Validation of Urban Freeway Models (2014) / Chapter Skim
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From page 217... ...
217 Overview This appendix presents the results of the recalibration of the L03 data-poor models and the development of new model forms for the 95th-, 90th-, and 80th-percentile predictive equations for the travel time index (TTI)
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218 Figures E.1 through E.7 evaluate the model performance from different perspectives. The scatter plot (Figure E.1)
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219 Figure E.2. Fit plot in x-axis log scale, recalibrated 95th-percentile TTI model, AllData.
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220 Figure E.4. Residual-by-predicted plot, recalibrated 95th-percentile TTI model, AllData.
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221 Figure E.6. Distribution of residuals, recalibrated 95th-percentile TTI model, AllData.
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222 Table E.6. Parameter Estimates, Recalibrated 95th-Percentile TTI Model, California Variable DF Parameter Estimate Standard Error t-Value Pr > t 95% Confidence Limits Intercept 1 1.00000 0 Infty <0.0001 1.00000 1.00000 ln(mean TTI)
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223 Figure E.9. Fit plot in x-axis log scale, recalibrated 95th-percentile TTI model, California.
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224 Figure E.11. Residual-by-predicted plot, recalibrated 95th-percentile TTI model, California.
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225 Figure E.13. Distribution of residuals, recalibrated 95th-percentile TTI model, California.
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226 Table E.9. Parameter Estimates, Recalibrated 95th-Percentile TTI Model, Minnesota Variable Label DF Parameter Estimate Standard Error t-Value Pr > t 95% Confidence Limits Intercept Intercept 1 1.00000 0 Infty <0.0001 1.00000 1.00000 MN_TTImean_ln ln(mean TTI)
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227 Figure E.16. Fit plot in x-axis log scale, recalibrated 95th-percentile TTI model, Minnesota.
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228 Figure E.18. Residual-by-predicted plot, recalibrated 95th-percentile TTI model, Minnesota.
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229 Figure E.20. Distribution of residuals, recalibrated 95th-percentile TTI model, Minnesota.
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230 Table E.12. Parameter Estimates, Recalibrated 90th-Percentile TTI Model, AllData Variable DF Parameter Estimate Standard Error t-Value Pr > t 95% Confidence Limits Intercept 1 1.00000 0 Infty <0.0001 1.00000 1.00000 ln(mean TTI)
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231 Figure E.22. Scatter plot in original scale, recalibrated 90th-percentile TTI model, AllData.
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232 Figure E.25. Residual-by-predicted plot, recalibrated 90th-percentile TTI model, AllData.
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233 Figure E.26. Outlier and leverage plot, recalibrated 90th-percentile TTI model, AllData.
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234 Table E.15. Parameter Estimates, Recalibrated 90th-Percentile TTI Model, California Variable Label DF Parameter Estimate Standard Error t-Value Pr > t 95% Confidence Limits Intercept Intercept 1 1.00000 0 Infty <0.0001 1.00000 1.00000 CA_TTImean_ln ln(mean TTI)
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235 Figure E.29. Fit plot in original scale, recalibrated 90th-percentile TTI model, California.
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236 Figure E.32. Residual-by-predicted plot, recalibrated 90th-percentile TTI model, California.
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237 Figure E.33. Outlier and leverage plot, recalibrated 90th-percentile TTI model, California.
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238 Table E.18. Parameter Estimates, Recalibrated 90th-Percentile TTI Model, Minnesota Variable Label DF Parameter Estimate Standard Error t-Value Pr > t 95% Confidence Limits Intercept Intercept 1 1.00000 0 Infty <0.0001 1.00000 1.00000 MN_TTImean_ln ln(mean TTI)
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239 Figure E.36. Fit plot in original scale, recalibrated 90th-percentile TTI model, Minnesota.
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240 Figure E.39. Residual-by-predicted plot, recalibrated 90th-percentile TTI model, Minnesota.
