Mutual Information for the Detection of Crush Conditions Thesis presented to the Faculty of Science and Engineering, in partial fulfillment of the Requirements for the Degree of Doctor of Philosophy September, 2012 Author: Supervisor: Peter Harding Prof. Martyn Amos Ind. Collaborator: Dr. Steven Gwynne Contents Table of Contents . i List of Tables . vi List of Figures . vii List of Equations . xv Declarations . xvii Acknowledgements . xviii Abstract . xix 1 Introduction 1 1.1 Scope of Study . 3 1.2 Identification of Crush Conditions in in silico Simulations . 3 1.3 Contributions . 4 1.4 Thesis Outline . 5 2 Evacuation and Crush 7 2.1 Introduction . 7 2.2 What is Evacuation? . 7 2.3 The Behaviour of Evacuating Crowds . 9 2.3.1 Fallacies of Crowd Behaviour . 9 2.4 What are Crush Conditions? . 13 2.5 What Causes Crush? . 14 2.5.1 Spatial . 14 2.5.2 Temporal . 15 2.5.3 Perceptual and Cognitive Factors . 17 2.5.4 Procedural . 18 2.5.5 Structural . 19 2.6 Types of Force in Evacuation Scenarios . 19 2.6.1 Pushing . 19 2.6.2 Stacking . 20 i 2.6.3 Leaning . 23 2.7 Historical Examples of Crush . 25 2.7.1 Hillsborough . 25 2.7.2 Rhode Island Nightclub . 32 2.7.3 Gothenburg Dancehall . 37 2.7.4 E2 Nightclub Incident . 42 2.7.5 Mihong Bridge Spring Festival Disaster . 46 2.8 A Diagnosis Issue in Crowd Crush Situations . 52 2.9 Detecting Crush Conditions via Phase Transitions - An Initial Idea . 54 2.10 Scope of this Study . 55 2.11 Summary . 56 3 Computational Studies of Evacuation 57 3.1 Introduction . 57 3.2 Computational Evacuation Modelling . 57 3.2.1 Representations of Physical Environment . 58 3.2.2 Behavioural/Movement Modelling . 65 3.3 Recent Trends in Research Activity . 67 3.3.1 Trends in Model Usage . 69 3.4 Modelling of Crush . 72 3.4.1 Implicit . 72 3.4.2 Explicit . 74 3.5 Difficulties in Modelling Injuries Caused by Force . 75 3.6 Our Proposed Approach . 76 3.7 Summary . 77 4 Social Forces Model 79 4.1 Introduction . 79 4.2 Background . 79 4.3 Description . 80 4.4 Mathematical Definition . 81 4.4.1 Movement . 82 4.4.2 Social Force . 82 4.4.3 Boundary Force . 85 4.4.4 Goal Finding . 85 4.5 Visual Example . 85 ii 4.6 Critical Analysis . 87 4.6.1 Limitations . 88 4.7 Time-lapsed Visual Example . 95 4.8 Differences in the FDS+Evac Implementation . 95 4.9 Summary . 98 5 Mutual Information 100 5.1 Introduction . 100 5.2 Background . 100 5.3 Joint Entropy . 102 5.3.1 Conditional Entropy . 102 5.4 Mutual Information . 103 5.4.1 Conditional Mutual Information . 104 5.4.2 Multi-variate Mutual Information . 105 5.4.3 The Logarithmic Base . 106 5.5 MI Example . 107 5.6 Information Theory . 108 5.7 Biological Stimuli-response Systems . 110 5.8 Chaotic Systems . 111 5.9 Medical Image Processing . 112 5.9.1 Normalised Mutual Information . 113 5.9.2 Higher Dimensional Mutual Information . 114 5.10 Complex Systems . 116 5.11 Summary . 121 6 Detection of a Phase Transition 122 6.1 Introduction . 122 6.2 Scalar Noise Model . 122 6.2.1 Phase Transition Point . 124 6.2.2 Susceptibility . 125 6.2.3 Methodology . 126 6.3 Development . 128 6.4 Experimentation . 130 6.4.1 Multivariate Mutual Information . 130 6.4.2 Conditional Mutual Information . 132 6.4.3 Normalised Mutual Information . 133 6.4.4 Studholme's Mutual Information . 134 iii 6.4.5 Wicks' Mutual Information . 135 6.5 Correlation Analysis . 137 6.6 Conclusions . 138 7 Initial Evacuation Simulations 140 7.1 Introduction . 140 7.2 Hypotheses . 140 7.3 Experimental Aims . 141 7.4 Expected Outcomes . 141 7.5 Methodology . 142 7.6 Experimentation . 145 7.6.1 Single Run . 148 7.6.2 Partial Data Analysis . 149 7.7 Calibration . 150 7.7.1 X and Y Discretisation . 151 7.7.2 Orientation Discretisation . 151 7.8 Negation of False Positives . 152 7.9 Results . 154 7.10 Hypothesis Testing . 155 7.11 Distribution of Force Across Agents . 155 7.12 Comparison with the Crowd Density Measure . 157 7.13 Summary . 158 8 Analysis of a Historic Event 160 8.1 Introduction . 160 8.2 Fire Dynamics Simulator . 160 8.3 The Station Nightclub Disaster . 161 8.4 Methodology . 161 8.4.1 Experimental Setup . 162 8.4.2 Validation . 165 8.4.3 Detection of Crush . 168 8.4.4 Idealised Scenario . 171 8.4.5 Realistic Scenario . 172 8.4.6 Correlation Analysis . 172 8.5 False Positives . 173 8.5.1 Specification . 174 8.5.2 Baseline . 174 iv 8.6 Summary . 178 9 Conclusions and Recommendations 179 9.1 Conclusions . 179 9.2 Recommendations for Further Research . 181 Bibliography 182 A Publications 198 Mutual Information for the Detection of Crush Conditions - Jour- nal article . 198 Mutual Information for the Detection of Crush Conditions - Con- ference Publication . 209 Prediction and Mitigation of Crush Conditions . 215 v List of Tables 2.1 Prisoners' Dilemma payoff matrix . 11 2.2 Pedestrian Levels of Service (LoS). Available space (s) is mea- sured in m2 per person [37]. 15 3.1 List of Evacuation models considered during literature review. Note: A model's inclusion in this table does not imply that it is widely used or actively developed, just that research papers exist which have explicitly referenced the model in question. 70 5.1 The inputs, X and Y ,.