of residuals versus the predicted values shows that the model consistently over- estimates the ln(95th-percentile TTI) except for one potential outlier with large negative residual located at the bottom of the figure, where ln(95th-percentile TTI) is around 0.14. All associated results and plots are included in the attachment. Midday—80th-Percentile TTI Midday—80th-peRcentile tti—all data The largest RMSE in the validation of the midday 80th- percentile TTI model comes from the CA data set at 10.86% (Table C.32). The smallest comes from the Salt Lake City data set, which is 3.6%. The RMSE for this model in the L03 report is 9.2%, which is smaller than the RMSE of the CA data set but larger than the RMSE values for the other three data sets. Based on the single criteria of RMSE, the research team may reach the conclusion that the model performs sat- isfactorily for the validation data sets. However, as discussed previously, the RMSE might easily lead to misleading con- clusions and further investigation of the validation details is required to evaluate whether the model assumptions are satisfied. The validation results show that the Student’s t-test rejects the null hypothesis of zero residual mean with a confidence level of 90%, and the Shapiro-Wilk normality test rejects the null hypothesis with the same threshold confidence level (Table C.33). The plot of residuals versus the predicted values (Figure C.46) manifests a problematic pattern in that the model tends to overestimate the response variable, resulting in more positive residuals. Additionally, potential outliers characterized by large negative residuals can be noted from the plot. The residual histogram and normality plots are shown in Figures C.47 and C.48. Midday—80th-peRcentile tti—califoRnia In this validation the Student’s t-test cannot reject the null hypothesis of zero residual mean with a confidence level of 90%, while the Shapiro-Wilk normality test rejects the null hypothesis of normal distribution with the threshold confi- dence level of 90%. The residual plot for the CA data set Table C.32. RMSE of Midday— 80th-Percentile TTI RMSE All Data CA MN Salt Lake City Mean TTI 8.99% 10.86% 6.61% 3.60% Table C.33. Residual Analysis of Midday— 80th-Percentile TTI—AllData Table C.33.a. Basic Statistical Measures Location Variability Mean 0.0116 Std deviation 0.0854 Median 0.0360 Variance 0.0073 Minimum -0.627 Range 0.7005 Maximum 0.0737 Interquartile range 0.0462 Table C.33.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0116 0.0008 0.0224 Std deviation 0.0854 0.0784 0.0938 Variance 0.0073 0.0062 0.0088 Table C.33.c. Tests for Location: Mu0 Test Statistic p-Value Student’s t t 2.1077 Pr > t 0.0361 Table C.33.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.5139 Pr < W <0.0001

91 shows similar problems noted in the validation results of the AllData set (i.e., more positive residuals and potential out- liers). All associated results and plots are included in the attachment. Midday—80th-peRcentile tti—MinneSota The validation of the midday 80th-percentile TTI model using the MN data set shows that the zero residual mean assumption and the normal distribution assumption are likely to be violated, as they fail to pass the respective statis- tical tests. The residuals versus the predicted values plot shows that the model tends to overestimate the response variable as there are more positive residuals than negative ones. There are also two potential outliers with large nega- tive residuals. All associated results and plots are included in the attachment. Midday—80th-peRcentile tti—Salt lake city In this validation the null hypothesis of zero residual mean is rejected by the Student’s t-test, and the null hypothesis of normal distribution of residual is also rejected in the normal- ity test with a confidence level of 90%. The plot of residuals versus the predicted values has an unusual pattern where the upper bound of residuals has a linearly increasing pattern. Such nonrandomness indicates that the model may not be able to adequately describe the relationship between the inde- pendent variables and the dependent variable. All associated results and plots are included in the attachment. Midday—50th-Percentile TTI Midday—50th-peRcentile tti—all data The RMSE for the midday 50th-percentile TTI model is 21.8% in the L03 report. This number is suspicious since it is larger than the RMSE for the midday 80th-percentile TTI model (9.2%) and equals to the RMSE for the midday 95th- percentile TTI model (21.8%). Since the pth-percentile TTI value generally decreases as the percentage p decreases, the large RMSE value for midday 50th-percentile TTI is un- expected. The RMSE values for the same model using the validation data sets are all smaller than 7% (Table C.34). If the Figure C.46. Residual plot of midday—80th-percentile TTI—AllData. Figure C.47. Residual histogram of midday— 80th-percentile TTI—AllData. Figure C.48. Residual normality plot of midday—80th-percentile TTI—AllData. Table C.34. RMSE of Midday— 50th-Percentile TTI RMSE All Data CA MN Salt Lake City Mean TTI 5.43% 6.93% 2.09% 2.08%

92 training data set used in L03 is similar to the validation data sets, then the RMSE value in the L03 report should not have been this large, which indicates that it could be erroneous. For the validation data sets the research team can see that the largest value still comes from the CA data set while the small- est ones come from the MN and the Salt Lake City sets. It should be noted again that such small RMSE values might not necessarily indicate good model performance. In this validation the zero residual mean hypothesis cannot be rejected with a confidence level of 90%, while the normal distribution hypothesis can be rejected with the same thresh- old confidence level (Table C.35). The histogram (Fig- ure C.50) and the normality plot (Figure C.51) demonstrate the fact that the residual distribution does not closely follow a normal distribution. Note that the zero residual mean hypothesis passing the Student’s t-test successfully could be because of the existence of potential outliers with large nega- tive residuals, which can be identified in the plot of residuals versus the predicted values (Figure C.49) and the histogram. Table C.35. Residual Analysis of Midday— 50th-Percentile TTI—AllData Table C.35.a. Basic Statistical Measures Location Variability Mean 0.0023 Std deviation 0.0529 Median 0.0184 Variance 0.0028 Minimum -0.471 Range 0.5043 Maximum 0.0333 Interquartile range 0.0226 Table C.35.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0023 -0.004 0.0090 Std deviation 0.0529 0.0486 0.0581 Variance 0.0028 0.0024 0.0034 Table C.35.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 0.6760 Pr > t 0.4997 Table C.35.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.4486 Pr < W <0.0001 Figure C.49. Residual plot of midday—50th-percentile TTI—AllData. Figure C.50. Residual histogram of midday— 50th-percentile TTI—AllData. From the plot of residuals versus the predicted values it is also evident that the upper bound of residuals increases linearly, which should not be the case if the regression model was behaving as expected. Midday—50th-peRcentile tti—califoRnia The validation of the midday 50th-percentile TTI model using the CA data set presents similar results to that using the AllData set. The null hypothesis of normal distribution fails to pass the Shapiro-Wilk test with a 90% confidence level, while the null hypothesis of zero residual mean cannot be rejected as the Student’s t-test yields a p-value larger than 0.1. The residual plots again present the previously identified problems: nonrandom residual pattern and potential outliers. The above discussion indicates that the model may not

93 perform satisfactorily. All associated results and plots are included in the attachment. Midday—50th-peRcentile tti—MinneSota In this validation the Student’s t-test and the Shapiro-Wilk test both reject their null hypothesis with a confidence level of 90%, indicating that the zero residual mean assumption and the normal distribution of residuals assumption are likely to be vio- lated. The plot of residuals versus the predicted values presents a nonrandom pattern of residuals, with the residuals increasing almost linearly on the upper bound. A well-behaving regression model should produce residuals randomly distributed along the zero reference line without any patterns. All associated results and plots are included in the attachment. Midday—50th-peRcentile tti—Salt lake city The Salt Lake City validation results show that the zero resid- ual mean hypothesis and the normal distribution hypothesis can be rejected with a confidence level of 90%. The residuals versus the predicted values plot indicates some nonrandom patterns on the upper bound of residuals. Hence, although the RMSE value for this data set is only 2.08%, the model does not perform satisfactorily. All associated results and plots are included in the attachment. Midday—10th-Percentile TTI Midday—10th-peRcentile tti—all data The RMSE value of the midday 10th-percentile TTI is 5.1% for the training data set in the L03 report. This is larger than all RMSE values for the validation data sets. The largest vali- dation RMSE still comes from the CA data set, while the smallest comes from the MN data set (Table C.36). In this validation the Student’s t-test rejects the null hypothesis of zero residual mean with a confidence level of 90%, and the normality test also rejects the null hypothesis of normal distribution (Table C.37). The residual plots (Fig- ure C.52) also reveal some problematic patterns. The histo- gram (Figure C.53) and the normality plot (Figure C.54) demonstrate the hypothesis-testing results, while the plot of residuals versus the predicted values shows an increasing upper bound of residuals as well as large negative residuals, indicating inadequacy of the model. Figure C.51. Residual normality plot of midday— 50th-percentile TTI—AllData. Table C.36. RMSE of Midday— 10th-Percentile TTI RMSE All Data CA MN Salt Lake City Mean TTI 1.81% 2.20% 0.80% 1.33% Table C.37. Residual Analysis of Midday— 10th-Percentile TTI—AllData Table C.37.a. Basic Statistical Measures Location Variability Mean 0.0021 Std deviation 0.0178 Median 0.0071 Variance 0.0003 Minimum -0.138 Range 0.1547 Maximum 0.0168 Interquartile range 0.0038 Table C.37.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0021 -17E-5 0.0044 Std deviation 0.0178 0.0164 0.0196 Variance 0.0003 0.0003 0.0004 Table C.37.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 1.8177 Pr > t 0.0704 Table C.37.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.4378 Pr < W <0.0001

94 Midday—10th-peRcentile tti—califoRnia The validation of the midday 10th-percentile TTI model using the CA data set presents similar results to that using the AllData set, except that zero residual mean hypothesis has passed the Student’s t-test. However, this success of passing the hypothesis test could be attributed to the exis- tence of large negative residuals, which can be seen from the residuals versus the predicted values plot. The normality test again rejects its null hypothesis. The upper bound of residuals in the plot of residuals versus the predicted values also presents a linearly increasing trend. These observations suggest that the model does not perform the prediction sat- isfactorily. All associated results and plots are included in the attachment. Midday—10th-peRcentile tti—MinneSota In this validation neither the null hypothesis of zero residual mean nor the null hypothesis of normal distribution pass their respective statistical tests, indicating the violation of basic regression model assumptions. The plot of residual versus the predicted value shows an unusual pattern. Most of the residu- als form an increasing linear shape above the zero reference line. However, the vertical axis value tells that these residuals are all approximately equal to 0.01, which is also revealed from the almost zero residual variance in the table included in the attachment. A well-performing model should have randomly distributed residuals rather than what is seen here. Midday—10th-peRcentile tti—Salt lake city In this validation the Student’s t-test yields a p-value of 0.5199, indicating that the null hypothesis of zero residual mean with a confidence level of 90% cannot be rejected. The Shapiro-Wilk normality test rejects the null hypothesis of normal distribu- tion. The plot of residual versus the predicted value presents a nonrandom pattern above the zero reference line, and there are some points below the zero reference line with relatively large negative residuals that may be potential outliers. All associated results and plots are included in the attachment. Conclusions of the Data-Rich Midday Model Validation The validation of the midday models does not present satisfying results. Violations of the basic regression model assumptions are consistently seen in the validation results. However, the most Figure C.53. Residual histogram of midday— 10th-percentile TTI—AllData. Figure C.52. Residual plot of midday—10th-percentile TTI—AllData. Figure C.54. Residual normality plot of midday— 10th-percentile TTI—AllData.

