Discrete Choice Analysis II

Transcription

1 Discrete Choice Analysis II Moshe Ben-Akiva / / ESD.210 Transportation Systems Analysis: Demand & Economics Fall 2008

2 Review Last Lecture Introduction to Discrete Choice Analysis A simple example route choice The Random Utility Model Systematic utility Random components Derivation of the Probit and Logit models Binary Probit Binary Logit Multinomial Logit 2

3 Outline This Lecture Model specification and estimation Aggregation and forecasting Independence from Irrelevant Alternatives (IIA) property Motivation for Nested Logit Nested Logit - specification and an example Appendix: Nested Logit model specification Advanced Choice Models 3

4 Specification of Systematic Components Types of Variables Attributes of alternatives: Z in, e.g., travel time, travel cost Characteristics of decision-makers: S n, e.g., age, gender, income, occupation Therefore: X in = h(z in, S n ) Examples: X in1 = Z in1 = travel cost X in2 = log(z in2 ) = log (travel time) X in3 = Z in1 /S n1 = travel cost / income Functional Form: Linear in the Parameters V in = β 1 X in1 + β 2 X in β k X ink V jn = β 1 X jn1 + β 2 X jn β k X jnk 4

5 Data Collection Data collection for each individual in the sample: Choice set: available alternatives Socio-economic characteristics Attributes of available alternatives Actual choice n Income Auto Time Transit Time Choice Auto Transit Auto Auto Transit 6 15 N/A 48.4 Transit 5

6 Model Specification Example V auto = β 0 + β 1 TT auto + β 2 ln(income) V transit = β 1 TT transit β 0 β 1 β 2 Auto 1 TT auto ln(income) Transit 0 TT transit 0 6

7 Probabilities of Observed Choices Individual 1: V auto = β 0 + β β 2 ln(35) V transit = β e β β 1 +ln(35)β 2 P(Auto) = e β β 1 +ln(35)β 2 + e 58.2β 1 Individual 2: V auto = β 0 + β β 2 ln(45) V transit = β e 31.0β 1 P(Transit) = e β β 1 +ln(45)β 2 + e 31.0β 1 7

8 Maximum Likelihood Estimation Find the values of β that are most likely to result in the choices observed in the sample: max L*(β) = P 1 (Auto)P 2 (Transit) P 6 (Transit) If y in = 1, if person n chose alternative i 0, if person n chose alternative j Then we maximize, over choices of {β 1, β 2, β k }, the following expression: N L * ( β 1, β 2,..., β k ) = P n (i ) y in P n ( j ) y jn n =1 β* = arg max β L* (β 1, β 2,, β k ) = arg max β log L* (β 1, β 2,, β k ) 8

9 Sources of Data on User Behavior Revealed Preferences Data Travel Diaries Field Tests Stated Preferences Data Surveys Simulators 9

10 Stated Preferences / Conjoint Experiments Used for product design and pricing For products with significantly different attributes When attributes are strongly correlated in real markets Where market tests are expensive or infeasible Uses data from survey trade-off experiments in which attributes of the product are systematically varied Applied in transportation studies since the early 1980s 10

11 Aggregation and Forecasting Objective is to make aggregate predictions from A disaggregate model, P( i X n ) Which is based on individual attributes and characteristics, X n Having only limited information about the explanatory variables 11

12 The Aggregate Forecasting Problem The fraction of population T choosing alt. i is: W( i ) = P ( i X ) p ( X dx X N T ( ), p(x) is the density function of X = 1 P i X n ), N T is the # in the population of interest N T n=1 Not feasible to calculate because: We never know each individual s complete vector of relevant attributes p(x) is generally unknown The problem is to reduce the required data 12

13 Sample Enumeration Use a sample to represent the entire population For a random sample: N s Wˆ (i) = 1 Pˆ(i x n ) where N s is the # of obs. in sample N s n=1 For a weighted sample: N s w n W ˆ ( i ) = Pˆ ˆp ( i x n ), where 1 is x n 's selection prob. n=1 w n w n n No aggregations bias, but there is sampling error 13

14 Disaggregate Prediction Generate a representative population Apply demand model Calculate probabilities or simulate decision for each decision maker Translate into trips Aggregate trips to OD matrices Assign traffic to a network Predict system performance 14

15 Generating Disaggregate Populations Household surveys Exogenous forecasts Census data Counts Data fusion (e.g., IPF, HH evolution) Representative Population 15

16 Review Empirical issues Model specification and estimation Aggregate forecasting Next More theoretical issues Independence from Irrelevant Alternatives (IIA) property Motivation for Nested Logit Nested Logit - specification and an example 16

17 Summary of Basic Discrete Choice Models Binary Probit: Binary Logit: Multinomial Logit: V n 1 2 P n n = Φ V n 1 ε e 2 dε (i C ) ( ) = P n ( n ) = 1+ 1 i C = P ( i C ) = e e V n V in n n V j C n e jn e V in 2π e V in e Vjn + 17

18 Independence from Irrelevant Alternatives (IIA) Property of the Multinomial Logit Model ε jn independent identically distributed (i.i.d.) ε jn ~ ExtremeValue(0,µ) j P i C e µv in n ( n ) = e µv jn j C n so P( i C 1 ) P(i C = P( ) P( j C 2 ) i, j, C 1, C 2 j C 1 2 ) such that i, j C 1, i, j C 2, C 1 C n and C 2 C n 18

19 Examples of IIA Route choice with an overlapping segment T-δ O Path 2 a b δ D Path 1 T e µt 1 P( 1 { 1, 2 a, 2b }) = P(2a { 1, 2 a, 2b}) = P(2 b { 1, 2 a,2b}) = = e µt 3 j { 1, 2 a,2b} 19

20 Red Bus / Blue Bus Paradox Consider that initially auto and bus have the same utility C n = {auto, bus} and V auto = V bus = V P(auto) = P(bus) = 1/2 Suppose that a new bus service is introduced that is identical to the existing bus service, except the buses are painted differently (red vs. blue) C n = {auto, red bus, blue bus}; V red bus = V blue bus = V Logit now predicts P(auto) = P(red bus) = P(blue bus) =1/3 We d expect P(auto) =1/2, P(red bus) = P(blue bus) =1/4 20

21 IIA and Aggregation Divide the population into two equally-sized groups: those who prefer autos, and those who prefer transit Mode shares before introducing blue bus: Population Auto Share Red Bus Share Auto people 90% 10% P(auto)/P(red bus) = 9 Transit people 10% 90% P(auto)/P(red bus) = 1/9 Total 50% 50% Auto and red bus share ratios remain constant for each group after introducing blue bus: Population Auto Share Red Bus Share Blue Bus Share Auto people 81.8% 9.1% 9.1% Transit people 5.2% 47.4% 47.4% Total 43.5% 28.25% 28.25% 21

22 Motivation for Nested Logit Overcome the IIA Problem of Multinomial Logit when Alternatives are correlated (e.g., red bus and blue bus) Multidimensional choices are considered (e.g., departure time and route) 22

23 Tree Representation of Nested Logit Example: Mode Choice (Correlated Alternatives) motorized non-motorized auto transit bicycle walk drive carpool bus metro alone 23

24 Tree Representation of Nested Logit Example: Route and Departure Time Choice (Multidimensional Choice) Route 1 Route 2 Route 3 8:10 8:20 8:30 8:40 8: :10 8:20 8:30 8:40 8:50 Route 1 Route 2 Route 3 24

25 Nested Model Estimation Logit at each node Utilities at lower level enter at the node as the inclusive value I NM = ln e i C NM V i Nonmotorized (NM) Motorized (M) Walk Bike Car Taxi Bus The inclusive value is often referred to as logsum 25

26 Nested Model Example Nonmotorized (NM) Motorized (M) Walk Bike Car Taxi Bus µ NM V e i P(i NM ) = e µ NM V Walk + e µ NM V Bike i = Walk, Bike I NM = 1 µ NM ln( e µ NM V Walk + e µ NM V Bike ) 26

27 Nested Model Example Nonmotorized (NM) Motorized (M) Walk Bike Car Taxi Bus µ M V e i P(i M ) = e µ M V Car + e µ MV Taxi + e µ MV Bus i = Car,Taxi, Bus I M = 1 ln( e µ MV Car + e µ MV Taxi + e µ MV Bus ) µ M 27

28 Nested Model Example Nonmotorized (NM) Motorized (M) Walk Bike Car Taxi Bus e µi NM P(NM ) = µi NM µi e + e M e µi M P(M ) = e µi NM + e µi M 28

29 Nested Model Example Calculation of choice probabilities P(Bus ) = P(Bus M) P(M) e µ M V Bus µim e = e µ M +e µ M +e µ M NM e +e V Car V Taxi V Bus µi µi e µ M V Bus e µ M = e µ M V Car +eµ M V Taxi +eµ M V Bus µ µ µ µ e NM VWalk e NM VBike e µ NM +e µ M M µ ln( e µ M VCar +e µ MVTaxi +e µ MVBus ) µ ln( + ) ln( e M VCar + e µ MVTaxi + e µ MVBus ) 29

