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Choice modelling in the age of machine learning -- discussion paper

Author

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  • S. Van Cranenburgh
  • S. Wang
  • A. Vij
  • F. Pereira
  • J. Walker
Abstract
Since its inception, the choice modelling field has been dominated by theory-driven modelling approaches. Machine learning offers an alternative data-driven approach for modelling choice behaviour and is increasingly drawing interest in our field. Cross-pollination of machine learning models, techniques and practices could help overcome problems and limitations encountered in the current theory-driven modelling paradigm, such as subjective labour-intensive search processes for model selection, and the inability to work with text and image data. However, despite the potential benefits of using the advances of machine learning to improve choice modelling practices, the choice modelling field has been hesitant to embrace machine learning. This discussion paper aims to consolidate knowledge on the use of machine learning models, techniques and practices for choice modelling, and discuss their potential. Thereby, we hope not only to make the case that further integration of machine learning in choice modelling is beneficial, but also to further facilitate it. To this end, we clarify the similarities and differences between the two modelling paradigms; we review the use of machine learning for choice modelling; and we explore areas of opportunities for embracing machine learning models and techniques to improve our practices. To conclude this discussion paper, we put forward a set of research questions which must be addressed to better understand if and how machine learning can benefit choice modelling.

Suggested Citation

  • S. Van Cranenburgh & S. Wang & A. Vij & F. Pereira & J. Walker, 2021. "Choice modelling in the age of machine learning -- discussion paper," Papers 2101.11948, arXiv.org, revised Nov 2021.
    • Handle: RePEc:arx:papers:2101.11948
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    1. Haghani, Milad & Sarvi, Majid, 2018. "Crowd behaviour and motion: Empirical methods," Transportation Research Part B: Methodological, Elsevier, vol. 107(C), pages 253-294.
    2. Mouter, Niek & van Cranenburgh, Sander & van Wee, Bert, 2017. "An empirical assessment of Dutch citizens' preferences for spatial equality in the context of a national transport investment plan," Journal of Transport Geography, Elsevier, vol. 60(C), pages 217-230.
    3. Parthasarathi, Pavithra & Levinson, David, 2010. "Post-construction evaluation of traffic forecast accuracy," Transport Policy, Elsevier, vol. 17(6), pages 428-443, November.
    4. Vij, Akshay & Krueger, Rico, 2017. "Random taste heterogeneity in discrete choice models: Flexible nonparametric finite mixture distributions," Transportation Research Part B: Methodological, Elsevier, vol. 106(C), pages 76-101.
    5. Mackie, Peter & Worsley, Tom & Eliasson, Jonas, 2014. "Transport appraisal revisited," Research in Transportation Economics, Elsevier, vol. 47(C), pages 3-18.
    6. Goulias, Konstadinos G., 1999. "Longitudinal analysis of activity and travel pattern dynamics using generalized mixed Markov latent class models," Transportation Research Part B: Methodological, Elsevier, vol. 33(8), pages 535-558, November.
    7. Fabian Bastin & Cinzia Cirillo & Philippe L. Toint, 2010. "Estimating Nonparametric Random Utility Models with an Application to the Value of Time in Heterogeneous Populations," Transportation Science, INFORMS, vol. 44(4), pages 537-549, November.
    8. Lorenzo Masiero & David Hensher, 2011. "Shift of reference point and implications on behavioral reaction to gains and losses," Transportation, Springer, vol. 38(2), pages 249-271, March.
    9. van Cranenburgh, Sander & Chorus, Caspar G., 2018. "Does the decision rule matter for large-scale transport models?," Transportation Research Part A: Policy and Practice, Elsevier, vol. 114(PB), pages 338-353.
    10. Fosgerau, Mogens & Bierlaire, Michel, 2007. "A practical test for the choice of mixing distribution in discrete choice models," Transportation Research Part B: Methodological, Elsevier, vol. 41(7), pages 784-794, August.
    11. Shenhao Wang & Baichuan Mo & Stephane Hess & Jinhua Zhao, 2021. "Comparing hundreds of machine learning classifiers and discrete choice models in predicting travel behavior: an empirical benchmark," Papers 2102.01130, arXiv.org.