95 problematic issue is the nonrandom patterns shown in the residual plots, which indicate that the model is not adequate. It may be mentioned again that the comparison of the L03 RMSE values and the current RMSE values is based on the assumption that they are defined in the same way. However, because of the limited knowledge on how the L03 models were built, the comparison of the current validation results with the L03 results cannot be ascertained completely. In addition, the RMSE of 21.8% for the midday 50th-percentile TTI provided in the L03 report seems to be erroneous. Weekday Models The weekly models either have two independent variables (the average D/C ratio and the ILHL), or only one (the aver- age D/C ratio), so the weather condition does not influence the prediction of the weekday travel time reliability metrics in the data-rich weekday models. The data-rich weekday 50th-percentile TTI model and the data-rich weekday 10th- percentile TTI model determine the response value using the average D/C ratio alone. Weekday—Mean TTI Weekday—Mean tti—all data The RMSE value for the data-rich weekday mean TTI model is 29.3% in the L03 report, which is only smaller than RMSE for the MN validation data set, as illustrated in Table C.38. The RMSE for the AllData set is 19.74, which is about 10% less. The smallest is from the Salt Lake City data set, which is only 5.95%. Since the RMSE alone cannot lead to confident conclusions, further investigation on the validation details is required. The validation of the weekday mean TTI model using the AllData set shows that the zero residual mean hypothesis can be rejected with a 90% confidence level in the Student’s t-test (Table C.39). The null hypothesis of normal distribution can be rejected with the same threshold confidence level in the Shapiro-Wilk normality test. The histogram (Figure C.56) and the normality plot (Figure C.57) also illustrate that the residual distribution is distorted and deviates from a normal distribu- tion. The plot of residuals versus the predicted values (Fig- ure C.55) shows a strong pattern, with the residual value increasing with the predicted value. Also, the model tends to overestimate the response. Table C.38. RMSE of Weekday—Mean TTI RMSE All Data CA MN Salt Lake City Mean TTI 19.74% 12.81% 35.99% 5.95% Table C.39. Residual Analysis of Weekday—Mean TTI—AllData Table C.39.a. Basic Statistical Measures Location Variability Mean 0.1212 Std deviation 0.1336 Median 0.1072 Variance 0.0178 Minimum -0.234 Range 1.3799 Maximum 1.1458 Interquartile range 0.0887 Table C.39.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.1212 0.1043 0.1381 Std deviation 0.1336 0.1226 0.1466 Variance 0.0178 0.0150 0.0215 Table C.39.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 14.150 Pr > t <0.0001 Table C.39.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.6754 Pr < W <0.0001 Figure C.55. Residual plot of weekday—mean TTI— AllData.

96 Weekday—Mean tti—califoRnia In the validation of the weekday mean TTI model using the CA data set, the null hypothesis of zero residual mean can be rejected with a confidence level of 90%, and the null hypothesis of normal distribution can be rejected with the same threshold confidence level. The plot of residuals ver- sus the predicted values shows problematic patterns, with the residuals increasing as the predicted values increase and that the model tending to overestimate the response vari- able. All associated results and plots are included in the attachment. Weekday—Mean tti—MinneSota In this validation both the Student’s t-test and the Shapiro- Wilk normality test reject the null hypotheses. The most sig- nificant problem is the nonrandom pattern shown in the residual versus the predicted value plot. The model is evi- dently biased toward the positive side, as all except one of the residual samples are above the zero reference line. The residu- als also increase with the predicted value, which should not happen if the model is well behaving. All associated results and plots are included in the attachment. Weekday—Mean tti—Salt lake city This validation shows that the model does not perform satis- factorily. The zero mean hypothesis and the normality hy pothesis are rejected in the statistical tests, but the most important unusual pattern is the nonrandomness shown in the plot of residuals versus the predicted values. The increas- ing trend and the overestimation tendency can be observed, which indicates that the model form may not be adequate. All associated results and plots are included in the attachment. Weekday—99th-Percentile TTI Weekday—99th-peRcentile tti—all data The RMSE value is 38.9% in the L03 report for the data- rich weekday 99th-percentile model. The RMSE for the AllData set is nearly twice as high, at 72.91%. The RMSE for the MN data set is the largest at 141.72%, which is highly influenced by the extreme values in the prediction. More details can be found in the validation analysis of the MN data set (Table C.40). Validating the weekday 99th-percentile TTI model using the AllData set shows that the model has the follow prob- lematic issues (Table C.41). The zero residual mean Stu- dent’s t-test yields a p-value less than 0.001, showing strong evidence that the null hypothesis can be rejected. The 95% confidence interval for the residual mean is [0.2732, 0.3842], implying that the model is biased toward the posi- tive side. The normality test also rejects the null hypothesis. The plot of residuals versus the predicted values (Fig- ure C.58) shows an overestimation tendency caused by more positive residual samples and an increasing trend. The residual histogram and normality plots are shown in Figures C.59 and C.60. Figure C.56. Residual histogram of weekday—mean TTI—AllData. Figure C.57. Residual normality plot of weekday—mean TTI—AllData. Table C.40. RMSE of Weekday— 99th-Percentile TTI RMSE All Data CA MN Salt Lake City Mean TTI 72.91% 50.04% 141.72% 30.87%

97 Weekday—99th-peRcentile tti—califoRnia In this validation both the 95% confidence limits for the zero residual mean are positive, which accords with the Student’s t-test results. The normality test also rejects the null hypoth- esis with a confidence level of 90%. The residual versus the predicted value plot manifests the overestimation tendency and the potential outliers. It should also be noted that the variance of residuals is relatively large, compared to the scale of the predicted value. These problems may render the model not reliable to predict unseen samples. All associated results and plots are included in the attachment. Figure C.59. Residual histogram of midday— 99th-percentile TTI—AllData. Figure C.60. Residual normality plot of midday— 99th-percentile TTI—AllData. Table C.41. Residual Analysis of Weekday—99th-Percentile TTI—AllData Table C.41.a. Basic Statistical Measures Location Variability Mean 0.3287 Std deviation 0.4389 Median 0.2902 Variance 0.1926 Minimum -0.480 Range 4.0280 Maximum 3.5480 Interquartile range 0.3872 Table C.41.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.3287 0.2732 0.3842 Std deviation 0.4389 0.4030 0.4818 Variance 0.1926 0.1624 0.2322 Table C.41.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 11.674 Pr > t <0.0001 Table C.41.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.7270 Pr < W <0.0001 Figure C.58. Residual plot of midday—99th-percentile TTI—AllData.

98 Weekday—99th-peRcentile tti—MinneSota In this validation the zero residual mean assumption and the normal distribution assumption seem to be violated as the Student’s t-test and the Shapiro-Wilk normality test reject their null hypotheses. The residual versus the predicted value plot renders unusual patterns, with the residuals increasing with the predicted values. There are also more positive resid- uals than negative ones. All associated results and plots are included in the attachment. Weekday—99th-peRcentile tti—Salt lake city In this validation the Student’s t-test and the normality test show strong evidence to reject the null hypotheses. The plot of residuals versus the predicted values generally shows a random pattern, except for there being more samples located in the upper left corner than anywhere else. In addition, the model still tends to yield more positive residuals than negative ones. All associated results and plots are included in the attachment. Weekday—95th-Percentile TTI Weekday—95th-peRcentile tti—all data The RMSE for the data-rich weekday 95th-percentile TTI is 31.8% for the L03 report, which is only larger than the RMSE value for the Salt Lake City data set. The MN RMSE is again influenced by the extreme values in the data set. The AllData RMSE is 83.32%, which is also influenced by the extreme values (Table C.42). The validation results show that null hypothesis of zero residual mean can be rejected with a confidence level of 90%, as the p-value is less than 0.0001, and the null hypothesis of normal distribution can also be rejected with the same threshold confidence level (Table C.43). The plot of residuals versus the predicted values (Figure C.61) shows similar prob- lems to those seen in the validation of the 99th-percentile TTI model. The increasing trend and the unbalanced pattern indicate that the model may be biased and that there may be an influential variable that is not included in the model. Note that the variance of residuals is relatively large given the scale of the predicted value. The residual histogram and normality plots are shown in Figures C.62 and C.63. Weekday—95th-peRcentile tti—califoRnia In this validation the null hypotheses in the Student’s t-test and the Shapiro-Wilk normality test are rejected. The plot of Table C.42. RMSE of Weekday— 95th-Percentile TTI RMSE All Data CA MN Salt Lake City Mean TTI 83.82% 40.46% 197.82% 22.85% Table C.43. Residual Analysis of Weekday—95th-Percentile TTI—AllData Table C.43.a. Basic Statistical Measures Location Variability Mean 0.3416 Std deviation 0.5050 Median 0.2780 Variance 0.2550 Minimum -0.406 Range 4.7819 Maximum 4.3758 Interquartile range 0.2729 Table C.43.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.3416 0.2778 0.4054 Std deviation 0.5050 0.4637 0.5544 Variance 0.2550 0.2150 0.3073 Table C.43.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 10.545 Pr > t <0.0001 Table C.43.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.6022 Pr < W <0.0001 Figure C.61. Residual plot of midday—95th-percentile TTI—AllData.

99 residuals versus the predicted values shows that the variation of residuals is relatively large given the scale of the predicted value. The increasing tendency and the fact that there are more positive residuals than negative ones indicate that the model may be biased and unreliable. All associated results and plots are included in the attachment. Weekday—95th-peRcentile tti—MinneSota The validation analysis of this data set is highly influenced by the potential outliers. The associated extremely large residuals make the RMSE value unusually large. The mean of residuals is also biased toward these extreme values. The statistical tests yield nearly zero p-values indicating strong evidence that the null hypotheses can be rejected with a confidence level of 90%. The plot of residuals versus the predicted values indicates potential outliers as well as other unusual patterns. There are more positive residuals than negative ones, and the residuals do not render a random pattern but rather an increasing trend. These unusual pat- terns demonstrate that the model may not be reliable for predicting unseen samples. All associated results and plots are included in the attachment. Weekday—95th-peRcentile tti—Salt lake city In this validation the Student’s t-test rejects the zero residual mean hypothesis with a confidence level of 90%, and the nor- mality test rejects the normal distribution hypothesis with the same threshold confidence level. The residual versus the predicted value plot display a generally random pattern except that there is a cluster of residuals located in the upper left of the plot that plays an influential role in the overall posi- tive residual mean. All associated results and plots are included in the attachment. Weekday—80th-Percentile TTI Weekday—80th-peRcentile tti—all data The RMSE value for the data-rich weekday 80th-percentile TTI model is 14.7% in the L03 report, which is almost the same as the RMSE for the CA validation data set. The RMSE for the Salt Lake City data set is much smaller, which may due to the small scale and variance of the validation data. The RMSE for the MN data set is the largest because of the extreme values (Table C.44). In the validation of the 80th-percentile TTI using the AllData set, the Student’ t-test yields a p-value less than 0.0001, indicating that the null hypothesis of zero residual mean can be rejected (Table C.45). The normality test also indi- cates strong evidence to reject the null hypothesis of nor- mal distribution. The plot of residuals versus the predicted values (Figure C.64) is still problematic as most of the resid- ual samples are above the zero reference line, and the increas- ing trend is obvious. Given these observations, the model does not perform satisfactorily for predicting the AllData set. The residual histogram and normality plots are shown in Figures C.65 and C.66. Figure C.62. Residual histogram of midday— 95th-percentile TTI—AllData. Figure C.63. Residual normality plot of midday— 95th-percentile TTI—AllData. Table C.44. RMSE of Weekday— 80th-Percentile TTI RMSE All Data CA MN Salt Lake City Mean TTI 29.28% 14.84% 59.43% 5.75%

100 Weekday—80th-peRcentile tti—califoRnia In this validation the Student’s t-test and the normality test reject their null hypotheses, indicating violation of the zero residual mean assumption and the normal distribution assump- tion. The plot of residuals versus predicted values reveals a non- constant residual variance problem; additionally, the residuals are unbalanced with more positive residuals than negative. All associated results and plots are included in the attachment. Weekday—80th-peRcentile tti—MinneSota The MN validation problems are similar to those of the vali- dation of the AllData set. The zero residual mean hypothesis Figure C.65. Residual histogram of midday— 80th-percentile TTI—AllData. Table C.45. Residual Analysis of Weekday—80th-Percentile TTI—AllData Table C.45.a. Basic Statistical Measures Location Variability Mean 0.1186 Std deviation 0.2282 Median 0.0958 Variance 0.0521 Minimum -0.387 Range 2.2166 Maximum 1.8296 Interquartile range 0.1441 Table C.45.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.1186 0.0897 0.1474 Std deviation 0.2282 0.2096 0.2506 Variance 0.0521 0.0439 0.0628 Table C.45.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 8.0997 Pr > t <0.0001 Table C.45.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.6720 Pr < W <0.0001 Figure C.64. Residual plot of midday—80th-percentile TTI—AllData. Figure C.66. Residual normality plot of midday— 80th-percentile TTI—AllData.

