Latent variables Michel Bierlaire Transport and Mobility Laboratory - PowerPoint PPT Presentation

Latent variables Michel Bierlaire Transport and Mobility Laboratory School of Architecture, Civil and Environmental Engineering Ecole Polytechnique F´ ed´ erale de Lausanne M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 1 / 47

Outline Outline Motivation 1 Modeling latent concepts 2 Estimation 3 Case studies 4 Conclusion 5 M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 2 / 47

Motivation Motivation Rationality? Standard random utility assumptions are often violated. Factors such as attitudes, perceptions, knowledge are not reflected. M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 3 / 47

Motivation Example: pain lovers Kahneman, D., Fredrickson, B., Schreiber, C.M., and Redelmeier, D., When More Pain Is Preferred to Less: Adding a Better End, Psychological Science, Vol. 4, No. 6, pp. 401-405, 1993. Short trial: immerse one hand in water at 14 ◦ for 60 sec. Long trial: immerse the other hand at 14 ◦ for 60 sec, then keep the hand in the water 30 sec. longer as the temperature of the water is gradually raised to 15 ◦ . Outcome: most people prefer the long trial. Explanation: duration plays a small role the peak and the final moments matter M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 4 / 47

Motivation Example: The Economist Subscription to The Economist Web only @ $59 Print only @ $125 Print and web @ $125 M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 5 / 47

Motivation Example: The Economist Subscription to The Economist Experiment 1 Experiment 2 Web only @ $59 Web only @ $59 Print only @ $125 Print and web @ $125 Print and web @ $125 M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 6 / 47

Motivation Example: The Economist Subscription to The Economist Experiment 1 Experiment 2 16 Web only @ $59 Web only @ $59 68 0 Print only @ $125 84 Print and web @ $125 Print and web @ $125 32 Source: Ariely (2008) Dominated alternative According to utility maximization, should not affect the choice But it affects the perception, which affects the choice. M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 7 / 47

Motivation Example: good or bad wine? Choose a bottle of wine... Experiment 1 Experiment 2 1 McFadden red at $10 McFadden red at $10 2 Nappa red at $12 Nappa red at $12 3 McFadden special reserve pinot noir at $60 Most would choose 2 Most would choose 1 Context plays a role on perceptions M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 8 / 47

Motivation Example: live and let die Population of 600 is threatened by a disease. Two alternative treatments to combat the disease have been proposed. Experiment 1 Experiment 2 # resp. = 152 # resp. = 155 Treatment A: Treatment C: 200 people saved 400 people die Treatment B: Treatment D: 600 people saved with 0 people die with prob. prob. 1/3 1/3 0 people saved with prob. 600 people die with prob. 2/3 2/3 M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 9 / 47

Motivation Example: live and let die Population of 600 is threatened by a disease. Two alternative treatments to combat the disease have been proposed. Experiment 1 Experiment 2 # resp. = 152 # resp. = 155 Treatment A: Treatment C: 72% 200 people saved 400 people die 22% Treatment B: Treatment D: 28% 600 people saved with 0 people die with prob. 78% prob. 1/3 1/3 0 people saved with prob. 600 people die with prob. 2/3 2/3 Source: Tversky & Kahneman (1986) M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 10 / 47

Motivation Example: to be free Choice between a fine and a regular chocolate Experiment 1 Experiment 2 Lindt $0.15 $0.14 Hershey $0.01 $0.00 Lindt chosen 73% 31% Hershey chosen 27% 69% Source: Ariely (2008) Predictably irrational, Harper Collins. M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 11 / 47

Modeling latent concepts Outline Motivation 1 Modeling latent concepts 2 Estimation 3 Case studies 4 Conclusion 5 M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 12 / 47

Modeling latent concepts Latent concepts Latent latent : potentially existing but not presently evident or realized (from Latin: lateo = lie hidden) Here: not directly observed Standard models are already based on a latent concept: utility Drawing convention Latent variable Observed variable structural relation: measurement: errors: M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 13 / 47

Modeling latent concepts Random utility Explanatory variables ε in V in = � k β ik x ikn Utility P n ( i ) = e V in / � j e V jn Choice M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 14 / 47