30 Extensions to Discrete Choice Modeling Multinomial Probit (MNP) Sampling and Estimation Methods Combined Data Sets Taste Heterogeneity Cross Nested Logit and GEV Models Mixed Logit and Probit (Hybrid Models) Latent Variables (e.g., Attitudes and Perceptions) Choice Set Generation 30

31 Summary Introduction to Discrete Choice Analysis A simple example The Random Utility Model Specification and Estimation of Discrete Choice Models Forecasting with Discrete Choice Models IIA Property - Motivation for Nested Logit Models Nested Logit 31

32 Additional Readings Ben-Akiva, M. and Bierlaire, M. (2003), Discrete Choice Models With Applications to Departure Time and Route Choice, The Handbook of Transportation Science, 2nd ed., (eds.) R.W. Hall, Kluwer, pp Ben-Akiva, M. and Lerman, S. (1985), Discrete Choice Analysis, MIT Press, Cambridge, Massachusetts. Train, K. (2003), Discrete Choice Methods with Simulation, Cambridge University Press, United Kingdom. And/Or take next semester! 32

33 Appendix Nested Logit model specification Cross-Nested Logit Logit Mixtures (Continuous/Discrete) Revealed + Stated Preferences

34 Nested Logit Model Specification Partition C n into M non-overlapping nests: C mn C m n = m m Deterministic utility term for nest C mn : V ~ V C = ~ V + 1 ln e µ m jn mn C mn µ m j C mn Model: P(i C n ) = P(C mn C n )P(i C mn ), i C mn C n where e µ ~ mv in e µ V Cmn P(C mn C n ) = and P(i C mn ) = ~ e µ V Cln l e µ mv jn j C mn 34

35 Continuous Logit Mixture Example: Combining Probit and Logit Error decomposed into two parts Probit-type portion for flexibility i.i.d. Extreme Value for tractability An intuitive, practical, and powerful method Correlations across alternatives Taste heterogeneity Correlations across space and time Requires simulation-based estimation 35

36 Cont. Logit Mixture: Error Component Illustration Utility: U auto U bus = β X auto = β X bus U subway = β X sub w ay Probability: + ξ + ν + ξ auto bus subw ay +ν auto bus + ξ + ν ν i.i.d. Extreme Value subw a y ε e.g. ξ ~ N(0,Σ) β X auto +ξ e auto Λ(auto X, ξ ) = e β X auto +ξ auto β X + e bus +ξ bus + e β X subway +ξ subway ξ unknown P(auto X ) = Λ(auto X, ξ ) f (ξ )d ξ ξ 36

37 Continuous Logit Mixture Random Taste Variation Logit: β is a constant vector Can segment, e.g. β low inc, β med inc, β high inc Logit Mixture: β can be randomly distributed Can be a function of personal characteristics Distribution can be Normal, Lognormal, Triangular, etc 37

38 Discrete Logit Mixture Latent Classes Main Postulate: Unobserved heterogeneity is generated by discrete or categorical constructs such as Different decision protocols adopted Choice sets considered may vary Segments of the population with varying tastes Above constructs characterized as latent classes 38

39 Latent Class Choice Model P( i ) = S Λ( i s) Q( s) s=1 Class-specific Choice Model Class Membership Model (probability of (probability of choosing i belonging to conditional on class s) belonging to class s) 39

40 Summary of Discrete Choice Models Logit NL/CNL Probit Logit Mixture Handles unobserved taste heterogeneity Flexible substitution pattern No No Yes Yes No Yes Yes Yes Handles panel data No No Yes Yes Requires error terms normally distributed Closed-form choice probabilities available Numerical approximation and/or simulation needed No No Yes No Yes Yes No No (cont.) Yes (discrete) No No Yes Yes (cont.) No (discrete) 40

41 6. Revealed and Stated Preferences Revealed Preferences Data Travel Diaries Field Tests Stated Preferences Data Surveys Simulators 41

42 Stated Preferences / Conjoint Experiments Used for product design and pricing For products with significantly different attributes When attributes are strongly correlated in real markets Where market tests are expensive or infeasible Uses data from survey trade-off experiments in which attributes of the product are systematically varied Applied in transportation studies since the early 1980s Can be combined with Revealed Preferences Data Benefit from strengths Correct for weaknesses Improve efficiency 42

43 Framework for Combining Data Attributes of Alternatives & Characteristics of Decision-Maker Utility Stated Preferences Revealed Preferences 43

44 MIT OpenCourseWare J / J / ESD.210J Transportation Systems Analysis: Demand and Economics Fall 2008 For information about citing these materials or our Terms of Use, visit: