discrete choice models for modal split overview. outline n general procedure for model application n...

Discrete Choice ModelsDiscrete Choice Modelsfor Modal Splitfor Modal Split

OverviewOverview

OutlineOutline

General procedure for model applicationGeneral procedure for model application Basic assumptions in Random Utility ModelBasic assumptions in Random Utility Model Uncertainty in choiceUncertainty in choice Utility & Logit modelUtility & Logit model Numerical exampleNumerical example Application issues in four step modelApplication issues in four step model SummarySummary

Choice Model Formulation

Predict Exogenous Explanatory & Policy Variables

Estimate Dissagregate Choice Model(s)

Individual & Travel Data

Apply Prediction Procedure

Aggregate (TAZ) Travel Prediction

Insert Predicted Proportions for Each Mode in the Four Step Sequence

Theory from microeconomicsTheory from microeconomics

We will skip the more theoretical description of We will skip the more theoretical description of principles, theorems, lemasprinciples, theorems, lemas

Emphasize practical aspectsEmphasize practical aspects Look at examplesLook at examples Note: Dan McFadden is Note: Dan McFadden is Professor of EconomicsProfessor of Economics

and and Nobel Laureate in EconomicsNobel Laureate in Economics http://emlab.berkeley.edu/users/mcfadden/http://emlab.berkeley.edu/users/mcfadden/

A site that contains a very good bibliography on A site that contains a very good bibliography on Random Utility ModelsRandom Utility Models

http://emlab.berkeley.edu/users/mcfadden/nobel00.html

http://emlab.berkeley.edu/users/mcfadden/

Basic Assumptions (1)Basic Assumptions (1) Suppose a trip maker i faces J options (choices or

alternatives) with index j=1,2,3…J. Assume that each trip maker associates with each

choice j=1,2,...,J a function called UTILITY representing the "convenience" of choosing mode j.

j=1,2,..., J is called the choice set. This is the set from which a decision maker chooses an option.

Note: Let’s assume that choice and consideration sets are the same.

Basic Assumptions ExampleBasic Assumptions Example– A person, i, needs to go to work from home to the A person, i, needs to go to work from home to the

downtown area. downtown area. – Suppose this person has three possible modes to Suppose this person has three possible modes to

choose from: Car (j=1), Bus (j=2), and her Bike choose from: Car (j=1), Bus (j=2), and her Bike (j=3). Total number of options (J=3).(j=3). Total number of options (J=3).

– One possible form of the person’s convenience One possible form of the person’s convenience function (called utility) is:function (called utility) is:

Ucar=F (car attributes, person characteristics, trip attributes)

Ubus=F (bus attributes, person characteristics, trip attributes)

Ubike=F (bike attributes, person characteristics, trip attributes)

Utility componentsUtility components Variables describing the individual --> this is an attempt to represent

"taste variation" from person to person. In our example if young persons have systematically differing preferences from the older individuals, then, age would be one of the variables.

Variables describing the choice characteristics (called choice attributes) in the choice set. For example, some travel modes are less expensive than others. Cost of the trip for each available mode would be another variable in the utility. Travel time is another key variable.

Variables describing the context such as the trip type, time of day, budget constraints.

Key Assumption (maximum utility)Key Assumption (maximum utility) Travelers (decision makers) formulate for each Travelers (decision makers) formulate for each

option a utility and they calculate its value.option a utility and they calculate its value. Then, they choose the option with the most Then, they choose the option with the most

advantageous utility (maximum utility). advantageous utility (maximum utility). Example: UExample: U(car,bus,bike)(car,bus,bike)=-0.5*cost-2*waiting time=-0.5*cost-2*waiting time

Cost by bus=$1, Waiting time=5 minutesCost by bus=$1, Waiting time=5 minutes Cost by car=$2.5, Waiting time=1 minuteCost by car=$2.5, Waiting time=1 minute Cost by bike=$0.2, Waiting time=0 minutesCost by bike=$0.2, Waiting time=0 minutes

Which mode is the most desirable, second less desirable, etc?

