meeghat habibian analysis of travel choice transportation demand analysis lecture note
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Meeghat Habibian
Analysis of Travel Choice
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Choice context in transportation
Destination choice
Mode choice
Route choice
AlsoTravel start time choice
Freight transportation agent choice
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Measurement of ChoiceChoice can be reflected by:
Number of people
Proportion of populationTherefore, it can be:
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Independent of the total number of population
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Choice Process
1. Deterministic
The Decision rule that is used by traveler is consistent and stable
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The Choice made will be consistently the same
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Deterministic Choice (Individual Level)
V(i)=Ai*Xi
V(i): choice functionXi : a vector of demand and supply variables
Ai : a vector of parameters that represent the effect of each variable
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The choice rule:
V(j)=max[V(i)]
Example
Route choiceThe choice function
V(i)=-0.2ti-1.0(ci/B)
ti: travel time (hours) of alternative i
ci: travel cost ($) of alternative i
B: annual income of individual (1000$)
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Example
Marginal rate of substitution between cost and time
As: Therefore:
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Value of time per hour=20% of annual income/1000
The time value of a person with annual income of $20000 would be $4/h
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𝑑𝑦𝑑𝑥
Choice Process
Due to:Behavior of choice makerAbsence of rational and consistent decision rule
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Provide a far superior means for predicting travel behavior than Deterministic one
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2. Stochastic
Stochastic Choice
A stochastic model is preferable because:
1. Idiosyncrasies of traveler behavior isn’t anticipated2. It is impossible to include all the variables in the
choice function3. Potential traveler don’t have full information about
system and alternatives
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Choice function is considered as a random function
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Stochastic ChoiceRandom 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
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Statistical assumptions are made regarding to the distributional nature of e(i)
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Stochastic Choice
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The probability that the alternative i is chosen:
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Stochastic Choice
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Therefore:
F: Joint distribution of the random componentfi(Φ): Density function of e(i)
Based on fi(Φ), different structures can be derived Due to lack of knowledge about the error term a number
of assumptions could be made (e.g., Normal distribution)
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The Probit Model
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The Probit Model Random Utilities [U(i),U(j),…] have a multivariate normal
distribution (MVN)
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n: Number of alternativesбij: Variance-covariance elements
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The Probit Model
This is equal to:
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ei follows: Multivariate normal distribution With zero mean A finite variance-covariance matrix
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The Probit Model
The resulted model is extensive and expensive No closed form
An approximated closed form is suggested by Clark
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Max [U(1),U(2),…,U(n)]~ N( Vmax, бmax2)
U(i) multivariate normal variableswith means V(i)and covariance elements бij
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The Probit Model Defining for any two normally distributed variable U1, U2
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ρ12: correlation coefficient
Φ: standard normal distributionØ: density function
Clark:
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+
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The Probit Model
And correlation between U3 and max of U1, U2
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Clark:
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The Probit Model
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And finally the choice probability is:
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The Probit Model For two alternatives from (5-5)
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For more than 3 alternatives:
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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
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Example 1
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Example 1from equation (5-12):
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In a similar manner:p(2)=0.81p(3)=0.05
p(1)+ p(2)+p(3)=0.15+0.81+0.05=1.01is sufficiently close to 1.0
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Binary Probit Model U(1),U(2) assumed independent and have normal distribution so:
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from Eq.(5-12):
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Binary Probit Model
Even V(1)<V(2) there is a non zero probability of choosing alternative 1
The larger utility function for an alternative the larger probability of its choice
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All or nothing vs. Probit:
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The Logit Model
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The Logit Model
This model is obtained by assuming that the random component, e(i), of choice utilities are IID:
Independent Identically distributed through a Gumbel distribution:
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ɵ= Scale parameter of Gumbel distribution
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The Logit Model
By combining:
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The Logit Model Formulation
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Multinomial Logit Model (MNL)
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The Logit Model
Advantages: Easier (than the Probit) in terms of
Parameters estimation Application Interpretation
Disadvantage: Restricted to the situations where alternatives have independent choices
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Route choice over a complex network
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Example
Choice vector: V=[-12,-10,-15]Direct application of Eq.(5-20):
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Comparison of Logit and
Probit Models
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Comparison of Logit and Probit Models
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Logit model
P(1)=0.12P(2)=0.875P(3)=0.005
Probit model
P(1)=0.15P(2)=0.81P(3)=0.05
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)
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Comparison of Logit and Probit Models Assuming independence of the utilities
There is not much difference between the results
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Binary logit and probit models of
mode choice
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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
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Alternative routes overlap on a link
Intercity travel mode choiceDestination choice
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Example
I: can represent a highway modeII: a bus mode with walking accessIII: a bus mode with walking access
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Three different routes between points A and B:
The overlap part: The alternatives ІІ and ІІІ are not independent
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The Extent of Error from Independence Assumption
x: measures the length of AC in comparison to AB
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X=0 total overlap between ІІ and ІІІX=1 independent alternatives
Assuming identical utility function V:
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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 shows a more realistic result
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Covariance of alternatives II and III is a function of x
The Independence of Irrelevant Alternatives (IIA)
Relative odds of choosing one alternative over another
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Relative odds between any two alternatives are independent of any other alternatives
Strength or weakness of model that have this property such as Logit
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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
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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:
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The Nested Logit Model
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The Nested Logit Model
Remember the problem: 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)
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The Nested Logit ModelDue to: Application of dependent alternatives Difficulty of Probit model
The Nested Logit structure is developed:
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NL
Bus
Blue Red
Car
MNL
Blue Bus Red Bus Car
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The Nested Logit Model
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k
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Varm ≤ Vark μk ≤ μm ɵk=μk /μm ≤1
μ=Scale parameter of Gumbel distribution
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Calibration of Choice Models
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Calibration of Choice Models
Estimate the parameters’ values
Performing the statistical test
Validating the model by comparing with observed
data
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Multinomial Disaggregate Models
V(n,m): choice function constructed for each individual n and
alternative m
xinm: ith variable for alternative m as measured for individual n
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Assume:
The choice model:
V(n,m)=∑i βimXinm
Pn (m)= f [V(n,m)]= g [∑i βimXinm]
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Multinomial Disaggregate Models
To estimate the model parameters βim, we define:
1 if individual n chooses alternative m 0 Otherwise
Nm=ΣnYnm
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Nm: number of individuals who choose alternative m in the sample
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Ynm=
The Likelihood Function
The likelihood of the observed sample is given by:
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To facilitate the procedure:
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The Likelihood Function
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i: originj: destination
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Multinomial Disaggregate Models
The maximum of LL can be obtained by:
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The confidence intervals for βim^ are asymptotically efficient.
ƏLL/Əβim=0
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Goodness of Fit
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Definitions
Logarithm of likelihood function is also presented as L(β)
Assuming no variable in model (all β=0) L(0)
Assuming market share (only constants in utility functions) L(c)
Note: L(0)<L(c)<L(β)<0
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Relations
L(0)=Σn Ln (Cn)– 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
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Goodness of Fit measures
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:
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Ratio of Likelihood Test
The Chi-square table is adopted:
-2[L(C)-L(β )]~χ²k
Generally:
-2[L(βr)-L(βu)]~χ²ku-kr
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Calibration Software Program
• NLOGIT 5.0
• BIOGEME
• LIMDEP
• GAUSS 1.49
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