# A Maximum Entropy Estimator for the Aggregate Hierarchical Logit Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Formulation and Estimation of Hierarchical Logit Model

#### 2.1. Formulation of Hierarchical Logit (HL) Model

_{agi}and V

_{gi}are the deterministic components of the utility perceived by a type i individual from alternative a and group g, respectively. We assume that both terms are linear functions of attributes (x and w) and parameters (β and γ) that are either generic or specific to an alternative or group and type of individual. Thus:

#### 2.2. Estimation of HL Model by Maximum Likelihood (ML)

_{k}for the alternatives chosen by the various individuals is equal to the sum predicted by the estimated choice probabilities. Then, the maximum likelihood estimators of a MNL model reproduce the average values of its explanatory variables (travel time, cost, etc.) and, if specific constants for each alternative are specified, the market (i.e., observed) modal shares.

#### 2.3. Estimation of HL Model by Maximum Entropy (ME)

## 3. Simulation Analysis of ML and ME Estimators

PARAMETER | VALUE (*) |
---|---|

${\beta}_{Car}^{0}$ | 0.9 |

${\beta}_{Bus}^{0}$ | 0 |

${\beta}_{Taxi}^{0}$ | 0.5 |

${\beta}_{Metro}^{0}$ | 0.4 |

${\beta}^{Time}$ | −0.25 |

${\beta}^{Cost}$ | −0.006 |

SVT (**) | 41.47 |

λ | 1 |

ϕ = λ/μ = 1/μ | 0.5 (***) |

MODE | VARIABLE | MEAN (*) | STD DEV (*) |
---|---|---|---|

Car | Travel time | 16 | 11 |

Cost | 2,031 | 138 | |

Taxi | Travel time | 17 | 11 |

Cost | 2,279 | 148 | |

Bus | Travel time | 54 | 12 |

Cost | 409 | 25 | |

Metro | Travel time | 45 | 7 |

Cost | 833 | 73 |

#### 3.1. Bias, Variance and Mean Squared Error (MSE)

**Figure 2.**Observed vs. modeled modal split using maximum likelihood estimation, for parameter value ϕ = 1/μ = 0.5.

METHOD | PARAMETER | SAMPLE SIZE | BIAS | VARIANCE | MSE (*) |
---|---|---|---|---|---|

ML | 1/μ | 500 | 0.16844 | 0.00096 | 0.02933 |

1/μ | 1,000 | 0.10912 | 0.00074 | 0.01265 | |

1/μ | 5,000 | 0.03067 | 0.00016 | 0.00110 | |

1/μ | 10,000 | 0.01655 | 0.00009 | 0.00036 | |

1/μ | 20,000 | 0.00821 | 0.00004 | 0.00011 | |

VST | 500 | 2.57679 | 47.61970 | 54.25957 | |

VST | 1,000 | 0.68741 | 13.87630 | 14.34884 | |

VST | 5,000 | 0.16172 | 3.01595 | 3.04210 | |

VST | 10,000 | 0.12386 | 1.23927 | 1.25461 | |

VST | 20,000 | 0.11537 | 0.71021 | 0.72352 | |

ME | 1/μ | 500 | 0.12867 | 0.00384 | 0.02039 |

1/μ | 1,000 | 0.03050 | 0.00250 | 0.00343 | |

1/μ | 5,000 | 0.00494 | 0.00047 | 0.00050 | |

1/μ | 10,000 | 0.00182 | 0.00017 | 0.00017 | |

1/μ | 20,000 | 0.00023 | 0.00006 | 0.00006 | |

VST | 500 | 1.70225 | 12.01048 | 14.90812 | |

VST | 1,000 | 0.35554 | 1.95599 | 2.08240 | |

VST | 5,000 | 0.19228 | 0.22072 | 0.25769 | |

VST | 10,000 | 0.08879 | 0.07469 | 0.08257 | |

VST | 20,000 | 0.03281 | 0.01565 | 0.01672 |

#### 3.2. Estimate of Consumer Surplus

_{gi}= EMU

_{g}, and can therefore estimate the average EMU as:

_{ag}is the number of individuals in group g that takes alternative a. We will use average EMU to make comparisons with the results generated by the simulations for various sample sizes (the sum of the t

_{ag}will therefore equal the size of the sample from which the parameters that give the EMU are estimated).

