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- 1. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you A discussion on the performance of the Coordinate Exchange Algorithm Tian Tian Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago 09 February 2016 Tian Tian MSCS, UIC A discussion on the performance of the Coordinate Exchange Algorithm
- 2. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you Background of DCEs Background A product/service of interest K attributes each with lk levels L = K k=1 lk alternatives/proﬁles Each choice set consists of J proﬁles Q = L J choice sets Each design consists of S choice sets M = Q S candidate designs (design space) Goal –to estimate the appeal of each attribute –to predict the behaviors of consumers –to emulate market decisions and forecast market demands Tian Tian MSCS, UIC A discussion on the performance of the Coordinate Exchange Algorithm
- 3. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you Background of DCEs Two critical components Parameter setup in the probabilistic model (we model the probability a consumer chooses proﬁle j in choice set s — pjs) An eﬃcient algorithm that ﬁnds the optimal discrete choice design(s) Tian Tian MSCS, UIC A discussion on the performance of the Coordinate Exchange Algorithm
- 4. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you Challenges and motivation Challenges First challenge: The probabilistic choice models are nonlinear ⇒ eﬃciency of the design depends on the unknown parameters Possible solutions: Linear optimal designs where parameters are set to zeros Locally optimal designs where parameters are assigned with nonzero values Bayesian optimal designs where parameters are introduced with a prior distribution Tian Tian MSCS, UIC A discussion on the performance of the Coordinate Exchange Algorithm
- 5. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you Challenges and motivation Challenges (Cont.) Second challenge: Diﬃcult (impossible) to theoretically construct locally/Bayesian op- timal designs under a nonlinear multi-dimensional probabilistic choice model. Existing algorithms: The mFA (modiﬁed Federov algorithm) The RSA (relabeling and swapping algorithm) The RSCA (relabeling, swapping, and cycling algorithm) The quadrature scheme The CEA (coordinate-exchange algorithm) Tian Tian MSCS, UIC A discussion on the performance of the Coordinate Exchange Algorithm
- 6. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you Challenges and motivation Motivation Question: Is the resulting design generated from the CEA indeed optimal, or at lease highly eﬃcient? An example: 3 × 3 × 2/3/8 ⇒ (3×3×2 3 ) 8 > 4 × 1018 Tian Tian MSCS, UIC A discussion on the performance of the Coordinate Exchange Algorithm
- 7. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you Review of the MNL model The MNL model The MNL (multinormial logit) probability pjs, j = 1, ..., J, s = 1, ..., S, is modeled as follows, pjs = exp(xjsβ) J i=1 exp(xisβ) – ˜K = K k=1(lk −1) is the total number of main eﬀect levels subject to the ”zero-sum” constraint – xjs is a ˜K × 1 vector denoting the attribute levels of proﬁle j in the sth choice set – β is a ˜K × 1 vector of parameter representing the main eﬀects of the attribute levels Tian Tian MSCS, UIC A discussion on the performance of the Coordinate Exchange Algorithm
- 8. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you Review of the CEA The CEA 1 Start with a randomly chosen starting design 2 Exchange levels for each attribute in each proﬁle in each choice set in the design 3 A level is updated when the new design results in a smaller criterion value 4 A complete cycle of iteration terminates when the algorithm has reached the last attribute in the last proﬁle in the last choice set 5 The algorithm goes back to the very ﬁrst attribute and contin- ues another round Tian Tian MSCS, UIC A discussion on the performance of the Coordinate Exchange Algorithm
- 9. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you Review of the CEA The CEA (Cont.) 6 A complete try of iteration terminates when no substitution is made in a whole cycle or other stopping rule is met 7 To avoid poorly local optima, T tries of the algorithm are re- peated each with a diﬀerent starting design 8 The optimal design is chosen from the T resulting designs. Tian Tian MSCS, UIC A discussion on the performance of the Coordinate Exchange Algorithm
- 10. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you Discrete and approximate design To take advantage of the structure under the continuous/approxi- mate design framework. Approximate/continuous: ˜ξ = {(Cq, wq)|q = 1, ..., Q}. Discrete/exact: ξ = {Cs|s = 1, ..., S} = {(Cq, wq)|q = 1, ..., Q} with wi = 1/S for i = 1, ..., S and wi = 0 for i = S + 1, ..., Q. Tian Tian MSCS, UIC A discussion on the performance of the Coordinate Exchange Algorithm
- 11. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you Rationale Eﬀ(ξCEA S ) = Obj(ξ∗ S) Obj(ξCEA S ) ≥ Obj(ξ∗ ) Obj(ξCEA S ) where Obj(·) is the objective criterion function. Tian Tian MSCS, UIC A discussion on the performance of the Coordinate Exchange Algorithm
- 12. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you Simulation study 3 × 3 × 2/2/12 3 × 3 × 2/2/12 Locally Bayesian Locally Bayesian D-optimal 99.65% 98.71% 99.80% 98.82% A-optimal 99.04% 96.83% 99.11% 97.15% V -optimal 98.69% 96.75% 97.52% 96.85% Table: Lower bound of the CEA eﬃciency under diﬀerent design setups Tian Tian MSCS, UIC A discussion on the performance of the Coordinate Exchange Algorithm
- 13. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you Simulation study (Cont.) 3 × 3 × 2 × 2 × 2/3/15 4 × 3 × 3 × 2/3/12 Locally Bayesian Locally Bayesian D-optimal 96.76% 98.67% 96.90% 97.78% A-optimal 92.84% 97.20% 95.11% 96.03% V -optimal 94.21% 97.30% 94.12% 96.77% Table: Lower bound of the CEA eﬃciency under diﬀerent design setups Tian Tian MSCS, UIC A discussion on the performance of the Coordinate Exchange Algorithm
- 14. Problem Introduction Review of the MNL model and the CEA Rationale of the approach Simulation study Thank you Tian Tian MSCS, UIC A discussion on the performance of the Coordinate Exchange Algorithm