................................ ................................ ................................ .................. vi LIST OF FIGURES ................................ ................................ ................................ ...... ix LIST OF TABL ES ................................ ................................ ................................ ......... x LIST OF ABBREVIATIONS ................................ ................................ ....................... xi ................................ ................................ ................................ .................. 1 INTRODUCTION ................................ ................................ ................................ ......... 1 1.1 Background a nd Motivations ................................ ................................ .......... 1 1.2 Research Objectives and Overview ................................ ................................ . 4 1.3 Scope of the Research ................................ ................................ ..................... 4 1.4 Organization of the thesis ................................ ................................ ................ 5 ................................ ................................ ................................ .................. 6 LITERATURE REVIEW ................................ ................................ .............................. 6 2.1 Introduction ................................ ................................ ................................ ..... 6 2.2 Hawkes P rocess in general ................................ ................................ .............. 6 2.2.1 Point process ................................ ................................ ......................... 6 2.2.2 Self-Exciting Point Process ................................ ................................ .. 7 2.2.3 Temporal Point Process ................................ ................................ ........ 7 2.2.4 Conditional Intensity Function ................................ ............................. 8 2.2.5 Hawkes process ................................ ................................ .................... 8 2.3 Hawkes Process in Traffic Crash System ................................ ........................ 9 2.3.1 Self-exciting Model for Traffic Crash System ................................ ..... 9 2.3.2 Stationary Condition ................................ ................................ ........... 11 2.4 Relevant Studies ................................ ................................ ............................ 12 2.5 Summary ................................ ................................ ................................ ........ 14 ................................ ................................ ................................ ................ 15 METHODOLOGY ................................ ................................ ................................ ...... 15 3.1 Introduction ................................ ................................ ................................ ... 15 3.2 Methodology Overview ................................ ................................ ................. 15 3.3 Study Area ................................ ................................ ................................ ..... 17 3.4 Data Collection ................................ ................................ .............................. 18 3.5 Data Processing ................................ ................................ ............................. 19 3.6 Most Likely Estimates of the Parameters: ................................ ..................... 21 3.6.1 Likelihood Equation ................................ ................................ ........... 21 3.6.2 Computational Methodology ................................ .............................. 22 3.7 Checking the Goodness of Fit ................................ ................................ ....... 24 3.8 Summary ................................ ................................ ................................ ........ 25 viii ................................ ................................ ................................ ................ 26 RESULTS AND DISCUS SION ................................ ................................ .................. 26 4.1 Introduction ................................ ................................ ................................ ... 26 4.2 Parameter Estimation ................................ ................................ ..................... 26 4.2.1 Crash Counting Process ................................ ................................ ...... 26 4.2.2 MLE of Parameters ................................ ................................ ............. 27 4.2.3 Conditional Intensity Function ................................ ........................... 28 4.3 Goodness of Fit ................................ ................................ .............................. 30 4.4 Stationary Condition ................................ ................................ ...................... 30 4.5 Discussions ................................ ................................ ................................ .... 31 4.5.1 Parameter Estimation ................................ ................................ .......... 31 4.5.2 Goodness of Fit Test ................................ ................................ ........... 32 4.5.3 Stationary Condition ................................ ................................ ........... 32 4.6 Summary ................................ ................................ ................................ ........ 33 ................................ ................................ ................................ ................ 34 CONCLUSION ................................ ................................ ................................ ............ 34 5.1 Introduction ................................ ................................ ................................ ... 34 5.2 Major Fi ndings of the Study ................................ ................................ .......... 34 5.3 Limitations of the Study ................................ ................................ ................ 35 5.4 Recommendations for Future Works ................................ ............................. 36 5.4.1 Future Studies ................................ ................................ ..................... 36 5.4.2 Road Safety Engineering ................................ ................................ .... 37 5.5 Concluding Remarks ................................ ................................ ..................... 37