Equivalence between the GNL models and entropy models

Equivalence in disaggregate level

Kei T. Akahashi, Takahiro Ohno

    Research output: Contribution to journalArticle

    Abstract

    This paper provides a proof that the parameter estimation problem in the generalized nested logit (GNL) models used in marketing science and transportation planning fields, is equivalent to information minimization problems with constraints in disaggregate level. To be specific, equivalence between log-likelihood maximization estimation problems of the GNL model and information minimization problems is proved using the two-stage optimization problem. In this problem, parameters in definite utility functions and allocation parameters correspond to an alternative level and similarity parameters correspond to a nest level. As part of the process to provide the proof, we show that constraints on allocation parameters are naturally considered in the log-likelihood maximization estimation problem of the GNL model. Using the properties of allocation parameters, we show new methods for parameter estimation in the GNL model, which rectify the heuristic methods proposed in Vovsha. First, we propose an estimation method using parameters in two-stage estimation as initial values in simultaneous estimation. Second, we propose a method using the primal-dual interior point method which utilizes duality in each stage of two-stage estimation in the GNL model. The GNL model includes the multinomial logit, the nested logit, the cross-nested logit and the pairwise combination logit models, and equivalence between all these models and entropy models are proved in this paper.

    Original languageEnglish
    Pages (from-to)9-20
    Number of pages12
    JournalJournal of Japan Industrial Management Association
    Volume64
    Issue number1
    Publication statusPublished - 2013

    Fingerprint

    Nested Models
    Logit Model
    Entropy
    Equivalence
    Two-stage Estimation
    Logit
    Minimization Problem
    Parameter Estimation
    Likelihood
    Model
    Multinomial Logit
    Parameter estimation
    Primal-dual Interior Point Method
    Simultaneous Estimation
    Nest
    Heuristic Method
    Nested logit model
    Utility Function
    Heuristic methods
    Pairwise

    Keywords

    • Disaggregate model
    • Duality
    • Entropy model
    • Nested logit model

    ASJC Scopus subject areas

    • Industrial and Manufacturing Engineering
    • Applied Mathematics
    • Management Science and Operations Research
    • Strategy and Management

    Cite this

    Equivalence between the GNL models and entropy models : Equivalence in disaggregate level. / Akahashi, Kei T.; Ohno, Takahiro.

    In: Journal of Japan Industrial Management Association, Vol. 64, No. 1, 2013, p. 9-20.

    Research output: Contribution to journalArticle

    @article{977adeadaed3481db598ed87830236cc,
    title = "Equivalence between the GNL models and entropy models: Equivalence in disaggregate level",
    abstract = "This paper provides a proof that the parameter estimation problem in the generalized nested logit (GNL) models used in marketing science and transportation planning fields, is equivalent to information minimization problems with constraints in disaggregate level. To be specific, equivalence between log-likelihood maximization estimation problems of the GNL model and information minimization problems is proved using the two-stage optimization problem. In this problem, parameters in definite utility functions and allocation parameters correspond to an alternative level and similarity parameters correspond to a nest level. As part of the process to provide the proof, we show that constraints on allocation parameters are naturally considered in the log-likelihood maximization estimation problem of the GNL model. Using the properties of allocation parameters, we show new methods for parameter estimation in the GNL model, which rectify the heuristic methods proposed in Vovsha. First, we propose an estimation method using parameters in two-stage estimation as initial values in simultaneous estimation. Second, we propose a method using the primal-dual interior point method which utilizes duality in each stage of two-stage estimation in the GNL model. The GNL model includes the multinomial logit, the nested logit, the cross-nested logit and the pairwise combination logit models, and equivalence between all these models and entropy models are proved in this paper.",
    keywords = "Disaggregate model, Duality, Entropy model, Nested logit model",
    author = "Akahashi, {Kei T.} and Takahiro Ohno",
    year = "2013",
    language = "English",
    volume = "64",
    pages = "9--20",
    journal = "Journal of Japan Industrial Management Association",
    issn = "0386-4812",
    publisher = "Nihon Keikei Kogakkai",
    number = "1",

    }

    TY - JOUR

    T1 - Equivalence between the GNL models and entropy models

    T2 - Equivalence in disaggregate level

    AU - Akahashi, Kei T.

    AU - Ohno, Takahiro

    PY - 2013

    Y1 - 2013

    N2 - This paper provides a proof that the parameter estimation problem in the generalized nested logit (GNL) models used in marketing science and transportation planning fields, is equivalent to information minimization problems with constraints in disaggregate level. To be specific, equivalence between log-likelihood maximization estimation problems of the GNL model and information minimization problems is proved using the two-stage optimization problem. In this problem, parameters in definite utility functions and allocation parameters correspond to an alternative level and similarity parameters correspond to a nest level. As part of the process to provide the proof, we show that constraints on allocation parameters are naturally considered in the log-likelihood maximization estimation problem of the GNL model. Using the properties of allocation parameters, we show new methods for parameter estimation in the GNL model, which rectify the heuristic methods proposed in Vovsha. First, we propose an estimation method using parameters in two-stage estimation as initial values in simultaneous estimation. Second, we propose a method using the primal-dual interior point method which utilizes duality in each stage of two-stage estimation in the GNL model. The GNL model includes the multinomial logit, the nested logit, the cross-nested logit and the pairwise combination logit models, and equivalence between all these models and entropy models are proved in this paper.

    AB - This paper provides a proof that the parameter estimation problem in the generalized nested logit (GNL) models used in marketing science and transportation planning fields, is equivalent to information minimization problems with constraints in disaggregate level. To be specific, equivalence between log-likelihood maximization estimation problems of the GNL model and information minimization problems is proved using the two-stage optimization problem. In this problem, parameters in definite utility functions and allocation parameters correspond to an alternative level and similarity parameters correspond to a nest level. As part of the process to provide the proof, we show that constraints on allocation parameters are naturally considered in the log-likelihood maximization estimation problem of the GNL model. Using the properties of allocation parameters, we show new methods for parameter estimation in the GNL model, which rectify the heuristic methods proposed in Vovsha. First, we propose an estimation method using parameters in two-stage estimation as initial values in simultaneous estimation. Second, we propose a method using the primal-dual interior point method which utilizes duality in each stage of two-stage estimation in the GNL model. The GNL model includes the multinomial logit, the nested logit, the cross-nested logit and the pairwise combination logit models, and equivalence between all these models and entropy models are proved in this paper.

    KW - Disaggregate model

    KW - Duality

    KW - Entropy model

    KW - Nested logit model

    UR - http://www.scopus.com/inward/record.url?scp=84923313198&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=84923313198&partnerID=8YFLogxK

    M3 - Article

    VL - 64

    SP - 9

    EP - 20

    JO - Journal of Japan Industrial Management Association

    JF - Journal of Japan Industrial Management Association

    SN - 0386-4812

    IS - 1

    ER -