101 and the normal distribution null hypotheses are rejected. The plot of residuals versus the predicted values shows an obvious increasing trend with mostly positive residuals. The range of the residuals is relatively large compared with the range of the predicted values. These observations indicate that the model does not perform satisfactorily. All associated results and plots are included in the attachment. Weekday—80th-peRcentile tti—Salt lake city In this validation the predicted value is within a small range, from around 0.04 to around 0.16, which should be kept in mind when evaluating the small RMSE and the small residual values. The Student’s t-test rejects the null hypothesis of zero residual mean with a confidence level of 90%, and the nor- mality test rejects the null hypothesis of normal distribution with the same threshold level of confidence. The residual plot shows an increasing trend of residuals. Generally the model does not perform satisfactorily enough. All associated results and plots are included in the attachment. Weekday—50th-Percentile TTI Weekday—50th-peRcentile tti—all data The data-rich weekday 50th-percentile TTI model does not include the ILHL as an independent variable, thus the extreme values in the ILHL data in MN does not influence the predic- tion of the weekday 50th-percentile TTI. The MN RMSE value is the smallest, at only 1.71%. The largest RMSE value comes from the CA data set, which is 5.92% (Table C.46). The RMSE in the L03 report for the same model is 4.7%. The residual plot indicates the presence of potential outli- ers with relatively large residuals below the zero reference line, showing a potential problem with increasing residual variance (Table C.47). Also, the residuals above the zero refer- ence line exhibit a slightly increasing linear trend. Given a maximum predicted value of 0.03, the variance of the residu- als below the zero reference line seems unusual. There is strong evidence shown in the Student’s t-test that the zero residual mean hypothesis cannot be rejected as the p-value is 0.8746. The research team noted that the residual mean is negatively close to zero, but the histogram (Figure C.68) and the plot of residuals versus the predicted values (Figure C.67) indicate that there are more positive residual samples than negative ones, which implies that the large negative residuals may have influenced the Student’s t-test results. The normality Table C.46. RMSE of Weekday— 50th-Percentile TTI RMSE All Data CA MN Salt Lake City Mean TTI 4.68% 5.92% 1.71% 2.16% Table C.47. Residual Analysis of Weekday—50th-Percentile TTI—AllData Table C.47.a. Basic Statistical Measures Location Variability Mean -46E-5 Std deviation 0.0458 Median 0.0146 Variance 0.0021 Minimum -0.399 Range 0.4271 Maximum 0.0282 Interquartile range 0.0197 Table C.47.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -46E-5 -0.006 0.0053 Std deviation 0.0458 0.0421 0.0503 Variance 0.0021 0.0018 0.0025 Table C.47.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t -0.158 Pr > t 0.8746 Table C.47.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.4735 Pr < W <0.0001 Figure C.67. Residual plot of midday—50th-percentile TTI—AllData.

102 test rejects the normal distribution assumption. The residual normality plot is shown in Figure C.69. Weekday—50th-peRcentile tti—califoRnia This validation presents similar results to the validation of the same model using the AllData set. The Student’s t-test yields a p-value large enough that the null hypothesis cannot be rejected with a confidence level of 90%. The normality test indicates that the null hypothesis of normal distribution can be rejected with the threshold confidence level. The plot of residuals versus the predicted values has some potential outliers below the zero reference line. All associated results and plots are included in the attachment. Weekday—50th-peRcentile tti—MinneSota In this validation the null hypothesis of zero residual mean and the null hypothesis of normal distribution are rejected by the statistical tests. The plot of residuals versus the predicted values shows that there are more positive residuals than nega- tive ones and that the upper bound of residuals shows a lin- early increasing trend. The residual scale is much smaller than that for the CA data set. All associated results and plots are included in the attachment. Weekday—50th-peRcentile tti—Salt lake city In this validation the standard deviation of the residual is 0.0215. Although seeming small, it is as large as the scale of the predicted values. The Student’s t-test cannot reject the null hypothesis of zero residual mean, as the p-value is larger than 10%. The normality test rejects the null hypothesis of normal distribution, as the p-value is less than 0.001. The plot of residuals versus the predicted values does not show satisfy- ing results as the upper bound of residuals follows a linear trend. The residual range is large compared with the range of the predicted value, affirming that the corresponding stan- dard deviation is relatively large. All associated results and plots are included in the attachment. Weekday—10th-Percentile TTI Weekday—10th-peRcentile tti—all data The RMSE value for the data-rich weekday 10th-percentile TTI model is 2.0% in the L03 report. The largest RMSE comes from the Salt Lake City data set. However, all the RMSE values in the validation investigation are smaller than the RMSE in the L03 report (Table C.48). In this validation the residual plots reveal several concerns (Table C.49). The histogram and the normality test results indicate violation of the normal distribution assumption. The plot of residuals versus the predicted values shows that most of the residuals are slightly larger than zero, and the upper bound shows a linear trend with slightly increasing val- ues (Figure C.70). There are also many samples with large negative residuals located below the zero reference line, com- pared with the small scale of predicted values. The Student’s t-test rejects the null hypothesis of zero residual mean. The Figure C.68. Residual histogram of midday— 50th-percentile TTI—AllData. Figure C.69. Residual normality plot of midday— 50th-percentile TTI—AllData. Table C.48. RMSE of Weekday— 10th-Percentile TTI RMSE All Data CA MN Salt Lake City Mean TTI 0.81% 0.74% 0.48% 1.30%

103 residual histogram and normality plots are shown in Fig- ures C.71 and C.72. Weekday—10th-peRcentile tti—califoRnia In this validation the Student’s t-test rejects the null hypoth- esis of zero residual mean with a confidence level of 90%, and the normality test rejects the null hypothesis of normal dis- tribution with the same threshold confidence level. The plot of residuals versus the predicted values illustrates similar pat- terns as observed from the validation of the AllData set. All associated results and plots are included in the attachment. Table C.49. Residual Analysis of Weekday—10th-Percentile TTI—AllData Table C.49.a. Basic Statistical Measures Location Variability Mean 0.0014 Std deviation 0.0080 Median 0.0040 Variance 0.0001 Minimum -0.062 Range 0.0685 Maximum 0.0066 Interquartile range 0.0017 Table C.49.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0014 0.0004 0.0024 Std deviation 0.0080 0.0073 0.0088 Variance 0.0001 54E-6 0.0001 Table C.49.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 2.6645 Pr > t 0.0082 Table C.49.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.5123 Pr < W <0.0001 Figure C.70. Residual plot of midday—10th-percentile TTI—AllData. Figure C.71. Residual histogram of midday— 10th-percentile TTI—AllData. Figure C.72. Residual normality plot of midday— 10th-percentile TTI—AllData.

104 Weekday—10th-peRcentile tti—MinneSota In this validation the null hypothesis of zero residual mean and the null hypothesis of normal distribution are rejected with a confidence level of 90%. The plot of residuals versus the predicted values shows an almost linearly increasing trend with the exception of three samples located in the lower right corner of the plot. It does not exhibit the random pat- tern expected from a good regression model. This indicates that the model may have missed some important indepen- dent variables. All associated results and plots are included in the attachment. Weekday—10th-peRcentile tti—Salt lake city In this validation the Student’s t-test shows that the zero residual mean hypothesis can be rejected with a confidence level of 90%, and the normality test also shows that the null hypothesis of normal distribution can be rejected with the threshold confidence level. The plot of residual versus the predicted value presents a nonrandom pattern with large negative residuals below the zero reference line, as the pre- dicted response is within a smaller range. All associated results and plots are included in the attachment. Conclusions of the Data-Rich Weekday Model Validation Most of the validations show an increasing trend of residu- als in the residual versus predicted value plot. These non- random patterns indicate that the assumption of the relationship between the dependent and independent vari- ables is questionable, regardless of the size of the RMSE val- ues. It is important to note that the RMSE criterion has inherited drawbacks and that the assumption of the rela- tionship between variables described by the adopted model is critical. The MN data set has several potential outliers caused pri- marily by extremely large ILHL values that yield large residu- als. These unusual residuals influence the statistical test results and the RMSE values. Overall, the data-rich weekday models do not perform satisfactorily. Conclusions The validation of the L03 data-rich models was performed on three regional data sets: California, Minnesota, and Salt Lake City, along with the combined overall data set (AllData). The main conclusion is that the average prediction errors (mea- sured by the RMSE) for each model are not acceptable across many of the regions. The RMSEs for each model and data set are presented in Table C.50. From a regional perspective, for all time slices except the weekday time period, the RMSEs are the highest when the models are applied to the California data set. During the weekday time period, the RMSEs are the highest when the models are applied to the Minnesota data set, and the lowest when applied to the Salt Lake City data set. The high RMSEs for the weekday time period in the Minnesota data set are caused by several potential outliers influenced by very high ILHL values. When the RMSEs are interpreted by the predicted measure, it is evident that, across all of the time periods, the highest RMSEs occur for the prediction of the 99th-percentile TTI. The RMSEs tend to decrease as the predicted TTI measure lowers (i.e., the RMSEs for the 50th- percentile models are lower than for the 80th-percentile mod- els, which are lower than for the 95th-percentile models, etc.). This is to be expected, as there is naturally more variability among the validation data sections at the higher moments of the travel time distribution. In particular, the RMSEs for the 10th-percentile travel time prediction are very low, especially during the midday and weekday time periods, when it is expected that the 10th-percentile travel time to be very close to the free-flow travel time. From a time period perspective, the highest RMSEs are seen during the peak period, which is defined specifically for each section to cover time periods of at least 75 min during which the average speeds fall below 45 mph. The RMSEs are lower, though still high, for the peak hour models. The peak hour and peak period models are predicted by the critical D/C ratio, the ILHL and, for some of the models, the precipi- tation factor. The RMSEs are the lowest during the midday period (11:00 a.m.–2:00 p.m.), during which congestion tends to be minimal. The midday period TTIs are predicted only by the critical D/C ratio. RMSEs during the weekday period (predicted by the average D/C ratio and, for some of the models, the ILHL), are slightly higher than they are for the midday period. Results also indicate that the models violate many of the assumptions of generalized regression and thus have room for enhancement. Generally, a good regression model is expected to present randomly scattered residuals without obvious trends. However, increasing trends and other non- random patterns were observed in the residual plots of many of the models. This indicates that the models may not be able to sufficiently describe the relationship between the indepen- dent variables and the dependent variable. Table C.51 sum- marizes the results of the t-test and normality test for each model as applied to the AllData set. These results indicate that the models tend to either system- atically overpredict the reliability measure (i.e., indicate that a section is less reliable than it actually is) or underpredict the reliability measure (i.e., indicate that a section is more reliable than it actually is). In all time periods except for the peak hour,

