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

m

Varm ≤ Vark μk ≤ μm ɵk=μk /μm ≤1

μ=Scale parameter of Gumbel distribution

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NL

Bus

Blue Red

Car

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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Finish59