    12. Yang Li & Asim Ansari, 2014. "A Bayesian Semiparametric Approach for Endogeneity and Heterogeneity in Choice Models," Management Science, INFORMS, vol. 60(5), pages 1161-1179, May.
    13. Shi, Haolun & Yin, Guosheng, 2018. "Boosting conditional logit model," Journal of choice modelling, Elsevier, vol. 26(C), pages 48-63.
    14. Burda, Martin & Harding, Matthew & Hausman, Jerry, 2008. "A Bayesian mixed logit-probit model for multinomial choice," Journal of Econometrics, Elsevier, vol. 147(2), pages 232-246, December.
    15. Griswold, Julia B. & Yu, Mengqiao & Filingeri, Victoria & Grembek, Offer & Walker, Joan L., 2018. "A behavioral modeling approach to bicycle level of service," Transportation Research Part A: Policy and Practice, Elsevier, vol. 116(C), pages 166-177.
    16. Meyerhoff, Jürgen, 2013. "Do turbines in the vicinity of respondents' residences influence choices among programmes for future wind power generation?," Journal of choice modelling, Elsevier, vol. 7(C), pages 58-71.
    17. Daniel Kahneman & Amos Tversky, 2013. "Prospect Theory: An Analysis of Decision Under Risk," World Scientific Book Chapters, in: Leonard C MacLean & William T Ziemba (ed.), HANDBOOK OF THE FUNDAMENTALS OF FINANCIAL DECISION MAKING Part I, chapter 6, pages 99-127, World Scientific Publishing Co. Pte. Ltd..
    18. Rico Krueger & Akshay Vij & Taha H. Rashidi, 2018. "A Dirichlet Process Mixture Model of Discrete Choice," Papers 1801.06296, arXiv.org.
    19. Arentze, Theo A. & Timmermans, Harry J. P., 2004. "A learning-based transportation oriented simulation system," Transportation Research Part B: Methodological, Elsevier, vol. 38(7), pages 613-633, August.
    20. Tversky, Amos & Kahneman, Daniel, 1992. "Advances in Prospect Theory: Cumulative Representation of Uncertainty," Journal of Risk and Uncertainty, Springer, vol. 5(4), pages 297-323, October.
    21. Daniel Kahneman & Jack L. Knetsch & Richard H. Thaler, 1991. "Anomalies: The Endowment Effect, Loss Aversion, and Status Quo Bias," Journal of Economic Perspectives, American Economic Association, vol. 5(1), pages 193-206, Winter.
    22. de Jong, Gerard & Daly, Andrew & Pieters, Marits & van der Hoorn, Toon, 2007. "The logsum as an evaluation measure: Review of the literature and new results," Transportation Research Part A: Policy and Practice, Elsevier, vol. 41(9), pages 874-889, November.
    23. Lhéritier, Alix & Bocamazo, Michael & Delahaye, Thierry & Acuna-Agost, Rodrigo, 2019. "Airline itinerary choice modeling using machine learning," Journal of choice modelling, Elsevier, vol. 31(C), pages 198-209.
    24. Francisco C. Pereira, 2019. "Rethinking travel behavior modeling representations through embeddings," Papers 1909.00154, arXiv.org.
    25. Kelvin J. Lancaster, 1966. "A New Approach to Consumer Theory," Journal of Political Economy, University of Chicago Press, vol. 74(2), pages 132-132.
    26. Loomes, Graham & Sugden, Robert, 1982. "Regret Theory: An Alternative Theory of Rational Choice under Uncertainty," Economic Journal, Royal Economic Society, vol. 92(368), pages 805-824, December.
    27. McFadden, Daniel, 1980. "Econometric Models for Probabilistic Choice among Products," The Journal of Business, University of Chicago Press, vol. 53(3), pages 13-29, July.
    28. Fifer, Simon & Rose, John & Greaves, Stephen, 2014. "Hypothetical bias in Stated Choice Experiments: Is it a problem? And if so, how do we deal with it?," Transportation Research Part A: Policy and Practice, Elsevier, vol. 61(C), pages 164-177.