Utility is actually an Indirect Conditional Utility

Uncertainty in utility (1)Uncertainty in utility (1) We (analysts) do not know all the factors that influence We (analysts) do not know all the factors that influence

choice behavior choice behavior Travelers (decision makers) do not always make choices Travelers (decision makers) do not always make choices

consistentlyconsistently We are not interested in including all possible variables We are not interested in including all possible variables

that affect behavior in our modelsthat affect behavior in our models We are interested in policy variables (taxes, fares, We are interested in policy variables (taxes, fares,

gasoline costs, waiting times) that we can “manipulate” gasoline costs, waiting times) that we can “manipulate” to find travelers reactionto find travelers reaction

We are also interested in social, demographic, and We are also interested in social, demographic, and economic traveler characteristics because these variables economic traveler characteristics because these variables allow us to link models to TAZsallow us to link models to TAZs

Incorporating uncertainty and Incorporating uncertainty and traveler/trip characteristics traveler/trip characteristics

The example becomes: UThe example becomes: U(bus,car,bike)(bus,car,bike)=-0.5*cost-=-0.5*cost-

2*waiting time + 2*waiting time + SOMETHING ELSESOMETHING ELSE

The “something else” is an indicator of “general The “something else” is an indicator of “general mode attractiveness” AND a random componentmode attractiveness” AND a random component

Let’s look at the details:Let’s look at the details:

Utility elements Utility elements

UUijij = = jj-0.5*cost-0.5*costjj-2*waiting -2*waiting

timetimejj + + jj * age * ageii + + ijij

Utility of person i for mode j



A constant for each mode j. Captures desirability of j

for unknown reasons




Waiting time is different for each mode j

Cost is different for each mode j




The effect of the age variable is different for each alternate mode

(Class: Let’s talk about behavioral meaning - bikes?)




The key indicator of uncertainty = our ignorance &

traveler variability for unknown reasons


In a similar way as for age we In a similar way as for age we can include other traveler and can include other traveler and trip characteristics (explanatory)trip characteristics (explanatory)

UUijij = = jj-0.5*cost-0.5*costjj-2*waiting time-2*waiting timejj

+ + jj * age * ageii + + ijij

In applications: These are parameters we estimate from data using regression methods




Can write as: UCan write as: Uijij = V = Vjjjj + + ijij


Systematic & measurable part

Random

Numerical exampleNumerical example(trip from home to work/school) (trip from home to work/school)

UUicaricar = = - 0.5*cost- 0.5*cost - 2*waiting time - 2*waiting time

+ + * age * ageii + + icaricar

UUibusibus = 5 - 0.5*cost = 5 - 0.5*cost - 2*waiting time - 2*waiting time

+ 0.25 * age+ 0.25 * ageii + + ibusibus

UUibikeibike = 12 - 0.5*cost = 12 - 0.5*cost - 2*waiting time - 2*waiting time

- 0.3 * age- 0.3 * ageii + + ibikeibikeNote: Different age coefficients - why?

Compare systematic part (V) Compare systematic part (V)

Compute for each person the systematic Compute for each person the systematic part of utility for each modepart of utility for each mode

Plot all V (syst. utilities) for all persons Plot all V (syst. utilities) for all persons Horizontal = ageHorizontal = age Vertical = V the systematic part of utility of Vertical = V the systematic part of utility of

each modeeach mode

Modal Utilities

-20

-10

0

10

20

0 20 40 60 80 100

Age

Uti

lity

Val

ue

Vcar

Vbus

Vbike

Probability of ChoiceProbability of Choice

We need to convert utilities to an estimate We need to convert utilities to an estimate of the chance to choose a modeof the chance to choose a mode

The specific equation to use depends on the The specific equation to use depends on the probability distribution of the random probability distribution of the random component (component () in the utility function ) in the utility function (U=V+(U=V+))

Ease of calculations should be considered in Ease of calculations should be considered in selecting a probability functionselecting a probability function

LOGIT ModelLOGIT Model Assume the random components (Assume the random components (ii) of the ) of the

utility are independent identically Gumbel utility are independent identically Gumbel distributed random variables then: distributed random variables then:

bikebus

carjij

icari

V

VcarP ,

)exp(

)exp()(

bikebus

carjij

ibusi

V

VbusP ,

)exp(

)exp()(

bikebus

carjij

ibikei

V

VbikeP ,

)exp(

)exp()(

Probability

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100

Age of Traveler

Pro

bab

ility

to

ch

oo

se a

m

od

e Pcar

Pbus

Pbike

ApplicationsApplications

Modal split (type of mode)Modal split (type of mode) Route choice (link by link or entire path)Route choice (link by link or entire path) Car ownership (type of car)Car ownership (type of car) Destination choice (shopping place)Destination choice (shopping place) Activity types (type of activity)Activity types (type of activity) Residential unit (size and type of home)Residential unit (size and type of home)

Practical issuesPractical issues

Choice set - consideration setChoice set - consideration set Variables to include in utilityVariables to include in utility Measurement of mode attributes (e.g.,in-Measurement of mode attributes (e.g.,in-

vehicle-travel-time)vehicle-travel-time) Need survey data and mode by mode Need survey data and mode by mode

attributes!attributes! Next: TAZ application and “complete” Next: TAZ application and “complete”

enumerationenumeration

Choice Model Formulation

Predict Exogenous Explanatory & Policy Variables

Estimate Dissagregate Choice Model(s)

Individual & Travel Data

Apply Prediction Procedure

Aggregate (TAZ) Travel Prediction

Insert in the Four Step Sequence

For the four step modal splitFor the four step modal split

We need aggregate TAZ proportions by We need aggregate TAZ proportions by each mode (% of trips by car, % trips by each mode (% of trips by car, % trips by bus, % trips by bike)bus, % trips by bike)

We have a disaggregate (individual) model We have a disaggregate (individual) model which tells us the likelihood (chance) of a which tells us the likelihood (chance) of a person to choose each modeperson to choose each mode

We need a procedure to go from We need a procedure to go from disaggregate predictions of chance to disaggregate predictions of chance to aggregate predictions of proportionsaggregate predictions of proportions

Taking Average TAZ Taking Average TAZ Characteristics Does Not WorkCharacteristics Does Not Work

(Pa+Pb)/2 is not the same as P ([Va+Vb]/2) - (Pa+Pb)/2 is not the same as P ([Va+Vb]/2) - a and b are value points for Va and b are value points for V

When the two are equated we have the Naïve When the two are equated we have the Naïve method of aggregationmethod of aggregation

Bias depends on how close the probability Bias depends on how close the probability function is to a linear function function is to a linear function

Following is an example from Probability to Following is an example from Probability to choose bus as an optionchoose bus as an option

Pbus

0

0.2

0.4

0.6

0.8

1

1.2

-5 0 5 10 15 20

Systematic Utility of Bus (Vbus)

Pro

abili

ty o

f C

ho

ice

Pbus

V=2 V=12

P(V=2)=0.034

P(V=12)=0.679

Consider a TAZ with two persons with V=2 &V=12

What is the correct TAZ What is the correct TAZ Proportion of Choosing the Bus?Proportion of Choosing the Bus?

(P(V=2)+P(V=12))/2(P(V=2)+P(V=12))/2 oror P((2+12)/2)=P(V=7)P((2+12)/2)=P(V=7)

Pbus

0

0.2

0.4

0.6

0.8

1

1.2

-5 0 5 10 15 20


Pro

abili

ty o

f C

ho

ice

Pbus

V=2 V=7 V=12

P(V=2)=0.034

P(V=12)=0.679

P(V=7)=0.223

The correct value is: [P(V=2)+P(V=12)]/2=0.357

Pbus

0

0.2

0.4

0.6

0.8

1

1.2

-5 0 5 10 15 20


Pro

abili

ty o

f C

ho

ice

Pbus

V=2 V=7 V=12

P(V=7)=0.223

[P(V=2)+P(V=12)]/2=0.357

Bias (see page 310 OW)

Naïve AggregationNaïve Aggregation

For each TAZ take the average value of For each TAZ take the average value of explanatory variablesexplanatory variables