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241 Figure E.41. Distribution of residuals, recalibrated 90th-percentile TTI model, Minnesota.
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242 Table E.19. Analysis of Variance, Recalibrated 80th-Percentile TTI Model, AllData Source DF Sum of Squares Mean Square F-Value Pr > F Model 1 573.17609 573.17609 122817 <0.0001 Error 322 1.50274 0.00467 Uncorrected total 323 574.67883 Table E.20.
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243 Table E.21. Parameter Estimates, Recalibrated 80th-Percentile TTI Model, AllData Variable Label DF Parameter Estimate Standard Error t-Value Pr > t 95% Confidence Limits Intercept Intercept 1 1.00000 0 Infty <0.0001 1.00000 1.00000 AllData_TTImean_ln ln(mean TTI)
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244 Figure E.46. Residual-by-predicted plot, recalibrated 80th-percentile TTI model, AllData.
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245 Figure E.47. Outlier and leverage plot, recalibrated 80th-percentile TTI model, AllData.
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246 Table E.24. Parameter Estimates, Recalibrated 80th-Percentile TTI Model, California Variable Label DF Parameter Estimate Standard Error t-Value Pr > t 95% Confidence Limits Intercept Intercept 1 1.00000 0 Infty <0.0001 1.00000 1.00000 CA_TTImean_ln ln(mean TTI)
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247 Figure E.50. Fit plot in original scale, recalibrated 80th-percentile TTI model, California.
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248 Figure E.53. Residual-by-predicted plot, recalibrated 80th-percentile TTI model, California.
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249 Figure E.54. Outlier and leverage plot, recalibrated 80th-percentile TTI model, California.
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250 Table E.27. Parameter Estimates, Recalibrated 80th-Percentile TTI Model, Minnesota Variable Label DF Parameter Estimate Standard Error t-Value Pr > t 95% Confidence Limits Intercept Intercept 1 1.00000 0 Infty <0.0001 1.00000 1.00000 MN_TTImean_ln ln(mean TTI)
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251 Figure E.57. Fit plot in original scale, recalibrated 80th-percentile TTI model, Minnesota.
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252 Figure E.60. Residual-by-predicted plot, recalibrated 80th-percentile TTI model, Minnesota.
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253 Figure E.61. Outlier and leverage plot, recalibrated 80th-percentile TTI model, Minnesota.
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254 Table E.29. Parameter Estimates, Recalibrated Standard Deviation TTI Model, AllData Parameter Estimate Approx.
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255 Figure E.64. Scatter plot in original scale, recalibrated standard deviation TTI model, AllData.
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256 Figure E.67. Residual-by-predicted plot, recalibrated standard deviation TTI model, AllData.
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257 Figure E.68. Distribution of raw residuals, recalibrated standard deviation TTI model, AllData.
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258 California The recalibrated model for the standard deviation of TTI with the CA set is ( ) = −StdDevTTI 0.6886 meanTTI 1CA 0.6444p The F-test (Table E.30)
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259 Figure E.71. Observed-by-predicted plot, recalibrated standard deviation TTI model, California.
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260 Figure E.74. Q-Q plot of residuals, recalibrated standard deviation TTI model, California.
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261 Table E.33. Parameter Estimates, Recalibrated Standard Deviation TTI Model, Minnesota Parameter Estimate Approx.
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262 Figure E.77. Residual-by-predicted plot, recalibrated standard deviation TTI model, Minnesota.
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263 Figure E.79. Q-Q plot of residuals, recalibrated standard deviation TTI model, Minnesota.
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264 Figure E.80. Scatter plot in original scale, recalibrated PctTripsOnTime50mph model, AllData.
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265 Figure E.81. Fit plot in original scale, recalibrated PctTripsOnTime50mph model, AllData.
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266 Figure E.83. Residual-by-predicted plot, recalibrated PctTripsOnTime50mph model, AllData.