Modeling latent concepts Attitudes Measuring attitudes Psychometric indicators Example: attitude towards the environment. For each question, response on a scale: strongly agree, agree, neutral, disagree, strongly disagree, no idea. The price of oil should be increased to reduce congestion and pollution More public transportation is necessary, even if it means additional taxes Ecology is a threat to minorities and small companies. People and employment are more important than the environment. I feel concerned by the global warming. Decisions must be taken to reduce the greenhouse gas emission. M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 15 / 47

Modeling latent concepts Indicators Model specification Indicators cannot be used as explanatory variables. Mainly two reasons: 1 Measurement errors Scale is arbitrary and discrete People may overreact Justification bias may produce exaggerated responses 2 No forecasting possibility No way to predict the indicators in the future M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 16 / 47

Modeling latent concepts Factor analysis ε i Latent variables X ∗ k I i = λ i + � k L ik X ∗ k Indicators M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 17 / 47

Modeling latent concepts Measurement equation Explanatory variables ε i X ∗ k = � j β j x j Latent variables X ∗ I i = λ i + � k L ik X ∗ k Indicators M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 18 / 47

Modeling latent concepts Measurement equation Continuous model: regression I = f ( X ∗ ; β ) + ε Discrete model: thresholds if − ∞ < X ∗ ≤ τ 1  1  if τ 1 < X ∗ ≤ τ 2  2    if τ 2 < X ∗ ≤ τ 3 3 I = if τ 3 < X ∗ ≤ τ 4 4    if τ 4 < X ∗ ≤ + ∞   5 M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 19 / 47

Modeling latent concepts Choice model Explanatory variables ε in ω in Utility Latent variables Choice Indicators M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 20 / 47

Estimation Outline Motivation 1 Modeling latent concepts 2 Estimation 3 Case studies 4 Conclusion 5 M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 21 / 47

Estimation Structural equations Explanatory variables ε in ω in Utility Latent variables Choice Indicators X ∗ n = h ( X n ; λ ) + ω n , ω n ∼ N (0 , Σ ω ) . M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 22 / 47

Estimation Structural equations Explanatory variables ε in ω in Utility Latent variables Choice Indicators U n = V ( X n , X ∗ n ; β ) + ε n , ε n ∼ EV(0 , µ ) . M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 23 / 47

Estimation Measurement equations Explanatory variables ε in ω in Utility Latent variables Choice Indicators Ordinal discrete variable: ordered probit model I ∗ n = m ( X ∗ n ; α ) + ν n , ν n ∼ N (0 , Σ ν ) M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 24 / 47

Estimation Ordered probit f ν n Pr( τ q − 1 ≤ I ∗ n ≤ τ q ) I ∗ I ∗ n − τ q n − τ q − 1 ν n M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 25 / 47

Estimation Measurement equations Explanatory variables ε in ω in Utility Latent variables Choice Indicators P ( I n = 1) = Pr( I ∗ n ≤ τ 1 ) P ( I n = 2) = Pr( I ∗ n ≤ τ 2 ) − Pr( I ∗ n ≤ τ 1 ) . . . P ( I n = 5) = 1 − Pr( I ∗ n ≤ τ 4 ) M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 26 / 47

Estimation Measurement equations Explanatory variables ε in ω in Utility Latent variables Choice Indicators P ( y in = 1) = Pr( U in ≥ U jn , ∀ j ) . M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 27 / 47

Estimation Estimation: likelihood Assuming ω n , ε n and ν n are independent, we have L n ( y n , I n | X n ; α, β, λ, Σ ω , Σ ν , µ, τ ) = � X ∗ P ( y n | X n , X ∗ ; β, µ ) P ( I n | X n , X ∗ ; α, Σ ν , τ ) f ( X ∗ | X n ; λ, Σ ω ) dX ∗ . Maximum likelihood estimation: � max log ( L n ( y n , I n | X n ; α, β, λ, Σ ω , Σ ν , µ, τ )) α,β,λ, Σ ε , Σ ν , Σ ω n Source: Walker (2001) M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 28 / 47

Case studies Outline Motivation 1 Modeling latent concepts 2 Estimation 3 Case studies 4 Conclusion 5 M. Bierlaire (TRANSP-OR ENAC EPFL) Latent variables 29 / 47