Jeffrey Hood Parsons Brinckerhoff, Inc. 4 th TRB Conference on Innovations in Travel Modeling Tampa, Florida April 30, 2012

Transcription

1 Jeffrey Hood Parsons Brinckerhoff, Inc. 4 th TRB Conference on Innovations in Travel Modeling Tampa, Florida April 30, 2012

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3 True Model y = x x 2 + N(0,2) Region A transfer sample, 100 obs. x 1 = N(0,1) x 2 = x 1 + N(0,0.5) Region B local sample, 10 obs. x 1 = N(0,1) x 2 = x 1 + N(0,0.5) Different Nearly Multicollinear Relationships Between Predictors

4 β Bayes = (Σ A -1 + Σ B -1 ) -1 (Σ A -1 β A + Σ B -1 β B ) and Σ Bayes = (Σ A -1 + Σ B -1 ) -1 where β Bayes = best estimate of parameters given all data β A = parameters estimated in region A β B = parameters estimated in region B Σ A = variance-covariance matrix for β A Σ B = variance-covariance matrix for β B Σ Bayes = variance-covariance matrix for β Bayes

5 Empirical Densit Region A Region B Bayes True Value Estimate of Beta x_1

6 Empirical Densit Region A Region B Bayes Better RMS Prediction Error

7 Sample estimation in Region A Variable Coefficient Std. Err. t-value p-value Intercept x x Region A prefers to omit x 2 Variable Coefficient Std. Err. t-value p-value Intercept x

8 Empirical Densit Estimate of Beta x_1

9 Empirical Densit RMS Prediction Error

10 Another transfer sample, 100 obs. x 1 = N(0,1) x 2 = x 1 + N(0,0.5) Different from both A & B

11 Sample with x 2 omitted Variable Coefficient Std. Err. t-value p-value Intercept x Region C Analyst introduces x 2 Variable Coefficient Std. Err. t-value p-value Intercept x x Region C

12 Empirical Densit Estimate of Beta x_1

13 Empirical Densit RMS Prediction Error

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15 District School School District School School District School School

16 Region Time period Time period Region Time period Time period Region Time period Time period

17 Response for indiv., region, time has density (,φ ) = h(,φ ) exp{( b( ) ) / a(φ )} Mean related to parameter by link function = -1 ( ) = b ( ) Parameter = modeled with linear predictor Coefficients contain fixed and random effects = individual region time

18 Function Normal Multinomial φ σ 2 1 a(φ) φ φ h(,φ) exp( 2 y / 2 ) / 2 K 1 I y k 1 k {0,1} b( ) θ 2 /2 log 1 K 1 k 1 exp( θ k ) ( ) μ ( ) θ k, K 1 log 1 1 μ exp( θ, K 1 k 1 k 1 k ) μ exp( θ k k,, )

19 Use entire data set to compare Model Contextual Effects Constants Variables Pooled Zero Zero Fixed Intercepts Fixed Zero Segmented Fixed Fixed Random Intercepts Random Zero Random Effects Random Random Not valid if # contexts is small (e.g. < six)

20 Context A Context B Context C Context D Estimate Apply Apply Estimate Estimate Apply Estimate Estimate Apply Estimate

21 A HB,s HB Rail Attraction 1 HB Rail Attraction 2 Bernoulli(r) NHB Rail Production 1 NHB Rail Production 2 P NHB,s HB Rail Attraction A HB,s Station s NHB Rail Production P NHB,s

22 Predicts all NHB rail trip ends at station from Single rate on all home-based trips attracted to station Employment density in station area Estimated w/ Washington D.C. Metro on-board survey Transferred to Honolulu for new system forecast On-board surveys from Los Angeles, 2001 (LA) Washington, D.C., 2008 (DC) Atlanta, 2010 (ATL)

23 Predictor Pooled (R 2 =0.82) Coef. t-stat. Home-Based attractions DC ATL Job density 3-10k / sq. mi DC ATL Job density 10-60k / sq. mi DC ATL Job density > 60k / sq. mi DC

24 Predictor Pooled (R 2 =0.82) Fixed (R 2 =0.87) Segm. (R 2 =0.89) Coef. t-stat. Coef. t-stat. Coef. t-stat. Home-Based attractions DC ATL Job density 3-10k / sq. mi DC ATL Job density 10-60k / sq. mi DC ATL Job density > 60k / sq. mi DC

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26 Weighted Residua Rail Accessibility

27 Predicts NHB-Work & NHB-Other trip ends from Separate rates on HBW Attractions HBW Productions HBO Attractions Rail accessibility to employment Walk accessibility to employment Income of HBW travelers Non-group-quarters college enrollment

28 Predictor Pooled (R 2 =0.96) Coef. t-stat. HBW attractions DC ATL Access, rail DC ATL Access, rail walk DC ATL % HBW attr. inc.>$25k DC ATL HBO attr. HBW:HBO ratio DC ATL HBW productions DC ATL Access, walk DC ATL

29 Predictor Pooled (R 2 =0.96) Fixed (R 2 =0.96) Segm. (R 2 =0.96) Coef. t-stat. Coef. t-stat. Coef. t-stat. HBW attractions DC ATL Access, rail DC ATL Access, rail walk DC ATL % HBW attr. inc.>$25k DC ATL HBO attr. HBW:HBO ratio DC ATL HBW productions DC ATL Access, walk DC ATL

30 NHBW Trip Ends Predictor LA/DC (R 2 =0.96) LA/ATL (R 2 =0.92) DC/ATL (R 2 =0.97) Coef. t-stat. Coef. t-stat. Coef. t-stat. HBW attractions Access, rail Access, rail walk % HBW attr. inc.>$25k HBO attr. HBW:HBO ratio HBW productions Access, walk

31 Restricted Model Pooled Free Model Fixed Intercepts Data Weighted DF RSS F- stat. Restr. Free Free Resid. p- val. All Pooled Segmented All LA/DC Segmented ATL LA/ATL Segmented DC ATL/DC Segmented LA

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33 Dawn McKinstry Mengzhao Hu Amar Sarvepalli Ben Stabler Sudhakar Athuru Robert Farley Chaushie Chu Guy Rousseau Steve Lewandowski Jonathan Nicholson Joel Freedman Rhett Fussell Bill Davidson Albyn Jones Jim Ryan Contact: