Introduction to mixtures in discrete choice models

Transcription

1 Introduction to mixtures in discrete choice models p. 1/42 Introduction to mixtures in discrete choice models Michel Bierlaire Transport and Mobility Laboratory

2 Introduction to mixtures in discrete choice models p. 2/42 Outline Mixtures Capturing correlation Alternative specific variance Taste heterogeneity Latent classes

3 Introduction to mixtures in discrete choice models p. 3/42 Mixtures In statistics, a mixture probability distribution function is a convex combination of other probability distribution functions. If f(ε, θ) is a distribution function, and if w(θ) is a non negative function such that w(θ)dθ = 1 then θ g(ε) = θ w(θ)f(ε, θ)dθ is also a distribution function. We say that g is a w-mixture of f. If f is a logit model, g is a continuous w-mixture of logit If f is a MEV model, g is a continuous w-mixture of MEV

4 Introduction to mixtures in discrete choice models p. 4/42 Mixtures Discrete mixtures are also possible. If w i, i = 1,...,n are non negative weights such that then g(ε) = n w i = 1 i=1 n w i f(ε, θ i ) i=1 is also a distribution function where θ i, i = 1,...,n are parameters. We say that g is a discrete w-mixture of f.

5 Introduction to mixtures in discrete choice models p. 5/42 Example: discrete mixture of normal distributions 2.5 N(5,0.16) N(8,1) 0.6 N(5,0.16) N(8,1)

6 Introduction to mixtures in discrete choice models p. 6/42 Example: discrete mixture of binary logit models P(1 s=1,x) P(1 s=2,x) 0. 4 P(1 s=1,x) P(1 s=2,x)

7 Introduction to mixtures in discrete choice models p. 7/42 Mixtures General motivation: generate flexible distributional forms For discrete choice: correlation across alternatives alternative specific variances taste heterogeneity...

8 Introduction to mixtures in discrete choice models p. 8/42 Continuous Mixtures of logit Combining probit and logit Error decomposed into two parts U in = V in + ξ + ν Normal distribution (probit): flexibility i.i.d EV (logit): tractability

9 Introduction to mixtures in discrete choice models p. 9/42 Logit Utility: ν i.i.d. extreme value Probability: U auto = βx auto + ν auto U bus = βx bus + ν bus U subway = βx subway + ν subway Λ(auto X) = e βx auto e βx auto + e βx bus + e βx subway

10 Introduction to mixtures in discrete choice models p. 10/42 Normal mixture of logit Utility: U auto = βx auto + ξ auto + ν auto U bus = βx bus + ξ bus + ν bus U subway = βx subway + ξ subway + ν subway ν i.i.d. extreme value, ξ N(0, Σ) Probability: Λ(auto X, ξ) = e βx auto+ξ auto e βx auto+ξ auto + e βx bus +ξ bus + e βx subway +ξ subway P(auto X) = ξ Λ(auto X, ξ)f(ξ)dξ

11 Introduction to mixtures in discrete choice models p. 11/42 Capturing correlations: nesting Utility: U auto = βx auto + ν auto U bus = βx bus + σ transit η transit + ν bus U subway = βx subway + σ transit η transit + ν subway ν i.i.d. extreme value, η transit N(0, 1), σ 2 transit =cov(bus,subway) Probability: Λ(auto X, η transit ) = e βx auto e βx auto + e βx bus +σ transit η transit + e βx subway+σ transit η transit P(auto X) = η Λ(auto X, ξ)f(η)dη

12 Introduction to mixtures in discrete choice models p. 12/42 Nesting structure Example: residential telephone ASC_BM ASC_SM ASC_LF ASC_EF BETA_C σ M σ F BM ln(cost(bm)) η M 0 SM ln(cost(sm)) η M 0 LF ln(cost(lf)) 0 η F EF ln(cost(ef)) 0 η F MF ln(cost(mf)) 0 η F

13 Introduction to mixtures in discrete choice models p. 13/42 Nesting structure Identification issues: If there are two nests, only one σ is identified If there are more than two nests, all σ s are identified Walker (2001) Results with 5000 draws..

