1. Analysis of Travel Choice
Meeghat Habibian

2. Potential Trip maker is faced with
Alternatives of trip attributes such as
Destination
Mode
Route

3. The Choice made will be consistently the same
Choice Process
1. Deterministic
Traveler faces repeatedly with the same set of Alternatives
The Decision rule that is used by traveler is consistent and stable
The Choice made will be consistently the same

4. Choice Process 2. Stochastic Due to: Behavior of choice maker
Absence of rational & consistent decision rule
Provide a far superior means for predicting travel behavior than Deterministic one

5. Independent of the total number of population
Measurement of Choice
Choice reflected by:
Number of people
Proportion of population
Independent of the total number of population

6. Deterministic Choice (individual level)
V(i)=Ai*Xi
V(i): choice function
Xi : a vector of demand and supply variables
Ai : a vector of parameters that represent the effect of each variable
Utility function

7. Deterministic Choice (individual level)
V(j)=max[V(i)]
Higher value of V(i)
Higher chance of being chosen

8. Example Route choosing ci: travel cost ($) of alternative i
ti: travel time (hours) of alternative i
ci: travel cost ($) of alternative i
B: annual income of individual (1000$)
V(i)=-0.2ti-1.0(ci/B)

9. Example Marginal rate of substitution between cost & time
Value of time per hour=20% of annual income
Te time value of a person with annual income of $20000 would be $4/h

10. Choice function is considered a random function
Stochastic Choice
A stochastic model is preferable because:
Idiosyncrasies of traveler behavior isn’t anticipated
It is impossible to include all the variables in the choice function
Potential traveler don’t have perfect information about system & alternatives
Choice function is considered a random function

11. Stochastic Choice Random utility model: U(i)=V(i)+e(i)
U(i)=choice function for alternative i
V(i)=deterministic function for alternative i
e(i)=a stochastic component
Statistical assumption are made regarding the distribution nature of e(i)

12. Stochastic Choice
The probability that the alternative i is chosen:

13. Stochastic Choice
So:
F is the joint distribution of the random component
fi(Φ) is the marginal density function of e(i)

14. the Probit Model
Random Utilities[U(i),U(j),…] have a multivariate normal distribution (MVN)
n :number of alternatives
бij :variance-covariance matrix

15. the Probit Model This is equal to: ei follows:
Multivariate normal distribution
With zero mean
Finite variance-covariance matrix

16. Max[U(1),U(2),…,U(n)]~N(Vmax,бmax^2)
the Probit Model
Probit model is obtained by combining (5-5),(5-6),(5-7):
1.Closed form
2.Approximate Closed form suggested by Clark
extensive & expensive
Max[U(1),U(2),…,U(n)]~N(Vmax,бmax^2)
U(i) multivariate normal variables
with means V(i)
and covariance бij

17. the Probit Model Clark:
Defining for any two normally distributed variable U1,U2
Clark:
ρ12: correlation coefficient
Φ: standard normal distribution
Ø: density function

18. the Probit Model
Clark:
And correlation between U3 and max of U1,U2

19. the Probit Model
And finally the choice probability is:

20. the Probit Model for two alternatives from (5-5)
for number of alternatives larger than 3:

21. Probit Model (example 1)
V=[-12,-10,-15] ;negative utilities such as cost or travel time ρ12=0.5 ;correlation of attribute 1,2 б12=0.5*2*2=2 ρ23=0

22. Probit Model (example)

23. Probit Model (example)
from equation (5-12):
In a similar manner:
p(2)=0.81
p(3)=0.05
p(1)+ p(2)+p(3)= =1.01
is sufficiently close to 1.0

24. Binary Probit Model
U(1),U(2) assumed independent and have normal distribution so:
from Eq.(5-12):

25. Binary Probit Model All or nothing vs. probit:
even with V(1)<V(2) there is a non zero probability of alternative 1
The larger utility function for a value the larger probability alternative choice

26. Logit Model
This model is obtained by assuming that random component e(i) of the choice utilities are IID:
Independent
Identically distributed with a Gumbel distribution:

27. Multinomial logit model (MNL)
By combining:
Multinomial logit model (MNL)

28. Route choice over complex network
Logit Model
Advantages:
Its parameters estimation, its application and interpretation is easier than the probit
Disadvantage:
Restricts to situations where alternatives have independent choice
Route choice over complex network

29. Logit Model (example)
Choice vector: V=[-12,-10,-15] Direct application of Eq.(5-20):

30. Comparison of Logit and Probit Models
Logit model
P(1)=0.12
P(2)=0.875
P(3)=0.005
The resulting logit model has the tendency to reduce the choice for low V as P(3) and increase it with high V as P(2)

31. Comparison of Logit and Probit Models
When the independence of the utilities is assumed, there isn’t much difference between the results
Binary logit and probit models of mode choice

32. Comparison of Logit and Probit Models
Alternatives with similar attributes, or with overlapping components that independence can’t be assumed, probit might be a better model
Alternatives that can be mutually exclusive, a logit model would be appropriate
Alternative routes overlap on a link
Intercity travel mode choice
Destination choice

33. Comparison of Logit and Probit Models(example)
Three different routes between point A & B:
if trip makers perceive the attributes randomly e(i) represent the difference between perceived & actual value of alternative
it is hardly likely that the difference between perceived & actual values for alternatives ІІ,ІІІ would be independent

34. The extent of error from independence assumption
x: measures the length of AC in comparison to AB
X=0 total overlap between ІІ and ІІІ
X=1 independent alternatives
So:

35. The extent of error from independence assumption
A choice model without dependence between e(ІІ), e(ІІІ) will always predict p(ІІІ)=0.33
A model with dependence term a more realistic result

36. The Independence of irrelevant alternatives (IIA)
Relative odds of choosing one alternative over another
Relative odds between any two alternatives are independent of any other alternatives
Considered a weakness of model that have this property such as Logit

37. The Independence of irrelevant alternatives (IIA)
for example in urban mode choice: Relative odds of taking automobile over taking a bus is independent of there is a train or not. But presence of a train as a third alternative affects probability of choosing the bus more

38. The Independence of irrelevant alternatives (IIA)
The binary logit model:
Can be derived as a deterministic choice model
This property leads to make logit model as intrinsically linear:

39. The Nested Logit model Mode choice problem including: Logit model:
Private car (0.33)
Red bus (0.33)
Blue bus (0.33)
Expectation:
Private car (0.50)
Red bus (0.25)
Blue bus (0.25)

40. The Nested Logit model Dependent alternatives
Difficulty of Probit model
The Nested Logit model developed:
MNL
Blue Bus
Red Bus
Car NL
Bus
Blue
Red
Car

41. The Nested Logit model NL
Bus
Blue
Red
Car k m

42. Calibration of choice models
Calibration process:
Estimate the parameters’ values
Evaluating the statistical signification of the estimates
Validating the model by comparing with observed date

43. Multinomial Disaggregate Models
Assume:
V(n,m)=∑i βimXinm
V(n,m): choice function constructed for each individual n and alternative m xinm: ith variable for alternative m as measured for individual n
The choice model:
Pn (m)= f [V(n,m)]= g [∑i βimXinm]

44. Multinomial Disaggregate Models
In estimating the model parameters βim, we observe a number of individuals:
Ynm=1; if individual n chooses alternative m
Ynm=0; Otherwise
Nm=ΣnYnm
Nm: number of individuals who choose alternative m in the observed sample

45. Multinomial Disaggregate Models
the likelihood of the observed sample is given by:
in order to find the maximum likelihood estimates of the parameter L, the logarithm function facilitate the procedure:

46. Multinomial Logit model
i: origin
j: destination

47. Multinomial Disaggregate Models
The maximum likelihood estimates of the parameter L can be obtained by:
ƏL/Əβim=0
The confidence intervals for βim^ thus estimated are asymptotically efficient, consistent, and normally distributed.

48. Goodness of fit
Logarithm of likelihood function is also presented as L(β)
Assuming no model (all β=0) the value of logarithm of likelihood function is presented by L(0)
Assuming market share (only constants in utility functions) the value of logarithm of likelihood function is presented by L(c)
Note: 0>L(β) > L(c) > L(0)

49. Goodness of fit L(0)=Σn Ln (Cn) L(c)=Σm Nm*Ln (Nm/N)
Cn : Number of choices for individual (market) n
L(c)=Σm Nm*Ln (Nm/N)
Nm: Number of individuals adopt alternative m
N: Sampled population

50. Goodness of fit Market share goodness of fit: Goodness of fit:
Goodness of fit regarding to market share
(Also known as Mc Fadden Goodness of fit):
Adjusted goodness of fit:
Adjusted goodness of fit regarding to market share:

51. Ratio of likelihood test
The Chi-square table is adopted:
Generally:
-2[L(βr)-L(βu)]~χ²ku-kr

52. Calibration softwares
NLOGIT 4.0
BIOGEM
LIMDEP
GAUSS 1.49