1. Trip Generation II Meeghat Habibian Transportation Demand Analysis
Lecture note
Trip Generation II
Meeghat Habibian

2. Content: Linear regression Statistical tests
Aggregate vs. Disaggregate approach
The dummy variable
Transferability and temporal stability of model
Accessibility
Transportation Demand Analysis- Lecture note

3. Introduction
Calibrating models to forecast trips produced from (attracted to) each zone in the future
Methods:
Graphs
Land use based factors (ITE)
Growth factor
Cross classification
Linear regression
Transportation Demand Analysis- Lecture note

4. Transportation Demand Analysis
Lecture note
Linear Regression

5. Linear Regression An approach for modeling a relationship between
Dependent variable (y) and
One (or more) independent variable(s) (xk)
y=β0+β1x1+β2x2 +β3x3 +β4x4 +…+βkxk +ε
Goal:
To calculate coefficients (βi), such to minimize sum of the squares of errors (differences between observations and estimation)
Transportation Demand Analysis- Lecture note

6. Matrix notation Y=X β+ ε n: number of observationβ0
Transportation Demand Analysis- Lecture note
n: number of observation
k: number of independent variables

7. Assumptions for error term
1- εi has normal distribution: εi ~ Normal
2- Mean of εi is 0: E(εi)=0
3- Expectation of εi2 is finite: E(εi2)=σ2
Var(εi)=E[εi-E(εi)]2=E[εi-0]2=E(εi2)= σ2
 Therefore: εi~ N(0 , σ2 )
Transportation Demand Analysis- Lecture note

8. Assumptions for error term
4- Errors are independent: ∀i≠j: E(εi εj)=0
E(εi εj)= E(εi) E(εj)=0 (Non auto regression)
5- Xi s are deterministic such to
6-Number of observations is grater than number of coefficients: n > k+1
7- Xi s are independent from each other
Transportation Demand Analysis- Lecture note

9. Society and Sample
Calculations are based on the sample but results should reflect the society
E(yi)=β0+β1xi
β 0~N(β0, σ2)
β 1~N(β1, σ2)
Transportation Demand Analysis- Lecture note

10. Dependent variable distribution (y)
y=β0+β1x1+β2x2 +β3x3 +β4x4 +…+βKxK + ε
E(y) =E(β0+ Σβkxk+ ε)=E(β0)+E(Σβkxk)+ E(ε)
=β0+Σ(E(βkxk)+0= β0+ ΣxkE(βk) = β0+ Σxkβk
Var(y)=E[y-E(y)]2=E[β0+ Σβkxk+ ε-E(β0+ Σβkxk)]2= E(ε2)= σ2
Therefore:
y~N(β0+β1x1+β2x2 +β3x3 +β4x4 +…+βKxK , σ2)
Transportation Demand Analysis- Lecture note

11. Question?
What are the mean and standard error of the estimated coefficients based on the sample (i.e., )?
Given (society):
y~N(β0+β1xi, σ2)
β 0~N(β0, σ2)
β 1~N(β1, σ2)
Transportation Demand Analysis- Lecture note

12. Remember from TP We were looking for estimation of β0 and β1:
Error sum of squares definition:
We calculated in order to minimize ESS:
Transportation Demand Analysis- Lecture note