Compute average value for each utility Compute average value for each utility function for each modefunction for each mode

Compute the corresponding probability and Compute the corresponding probability and use it as the TAZ proportion choosing each use it as the TAZ proportion choosing each modemode

Market SegmenationMarket Segmenation

Divide the residents in each TAZ into Divide the residents in each TAZ into relatively homogeneous segmentsrelatively homogeneous segments

Apply Naïve aggregation to each segment Apply Naïve aggregation to each segment and get proportions for each modeand get proportions for each mode

Compute the TAZ proportion either as Compute the TAZ proportion either as average segment-specific proportion or average segment-specific proportion or weighted segment-specific proportionweighted segment-specific proportion

Complete EnumerationComplete Enumeration

Compute for each person and for each mode Compute for each person and for each mode the probability to choose a modethe probability to choose a mode

Compute the proportion for each mode as Compute the proportion for each mode as an average of the individual probabilitiesan average of the individual probabilities

Stochastic microsimulation is a method Stochastic microsimulation is a method derived from this - see also Chapter 12 of derived from this - see also Chapter 12 of Goulias, 2003 (red book)Goulias, 2003 (red book)

Example Example (TAZ with four persons)(TAZ with four persons)

Age Vcar Vbus VbikeSegment 1 45 7.500 5.750 -1.600Segment 2 21 3.900 -0.250 5.600Segment 2 20 3.750 -0.500 5.900Segment 3 79 12.600 14.250 -11.800Average 41.25 6.938 4.813 -0.475Exp (U) 1030.192 123.039 0.622Naïve Prob 0.893 0.107 0.001

Vicar = - 0.5*cost - 2*waiting time + * agei

Vibus = 5 - 0.5*cost - 2*waiting time + 0.25 * agei

Vibike = 12 - 0.5*cost - 2*waiting time - 0.3 * agei

Compare values of the three Compare values of the three methodsmethods

Pcar Pbus PbikeAverage of Segments 0.372 0.329 0.298Weigthed Averageof Segments 0.305 0.247 0.447

Naïve Aggregation 0.893 0.107 0.001

Complete Enumeration 0.318 0.248 0.434

Theoretical issuesTheoretical issues

Gumbel IID convenient but is it realistic?Gumbel IID convenient but is it realistic? IID components imply unrelated options in IID components imply unrelated options in

the unobserved components - new models the unobserved components - new models account for relationsaccount for relations

Trips are related - different formulationsTrips are related - different formulations See CE 523See CE 523

Additional sourcesAdditional sources

http://www.bts.gov/ntl/DOCS/SICM.html http://www.bts.gov/ntl/DOCS/SICM.html (Spear’s report on how to apply models)(Spear’s report on how to apply models)

http://www.bts.gov/ntl/DOCS/UT.html http://www.bts.gov/ntl/DOCS/UT.html (self-instructional overview with examples)(self-instructional overview with examples)

http://www.tfhrc.gov///////safety/pedbike/http://www.tfhrc.gov///////safety/pedbike/vol2/sec2.5.htm (simple description of most vol2/sec2.5.htm (simple description of most of the key issues)of the key issues)

All sites accessed September 22, 2003All sites accessed September 22, 2003

SummarySummary Rational economic behaviorRational economic behavior Utility linear in systematic and random Utility linear in systematic and random

componentscomponents Choice probability is function of utilities – Choice probability is function of utilities –

non linear function!non linear function! Application by enumeration is best - Application by enumeration is best -

weighted average by market segments may weighted average by market segments may be good - depends on application!be good - depends on application!

Aggregate models are also available – Aggregate models are also available – approximate! approximate!

Surveys must be used for this stepSurveys must be used for this step

Additional reading suggestionsAdditional reading suggestions(for future reference)(for future reference)

Ortuzar Willumsen - Chapter 8 (8.1, 8.2, Ortuzar Willumsen - Chapter 8 (8.1, 8.2, 8.3)8.3)

Ortuzar Willumsen - Chapter 9 (9.1, 9.2, Ortuzar Willumsen - Chapter 9 (9.1, 9.2, 9.3)9.3)

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