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267 California The recalibrated model for the PctTripsOnTime50mph with the CA set is = [ ] − −PctTripsOnTime50mphCA 2.2663 meanTTI 1 e The F-test results (Table E.36)
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268 Figure E.86. Fit plot in original scale, recalibrated PctTripsOnTime50mph model, California.
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269 Figure E.88. Residual-by-predicted plot, recalibrated PctTripsOnTime50mph model, California.
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270 Figure E.90. Q-Q plot of residuals, recalibrated PctTripsOnTime50mph model, California.
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271 Figure E.91. Fit plot in original scale, recalibrated PctTripsOnTime50mph model, Minnesota.
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272 Figure E.93. Residual-by-predicted plot, recalibrated PctTripsOnTime50mph model, Minnesota.
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273 PctTripsOnTime45mph The recalibration of the PctTripsOnTime45mph model is conducted by building a nonlinear model between the response variable, PctTripsOnTime45mph, and the independent variable, mean TTI. The model form is the same as the L03 datapoor model form, described as = [ ]
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274 Figure E.96. Scatter plot in original scale, recalibrated PctTripsOnTime45mph model, AllData.
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275 Figure E.98. Observed-by-predicted plot, recalibrated PctTripsOnTime45mph model, AllData.
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276 Figure E.100. Distribution of raw residuals, recalibrated PctTripsOnTime45mph model, AllData.
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277 California The recalibrated model for the PctTripsOnTime45mph with the CA set is = [ ] − −PctTripsOnTime45mphCA 1.6125 meanTTI 1 e The recalibrated model for the CA data set presents similar results to that for the AllData set.
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278 Figure E.103. Observed-by-predicted plot, recalibrated PctTripsOnTime45mph model, California.
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279 Figure E.105. Distribution of raw residuals, recalibrated PctTripsOnTime45mph model, California.
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280 Table E.45. Parameter Estimates, Recalibrated PctTripsOnTime45mph Model, Minnesota Parameter Estimate Approx.
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281 Figure E.108. Observed-by-predicted plot, recalibrated PctTripsOnTime45mph model, Minnesota.
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282 Figure E.110. Distribution of raw residuals, recalibrated PctTripsOnTime45mph model, Minnesota.
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283 PctTripsOnTime30mph The recalibration of the PctTripsOnTime30mph model is conducted by building a nonlinear model between the response variable, PctTripsOnTime30mph, and the independent variable, mean TTI. The model form is the same as the L03 datapoor model form, described as [ ]
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284 Figure E.113. Fit plot in original scale, recalibrated PctTripsOnTime30mph model, AllData.
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285 Figure E.115. Residual-by-predicted plot, recalibrated PctTripsOnTime30mph model, AllData.
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286 California The recalibrated model for the PctTripsOnTime30mph with the CA set is [ ]
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287 Figure E.118. Fit plot in original scale, recalibrated PctTripsOnTime30mph model, California.
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288 Figure E.120. Residual-by-predicted plot, recalibrated PctTripsOnTime30mph model, California.
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289 Figure E.122. Q-Q plot of residuals, recalibrated PctTripsOnTime30mph model, California.
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290 Figure E.124. Observed-by-predicted plot, recalibrated PctTripsOnTime30mph model, Minnesota.
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291 Figure E.125. Residual-by-predicted plot, recalibrated PctTripsOnTime30mph model, Minnesota.
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292 new Models This chapter presents the results of the new model development for the 95th-, 90th-, and 80th-percentile TTI SHRP 2 L03 models. The new model development was performed on three of the L33 data sets: California, Minnesota, and AllData (representing roadway sections in both CA and MN)
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293 Table E.52. MSE Summary Table (Comparable)
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294 Figure E.129. Fit plot, new model: power form with a single parameter, 95th-percentile TTI, AllData.
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295 Figure E.130. Observed-by-predicted plot, new model: power form with a single parameter, 95th-percentile TTI, AllData.
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296 Figure E.133. Q-Q plot of residuals, new model: power form with a single parameter, 95th-percentile TTI, AllData.
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297 Table E.55. Analysis of Variance, New Model: Power Form with Two Parameters, 95th-Percentile TTI, AllData Source DF Sum of Squares Mean Square F-Value Approx.
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298 Figure E.135. Fit plot, new model: power form with two parameters, 95th-percentile TTI, AllData.
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299 Figure E.137. Residual-by-predicted plot, new model: power form with two parameters, 95th-percentile TTI, AllData.
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300 Figure E.139. Q-Q plot of residuals, new model: power form with two parameters, 95th-percentile TTI, AllData.
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301 Figure E.140. Fit plot by region, new model: polynomial form with two parameters, 95th-percentile TTI, AllData.
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302 Figure E.142. Observed-by-predicted plot, new model: polynomial form with two parameters, 95th-percentile TTI, AllData.
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303 Figure E.144. Distribution of residuals, new model: polynomial form with two parameters, 95th-percentile TTI, AllData.
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304 California poWeR foRM Model With a Single paRaMeteR Model: =95th-percentile TTI meanTTICA 1.8922 poWeR foRM Model With tWo paRaMeteRS Model: =95th-percentile TTI 1.0363 meanTTICA 1.8232p polynoMial foRM Model With tWo paRaMeteRS Model: = + 95th-percentile TTI 0.2245 meanTTI 0.8119 meanTTI CA 2 p p Minnesota poWeR foRM Model With a Single paRaMeteR Model: =95th-percentile TTI meanTTIMN 2.0516 poWeR foRM Model With tWo paRaMeteRS Model: =95th-percentile TTI 1.0871 meanTTIMN 1.9081p Table E.59. Analysis of Variance, New Model: Power Form with a Single Parameter, 95th-Percentile TTI, California Source DF Sum of Squares Mean Square F-Value Approx.
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305 Figure E.147. Observed-by-predicted plot, new model: power form with a single parameter, 95th-percentile TTI, California.
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306 Figure E.149. Distribution of residuals, new model: power form with a single parameter, 95th-percentile TTI, California.
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307 Table E.61. Analysis of Variance, New Model: Power Form with Two Parameters, 95th-Percentile TTI, California Source DF Sum of Squares Mean Square F-Value Approx.
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308 Figure E.152. Observed-by-predicted plot, new model: power form with two parameters, 95th-percentile TTI, California.
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309 Figure E.154. Distribution of residuals, new model: power form with two parameters, 95th-percentile TTI, California.
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310 Table E.63. Analysis of Variance, New Model: Polynomial Form with Two Parameters, 95th-Percentile TTI, California Source DF Sum of Squares Mean Square F-Value Approx.
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311 Figure E.157. Observed-by-predicted plot, new model: polynomial form with two parameters, 95th-percentile TTI, California.
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312 Figure E.159. Distribution of residuals, new model: polynomial form with two parameters, 95th-percentile TTI, California.
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313 Table E.65. Analysis of Variance, New Model: Power Form with a Single Parameter, 95th-Percentile TTI, Minnesota Source DF Sum of Squares Mean Square F-Value Approx.
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314 Figure E.162. Observed-by-predicted plot, new model: power form with a single parameter, 95th-percentile TTI, Minnesota.
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315 Figure E.164. Distribution of residuals, new model: power form with a single parameter, 95th-percentile TTI, Minnesota.
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316 Table E.67. Analysis of Variance, New Model: Power Form with Two Parameters, 95th-Percentile TTI, Minnesota Source DF Sum of Squares Mean Square F-Value Approx.
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317 Figure E.167. Observed-by-predicted plot, new model: power form with two parameters, 95th-percentile TTI, Minnesota.
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318 Figure E.169. Distribution of residuals, new model: power form with two parameters, 95th-percentile TTI, Minnesota.
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319 Table E.69. Analysis of Variance, New Model: Polynomial Form with Two Parameters, 95th-Percentile TTI, Minnesota Source DF Sum of Squares Mean Square F-Value Approx.
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320 Figure E.172. Observed-by-predicted plot, new model: polynomial form with two parameters, 95th-percentile TTI, Minnesota.
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321 Figure E.174. Distribution of residuals, new model: polynomial form with two parameters, 95th-percentile TTI, Minnesota.
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322 90th-Percentile TTI Summary For the 90th-percentile TTI models, the MSE table (Table E.71) shows that the MSE values for the AllData and the CA data sets are smaller with the new models.
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323 Figure E.176. Fit plot by region, new model: power form with a single parameter, 90th-percentile TTI, AllData.
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324 Figure E.178. Observed-by-predicted plot, new model: power form with a single parameter, 90th-percentile TTI, AllData.
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325 Figure E.180. Distribution of residuals, new model: power form with a single parameter, 90th-percentile TTI, AllData.
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326 Table E.74. Analysis of Variance, New Model: Power Form with Two Parameters, 90th-Percentile TTI, AllData Source DF Sum of Squares Mean Square F-Value Approx.
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327 Figure E.183. Fit plot, new model: power form with two parameters, 90th-percentile TTI, AllData.
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328 Figure E.185. Residual-by-predicted plot, new model: power form with two parameters, 90th-percentile TTI, AllData.
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329 Figure E.187. Q-Q plot of residuals, new model: power form with two parameters, 90th-percentile TTI, AllData.
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330 Figure E.188. Fit plot by region, new model: polynomial form with two parameters, 90th-percentile TTI, AllData.
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331 Figure E.190. Observed-by-predicted plot, new model: polynomial form with two parameters, 90th-percentile TTI, AllData.
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332 Figure E.192. Distribution of residuals, new model: polynomial form with two parameters, 90th-percentile TTI, AllData.
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333 Table E.78. Analysis of Variance, New Model: Power Form with a Single Parameter, 90th-Percentile TTI, California Source DF Sum of Squares Mean Square F-Value Approx.
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334 Figure E.195. Observed-by-predicted plot, new model: power form with a single parameter, 90th-percentile TTI, California.
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335 Figure E.197. Distribution of residuals, new model: power form with a single parameter, 90th-percentile TTI, California.
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336 Table E.80. Analysis of Variance, New Model: Power Form with Two Parameters, 90th-Percentile TTI, California Source DF Sum of Squares Mean Square F-Value Approx.
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337 Figure E.200. Observed-by-predicted plot, new model: power form with two parameters, 90th-percentile TTI, California.
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338 Figure E.202. Distribution of residuals, new model: power form with two parameters, 90th-percentile TTI, California.
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339 Table E.82. Analysis of Variance, New Model: Polynomial Form with Two Parameters, 90th-Percentile TTI, California Source DF Sum of Squares Mean Square F-Value Approx.
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340 Figure E.205. Observed-by-predicted plot, new model: polynomial form with two parameters, 90th-percentile TTI, California.
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341 Figure E.207. Distribution of residuals, new model: polynomial form with two parameters, 90th-percentile TTI, California.
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342 Table E.85. Parameter Estimates, New Model: Power Form with a Single Parameter, 90th-Percentile TTI, Minnesota Parameter Estimate Approx.
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343 Figure E.210. Observed-by-predicted plot, new model: power form with a single parameter, 90th-percentile TTI, Minnesota.
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344 Figure E.212. Distribution of residuals, new model: power form with a single parameter, 90th-percentile TTI, Minnesota.
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345 Table E.86. Analysis of Variance, New Model: Power Form with Two Parameters, 90th-Percentile TTI, Minnesota Source DF Sum of Squares Mean Square F-Value Approx.
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346 Figure E.215. Observed-by-predicted plot, new model: power form with two parameters, 90th-percentile TTI, Minnesota.
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347 Figure E.217. Distribution of residuals, new model: power form with two parameters, 90th-percentile TTI, Minnesota.