105 Table C.50. Summary of RMSE Values for Each Model by Region Model Details RMSE Value by Region Analysis Time Slice Model All Data CA MN Salt Lake City Peak period Mean TTI 96.94% 127.55% 21.59% - 99th Percentile 403.44% 607.76% 63.67% - 95th Percentile 251.95% 359.19% 45.85% - 80th Percentile 151.95% 206.54% 30.95% - 50th Percentile 89.55% 116.63% 23.15% - 10th Percentile 12.13% 14.43% 6.23% - Peak hour Mean TTI 25.45% 26.97% 24.68% - 99th Percentile 50.74% 52.78% 47.46% - 95th Percentile 38.38% 40.19% 37.27% - 80th Percentile 35.13% 36.89% 34.06% - 50th Percentile 28.85% 32.41% 24.22% - 10th Percentile 18.50% 22.24% 12.14% - Midday Mean TTI 6.24% 7.57% 4.07% 3.52% 99th Percentile 32.32% 34.95% 25.86% 34.01% 95th Percentile 15.62% 17.29% 14.01% 12.55% 80th Percentile 8.99% 10.86% 6.61% 3.60% 50th Percentile 5.43% 6.93% 2.09% 2.08% 10th Percentile 1.81% 2.20% 0.80% 1.33% Weekday Mean TTI 19.74% 12.81% 35.99% 5.95% 99th Percentile 72.91% 50.04% 141.72% 30.87% 95th Percentile 83.82% 40.46% 197.82% 22.85% 80th Percentile 29.28% 14.84% 59.43% 5.75% 50th Percentile 4.68% 5.92% 1.71% 2.16% 10th Percentile 0.81% 0.74% 0.48% 1.30% the null hypothesis of the normality test was rejected for all models, indicating that the residuals are nonnormally distrib- uted. During the peak period, midday, and weekday time peri- ods, the null hypothesis of the t-test was rejected for nearly all of the models, indicating that the mean of the residuals is non- zero. The null hypotheses of these tests were most frequently not rejected during the peak hour time period, but only the mean TTI model of this time period performed satisfactorily well when considering the pattern of residuals. Further investigation of the residual patterns by time period shows that during the peak period each of the models tends to underpredict the measured TTI at low TTI values and overpredict it at higher TTIs. The residuals also tend to get larger as the TTI increases. The peak hour models also exhibit a tendency to underpredict the measured TTI at low TTI values and overpredict it at higher TTIs. During the mid- day period the main problem with the residuals is that they exhibit nonconstant variance, as seen through the cone- shaped residual plots of the models, and have a tendency to overpredict the TTI. During the weekday period, the residu- als of the models are mostly positive, indicating a tendency to consistently overpredict the TTI.

106 Table C.51. Summary of Student’s t-Test and Shapiro-Wilk Normality Test Results for AllData Set Model Details Null Hypothesis Result Analysis Time Slice Model t-Testa Normality Testb Peak period Mean TTI Reject Reject 99th Percentile Reject Reject 95th Percentile Reject Reject 80th Percentile Reject Reject 50th Percentile Reject Reject 10th Percentile Reject Reject Peak hour Mean TTI Cannot Reject Cannot Reject 99th Percentile Cannot Reject Reject 95th Percentile Cannot Reject Cannot Reject 80th Percentile Reject Cannot Reject 50th Percentile Reject Cannot Reject 10th Percentile Cannot Reject Reject Midday Mean TTI Reject Reject 99th Percentile Reject Reject 95th Percentile Reject Reject 80th Percentile Reject Reject 50th Percentile Cannot Reject Reject 10th Percentile Reject Reject Weekday Mean TTI Reject Reject 99th Percentile Reject Reject 95th Percentile Reject Reject 80th Percentile Reject Reject 50th Percentile Cannot Reject Reject 10th Percentile Reject Reject a t-test results indicate whether the null hypothesis assumption that the residuals satisfy zero residual mean can be rejected or not with a certain confidence level. b Normality test results indicate whether the null hypothesis assumption that the residuals satisfy the normal distribution can be rejected or not with a certain confidence level.

107 Peak Period Mean TTI Model Peak Period—Mean TTI Model—California Appendix C Attachment Table C.52. Residual Analysis of Peak Period—Mean TTI—California Table C.52.a. Basic Statistical Measures Location Variability Mean 0.3287 Std deviation 0.7626 Median 0.0533 Variance 0.5815 Minimum -0.3479 Range 3.0176 Maximum 2.6696 Interquartile range 0.2987 Table C.52.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.3287 0.0940 0.5634 Std deviation 0.7626 0.6288 0.9692 Variance 0.5815 0.3954 0.9394 Table C.52.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 2.8263 Pr > t 0.0072 Table C.52.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.6453 Pr < W <0.0001 Figure C.73. Residual plot of peak period—mean TTI—California. Figure C.74. Residual histogram of peak period— mean TTI—California.

108 MinneSota Figure C.75. Residual normality plot of peak period—mean TTI—California. Figure C.78. Residual normality plot of peak period—mean TTI—Minnesota. Figure C.77. Residual histogram of peak period— mean TTI—Minnesota. Figure C.76. Residual plot of peak period—mean TTI—Minnesota. Table C.53.c. Tests for Location: Mu0 Test Statistic p-Value Student’s t t -0.205 Pr > t 0.8401 Table C.53.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.8416 Pr < W 0.0049 Table C.53. Residual Analysis of Peak Period—Mean TTI—Minnesota Table C.53.a. Basic Statistical Measures Location Variability Mean -0.009 Std deviation 0.2006 Median -0.073 Variance 0.0402 Minimum -0.247 Range 0.6507 Maximum 0.4042 Interquartile range 0.2395 Table C.53.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.009 -0.106 0.0873 Std deviation 0.2006 0.1516 0.2966 Variance 0.0402 0.0230 0.0880

109 99th-Percentile TTI Model California Table C.54. Residual Analysis of Peak Period—99th-Percentile TTI—California Table C.54.a. Basic Statistical Measures Location Variability Mean 0.8780 Std deviation 1.7696 Median 0.2508 Variance 3.1316 Minimum -0.725 Range 7.2825 Maximum 6.5571 Interquartile range 0.8412 Table C.54.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.8780 0.3334 1.4226 Std deviation 1.7696 1.4591 2.2492 Variance 3.1316 2.1291 5.0590 Table C.54.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 3.2534 Pr > t 0.0023 Table C.54.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.6600 Pr < W <0.0001 Figure C.79. Residual plot of peak period— 99th-percentile TTI—California. Figure C.80. Residual histogram of peak period— 99th-percentile TTI—California. Figure C.81. Residual normality plot of peak period—99th-percentile TTI—California.

110 Minnesota Table C.55. Residual Analysis of Peak Period—99th-Percentile TTI—Minnesota Table C.55.a. Basic Statistical Measures Location Variability Mean -0.201 Std deviation 0.4622 Median -0.332 Variance 0.2137 Minimum -0.861 Range 1.8231 Maximum 0.9624 Interquartile range 0.4244 Table C.55.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.201 -0.424 0.0221 Std deviation 0.4622 0.3493 0.6836 Variance 0.2137 0.1220 0.4673 Table C.55.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t -1.893 Pr > t 0.0746 Table C.55.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9208 Pr < W 0.1171 Figure C.82. Residual plot of peak period— 99th-percentile TTI—Minnesota. Figure C.83. Residual histogram of peak period— 99th-percentile TTI—Minnesota. Figure C.84. Residual normality plot of peak period—99th-percentile TTI—Minnesota.

111 95th-Percentile TTI Model California Table C.56. Residual Analysis of Peak Period—95th-Percentile TTI—California Table C.56.a. Basic Statistical Measures Location Variability Mean 0.6326 Std deviation 1.4032 Median 0.1346 Variance 1.9691 Minimum -0.684 Range 5.7559 Maximum 5.0720 Interquartile range 0.6441 Table C.56.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.6326 0.2007 1.0645 Std deviation 1.4032 1.1570 1.7835 Variance 1.9691 1.3387 3.1810 Table C.56.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 2.9562 Pr > t 0.0051 Table C.56.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.6563 Pr < W <0.0001 Figure C.85. Residual plot of peak period— 95th-percentile TTI—California. Figure C.86. Residual histogram of peak period— 95th-percentile TTI—California. Figure C.87. Residual normality plot of peak period—95th-percentile TTI—California.

112 Minnesota Table C.57. Residual Analysis of Peak Period—95th-Percentile TTI—Minnesota Table C.57.a. Basic Statistical Measures Location Variability Mean -0.161 Std deviation 0.3506 Median -0.245 Variance 0.1229 Minimum -0.566 Range 1.2796 Maximum 0.7138 Interquartile range 0.3646 Table C.57.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.161 -0.330 0.0077 Std deviation 0.3506 0.2649 0.5184 Variance 0.1229 0.0702 0.2688 Table C.57.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t -2.005 Pr > t 0.0602 Table C.57.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.8942 Pr < W 0.0383 Figure C.88. Residual plot of peak period— 95th-percentile TTI—Minnesota. Figure C.89. Residual histogram of peak period— 95th-percentile TTI—Minnesota. Figure C.90. Residual normality plot of peak period—95th-percentile TTI—Minnesota.

113 80th-Percentile TTI Model California Table C.58. Residual Analysis of Peak Period—80th-Percentile TTI—California Table C.58.a. Basic Statistical Measures Location Variability Mean 0.4384 Std deviation 1.0430 Median 0.0584 Variance 1.0879 Minimum -0.381 Range 4.0617 Maximum 3.6805 Interquartile range 0.4264 Table C.58.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.4384 0.1174 0.7594 Std deviation 1.0430 0.8600 1.3257 Variance 1.0879 0.7396 1.7574 Table C.58.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 2.7565 Pr > t 0.0086 Table C.58.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.6391 Pr < W <0.0001 Figure C.91. Residual plot of peak period— 80th-percentile TTI—California. Figure C.92. Residual histogram of peak period—80th-percentile TTI—California. Figure C.93. Residual normality plot of peak period—80th-percentile TTI—California.