SAMPLE SIZE | SIMULATION | ML | Δ% ML (*) | ME | Δ% ME (*) |
---|---|---|---|---|---|

500 | 7.0574 | 4.7481 | 32.7% | 5.4107 | 23.3% |

1,000 | 7.0601 | 5.5514 | 21.4% | 5.8450 | 17.2% |

5,000 | 7.0918 | 6.6807 | 5.8% | 6.8073 | 4.0% |

10,000 | 7.0755 | 6.8564 | 3.1% | 6.9500 | 1.8% |

20,000 | 7.0679 | 6.9539 | 1.6% | 7.0223 | 0.6% |

#### 3.3. Out-of-Sample Prediction

## 4. Conclusions

## References

- Anas, A. Discrete choice theory, information theory and the multinomial logit and gravity models. Transp. Res.
**1983**, 17, 13–23. [Google Scholar] [CrossRef] - Williams, H.C.W.L. On the formation of travel demand models and economic evaluation measures of user benefit. Environ. Plan.
**1977**, 9, 285–344. [Google Scholar] [CrossRef] - Wilson, A.G. Entropy in Urban and Regional Modeling; Pion: London, UK, 1970. [Google Scholar]
- Morrison, W.; Thumann, R. Lagrangian multiplier approach to the solution of a special constrained matrix problem. J. Reg. Sci.
**1980**, 20, 279–292. [Google Scholar] [CrossRef] - Fotheringham, A. A new set of spatial interaction models: The theory of competing destinations. Environ. Plan.
**1983**, 15A, 15–36. [Google Scholar] [CrossRef] - Fotheringham, A. Modeling hierarchical destination choice. Environ. Plan.
**1986**, 18, 401–418. [Google Scholar] [CrossRef] - Fang, S.; Tsao, J. Linearly-constrained entropy maximization problem with quadratic cost and its applications to transportation planning problems. Transp. Sci.
**1995**, 29, 353–365. [Google Scholar] [CrossRef] - Thorsen, I.; Gitlesen, J.P. Empirical evaluation of alternative model specifications to predict commuting flows. J. Reg. Sci.
**1998**, 38, 273–292. [Google Scholar] [CrossRef] - De Grange, L; Ibeas, A; Gonzalez, F. A hierarchical gravity model with spatial correlation: Mathematical formulation and parameter estimation. Netw. Spat. Econ.
**2009**. [Google Scholar] [CrossRef] - De Grange, L.; Fernandez, J.E.; De Cea, J. A consolidated model of trip distribution. Transp. Res.
**2010**, 46, 61–75. [Google Scholar] [CrossRef] - Boyce, D.; LeBlanc, L.; Chon, K.; Lee, Y.; Lin, K. Implementation and computational issues for combined models of location, destination, mode and route choice. Environ. Plan.
**1983**, 15, 1219–1230. [Google Scholar] [CrossRef] - Boyce, D.; LeBlanc, L.; Chon, K. Network equilibrium models of urban location and travel choices: A retrospective survey. J. Reg. Sci.
**1988**, 28, 159–183. [Google Scholar] [CrossRef] - Safwat, K.; Magnanti, T. A combined trip generation, trip distribution, modal split and traffic assignment model. Transp. Sci.
**1988**, 22, 14–30. [Google Scholar] [CrossRef] - Brice, S. Derivation of nested transport models within a mathematical programming framework. Transp. Res.
**1989**, 23, 19–28. [Google Scholar] [CrossRef] - Fernandez, J.E.; De Cea, J.; Florian, M.; Cabrera, E. Network equilibrium models with combined modes. Transp. Sci.
**1994**, 28, 182–192. [Google Scholar] [CrossRef] - Oppenheim, N. Urban Travel Demand Modeling; John Wiley & Sons: New York, NY, USA, 1995. [Google Scholar]
- Abrahamsson, T.; Lundqvist, L. Formulation and estimation of combined network equilibrium models with applications to stockholm. Transp. Sci.
**1999**, 33, 80–100. [Google Scholar] [CrossRef] - Boyce, D.; Bar-Gera, H. Validation of multiclass urban travel forecasting models combining origin-destination, mode, and route choices. J. Reg. Sci.
**2003**, 43, 517–540. [Google Scholar] [CrossRef] - Ham, H.; Tschangho, J.; Boyce, D. Implementation and estimation of a combined model of interregional, multimodal commodity shipments and transportation network flows. Transp. Res.
**2005**, 39, 65–79. [Google Scholar] [CrossRef] - Garcia, R.; Marin, A. Network equilibrium with combined modes: Models and solution algorithms. Transp. Res.
**2005**, 39, 223–254. [Google Scholar] [CrossRef] - De Cea, J.; Fernandez, J.E.; De Grange, L. Combined models with hierarchical demand choices: A multi-objective entropy optimization approach. Transp. Rev.
**2008**, 28, 415–438. [Google Scholar] [CrossRef] - Donoso, P.; De Grange, L. A Microeconomic interpretation of the maximum entropy estimator of multinomial logit models and its equivalence to the maximum likelihood estimator. Entropy
**2010**, 12, 2077–2084. [Google Scholar] [CrossRef] - SECTRA. Encuesta Origen Destino de Viajes 2001 para el Gran Santiago. Secretaría Interministerial de Planificación de Transporte, 2002, Santiago de Chile. Available online: http://www.sectra.cl/transporte/transporte_urbano_eod_frm.html (accessed on 21 March 2010).
- Ortuzar, J.d.D.; Willumsen, L.G. Modeling Transport; John Wiley & Sons: Chichester, UK, 2001. [Google Scholar]

## Appendices

## Appendix A: Microeconomic Interpretation of the Entropy Maximization Dual Problem

_{i}) of an individual i under a hierarchical logit choice structure is given by

## Appendix B: Estimates of Bias, Variance and Mean Square Error for Parameter Values ϕ = 1/μ = 0.2 and ϕ = 1/μ = 0.9

METHOD | PARAMETER | SAMPLE SIZE | BIAS | VARIANCE | MSE (*) |
---|---|---|---|---|---|

ML | 1/μ | 500 | 0.09615 | 0.00198 | 0.01122 |

1/μ | 1,000 | 0.06589 | 0.00030 | 0.00464 | |

1/μ | 5,000 | 0.00966 | 0.00011 | 0.00020 | |

1/μ | 10,000 | 0.00716 | 0.00005 | 0.00010 | |

1/μ | 20,000 | 0.00274 | 0.00001 | 0.00002 | |

VST | 500 | 1.10301 | 46.22001 | 47.43665 | |

VST | 1,000 | 0.45788 | 28.97585 | 29.18551 | |

VST | 5,000 | 0.23895 | 3.70613 | 3.76323 | |

VST | 10,000 | 0.07104 | 2.73371 | 2.73876 | |

VST | 20,000 | 0.04468 | 0.91350 | 0.91550 | |

ΜΕ | 1/μ | 500 | 0.12461 | 0.00017 | 0.01570 |

1/μ | 1,000 | 0.05836 | 0.00048 | 0.00389 | |

1/μ | 5,000 | 0.00517 | 0.00007 | 0.00010 | |

1/μ | 10,000 | 0.00410 | 0.00004 | 0.00006 | |

1/μ | 20,000 | 0.00300 | 0.00003 | 0.00004 | |

VST | 500 | 0.81525 | 5.40041 | 6.06504 | |

VST | 1,000 | 0.64647 | 3.43284 | 3.85076 | |

VST | 5,000 | 0.20451 | 0.28043 | 0.32226 | |

VST | 10,000 | 0.09542 | 0.12176 | 0.13087 | |

VST | 20,000 | 0.04107 | 0.07484 | 0.07653 |

METHOD | PARAMETER | SAMPLE SIZE | BIAS | VARIANCE | MSE (*) |
---|---|---|---|---|---|

ML | 1/μ | 500 | 0.22624 | 0.00432 | 0.05550 |

1/μ | 1,000 | 0.12514 | 0.00285 | 0.01851 | |

1/μ | 5,000 | 0.03456 | 0.00049 | 0.00168 | |

1/μ | 10,000 | 0.01835 | 0.00033 | 0.00067 | |

1/μ | 20,000 | 0.01331 | 0.00016 | 0.00034 | |

VST | 500 | 1.27926 | 28.32571 | 29.96221 | |

VST | 1,000 | 1.16837 | 11.69802 | 13.06312 | |

VST | 5,000 | 0.12918 | 1.92596 | 1.94265 | |

VST | 10,000 | 0.08153 | 0.96521 | 0.97186 | |

VST | 20,000 | 0.05641 | 0.38886 | 0.39204 | |

ΜΕ | 1/μ | 500 | 0.23444 | 0.00784 | 0.06280 |

1/μ | 1,000 | 0.09184 | 0.00452 | 0.01296 | |

1/μ | 5,000 | 0.01017 | 0.00074 | 0.00084 | |

1/μ | 10,000 | 0.00754 | 0.00027 | 0.00032 | |

