A Maximum Likelihood Method with Penalty to Estimate
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A MAXIMUM LIKELIHOOD METHOD WITH PENALTY TO ESTIMATE LINK TRAVEL TIME BASED ON TRIP ITINERARY DATA A Thesis by CHUJUN ZHONG Submitted to the Office of Graduate and Professional Studies of Texas A&M University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Chair of Committee, Xiubin Bruce Wang Committee Members, Luca Quadrifoglio Kenneth Joh Department Head, Robin Autenrieth December 2014 Major Subject: Civil Engineering Copyright 2014 Chujun Zhong ABSTRACT Travel time is an important network performance measure. It is a challenging subject due to the fluctuations in traffic characteristics, such as traffic flow. This study proposes a maximum likelihood method with penalty to estimate link travel time based on trip itinerary data from a statistical point. Three penalized models, which are Lasso penalized model, Ridge penalized model and Revised-Lasso penalized model, are introduced. The models are discussed and compared with the basic model which is a maximum likelihood function without penalty. First, the predictive performance of the basic model and three penalized models are evaluated based on the data of three simulated networks. Results suggest that Revised-Lasso penalized model outperforms other models. In this research, Revised-Lasso penalized model is applied to a simplified Sioux Falls network. This study also provides a detailed procedure to estimate link travel time parameters in the simplified Sioux Falls network. Finally, the effect of the sample size on estimation accuracy is tested. The results show that sample size has a significant effect on the basic model estimation, but it has little effect on the Revised-Lasso penalized model estimation. This study provides an efficient and accurate way to estimate link travel time distribution. ii DEDICATION To my father,Bo Zhong and my mother, Lihua Yan. iii ACKNOWLEDGEMENTS First, I would like to express my appreciation to my advisor, Dr. Bruce Wang. I would like to thank him for introducing me to this topic, for providing financial support and for his guidance. I also would like to thank Dr. Luca Quadrifoglio and Dr. Kenneth Joh for being on my committee. And thank Dr. Yunlong Zhang for substituting for Dr. Quadrifoglio at my defense. I would thank Kai Yin and Wen Wang for their mentoring. In particular, Kai gave me some initial ideas and helped to improve. Finally, I would like to thank the National Center for Freight and Infrastructure Research and Education (CFIRE) at the University of Wisconsin for giving me the financial support. iv TABLE OF CONTENTS Page ABSTRACT .......................................................................................................................ii DEDICATION ................................................................................................................. iii ACKNOWLEDGEMENTS .............................................................................................. iv TABLE OF CONTENTS ................................................................................................... v LIST OF FIGURES ..........................................................................................................vii LIST OF TABLES ......................................................................................................... viii CHAPTER I INTRODUCTION ........................................................................................ 1 1.1 Problem Statement ................................................................................................ 4 1.2 Research Tasks ...................................................................................................... 4 1.3 Thesis Organization .............................................................................................. 5 CHAPTER II LITERATURE REVIEW ............................................................................ 7 2.1 Reviews of Travel Time Estimation Study ........................................................... 7 2.2 Reviews of Maximum Likelihood ...................................................................... 14 2.3 Reviews of Penalized Function ........................................................................... 15 CHAPTER III MODEL DESCRIPTION AND ALGORITHM ...................................... 17 3.1 Model Description ............................................................................................... 17 3.2 Estimation of Mean Travel Time ........................................................................ 20 3.3 Estimation of Link Travel Time Variance .......................................................... 21 CHAPTER IV MODEL VALIDATION ......................................................................... 24 4.1 Data Collection.................................................................................................... 24 4.2 Test Networks Description .................................................................................. 26 4.3 Estimation Results ............................................................................................... 29 4.4 Sioux Falls Network Test and Results ................................................................ 41 4.5 Effect of Sample Size on Estimation .................................................................. 48 v CHAPTER V APPLICATION ......................................................................................... 53 5.1 Application Problem Description ........................................................................ 53 5.2 Model Application .............................................................................................. 54 5.3 Application Results Analysis .............................................................................. 57 CHAPTER VI CONCLUSIONS AND DISCUSSIONS ................................................. 58 REFERENCES ................................................................................................................. 61 APPENDIX A .................................................................................................................. 64 APPENDIX B .................................................................................................................. 67 APPENDIX C .................................................................................................................. 68 APPENDIX D .................................................................................................................. 73 APPENDIX E ................................................................................................................... 77 APPENDIX F ................................................................................................................... 90 APPENDIX G .................................................................................................................. 92 APPENDIX H ................................................................................................................ 104 vi LIST OF FIGURES Page Figure 1. Travel Time on DMS in Portland, Oregon ......................................................... 2 Figure 2. Chicago Travel Times on Web ........................................................................... 2 Figure 3. The Procedure of Data Collection Method ....................................................... 25 Figure 4. Standard Deviation Estimation Comparison in the 10-link Network ............... 34 Figure 5. Standard Deviation Estimation Comparison in the 20-link Network ............... 36 Figure 6. Standard Deviation Estimation Comparison in the 50-link Network ............... 38 Figure 7. Simplified Sioux Falls Network with 27 Links ................................................ 42 Figure 8. Sample Size Effect on RMSE ........................................................................... 51 Figure 9. Sample Size Effect on MAPE ........................................................................... 52 Figure 10. Simulation of Selected Network in College Station ....................................... 54 vii LIST OF TABLES Page Table 1. Trip-Link Matrix in the 10-link Network ........................................................... 26 Table 2. Real Mean and Standard Deviation in the 10-link Network .............................. 27 Table 3. Observed Trip Itinerary in the10-link Network ................................................. 27 Table 4. Real Mean and Standard Deviation in the 20-link Network .............................. 28 Table 5. Real Mean and Standard Deviation in the 50-link Network .............................. 28 Table 6. Mean Estimation Result in the 10-link Network ................................................ 30 Table 7. Mean Estimation Result in the 20-link Network ................................................ 31 Table 8. Mean Estimation Result in the 50-link Network ................................................ 31 Table 9. Standard Deviation Estimation Result in the 10-link Network .......................... 33 Table 10. Standard Deviation Estimation Result in the 20-link Network ........................ 35 Table 11. Standard Deviation Estimation Result in the 50-link Network ........................ 37 Table 12. Improved Rate of Penalized Models ................................................................ 39 Table 13. OD Selected in the Sioux Falls Network ......................................................... 43 Table 14. Real Mean and Standard Deviation in the Sioux Falls