114 Minnesota Table C.59. Residual Analysis of Peak Period—80th-Percentile TTI—Minnesota Table C.59.a. Basic Statistical Measures Location Variability Mean -0.089 Std deviation 0.2616 Median -0.188 Variance 0.0684 Minimum -0.404 Range 0.9128 Maximum 0.5086 Interquartile range 0.3141 Table C.59.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.089 -0.215 0.0372 Std deviation 0.2616 0.1976 0.3868 Variance 0.0684 0.0391 0.1496 Table C.59.c. Tests for Location: Mu0 Test Statistic p-Value Student’s t t -1.481 Pr > t 0.1560 Table C.59.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.8741 Pr < W 0.0169 Figure C.94. Residual plot of peak period— 80th-percentile TTI—Minnesota. Figure C.95. Residual histogram of peak period— 80th-percentile TTI—Minnesota. Figure C.96. Residual normality plot of peak period—80th-percentile TTI—Minnesota.

115 50th-Percentile TTI Model California Table C.60. Residual Analysis of Peak Period—50th-Percentile TTI—California Table C.60.a. Basic Statistical Measures Location Variability Mean 0.2892 Std deviation 0.7254 Median 0.0125 Variance 0.5261 Minimum -0.368 Range 2.8998 Maximum 2.5315 Interquartile range 0.3859 Table C.60.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.2892 0.0660 0.5125 Std deviation 0.7254 0.5981 0.9219 Variance 0.5261 0.3577 0.8500 Table C.60.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 2.6147 Pr > t 0.0124 Table C.60.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.6624 Pr < W <0.0001 Figure C.97. Residual plot of peak period— 50th-percentile TTI—California. Figure C.98. Residual histogram of peak period— 50th-percentile TTI—California. Figure C.99. Residual normality plot of peak period—50th-percentile TTI—California.

116 Minnesota Table C.61. Residual Analysis of Peak Period—50th-Percentile TTI—Minnesota Table C.61.a. Basic Statistical Measures Location Variability Mean 0.0574 Std deviation 0.2056 Median -0.033 Variance 0.0423 Minimum -0.158 Range 0.7232 Maximum 0.5649 Interquartile range 0.2754 Table C.61.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0574 -0.042 0.1565 Std deviation 0.2056 0.1554 0.3041 Variance 0.0423 0.0241 0.0925 Table C.61.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 1.2169 Pr > t 0.2394 Table C.61.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.8309 Pr < W 0.0033 Figure C.100. Residual plot of peak period— 50th-percentile TTI—Minnesota. Figure C.101. Residual histogram of peak period— 50th-percentile TTI—Minnesota. Figure C.102. Residual normality plot of peak period—50th-percentile TTI—Minnesota.

117 10th-Percentile TTI Model California Table C.62. Residual Analysis of Peak Period—10th-Percentile TTI—California Table C.62.a. Basic Statistical Measures Location Variability Mean 0.0276 Std deviation 0.1335 Median 0.0234 Variance 0.0178 Minimum -0.344 Range 0.7344 Maximum 0.3906 Interquartile range 0.0854 Table C.62.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0276 -0.013 0.0687 Std deviation 0.1335 0.1100 0.1696 Variance 0.0178 0.0121 0.0288 Table C.62.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 1.3572 Pr > t 0.1820 Table C.62.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.8876 Pr < W 0.0005 Figure C.103. Residual plot of peak period— 10th-percentile TTI—California. Figure C.104. Residual histogram of peak period— 10th-percentile TTI—California. Figure C.105. Residual normality plot of peak period—10th-percentile TTI—California.

118 Minnesota Table C.63. Residual Analysis of Peak Period—10th-Percentile TTI—Minnesota Table C.63.a. Basic Statistical Measures Location Variability Mean 0.0521 Std deviation 0.0314 Median 0.0553 Variance 0.0010 Minimum -0.023 Range 0.1273 Maximum 0.1046 Interquartile range 0.0457 Table C.63.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0521 0.0370 0.0673 Std deviation 0.0314 0.0238 0.0465 Variance 0.0010 0.0006 0.0022 Table C.63.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 7.2260 Pr > t <0.0001 Table C.63.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9643 Pr < W 0.6593 Figure C.106. Residual plot of peak period— 10th-percentile TTI—Minnesota. Figure C.107. Residual histogram of peak period— 10th-percentile TTI—Minnesota. Figure C.108. Residual normality plot of peak period—10th-percentile TTI—Minnesota.

119 Peak Hour Mean TTI Model California Table C.64. Residual Analysis of Peak Hour—Mean TTI—California Table C.64.a. Basic Statistical Measures Location Variability Mean -0.085 Std deviation 0.2258 Median -0.055 Variance 0.0510 Minimum -0.547 Range 0.9459 Maximum 0.3987 Interquartile range 0.2853 Table C.64.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.085 -0.155 -0.016 Std deviation 0.2258 0.1862 0.2870 Variance 0.0510 0.0347 0.0824 Table C.64.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t -2.470 Pr > t 0.0177 Table C.64.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9801 Pr < W 0.6519 Figure C.109. Residual plot of peak hour—mean TTI—California. Figure C.110. Residual histogram of peak hour— mean TTI—California. Figure C.111. Residual normality plot of peak hour—mean TTI—California.

120 Minnesota Table C.65.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9361 Pr < W 0.1203 Table C.65.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 1.2588 Pr > t 0.2202 Table C.65.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0549 -0.035 0.1449 Std deviation 0.2181 0.1703 0.3034 Variance 0.0476 0.0290 0.0920 Table C.65.a. Basic Statistical Measures Location Variability Mean 0.0549 Std deviation 0.2181 Median 0.1414 Variance 0.0476 Minimum -0.422 Range 0.7718 Maximum 0.3501 Interquartile range 0.2612 Table C.65. Residual Analysis of Peak Hour—Mean TTI—Minnesota Figure C.112. Residual plot of peak hour—mean TTI—Minnesota. Figure C.113. Residual histogram of peak hour— mean TTI—Minnesota. Figure C.114. Residual normality plot of peak hour—mean TTI—Minnesota.

121 99th-Percentile TTI Model California Table C.66. Residual Analysis of Peak Hour—99th-Percentile TTI—California Table C.66.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9148 Pr < W 0.0036 Table C.66.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 0.2594 Pr > t 0.7966 Table C.66.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0170 -0.115 0.1488 Std deviation 0.4285 0.3533 0.5446 Variance 0.1836 0.1248 0.2966 Table C.66.a. Basic Statistical Measures Location Variability Mean 0.0170 Std deviation 0.4285 Median -0.046 Variance 0.1836 Minimum -0.832 Range 2.2377 Maximum 1.4057 Interquartile range 0.3914 Figure C.115. Residual plot of peak hour— 99th-percentile TTI—California. Figure C.116. Residual histogram of peak hour— 99th-percentile TTI—California. Figure C.117. Residual normality plot of peak hour—99th-percentile TTI—California.

122 Minnesota Table C.67. Residual Analysis of Peak Hour—99th-Percentile TTI—Minnesota Table C.67.a. Basic Statistical Measures Location Variability Mean -0.156 Std deviation 0.3630 Median -0.114 Variance 0.1318 Minimum -0.912 Range 1.3634 Maximum 0.4517 Interquartile range 0.3377 Table C.67.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9409 Pr < W 0.1552 Table C.67.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t -2.147 Pr > t 0.0421 Table C.67.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.156 -0.306 -0.006 Std deviation 0.3630 0.2835 0.5051 Variance 0.1318 0.0804 0.2551 Figure C.118. Residual plot of peak hour— 99th-percentile TTI—Minnesota. Figure C.119. Residual histogram of peak hour— 99th-percentile TTI—Minnesota. Figure C.120. Residual normality plot of peak hour—99th-percentile TTI—Minnesota.

123 95th-Percentile TTI Model California Table C.68.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t -0.877 Pr > t 0.3854 Table C.68.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.045 -0.150 0.0589 Std deviation 0.3387 0.2793 0.4305 Variance 0.1147 0.0780 0.1853 Table C.68. Residual Analysis of Peak Hour—95th-Percentile TTI—California Table C.68.a. Basic Statistical Measures Location Variability Mean -0.045 Std deviation 0.3387 Median -0.057 Variance 0.1147 Minimum -0.623 Range 1.5815 Maximum 0.9590 Interquartile range 0.4299 Table C.68.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9664 Pr < W 0.2357 Figure C.121. Residual plot of peak hour— 95th-percentile TTI—California. Figure C.123. Residual normality plot of peak hour—95th-percentile TTI—California. Figure C.122. Residual histogram of peak hour— 95th-percentile TTI—California.

124 Minnesota Table C.69. Residual Analysis of Peak Hour—95th-Percentile TTI—Minnesota Table C.69.a. Basic Statistical Measures Location Variability Mean -0.012 Std deviation 0.3231 Median 0.0493 Variance 0.1044 Minimum -0.757 Range 1.2923 Maximum 0.5351 Interquartile range 0.3942 Table C.69.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9464 Pr < W 0.2083 Table C.69.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t -0.188 Pr > t 0.8522 Table C.69.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.012 -0.146 0.1212 Std deviation 0.3231 0.2523 0.4494 Variance 0.1044 0.0636 0.2020 Figure C.124. Residual plot of peak hour— 95th-percentile TTI—Minnesota. Figure C.125. Residual histogram of peak hour— 95th-percentile TTI—Minnesota. Figure C.126. Residual normality plot of peak hour—95th-percentile TTI—Minnesota.

125 80th-Percentile TTI Model California Table C.70.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.127 -0.216 -0.037 Std deviation 0.2908 0.2398 0.3696 Variance 0.0846 0.0575 0.1366 Table C.70.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t -2.855 Pr > t 0.0067 Table C.70.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9143 Pr < W 0.0035 Table C.70. Residual Analysis of Peak Hour—80th-Percentile TTI—California Table C.70.a. Basic Statistical Measures Location Variability Mean -0.127 Std deviation 0.2908 Median -0.187 Variance 0.0846 Minimum -0.579 Range 1.2892 Maximum 0.7099 Interquartile range 0.2590 Figure C.127. Residual plot of peak hour— 80th-percentile TTI—California. Figure C.128. Residual histogram of peak hour— 80th-percentile TTI—California. Figure C.129. Residual normality plot of peak hour—80th-percentile TTI—California.

126 Minnesota Table C.71. Residual Analysis of Peak Hour—80th-Percentile TTI—Minnesota Table C.71.a. Basic Statistical Measures Location Variability Mean -0.102 Std deviation 0.2805 Median -0.076 Variance 0.0787 Minimum -0.738 Range 1.0207 Maximum 0.2824 Interquartile range 0.3547 Table C.71.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.102 -0.218 0.0139 Std deviation 0.2805 0.2190 0.3902 Variance 0.0787 0.0480 0.1523 Table C.71.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9279 Pr < W 0.0779 Table C.71.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t -1.816 Pr > t 0.0818 Figure C.130. Residual plot of peak hour— 80th-percentile TTI—Minnesota. Figure C.131. Residual histogram of peak hour— 80th-percentile TTI—Minnesota. Figure C.132. Residual normality plot of peak hour—80th-percentile TTI—Minnesota.

127 50th-Percentile TTI Model California Table C.72.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9844 Pr < W 0.8187 Table C.72.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t -3.750 Pr > t 0.0005 Table C.72.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.141 -0.216 -0.065 Std deviation 0.2458 0.2027 0.3125 Variance 0.0604 0.0411 0.0976 Table C.72. Residual Analysis of Peak Hour—50th-Percentile TTI—California Table C.72.a. Basic Statistical Measures Location Variability Mean -0.141 Std deviation 0.2458 Median -0.161 Variance 0.0604 Minimum -0.624 Range 0.9830 Maximum 0.3588 Interquartile range 0.3255 Figure C.133. Residual plot of peak hour— 50th-percentile TTI—California. Figure C.134. Residual histogram of peak hour— 50th-percentile TTI—California. Figure C.135. Residual normality plot of peak hour—50th-percentile TTI—California.