1/μ | 20,000 | 0.00450 | 0.00016 | 0.00018 | |

VST | 500 | 0.70652 | 5.28093 | 5.78010 | |

VST | 1,000 | 0.65276 | 2.30125 | 2.72735 | |

VST | 5,000 | 0.16269 | 0.18473 | 0.21119 | |

VST | 10,000 | 0.09341 | 0.09402 | 0.10275 | |

VST | 20,000 | 0.00434 | 0.00270 | 0.00272 |

## Appendix C: Distribution of DMS with ML Estimation for Parameter Values ϕ = 1/μ = 0.2 and ϕ = 1/μ = 0.9

## Appendix D: Estimates of EMU for Parameter Values ϕ = 1/μ = 0.2 and ϕ = 1/μ = 0.9

SAMPLE SIZE | SIMULATION | ML | Δ% ML (*) | ME | Δ% ME (*) | Δ% ML/ME (**) |
---|---|---|---|---|---|---|

500 | 3.4436 | 1.7934 | 47.9% | 2.1346 | 38.0% | 26.1% |

1,000 | 3.4765 | 2.3030 | 33.8% | 2.5821 | 25.7% | 31.2% |

5,000 | 3.4075 | 3.2077 | 5.9% | 3.2767 | 3.8% | 52.8% |

10,000 | 3.4228 | 3.2882 | 3.9% | 3.3239 | 2.9% | 36.1% |

20,000 | 3.4205 | 3.3780 | 1.2% | 3.3936 | 0.8% | 58.1% |

SAMPLE SIZE | SIMULATION | ML | Δ% ML (*) | ME | Δ% ME (*) | Δ% ML/ME (**) |
---|---|---|---|---|---|---|

500 | 11.7592 | 8.8470 | 24.8% | 9.6120 | 18.3% | 35.6% |

1,000 | 11.7305 | 10.0668 | 14.2% | 10.6354 | 9.3% | 51.9% |

5,000 | 11.7876 | 11.3636 | 3.6% | 11.5302 | 2.2% | 64.8% |

10,000 | 11.7835 | 11.5360 | 2.1% | 11.6765 | 0.9% | 131.2% |

20,000 | 11.7728 | 11.6164 | 1.3% | 11.6959 | 0.7% | 103.3% |

## Appendix E: Estimates of ΔEMU for a 10% Reduction in Travel Time for the Parameters ϕ = 1/μ = 0.2 and ϕ = 1/μ = 0.9

© 2011 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Donoso, P.; De Grange, L.; González, F.
A Maximum Entropy Estimator for the Aggregate Hierarchical Logit Model. *Entropy* **2011**, *13*, 1425-1445.
https://doi.org/10.3390/e13081425

**AMA Style**

Donoso P, De Grange L, González F.
A Maximum Entropy Estimator for the Aggregate Hierarchical Logit Model. *Entropy*. 2011; 13(8):1425-1445.
https://doi.org/10.3390/e13081425

**Chicago/Turabian Style**

Donoso, Pedro, Louis De Grange, and Felipe González.
2011. "A Maximum Entropy Estimator for the Aggregate Hierarchical Logit Model" *Entropy* 13, no. 8: 1425-1445.
https://doi.org/10.3390/e13081425