14 NL NML NML NML NML σ F = 0 σ M = 0 σ F = σ M L Value Scaled Value Scaled Value Scaled Value Scaled Value Scaled ASC BM ASC EF ASC LF ASC SM B LOGCOST FLAT MEAS σ F σ M σ 2 F + σ2 M

15 Introduction to mixtures in discrete choice models p. 14/42 Comments The scale of the parameters is different between NL and the mixture model Normalization can be performed in several ways σ F = 0 σ M = 0 σ F = σ M Final log likelihood should be the same But... estimation relies on simulation Only an approximation of the log likelihood is available Final log likelihood with draws: Unnormalized: σ M = σ F : σ F = 0: σ M = 0:

16 Introduction to mixtures in discrete choice models p. 15/42 Cross nesting 5 5Nest 1 5Nest Bus Train Car Ped. Bike U bus = V bus +ξ 1 +ε bus U train = V train +ξ 1 +ε train U car = V car +ξ 1 +ξ 2 +ε car U ped = V ped +ξ 2 +ε ped U bike = V bike +ξ 2 +ε bike P(car) = P(car ξ 1, ξ 2 )f(ξ 1 )f(ξ 2 )dξ 2 dξ 1 ξ 2 ξ 1

17 Introduction to mixtures in discrete choice models p. 16/42 Identification issue Not all parameters can be identified For logit, one ASC has to be constrained to zero Identification of NML is important and tricky See Walker, Ben-Akiva & Bolduc (2007) for a detailed analysis

18 Introduction to mixtures in discrete choice models p. 17/42 Alternative specific variance Error terms in logit are i.i.d. and, in particular, have the same variance U in = β T x in + ASC i + ε in ε in i.i.d. extreme value Var(ε in ) = π 2 /6µ 2 In order allow for different variances, we use mixtures U in = β T x in + ASC i + σ i ξ i + ε in where ξ i N(0, 1) Variance: Var(σ i ξ i + ε in ) = σ 2 i + π2 6µ 2

19 Introduction to mixtures in discrete choice models p. 18/42 Alternative specific variance Identification issue: Not all σs are identified One of them must be constrained to zero Not necessarily the one associated with the ASC constrained to zero In theory, the smallest σ must be constrained to zero In practice, we don t know a priori which one it is Solution: 1. Estimate a model with a full set of σs 2. Identify the smallest one and constrain it to zero.

20 Introduction to mixtures in discrete choice models p. 19/42 Alternative specific variance Example with Swissmetro ASC_CAR ASC_SBB ASC_SM B_COST B_FR B_TIME Car cost 0 time Train cost freq. time Swissmetro cost freq. time + alternative specific variance

21 Logit ASV ASV norm. L Value Scaled Value Scaled Value Scaled ASC CAR ASC SM B COST B FR B TIME SIGMA CAR SIGMA TRAIN SIGMA SM

22 Introduction to mixtures in discrete choice models p. 20/42 Taste heterogeneity Population is heterogeneous Taste heterogeneity is captured by segmentation Deterministic segmentation is desirable but not always possible Distribution of a parameter in the population

23 Introduction to mixtures in discrete choice models p. 21/42 Random parameters U i = β t T i + β c C i + ε i U j = β t T j + β c C j + ε j Let β t N( β t, σt 2 ), or, equivalently, β t = β t + σ t ξ, with ξ N(0, 1). U i = β t T i + σ t ξt i + β c C i + ε i U j = β t T j + σ t ξt j + β c C j + ε j If ε i and ε j are i.i.d. EV and ξ is given, we have P(i ξ) = e β t T i +σ t ξt i +β c C i, and e β t T i +σ t ξt i +β c C i + e βt T j +σ t ξt j +β c C j P(i) = P(i ξ)f(ξ)dξ. ξ

24 Introduction to mixtures in discrete choice models p. 22/42 Random parameters Example with Swissmetro ASC_CAR ASC_SBB ASC_SM B_COST B_FR B_TIME Car cost 0 time Train cost freq. time Swissmetro cost freq. time B_TIME randomly distributed across the population, normal distribution

25 Introduction to mixtures in discrete choice models p. 23/42 Random parameters Logit RC L ASC_CAR_SP ASC_SM_SP B_COST B_FR B_TIME S_TIME Prob(B_TIME 0) 8.8% χ