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348 Table E.88. Analysis of Variance, New Model: Polynomial Form with Two Parameters, 90th-Percentile TTI, Minnesota Source DF Sum of Squares Mean Square F-Value Approx.
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349 Figure E.220. Observed-by-predicted plot, new model: polynomial form with two parameters, 90th-percentile TTI, Minnesota.
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350 Figure E.222. Distribution of residuals, new model: polynomial form with two parameters, 90th-percentile TTI, Minnesota.
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351 80th-Percentile TTI Summary For the 80th-percentile TTI, the new models all have smaller MSE values than the recalibrated model. The MSE for the AllData set is reduced by almost 50% with the new models (Table E.90)
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352 Figure E.224. Fit plot, new model: power form with a single parameter, 80th-percentile TTI, AllData.
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353 Figure E.226. Residual-by-predicted plot, new model: power form with a single parameter, 80th-percentile TTI, AllData.
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354 Figure E.228. Q-Q plot of residuals, new model: power form with a single parameter, 80th-percentile TTI, AllData.
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355 Figure E.229. Fit plot, new model: power form with two parameters, 80th-percentile TTI, AllData.
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356 Figure E.231. Residual-by-predicted plot, new model: power form with two parameters, 80th-percentile TTI, AllData.
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357 Figure E.233. Q-Q plot of residuals, new model: power form with two parameters, 80th-percentile TTI, AllData.
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358 Figure E.234. Fit plot, new model: polynomial form with two parameters, 80th-percentile TTI, AllData.
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359 Figure E.236. Residual-by-predicted plot, new model: polynomial form with two parameters, 80th-percentile TTI, AllData.
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360 Figure E.238. Q-Q plot of residuals, new model: polynomial form with two parameters, 80th-percentile TTI, AllData.
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361 Figure E.239. Fit plot, new model: power form with a single parameter, 80th-percentile TTI, California.
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362 Figure E.241. Residual-by-predicted plot, new model: power form with a single parameter, 80th-percentile TTI, California.
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363 Figure E.243. Q-Q plot of residuals, new model: power form with a single parameter, 80th-percentile TTI, California.
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364 Figure E.244. Fit plot, new model: power form with two parameters, 80th-percentile TTI, California.
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365 Figure E.246. Residual-by-predicted plot, new model: power form with two parameters, 80th-percentile TTI, California.
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366 Figure E.248. Q-Q plot of residuals, new model: power form with two parameters, 80th-percentile TTI, California.
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367 Figure E.249. Fit plot, new model: polynomial form with two parameters, 80th-percentile TTI, California.
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368 Figure E.251. Residual-by-predicted plot, new model: polynomial form with two parameters, 80th-percentile TTI, California.
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369 Figure E.253. Q-Q plot of residuals, new model: polynomial form with two parameters, 80th-percentile TTI, California.
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370 Figure E.254. Fit plot, new model: power form with a single parameter, 80th-percentile TTI, Minnesota.
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371 Figure E.256. Residual-by-predicted plot, new model: power form with a single parameter, 80th-percentile TTI, Minnesota.
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372 Figure E.258. Q-Q plot of residuals, new model: power form with a single parameter, 80th-percentile TTI, Minnesota.
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373 Figure E.259. Fit plot, new model: power form with two parameters, 80th-percentile TTI, Minnesota.
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374 Figure E.261. Residual-by-predicted plot, new model: power form with two parameters, 80th-percentile TTI, Minnesota.
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375 Figure E.263. Q-Q plot of residuals, new model: power form with two parameters, 80th-percentile TTI, Minnesota.
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376 Figure E.264. Fit plot, new model: polynomial form with two parameters, 80th-percentile TTI, Minnesota.
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377 Figure E.266. Residual-by-predicted plot, new model: polynomial form with two parameters, 80th-percentile TTI, Minnesota.
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378 Figure E.268. Q-Q plot of residuals, new model: polynomial form with two parameters, 80th-percentile TTI, Minnesota.
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