    29. Rothenberg, Thomas J, 1971. "Identification in Parametric Models," Econometrica, Econometric Society, vol. 39(3), pages 577-591, May.
    30. Alwosheel, Ahmad & van Cranenburgh, Sander & Chorus, Caspar G., 2018. "Is your dataset big enough? Sample size requirements when using artificial neural networks for discrete choice analysis," Journal of choice modelling, Elsevier, vol. 28(C), pages 167-182.
    31. Hess, Stephane & Bierlaire, Michel & Polak, John W., 2005. "Estimation of value of travel-time savings using mixed logit models," Transportation Research Part A: Policy and Practice, Elsevier, vol. 39(2-3), pages 221-236.
    32. van Cranenburgh, Sander & Bliemer, Michiel C.J., 2019. "Information theoretic-based sampling of observations," Journal of choice modelling, Elsevier, vol. 31(C), pages 181-197.
    33. Correia, Gonçalo Homem de Almeida & Looff, Erwin & van Cranenburgh, Sander & Snelder, Maaike & van Arem, Bart, 2019. "On the impact of vehicle automation on the value of travel time while performing work and leisure activities in a car: Theoretical insights and results from a stated preference survey," Transportation Research Part A: Policy and Practice, Elsevier, vol. 119(C), pages 359-382.
    34. Louviere,Jordan J. & Hensher,David A. & Swait,Joffre D., 2000. "Stated Choice Methods," Cambridge Books, Cambridge University Press, number 9780521788304, September.
    35. Sifringer, Brian & Lurkin, Virginie & Alahi, Alexandre, 2020. "Enhancing discrete choice models with representation learning," Transportation Research Part B: Methodological, Elsevier, vol. 140(C), pages 236-261.
    36. Hillel, Tim & Bierlaire, Michel & Elshafie, Mohammed Z.E.B. & Jin, Ying, 2021. "A systematic review of machine learning classification methodologies for modelling passenger mode choice," Journal of choice modelling, Elsevier, vol. 38(C).
    37. Paz, Alexander & Arteaga, Cristian & Cobos, Carlos, 2019. "Specification of mixed logit models assisted by an optimization framework," Journal of choice modelling, Elsevier, vol. 30(C), pages 50-60.
    38. Christopher Allen & David M A Mehler, 2019. "Open science challenges, benefits and tips in early career and beyond," PLOS Biology, Public Library of Science, vol. 17(5), pages 1-14, May.
    39. Brathwaite, Timothy & Walker, Joan L., 2018. "Asymmetric, closed-form, finite-parameter models of multinomial choice," Journal of choice modelling, Elsevier, vol. 29(C), pages 78-112.
    40. Sebastian Bach & Alexander Binder & Grégoire Montavon & Frederick Klauschen & Klaus-Robert Müller & Wojciech Samek, 2015. "On Pixel-Wise Explanations for Non-Linear Classifier Decisions by Layer-Wise Relevance Propagation," PLOS ONE, Public Library of Science, vol. 10(7), pages 1-46, July.
    41. Daniel McFadden, 2001. "Economic Choices," American Economic Review, American Economic Association, vol. 91(3), pages 351-378, June.
    42. Glerum, Aurélie & Atasoy, Bilge & Bierlaire, Michel, 2014. "Using semi-open questions to integrate perceptions in choice models," Journal of choice modelling, Elsevier, vol. 10(C), pages 11-33.
    43. Hess, Stephane & Palma, David, 2019. "Apollo: A flexible, powerful and customisable freeware package for choice model estimation and application," Journal of choice modelling, Elsevier, vol. 32(C), pages 1-1.
    44. Andrew Daly & James Fox & Jan Gerrit Tuinenga, 2005. "Pivot-Point Procedures in Practical Travel Demand Forecasting," ERSA conference papers ersa05p784, European Regional Science Association.
    45. Mariel, Petr & Meyerhoff, Jürgen & Hess, Stephane, 2015. "Heterogeneous preferences toward landscape externalities of wind turbines – combining choices and attitudes in a hybrid model," Renewable and Sustainable Energy Reviews, Elsevier, vol. 41(C), pages 647-657.
    46. Angel Bujosa & Antoni Riera & Robert Hicks, 2010. "Combining Discrete and Continuous Representations of Preference Heterogeneity: A Latent Class Approach," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 47(4), pages 477-493, December.
    47. Manski, Charles F & Lerman, Steven R, 1977. "The Estimation of Choice Probabilities from Choice Based Samples," Econometrica, Econometric Society, vol. 45(8), pages 1977-1988, November.
    48. Junyi Shen, 2009. "Latent class model or mixed logit model? A comparison by transport mode choice data," Applied Economics, Taylor & Francis Journals, vol. 41(22), pages 2915-2924.
    49. Aiste Ruseckaite & Dennis Fok & Peter Goos, 2020. "Flexible Mixture-Amount Models Using Multivariate Gaussian Processes," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 38(2), pages 257-271, April.
    50. De Borger, Bruno & Fosgerau, Mogens, 2008. "The trade-off between money and travel time: A test of the theory of reference-dependent preferences," Journal of Urban Economics, Elsevier, vol. 64(1), pages 101-115, July.
    51. Michael Keane & Nada Wasi, 2013. "Comparing Alternative Models Of Heterogeneity In Consumer Choice Behavior," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 28(6), pages 1018-1045, September.
    52. Stephane Hess & Andrew Daly & Richard Batley, 2018. "Revisiting consistency with random utility maximisation: theory and implications for practical work," Theory and Decision, Springer, vol. 84(2), pages 181-204, March.
    53. van Cranenburgh, Sander & Collins, Andrew T., 2019. "New software tools for creating stated choice experimental designs efficient for regret minimisation and utility maximisation decision rules," Journal of choice modelling, Elsevier, vol. 31(C), pages 104-123.
    54. Fosgerau, Mogens & Hess, Stephane, 2009. "A comparison of methods for representing random taste heterogeneity in discrete choice models," European Transport \ Trasporti Europei, ISTIEE, Institute for the Study of Transport within the European Economic Integration, issue 42, pages 1-25.
    55. Hensher, David A. & Ton, Tu T., 2000. "A comparison of the predictive potential of artificial neural networks and nested logit models for commuter mode choice," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 36(3), pages 155-172, September.
    56. Yafei Han & Francisco Camara Pereira & Moshe Ben-Akiva & Christopher Zegras, 2020. "A Neural-embedded Choice Model: TasteNet-MNL Modeling Taste Heterogeneity with Flexibility and Interpretability," Papers 2002.00922, arXiv.org, revised Jul 2022.
    57. Alwosheel, Ahmad & van Cranenburgh, Sander & Chorus, Caspar G., 2019. "‘Computer says no’ is not enough: Using prototypical examples to diagnose artificial neural networks for discrete choice analysis," Journal of choice modelling, Elsevier, vol. 33(C).
    58. Small, Kenneth A., 2012. "Valuation of travel time," Economics of Transportation, Elsevier, vol. 1(1), pages 2-14.
    59. Yadong Wang & Qiang Meng, 2019. "Integrated method for forecasting container slot booking in intercontinental liner shipping service," Flexible Services and Manufacturing Journal, Springer, vol. 31(3), pages 653-674, September.
    60. Arentze, Theo & Timmermans, Harry, 2007. "Parametric action decision trees: Incorporating continuous attribute variables into rule-based models of discrete choice," Transportation Research Part B: Methodological, Elsevier, vol. 41(7), pages 772-783, August.
    61. Chenfeng Xiong & Xiqun Chen & Xiang He & Wei Guo & Lei Zhang, 2015. "The analysis of dynamic travel mode choice: a heterogeneous hidden Markov approach," Transportation, Springer, vol. 42(6), pages 985-1002, November.
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    7. Teodóra Szép & Sander Cranenburgh & Caspar Chorus, 2024. "Moral rhetoric in discrete choice models: a Natural Language Processing approach," Quality & Quantity: International Journal of Methodology, Springer, vol. 58(1), pages 179-206, February.
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