26 Introduction to mixtures in discrete choice models p. 24/42 Random parameters 25 Distribution of B_TIME

27 Introduction to mixtures in discrete choice models p. 25/42 Random parameters Example with Swissmetro ASC_CAR ASC_SBB ASC_SM B_COST B_FR B_TIME Car cost 0 time Train cost freq. time Swissmetro cost freq. time B_TIME randomly distributed across the population, log normal distribution

28 Introduction to mixtures in discrete choice models p. 26/42 Random parameters [Utilities] 11 SBB_SP TRAIN_AV_SP ASC_SBB_SP * one + B_COST * TRAIN_COST + B_FR * TRAIN_FR 21 SM_SP SM_AV ASC_SM_SP * one + B_COST * SM_COST + B_FR * SM_FR 31 Car_SP CAR_AV_SP ASC_CAR_SP * one + B_COST * CAR_CO [GeneralizedUtilities] 11 - exp( B_TIME [ S_TIME ] ) * TRAIN_TT 21 - exp( B_TIME [ S_TIME ] ) * SM_TT 31 - exp( B_TIME [ S_TIME ] ) * CAR_TT

29 Introduction to mixtures in discrete choice models p. 27/42 Random parameters Logit RC-norm. RC-logn ASC_CAR_SP ASC_SM_SP B_COST B_FR B_TIME S_TIME Prob(β > 0) 8.8% 0.0% χ

30 Introduction to mixtures in discrete choice models p. 28/42 Random parameters 40 Lognormal Distribution of B_TIME Normal Distribution of B_TIME Lognormal mean Normal mean MNL

31 Introduction to mixtures in discrete choice models p. 29/42 Random parameters Example with Swissmetro ASC_CAR ASC_SBB ASC_SM B_COST B_FR B_TIME Car cost 0 time Train cost freq. time Swissmetro cost freq. time B_TIME randomly distributed across the population, discrete distribution P(β time = ˆβ) = ω 1 P(β time = 0) = ω 2 = 1 ω 1

32 Introduction to mixtures in discrete choice models p. 30/42 Random parameters [DiscreteDistributions] B_TIME < B_TIME_1 ( W1 ) B_TIME_2 ( W2 ) > [LinearConstraints] W1 + W2 = 1.0

33 Introduction to mixtures in discrete choice models p. 31/42 Random parameters Logit RC-norm. RC-logn. RC-disc ASC_CAR_SP ASC_SM_SP B_COST B_FR B_TIME S_TIME W W Prob(β > 0) 8.8% 0.0% 0.0% χ

34 Introduction to mixtures in discrete choice models p. 32/42 Latent classes Latent classes capture unobserved heterogeneity They can represent different: Choice sets Decision protocols Tastes Model structures etc.

35 Introduction to mixtures in discrete choice models p. 33/42 Latent classes P(i) = S Λ(i s)q(s) s=1 Λ(i s) is the class-specific choice model probability of choosing i given that the individual belongs to class s Q(s) is the class membership model probability of belonging to class s

36 Introduction to mixtures in discrete choice models p. 34/42 Example: residential location Hypothesis Lifestyle preferences exist (e.g., suburb vs. urban) Lifestyle differences lead to differences in considerations, criterion, and preferences for residential location choices Infer lifestyle preferences from choice behavior using latent class choice model Latent classes = lifestyle Choice model = location decisions

37 Example: residential location Introduction to mixtures in discrete choice models p. 35/42

38 Introduction to mixtures in discrete choice models p. 36/42 Latent lifestyle segmentation Class 1 Class 2 Class 3 Suburban, school, Transit, school, High density, urban auto affluent, less affluent, activity, older, more established younger families non-family, professionals families

39 Introduction to mixtures in discrete choice models p. 37/42 Summary Logit mixtures models Computationally more complex than MEV Allow for more flexibility than MEV Continuous mixtures: alternative specific variance, nesting structures, random parameters P(i) = ξ Λ(i ξ)f(ξ)dξ Discrete mixtures: well-defined latent classes of decision makers S P(i) = Λ(i s)q(s). s=1

40 Introduction to mixtures in discrete choice models p. 38/42 Tips for applications Be careful: simulation can mask specification and identification issues Do not forget about the systematic portion