13. Remember from TP
Transportation Demand Analysis- Lecture note

14. Remember from TP By solving the recent equations:
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15. Definitions
Therefore:
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16. Expectation of 𝛽 1 (Mean)
𝛽 1 = 𝑆 𝑥𝑦 𝑆 𝑥𝑥 = 𝑖 𝑥 𝑖 − 𝑥 ∗ 𝑥 𝑖 − 𝑦 𝑖 𝑥 𝑖 − 𝑥 2
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17. Expectation of 𝛽 1 (Mean)
Demonstration: 𝛽 1 = 𝑆 𝑥𝑦 𝑆 𝑥𝑥 = 𝑖 𝑥 𝑖 − 𝑥 ∗ 𝑥 𝑖 − 𝑦 𝑖 𝑥 𝑖 − 𝑥 2
𝐸 𝛽 1 =𝐸 𝑖 𝑥 𝑖 − 𝑥 𝑦 𝑖 − 𝑦 𝑆 𝑥𝑥
= 1 𝑆 𝑥𝑥 𝐸( 𝑖 𝑥 𝑖 − 𝑥 𝑦 𝑖 − 𝑦 )
= 1 𝑆 𝑥𝑥 ( 𝑖 𝑥 𝑖 − 𝑥 𝐸 𝑦 𝑖 − 𝑦 )
Transportation Demand Analysis- Lecture note

18. Expectation of 𝛽 1 (Mean)
= 1 𝑆 𝑥𝑥 ( 𝑖 𝑥 𝑖 − 𝑥 𝐸 𝑦 𝑖 −𝐸 𝑦
= 1 𝑆 𝑥𝑥 ( 𝑖 𝑥 𝑖 − 𝑥 ( 𝛽 0 + 𝛽 1 𝑥 𝑖 − 𝛽 0 + 𝛽 1 𝑥 ) )
= 1 𝑆 𝑥𝑥 𝑖 𝑥 𝑖 − 𝑥 𝛽 1 𝑥 𝑖 − 𝑥
= 𝛽 1 𝑆 𝑥𝑥 𝑖 𝑥 𝑖 − 𝑥 = 𝛽 1 𝑆 𝑥𝑥 𝑆 𝑥𝑥 = 𝛽 1
Transportation Demand Analysis- Lecture note
Therefore: E( 𝛽 1 )= β1

19. Variation of 𝛽 1 (Variance)
We can rewrite 𝛽 1 in form of linear combination of 𝑦 𝑖 as follows:
𝛽 1 = 𝑖 𝑐 𝑖 𝑦 𝑖
𝛽 1 = 𝑖 𝑥 𝑖 − 𝑥 𝑦 𝑖 − 𝑦 𝑆 𝑥𝑥
= 𝑖 𝑥 𝑖 − 𝑥 𝑦 𝑖 − 𝑖 𝑥 𝑖 − 𝑥 𝑦 𝑆 𝑥𝑥
= 𝑖 𝑥 𝑖 − 𝑥 𝑦 𝑖 − 𝑦 𝑖 𝑥 𝑖 − 𝑥 𝑆 𝑥𝑥
Transportation Demand Analysis- Lecture note

20. Variation of 𝛽 1 (Variance)
𝑖 𝑥 𝑖 − 𝑥 𝑦 𝑖 𝑆 𝑥𝑥
= 𝑖 𝑥 𝑖 − 𝑥 𝑆 𝑥𝑥 𝑦 𝑖
= 𝑖 𝑐 𝑖 𝑦 𝑖 ; 𝐶 𝑖 = 𝑥 𝑖 − 𝑥 𝑆 𝑥𝑥
Transportation Demand Analysis- Lecture note

21. Variation of 𝛽 1 (Variance)
=E [Σciεi]2
=E [Σci2εi2+2ΣiΣjcicjεiεj]
Transportation Demand Analysis- Lecture note

22. Variation of 𝛽 1 (Variance)
𝜎 2 𝑆 𝑥𝑥 2 𝑖 𝑥 𝑖 − 𝑥 2 = 𝜎 2 𝑆 𝑥𝑥 2 𝑆 𝑥𝑥 = 𝜎 2 𝑆 𝑥𝑥
→ 𝛽 1 ~𝑁( 𝛽 1 , 𝜎 2 𝑖 𝑥 𝑖 − 𝑥 )
Transportation Demand Analysis- Lecture note

23. Expectation of 𝛽 0 (Mean)
𝛽 0 = 𝑦 − 𝛽 1 𝑥
→𝐸 𝛽 0 =𝐸 𝑦 − 𝛽 1 𝑥
=𝐸 𝑦 − 𝑥 𝐸 𝛽 1
= 𝛽 𝛽 1 𝑥 − 𝛽 1 𝑥
= 𝛽 0
Transportation Demand Analysis- Lecture note

24. Variation of 𝛽 0 (Variance)
Demonstrate that 𝛽 0 is linear combination of 𝑦 𝑖 :
𝛽 0 = 𝑖 𝑑 𝑖 𝑦 𝑖
𝛽 0 = 𝑦 − 𝛽 1 𝑥 = 1 𝑛 𝑖 𝑦 𝑖 −( 𝑖 𝑐 𝑖 𝑦 𝑖 ) 𝑥 = 1 𝑛 𝑖 𝑦 𝑖 − 𝑥 𝑖 𝑥 𝑖 − 𝑥 𝑆 𝑥𝑥 𝑦 𝑖 = 1 𝑛 𝑖 𝑦 𝑖 − 𝑥 𝑆 𝑥𝑥 𝑖 𝑥 𝑖 − 𝑥 𝑦 𝑖
Transportation Demand Analysis- Lecture note

25. Variation of 𝛽 0 (Variance)
= 𝑖 1 𝑛 − 𝑥 𝑆 𝑥𝑥 ( 𝑥 𝑖 − 𝑥 ) 𝑦 𝑖 = 𝑖 𝑑 𝑖 𝑦 𝑖
𝑑 𝑖 = 1 𝑛 − 𝑥 𝑆 𝑥𝑥 𝑥 𝑖 − 𝑥
𝑉𝑎𝑟 𝛽 0 =𝑉𝑎𝑟 𝑖 𝑑 𝑖 𝑦 𝑖 =𝐸 𝑖 𝑑 𝑖 𝑦 𝑖 −𝐸( 𝑖 𝑑 𝑖 𝑦 𝑖 ) 2 =𝐸 𝑖 𝑑 𝑖 𝑦 𝑖 − 𝑖 𝑑 𝑖 𝐸( 𝑦 𝑖 ) 2
Transportation Demand Analysis- Lecture note

26. Variation of 𝛽 0 (Variance)
2 2 =
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27. Variation of 𝛽 0 (Variance)
= 𝑖 𝑑 𝑖 2 𝐸 𝜀 𝑖 𝑖≠𝑗 𝑑 𝑖 𝑑 𝑗 𝐸 𝜀 𝑖 𝜀 𝑗
= 𝑖 𝑑 𝑖 2 𝜎 2 = 𝜎 2 𝑖 𝑑 𝑖 2 = 𝜎 2 𝑖 𝑛 − 𝑥 𝑆 𝑥𝑥 𝑥 𝑖 − 𝑥 = 𝜎 𝑛 − 𝑥 2 𝑆 𝑥𝑥
→ 𝛽 0 ~ 𝑁 𝛽 0 , 𝜎 𝑛 + 𝑥 2 𝑖 𝑥 𝑖 − 𝑥 2 +
Transportation Demand Analysis- Lecture note

28. Conclusion ( 𝛽 )
Least square estimators have normal Distribution as follows:
Transportation Demand Analysis- Lecture note
Best Linear Unbiased Estimator = BLUE

29. Sample standard error σ is measure of variability of y
lower values for σ2 show that observations are closer to regression line
For 1 independent variable:
For k independent variables:
Transportation Demand Analysis- Lecture note

30. Transportation Demand Analysis
Lecture note
Statistical Tests

31. Correlation Coefficient
Square of correlation coefficient is called Coefficient of Determination, R2
-1≤R≤1
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32. Goodness of fit measure
Define:
TSS=RSS+ESS
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33. Coefficient significance
T-student test (T-Test)
T-Table Tcritical (α,n-k-1)
T0<Tcritical  H0 can not be rejected at α level
T0>Tcritical  H0 is rejected at α level
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34. Statistical Hypothesis:
F test:
For checking whole model:
H0:R2= (RSS=0)
K: Number of independent variables
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F0<Fcritical  H0 can not be rejected at α level
F0>Fcritical  H0 is rejected at α level

35. Choosing Best Model 1- Low correlation of independent variables (IVs)
2- Intuitive sign of coefficients
3- T-Student test
4- Smaller values for β0
5- Goodness of fit value
6- F test
7- Lower number of variables
Transportation Demand Analysis- Lecture note

36. Aggregate vs. Disaggregate Approach
Transportation Demand Analysis
Lecture note
Aggregate vs. Disaggregate Approach

37. Modeling Approaches Disaggregate models Lower sample size
Aggregate total (zonal average level)
Aggregate rate (household-based average level)
Higher variation
Lower goodness of fit
Makes more sense
Transportation Demand Analysis- Lecture note

38. Aggregate & disaggregate:
Values of R2 for aggregate models are much higher than disaggregate models
Coefficient of aggregate models have grater variance than disaggregate models
A class lecture on variance of above models
Transportation Demand Analysis- Lecture note

39. Transportation Demand Analysis
Lecture note
The Dummy Variable

40. Why we need dummy variables?
Non quantifiable variables
e.g., Gender, Marital status, …
2. Non uniform effect (in different intervals) variables
e.g., Age, distance, …
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Uniform effect
Non-uniform effect

41. Structure Dummy variable = Example: D1= D2=
1, if a condition is satisfied
0, otherwise
Dummy variable =
Example:
D1=
D2=
1, if a household has 1 car
0, otherwise
Transportation Demand Analysis- Lecture note
1, if a household has 2 or more car
0, otherwise

42. Example (Trip generation)
An ordinary regression model:
Trip number = 𝛽 0+ 𝛽 1 (Car ownership)+ 𝛽 2 (HHSZ)
Trips
Cars
HHSZ=2
HHSZ=1
HHSZ=0 β1
Transportation Demand Analysis- Lecture note
𝛽 0+2 𝛽 2
𝛽 0+ 𝛽 2
𝛽 0

43. Example (Trip generation)
An ordinary regression model:
Trip number = 𝛽 0+ 𝛽 1 (Car ownership)+ 𝛽 2 (HHSZ)
A dummy enhanced model:
Trip number = 𝛽 0+ 𝛽 1X2i+…+ 𝛽 5X6i+ 𝛽 6Z2i+ 𝛽 7Z3i+εi
X1i= Z1i=
X2i= Z2i= …
X6i= Z3i=
Therefore: Household i:
1, household i has 1 member
0, otherwise
1, household i has not car
0, otherwise
Transportation Demand Analysis- Lecture note
1, household i has 2 members
0, otherwise
1, household i has 1 car
0, otherwise
1, household i has 6 members
0, otherwise
1, household i has 2+ cars
0, otherwise
X1i+X2i+X3i+X4i+X5i+X6i=1 Z1i+Z2i+Z3i=1

44. Example (Trip generation)
Dummy enhanced model:
Trip number = 𝛽 0+ 𝛽 1X2i+…+ 𝛽 5X6i+ 𝛽 6Z2i+ 𝛽 7Z3i+εi
X1i= Z1i=
X2i= Z2i= …
X6i= Z3i=
Examples:
Household containing 1 member and no car: 𝑌 = 𝛽 0
Household containing 1 member and 1 car: 𝑌 = 𝛽 0+ 𝛽 6
Household containing 3 members and 3 cars: 𝑌 = 𝛽 0+ 𝛽 3+ 𝛽 7
1, household i has 1 member
0, otherwise
1, household i has not car
0, otherwise
1, household i has 2 members
0, otherwise
1, household i has 1 car
0, otherwise
Transportation Demand Analysis- Lecture note
1, household i has 6 members
0, otherwise
1, household i has 2+ cars
0, otherwise

45. Notes Each pair of dummy variables must not overlapped
Union of all levels must be the Universal set
At least, one level (known as base level) should excluded from the modeling
A multiplicative dummy may also use (e.g., XZ) which would be a more complicated case
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46. Transferability and Temporal Stability of Model
Transportation Demand Analysis
Lecture note