128 Minnesota Table C.73.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0219 -0.069 0.1128 Std deviation 0.2202 0.1719 0.3063 Variance 0.0485 0.0296 0.0938 Table C.73.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 0.4982 Pr > t 0.6228 Table C.73.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9545 Pr < W 0.3161 Table C.73. Residual Analysis of Peak Hour—50th-Percentile TTI—Minnesota Table C.73.a. Basic Statistical Measures Location Variability Mean 0.0219 Std deviation 0.2202 Median 0.0730 Variance 0.0485 Minimum -0.426 Range 0.8297 Maximum 0.4035 Interquartile range 0.3632 Figure C.136. Residual plot of peak hour— 50th-percentile TTI—Minnesota. Figure C.137. Residual histogram of peak hour— 50th-percentile TTI—Minnesota. Figure C.138. Residual normality plot of peak hour—50th-percentile TTI—Minnesota.

129 10th-Percentile TTI Model California Table C.74.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.8385 Pr < W <0.0001 Table C.74.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t -2.603 Pr > t 0.0127 Table C.74.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.075 -0.133 -0.017 Std deviation 0.1886 0.1555 0.2397 Variance 0.0356 0.0242 0.0574 Table C.74. Residual Analysis of Peak Hour—10th-Percentile TTI—California Table C.74.a. Basic Statistical Measures Location Variability Mean -0.075 Std deviation 0.1886 Median -0.027 Variance 0.0356 Minimum -0.640 Range 0.7884 Maximum 0.1486 Interquartile range 0.1986 Figure C.139. Residual plot of peak hour— 10th-percentile TTI—California. Figure C.141. Residual normality plot of peak hour—10th-percentile TTI—California. Figure C.140. Residual histogram of peak hour— 10th-percentile TTI—California.

130 Minnesota Table C.75. Residual Analysis of Peak Hour—10th-Percentile TTI—Minnesota Table C.75.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0770 0.0412 0.1128 Std deviation 0.0866 0.0677 0.1205 Variance 0.0075 0.0046 0.0145 Table C.75.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 4.4440 Pr > t 0.0002 Table C.75.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.7271 Pr < W <0.0001 Table C.75.a. Basic Statistical Measures Location Variability Mean 0.0770 Std deviation 0.0866 Median 0.1010 Variance 0.0075 Minimum -0.261 Range 0.4352 Maximum 0.1739 Interquartile range 0.0461 Figure C.142. Residual plot of peak hour— 10th-percentile TTI—Minnesota. Figure C.144. Residual normality plot of peak hour—10th-percentile TTI—Minnesota. Figure C.143. Residual histogram of peak hour— 10th-percentile TTI—Minnesota.

131 Midday Mean TTI Model California Table C.76.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0164 0.0045 0.0283 Std deviation 0.0714 0.0639 0.0809 Variance 0.0051 0.0041 0.0065 Table C.76.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.5476 Pr < W <0.0001 Table C.76.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 2.7157 Pr > t 0.0075 Table C.76. Residual Analysis of Midday— Mean TTI—California Table C.76.a. Basic Statistical Measures Location Variability Mean 0.0164 Std deviation 0.0714 Median 0.0377 Variance 0.0051 Minimum -0.443 Range 0.5167 Maximum 0.0741 Interquartile range 0.0370 Figure C.145. Residual plot of midday—mean TTI— California. Figure C.146. Residual histogram of midday—mean TTI—California. Figure C.147. Residual normality plot of midday— mean TTI—California.

132 Minnesota Table C.77. Residual Analysis of Midday— Mean TTI—Minnesota Table C.77.a. Basic Statistical Measures Location Variability Mean 0.0273 Std deviation 0.0294 Median 0.0339 Variance 0.0009 Minimum -0.138 Range 0.1943 Maximum 0.0560 Interquartile range 0.0259 Table C.77.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0273 0.0197 0.0349 Std deviation 0.0294 0.0249 0.0358 Variance 0.0009 0.0006 0.0013 Table C.77.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 7.2065 Pr > t <0.0001 Table C.77.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.6904 Pr < W <0.0001 Figure C.148. Residual plot of midday—mean TTI—Minnesota. Figure C.150. Residual normality plot of midday— mean TTI—Minnesota. Figure C.149. Residual histogram of midday—mean TTI—Minnesota.

133 Salt Lake City Table C.78. Residual Analysis of Midday— Mean TTI—Salt Lake City Table C.78.a. Basic Statistical Measures Location Variability Mean 0.0281 Std deviation 0.0206 Median 0.0342 Variance 0.0004 Minimum -0.016 Range 0.0670 Maximum 0.0507 Interquartile range 0.0207 Table C.78.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.8171 Pr < W <0.0001 Table C.78.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 7.7165 Pr > t <0.0001 Table C.78.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0281 0.0207 0.0355 Std deviation 0.0206 0.0165 0.0274 Variance 0.0004 0.0003 0.0007 Figure C.151. Residual plot of midday—mean TTI— Salt Lake City. Figure C.153. Residual normality plot of midday— mean TTI—Salt Lake City. Figure C.152. Residual histogram of midday—mean TTI—Salt Lake City.

134 99th-Percentile TTI Model California Table C.79. Residual Analysis of Midday— 99th-Percentile TTI—California Table C.79.a. Basic Statistical Measures Location Variability Mean 0.2094 Std deviation 0.2152 Median 0.2689 Variance 0.0463 Minimum -0.535 Range 1.2456 Maximum 0.7110 Interquartile range 0.2254 Table C.79.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.2094 0.1735 0.2454 Std deviation 0.2152 0.1926 0.2438 Variance 0.0463 0.0371 0.0595 Table C.79.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 11.517 Pr > t <0.0001 Table C.79.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.8888 Pr < W <0.0001 Figure C.154. Residual plot of midday— 99th-percentile TTI—California. Figure C.156. Residual normality plot of midday— 99th-percentile TTI—California. Figure C.155. Residual histogram of midday— 99th-percentile TTI—California.

135 Minnesota Table C.80.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.1419 0.0948 0.1891 Std deviation 0.1825 0.1547 0.2225 Variance 0.0333 0.0239 0.0495 Table C.80.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 6.0250 Pr > t <0.0001 Table C.80.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.8145 Pr < W <0.0001 Table C.80. Residual Analysis of Midday— 99th-Percentile TTI—Minnesota Table C.80.a. Basic Statistical Measures Location Variability Mean 0.1419 Std deviation 0.1825 Median 0.1902 Variance 0.0333 Minimum -0.457 Range 0.8090 Maximum 0.3525 Interquartile range 0.1319 Figure C.157. Residual plot of midday— 99th-percentile TTI—Minnesota. Figure C.158. Residual histogram of midday— 99th-percentile TTI—Minnesota. Figure C.159. Residual normality plot of midday— 99th-percentile TTI—Minnesota.

136 Salt Lake City Table C.81. Residual Analysis of Midday— 99th-Percentile TTI—Salt Lake City Table C.81.a. Basic Statistical Measures Location Variability Mean 0.2697 Std deviation 0.1158 Median 0.3032 Variance 0.0134 Minimum -0.257 Range 0.6388 Maximum 0.3821 Interquartile range 0.0960 Table C.81.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.2697 0.2279 0.3114 Std deviation 0.1158 0.0928 0.1539 Variance 0.0134 0.0086 0.0237 Table C.81.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 13.175 Pr > t <0.0001 Table C.81.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.6933 Pr < W <0.0001 Figure C.160. Residual plot of midday— 99th-percentile TTI—Salt Lake City. Figure C.162. Residual normality plot of midday— 99th-percentile TTI—Salt Lake City. Figure C.161. Residual histogram of midday— 99th-percentile TTI—Salt Lake City.

137 95th-Percentile TTI Model California Table C.82. Residual Analysis of Midday— 95th-Percentile TTI—California Table C.82.a. Basic Statistical Measures Location Variability Mean 0.0787 Std deviation 0.1392 Median 0.1156 Variance 0.0194 Minimum -0.662 Range 0.9157 Maximum 0.2537 Interquartile range 0.0949 Table C.82.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0787 0.0555 0.1020 Std deviation 0.1392 0.1246 0.1577 Variance 0.0194 0.0155 0.0249 Table C.82c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 6.6931 Pr > t <0.0001 Table C.82.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.6580 Pr < W <0.0001 Figure C.163. Residual plot of midday— 95th-percentile TTI—California. Figure C.165. Residual normality plot of midday— 95th-percentile TTI—California. Figure C.164. Residual histogram of midday— 95th-percentile TTI—California.

138 Minnesota Table C.83. Residual Analysis of Midday— 95th-Percentile TTI—Minnesota Table C.83.a. Basic Statistical Measures Location Variability Mean 0.0615 Std deviation 0.1168 Median 0.0957 Variance 0.0136 Minimum -0.395 Range 0.5632 Maximum 0.1684 Interquartile range 0.0782 Table C.83.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0615 0.0313 0.0916 Std deviation 0.1168 0.0990 0.1424 Variance 0.0136 0.0098 0.0203 Table C.83.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 4.0784 Pr > t 0.0001 Table C.83.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.7090 Pr < W <0.0001 Figure C.166. Residual plot of midday— 95th-percentile TTI—Minnesota. Figure C.168. Residual normality plot of midday— 95th-percentile TTI—Minnesota. Figure C.167. Residual histogram of midday— 95th-percentile TTI—Minnesota.

139 Salt Lake City Table C.84. Residual Analysis of Midday— 95th-Percentile TTI—Salt Lake City Table C.84.a. Basic Statistical Measures Location Variability Mean 0.1034 Std deviation 0.0583 Median 0.1182 Variance 0.0034 Minimum -0.170 Range 0.3232 Maximum 0.1527 Interquartile range 0.0325 Table C.84.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.1034 0.0824 0.1244 Std deviation 0.0583 0.0467 0.0775 Variance 0.0034 0.0022 0.0060 Table C.84.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 10.036 Pr > t <0.0001 Table C.84.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.6238 Pr < W <0.0001 Figure C.169. Residual plot of midday— 95th-percentile TTI—Salt Lake City. Figure C.170. Residual histogram of midday— 95th-percentile TTI—Salt Lake City. Figure C.171. Residual normality plot of midday— 95th-percentile TTI—Salt Lake City.

140 80th-Percentile TTI Model California Table C.85. Residual Analysis of Midday— 80th-Percentile TTI—California Table C.85.a. Basic Statistical Measures Location Variability Mean 0.0047 Std deviation 0.1033 Median 0.0368 Variance 0.0107 Minimum -0.627 Range 0.7005 Maximum 0.0737 Interquartile range 0.0517 Table C.85.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0047 -0.013 0.0220 Std deviation 0.1033 0.0925 0.1171 Variance 0.0107 0.0086 0.0137 Table C.85.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 0.5383 Pr > t 0.5912 Table C.85.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.5368 Pr < W <0.0001 Figure C.172. Residual plot of midday— 80th-percentile TTI—California. Figure C.174. Residual normality plot of midday— 80th-percentile TTI—California. Figure C.173. Residual histogram of midday— 80th-percentile TTI—California.

141 Minnesota Table C.86. Residual Analysis of Midday— 80th-Percentile TTI—Minnesota Table C.86.a. Basic Statistical Measures Location Variability Mean 0.0231 Std deviation 0.0602 Median 0.0365 Variance 0.0036 Minimum -0.360 Range 0.4228 Maximum 0.0631 Interquartile range 0.0276 Table C.86.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0231 0.0075 0.0386 Std deviation 0.0602 0.0510 0.0734 Variance 0.0036 0.0026 0.0054 Table C.86.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 2.9669 Pr > t 0.0043 Table C.86.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.4994 Pr < W <0.0001 Figure C.175. Residual plot of midday— 80th-percentile TTI—Minnesota. Figure C.176. Residual histogram of midday— 80th-percentile TTI—Minnesota. Figure C.177. Residual normality plot of midday— 80th-percentile TTI—Minnesota.

142 Salt Lake City Figure C.178. Residual plot of midday— 80th-percentile TTI—Salt Lake City. Table C.87. Residual Analysis of Midday— 80th-Percentile TTI—Salt Lake City Table C.87.a. Basic Statistical Measures Location Variability Mean 0.0265 Std deviation 0.0238 Median 0.0350 Variance 0.0006 Minimum -0.025 Range 0.0761 Maximum 0.0511 Interquartile range 0.0249 Table C.87.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0265 0.0179 0.0350 Std deviation 0.0238 0.0191 0.0316 Variance 0.0006 0.0004 0.0010 Table C.87.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 6.3027 Pr > t <0.0001 Table C.87.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.8161 Pr < W <0.0001 Figure C.179. Residual histogram of midday— 80th-percentile TTI—Salt Lake City. Figure C.180. Residual normality plot of midday— 80th-percentile TTI—Salt Lake City.

143 50th-Percentile TTI Model California Figure C.181. Residual plot of midday— 50th-percentile TTI—California. Table C.88. Residual Analysis of Midday— 50th-Percentile TTI—California Table C.88.a. Basic Statistical Measures Location Variability Mean -0.004 Std deviation 0.0671 Median 0.0181 Variance 0.0045 Minimum -0.471 Range 0.5043 Maximum 0.0333 Interquartile range 0.0258 Table C.88.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.004 -0.015 0.0070 Std deviation 0.0671 0.0601 0.0761 Variance 0.0045 0.0036 0.0058 Table C.88.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t -0.740 Pr > t 0.4607 Table C.88.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.4886 Pr < W <0.0001 Figure C.182. Residual histogram of midday— 50th-percentile TTI—California. Figure C.183. Residual normality plot of midday— 50th-percentile TTI—California.

144 Minnesota Table C.89. Residual Analysis of Midday— 50th-Percentile TTI—Minnesota Table C.89.a. Basic Statistical Measures Location Variability Mean 0.0177 Std deviation 0.0108 Median 0.0208 Variance 0.0001 Minimum -0.020 Range 0.0485 Maximum 0.0283 Interquartile range 0.0081 Table C.89.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0177 0.0150 0.0205 Std deviation 0.0108 0.0091 0.0131 Variance 0.0001 0.0001 0.0002 Table C.89.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 12.764 Pr > t <0.0001 Table C.89.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.7497 Pr < W <0.0001 Figure C.184. Residual plot of midday— 50th-percentile TTI—Minnesota. Figure C.185. Residual histogram of midday— 50th-percentile TTI—Minnesota. Figure C.186. Residual normality plot of midday— 50th-percentile TTI—Minnesota.

145 Salt Lake City Table C.90. Residual Analysis of Midday— 50th-Percentile TTI—Salt Lake City Table C.90.a. Basic Statistical Measures Location Variability Mean 0.0062 Std deviation 0.0199 Median 0.0152 Variance 0.0004 Minimum -0.036 Range 0.0586 Maximum 0.0224 Interquartile range 0.0230 Table C.90.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0062 -93E-5 0.0134 Std deviation 0.0199 0.0159 0.0264 Variance 0.0004 0.0003 0.0007 Table C.90.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 1.7763 Pr > t 0.0855 Table C.90.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.7216 Pr < W <0.0001 Figure C.187. Residual plot of midday— 50th-percentile TTI—Salt Lake City. Figure C.188. Residual histogram of midday— 50th-percentile TTI—Salt Lake City. Figure C.189. Residual normality plot of midday— 50th-percentile TTI—Salt Lake City.

146 10th-Percentile TTI Model California Table C.91. Residual Analysis of Midday— 10th-Percentile TTI—California Table C.91.a. Basic Statistical Measures Location Variability Mean 0.0016 Std deviation 0.0218 Median 0.0075 Variance 0.0005 Minimum -0.138 Range 0.1547 Maximum 0.0168 Interquartile range 0.0050 Table C.91.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0016 -0.002 0.0052 Std deviation 0.0218 0.0195 0.0247 Variance 0.0005 0.0004 0.0006 Table C.91.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 0.8584 Pr > t 0.3922 Table C.91.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.4094 Pr < W <0.0001 Figure C.190. Residual plot of midday— 10th-percentile TTI—California. Figure C.192. Residual normality plot of midday— 10th-percentile TTI—California. Figure C.191. Residual histogram of midday— 10th-percentile TTI—California.

147 Minnesota Table C.92. Residual Analysis of Midday— 10th-Percentile TTI—Minnesota Table C.92.a. Basic Statistical Measures Location Variability Mean 0.0069 Std deviation 0.0041 Median 0.0079 Variance 171E-7 Minimum -0.011 Range 0.0208 Maximum 0.0097 Interquartile range 0.0018 Table C.92.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0069 0.0058 0.0079 Std deviation 0.0041 0.0035 0.0050 Variance 171E-7 123E-7 254E-7 Table C.92.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 12.851 Pr > t <0.0001 Table C.92.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.5066 Pr < W <0.0001 Figure C.193. Residual plot of midday— 10th-percentile TTI—Minnesota. Figure C.194. Residual histogram of midday— 10th-percentile TTI—Minnesota. Figure C.195. Residual normality plot of midday— 10th-percentile TTI—Minnesota.

148 Salt Lake City Table C.93. Residual Analysis of Midday— 10th-Percentile TTI—Salt Lake City Table C.93.a. Basic Statistical Measures Location Variability Mean -0.002 Std deviation 0.0133 Median 0.0053 Variance 0.0002 Minimum -0.030 Range 0.0381 Maximum 0.0077 Interquartile range 0.0138 Table C.93.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.002 -0.006 0.0033 Std deviation 0.0133 0.0107 0.0177 Variance 0.0002 0.0001 0.0003 Table C.93.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t -0.651 Pr > t 0.5199 Table C.93.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.6587 Pr < W <0.0001 Figure C.198. Residual normality plot of midday— 10th-percentile TTI—Salt Lake City. Figure C.197. Residual histogram of midday— 10th-percentile TTI—Salt Lake City. Figure C.196. Residual plot of midday— 10th-percentile TTI—Salt Lake City.

149 Weekday Mean TTI Model California Table C.94. Residual Analysis of Weekday—Mean TTI—California Table C.94.a. Basic Statistical Measures Location Variability Mean 0.0936 Std deviation 0.0761 Median 0.0970 Variance 0.0058 Minimum -0.234 Range 0.5556 Maximum 0.3215 Interquartile range 0.0605 Table C.94.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0936 0.0810 0.1063 Std deviation 0.0761 0.0682 0.0862 Variance 0.0058 0.0046 0.0074 Table C.94.c. Tests for Location: Mu0 Test Statistic p-Value Student’s t t 14.660 Pr > t <0.0001 Table C.94.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9011 Pr < W <0.0001 Figure C.201. Residual normality plot of weekday—mean TTI—California. Figure C.200. Residual histogram of weekday— mean TTI—California. Figure C.199. Residual plot of weekday—mean TTI—California.

150 Minnesota Table C.95. Residual Analysis of Weekday—Mean TTI—Minnesota Table C.95.a. Basic Statistical Measures Location Variability Mean 0.2352 Std deviation 0.1997 Median 0.1658 Variance 0.0399 Minimum -0.002 Range 1.1476 Maximum 1.1458 Interquartile range 0.0740 Table C.95.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.2352 0.1836 0.2867 Std deviation 0.1997 0.1692 0.2435 Variance 0.0399 0.0286 0.0593 Table C.95.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 9.1235 Pr > t <0.0001 Table C.95.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.5745 Pr < W <0.0001 Figure C.202. Residual plot of weekday—mean TTI—Minnesota. Figure C.204. Residual normality plot of weekday—mean TTI—Minnesota. Figure C.203. Residual histogram of weekday— mean TTI—Minnesota.

151 Salt Lake City Table C.96. Residual Analysis of Weekday—Mean TTI—Salt Lake City Table C.96.a. Basic Statistical Measures Location Variability Mean 0.0503 Std deviation 0.0290 Median 0.0597 Variance 0.0008 Minimum -0.000 Range 0.1074 Maximum 0.1072 Interquartile range 0.0423 Table C.96.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0503 0.0394 0.0611 Std deviation 0.0290 0.0231 0.0390 Variance 0.0008 0.0005 0.0015 Table C.96.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 9.4868 Pr > t <0.0001 Table C.96.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.8662 Pr < W 0.0014 Figure C.205. Residual plot of weekday—mean TTI—Salt Lake City. Figure C.206. Residual histogram of weekday— mean TTI—Salt Lake City. Figure C.207. Residual normality plot of weekday—mean TTI—Salt Lake City.

152 99th-Percentile TTI Model California Table C.97. Residual Analysis of Weekday—99th-Percentile TTI—California Table C.97.a. Basic Statistical Measures Location Variability Mean 0.2965 Std deviation 0.2779 Median 0.2891 Variance 0.0772 Minimum -0.362 Range 1.4728 Maximum 1.1110 Interquartile range 0.3403 Table C.97.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.2965 0.2504 0.3426 Std deviation 0.2779 0.2489 0.3146 Variance 0.0772 0.0620 0.0990 Table C.97.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 12.714 Pr > t <0.0001 Table C.97.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9824 Pr < W 0.0650 Figure C.210. Residual normality plot of weekday—99th-percentile TTI—California. Figure C.209. Residual histogram of weekday— 99th-percentile TTI—California. Figure C.208. Residual plot of weekday— 99th-percentile TTI—California.

153 Minnesota Table C.98. Residual Analysis of Weekday—99th-Percentile TTI—Minnesota Table C.98.a. Basic Statistical Measures Location Variability Mean 0.5043 Std deviation 0.7305 Median 0.5237 Variance 0.5336 Minimum -0.480 Range 4.0280 Maximum 3.5480 Interquartile range 0.5125 Table C.98.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.5043 0.3156 0.6930 Std deviation 0.7305 0.6192 0.8909 Variance 0.5336 0.3834 0.7938 Table C.98.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 5.3476 Pr > t <0.0001 Table C.98.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.7169 Pr < W <0.0001 Figure C.213. Residual normality plot of weekday—99th-percentile TTI—Minnesota. Figure C.212. Residual histogram of weekday— 99th-percentile TTI—Minnesota. Figure C.211. Residual plot of weekday— 99th-percentile TTI—Minnesota.

154 Salt Lake City Table C.99. Residual Analysis of Weekday— 99th-Percentile TTI—Salt Lake City Table C.99.a. Basic Statistical Measures Location Variability Mean 0.2044 Std deviation 0.1778 Median 0.2447 Variance 0.0316 Minimum -0.261 Range 0.6747 Maximum 0.4139 Interquartile range 0.1944 Table C.99.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.2044 0.1380 0.2708 Std deviation 0.1778 0.1416 0.2390 Variance 0.0316 0.0201 0.0571 Table C.99.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 6.2975 Pr > t <0.0001 Table C.99.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.8601 Pr < W 0.0010 Figure C.214. Residual plot of weekday— 99th-percentile TTI—Salt Lake City. Figure C.216. Residual normality plot of weekday—99th-percentile TTI—Salt Lake City. Figure C.215. Residual histogram of weekday— 99th-percentile TTI—Salt Lake City.

155 95th-Percentile TTI Model California Table C.100. Residual Analysis of Weekday—95th-Percentile TTI—California Table C.100.a. Basic Statistical Measures Location Variability Mean 0.2369 Std deviation 0.2444 Median 0.2792 Variance 0.0597 Minimum -0.406 Range 1.4594 Maximum 1.0533 Interquartile range 0.2651 Table C.100.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.2369 0.1963 0.2774 Std deviation 0.2444 0.2189 0.2767 Variance 0.0597 0.0479 0.0766 Table C.100.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 11.549 Pr > t <0.0001 Table C.100.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9378 Pr < W <0.0001 Figure C.217. Residual plot of weekday— 95th-percentile TTI—California. Figure C.218. Residual histogram of weekday— 95th-percentile TTI—California. Figure C.219. Residual normality plot of weekday—95th-percentile TTI—California.

156 Minnesota Table C.101. Residual Analysis of Weekday—95th-Percentile TTI—Minnesota Table C.101.a. Basic Statistical Measures Location Variability Mean 0.6993 Std deviation 0.8449 Median 0.5499 Variance 0.7138 Minimum -0.221 Range 4.5969 Maximum 4.3758 Interquartile range 0.4778 Table C.101.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.6993 0.4811 0.9176 Std deviation 0.8449 0.7161 1.0304 Variance 0.7138 0.5128 1.0618 Table C.101.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 6.4118 Pr > t <0.0001 Table C.101.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.6407 Pr < W <0.0001 Figure C.222. Residual normality plot of weekday—95th-percentile TTI—Minnesota. Figure C.221. Residual histogram of weekday— 95th-percentile TTI—Minnesota. Figure C.220. Residual plot of weekday— 95th-percentile TTI—Minnesota.

157 Salt Lake City Table C.102. Residual Analysis of Weekday—95th-Percentile TTI— Salt Lake City Table C.102.a. Basic Statistical Measures Location Variability Mean 0.1696 Std deviation 0.1185 Median 0.1951 Variance 0.0140 Minimum -0.167 Range 0.5681 Maximum 0.4012 Interquartile range 0.1022 Table C.102.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.1696 0.1254 0.2139 Std deviation 0.1185 0.0944 0.1593 Variance 0.0140 0.0089 0.0254 Table C.102.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 7.8406 Pr > t <0.0001 Table C.102.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.8312 Pr < W 0.0003 Figure C.225. Residual normality plot of weekday—95th-percentile TTI—Salt Lake City. Figure C.224. Residual histogram of weekday— 95th-percentile TTI—Salt Lake City. Figure C.223. Residual plot of weekday— 95th-percentile TTI—Salt Lake City.

158 80th-Percentile TTI Model California Table C.103. Residual Analysis of Weekday—80th-Percentile TTI—California Table C.103.a. Basic Statistical Measures Location Variability Mean 0.0564 Std deviation 0.1267 Median 0.0883 Variance 0.0161 Minimum -0.387 Range 0.7396 Maximum 0.3527 Interquartile range 0.1138 Table C.103.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0564 0.0354 0.0774 Std deviation 0.1267 0.1135 0.1435 Variance 0.0161 0.0129 0.0206 Table C.103.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 5.3045 Pr > t <0.0001 Table C.103.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9180 Pr < W <0.0001 Figure C.226. Residual plot of weekday— 80th-percentile TTI—California. Figure C.228. Residual normality plot of weekday—80th-percentile TTI—California. Figure C.227. Residual histogram of weekday— 80th-percentile TTI—California.

159 Minnesota Table C.104. Residual Analysis of Weekday—80th-Percentile TTI—Minnesota Table C.104.a. Basic Statistical Measures Location Variability Mean 0.3134 Std deviation 0.3484 Median 0.2155 Variance 0.1214 Minimum -0.078 Range 1.9074 Maximum 1.8296 Interquartile range 0.1652 Table C.104.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.3134 0.2234 0.4034 Std deviation 0.3484 0.2953 0.4249 Variance 0.1214 0.0872 0.1805 Table C.104.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 6.9691 Pr > t <0.0001 Table C.104.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.6274 Pr < W <0.0001 Figure C.229. Residual plot of weekday— 80th-percentile TTI—Minnesota. Figure C.230. Residual histogram of weekday— 80th-percentile TTI—Minnesota. Figure C.231. Residual normality plot of weekday—80th-percentile TTI—Minnesota.

160 Salt Lake City Table C.105. Residual Analysis of Weekday—80th-Percentile TTI— Salt Lake City Table C.105.a. Basic Statistical Measures Location Variability Mean 0.0432 Std deviation 0.0360 Median 0.0533 Variance 0.0013 Minimum -0.027 Range 0.1305 Maximum 0.1036 Interquartile range 0.0466 Table C.105.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0432 0.0298 0.0567 Std deviation 0.0360 0.0287 0.0485 Variance 0.0013 0.0008 0.0023 Table C.105.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 6.5668 Pr > t <0.0001 Table C.105.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.9118 Pr < W 0.0165 Figure C.234. Residual normality plot of weekday—80th-percentile TTI—Salt Lake City. Figure C.233. Residual histogram of weekday— 80th-percentile TTI—Salt Lake City. Figure C.232. Residual plot of weekday— 80th-percentile TTI—Salt Lake City.

161 50th-Percentile TTI Model California Table C.106. Residual Analysis of Weekday—50th-Percentile TTI—California Table C.106.a. Basic Statistical Measures Location Variability Mean -0.005 Std deviation 0.0575 Median 0.0152 Variance 0.0033 Minimum -0.399 Range 0.4271 Maximum 0.0282 Interquartile range 0.0247 Table C.106.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.005 -0.015 0.0042 Std deviation 0.0575 0.0515 0.0651 Variance 0.0033 0.0027 0.0042 Table C.106.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t -1.099 Pr > t 0.2735 Table C.106.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.4980 Pr < W <0.0001 Figure C.237. Residual normality plot of weekday—50th-percentile TTI—California. Figure C.236. Residual histogram of weekday— 50th-percentile TTI—California. Figure C.235. Residual plot of weekday— 50th-percentile TTI—California.

162 Minnesota Table C.107. Residual Analysis of Weekday—50th-Percentile TTI—Minnesota Table C.107.a. Basic Statistical Measures Location Variability Mean 0.0153 Std deviation 0.0074 Median 0.0178 Variance 543E-7 Minimum -0.024 Range 0.0449 Maximum 0.0213 Interquartile range 0.0050 Table C.107.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0153 0.0134 0.0172 Std deviation 0.0074 0.0062 0.0090 Variance 543E-7 39E-6 0.0001 Table C.107.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 16.049 Pr > t <0.0001 Table C.107.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.6674 Pr < W <0.0001 Figure C.238. Residual plot of weekday— 50th-percentile TTI—Minnesota. Figure C.240. Residual normality plot of weekday—50th-percentile TTI—Minnesota. Figure C.239. Residual histogram of weekday— 50th-percentile TTI—Minnesota.

163 Salt Lake City Table C.108. Residual Analysis of Weekday—50th-Percentile TTI— Salt Lake City Table C.108.a. Basic Statistical Measures Location Variability Mean -0.003 Std deviation 0.0215 Median 0.0102 Variance 0.0005 Minimum -0.047 Range 0.0620 Maximum 0.0152 Interquartile range 0.0282 Table C.108.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.003 -0.011 0.0046 Std deviation 0.0215 0.0171 0.0289 Variance 0.0005 0.0003 0.0008 Table C.108.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t -0.865 Pr > t 0.3942 Table C.108.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.7669 Pr < W <0.0001 Figure C.241. Residual plot of weekday— 50th-percentile TTI—Salt Lake City. Figure C.242. Residual histogram of weekday— 50th-percentile TTI—Salt Lake City. Figure C.243. Residual normality plot of weekday—50th-percentile TTI—Salt Lake City.

164 10th-Percentile TTI Model California Table C.109. Residual Analysis of Weekday—10th-Percentile TTI—California Table C.109.a. Basic Statistical Measures Location Variability Mean 0.0028 Std deviation 0.0069 Median 0.0045 Variance 478E-7 Minimum -0.062 Range 0.0685 Maximum 0.0066 Interquartile range 0.0013 Table C.109.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0028 0.0016 0.0039 Std deviation 0.0069 0.0062 0.0078 Variance 478E-7 383E-7 0.0001 Table C.109.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 4.7561 Pr > t <0.0001 Table C.109.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.3795 Pr < W <0.0001 Figure C.244. Residual plot of weekday— 10th-percentile TTI—California. Figure C.245. Residual histogram of weekday— 10th-percentile TTI—California. Figure C.246. Residual normality plot of weekday—10th-percentile TTI—California.

165 Minnesota Table C.110. Residual Analysis of Weekday—10th-Percentile TTI—Minnesota Table C.110.a. Basic Statistical Measures Location Variability Mean 0.0032 Std deviation 0.0036 Median 0.0041 Variance 13E-6 Minimum -0.014 Range 0.0184 Maximum 0.0049 Interquartile range 0.0006 Table C.110.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean 0.0032 0.0023 0.0042 Std deviation 0.0036 0.0031 0.0044 Variance 13E-6 932E-8 193E-7 Table C.110.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t 6.9862 Pr > t <0.0001 Table C.110.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.3382 Pr < W <0.0001 Figure C.249. Residual normality plot of weekday—10th-percentile TTI—Minnesota. Figure C.248. Residual histogram of weekday— 10th-percentile TTI—Minnesota. Figure C.247. Residual plot of weekday— 10th-percentile TTI—Minnesota.

166 Salt Lake City Table C.111.d. Tests for Normality Test Statistic p-Value Shapiro-Wilk W 0.7251 Pr < W <0.0001 Table C.111. Residual Analysis of Weekday—10th-Percentile TTI— Salt Lake City Table C.111.a. Basic Statistical Measures Location Variability Mean -0.006 Std deviation 0.0119 Median 0.0023 Variance 0.0001 Minimum -0.031 Range 0.0343 Maximum 0.0034 Interquartile range 0.0208 Table C.111.b. Basic Confidence Limits Assuming Normality Parameter Estimate 95% Confidence Limits Mean -0.006 -0.010 -0.001 Std deviation 0.0119 0.0095 0.0160 Variance 0.0001 0.0001 0.0003 Table C.111.c. Tests for Location: Mu=0 Test Statistic p-Value Student’s t t -2.567 Pr > t 0.0157 Figure C.250. Residual plot of weekday— 10th-percentile TTI—Salt Lake City. Figure C.252. Residual normality plot of weekday—10th-percentile TTI—Salt Lake City. Figure C.251. Residual histogram of weekday— 10th-percentile TTI—Salt Lake City.