47. Model Transferability
Aggregate model
Aggregate Total
Different zone size, topography, …
Aggregate rate
Households behavior can be similar
Disaggregate model
Individuals may behave more similar
E.g., New mass transit choice for new towns
Transportation Demand Analysis- Lecture note

48. Temporal Stability Use a model after a long period in the future
Shall a new dataset for new time (t2) be collected?
Two issues should be checked (small dataset)
Macro
Observation – estimation graph
Micro
Statistical assessment of coefficients
Transportation Demand Analysis- Lecture note

49. Macro (Observation – estimation graph)
Use an observation-estimation graph
Yt2=α+βY^t1 α
Statistically, α should be 0 and β should be 1
Yt2 β
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Y^t1

50. Micro (Statistical assessment of coefficients)
Differences of respective βs should be statistically 0
As βs have normal distributions, their respective difference has also normal distribution:
βt1-βt2 ~ N( 0, SE(βt1-βt2) )
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51. Micro (Example)
Trip generation data for 357 household is available for times t1 and t2
Trip number = β0+β1 (Car ownership)+ β2 (HHSZ)
Two models have been calibrated as follows:
Transportation Demand Analysis- Lecture note t1 t2 β0
-0.45
-0.19 β1
1.40 (0.13)
1.46 (0.15) β2
1.92 (0.31)
1.52 (0.27) R2
0.34
0.36

52. Micro (Example) Assessing β1
Trip number = β0+β1 (Car ownership)+ β2 (HHSZ)
H0: βt1-βt2 =0
T=(βt1-βt2 )/SE(βt1-βt2 )
Remember: Var(X-Y)=Var(X)+Var(Y)-2Cov(X,Y)
Independent samples: Cov(βt1,βt2 )=0
SE2=(0.13)2+(0.15)2-2*0=  SE=0.198
T=( )/0.198=  H0 can not be rejected t1 t2 β0
-0.45
-0.19
Β1 (SE)
1.40 (0.13)
1.46 (0.15)
Β2 (SE)
1.92 (0.31)
1.52 (0.27) R2
0.34
0.36
Transportation Demand Analysis- Lecture note

53. Micro (Example) Assessing β2
Trip number = β0+β1 (Car ownership)+ β2 (HHSZ)
H0: βt1-βt2 =0
T=(βt1-βt2 )/SE(βt1-βt2 )
Remember: Var(X-Y)=Var(X)+Var(Y)-2Cov(X,Y)
Independent samples: Cov(βt1,βt2 )=0
SE2=(0.31)2+(0.27)2-2*0=0.169  SE=0.411
T=( )/0.411=0.97  H0 can not be rejected t1 t2 β0
-0.45
-0.19
Β1 (SE)
1.40 (0.13)
1.46 (0.15)
Β2 (SE)
1.92 (0.31)
1.52 (0.27) R2
0.34
0.36
Transportation Demand Analysis- Lecture note

54. Notes No change in the role of variables during the studied period
E.g., Car as a proxy of income vs. Car as an essential instrument in individual’s lifestyle
Pooling the data together and use a respective dummy is also recommended to calibrate a model for a long period with two datasets
Transportation Demand Analysis- Lecture note

55. Transportation Demand Analysis
Lecture note
Accessibility

56. Necessity Trip generation models are not sensitive to policy making
Because they are not sensitive to attributes of the transportation system (e.g., time and cost)
Note:
Time and cost depend on both origin and destination which is not known in trip generation stage
Transportation Demand Analysis- Lecture note

57. Definition Acci=ΣjcijAj Acci: Accessibility index for zone i
Cij: Cost of travel between I and j
Aj: Opportunities at zone j (e.g., Population, Student number, School number)
Transportation Demand Analysis- Lecture note

58. Transportation Demand Analysis- Lecture note
Finish