Approaches to Geo-mathematical Modelling

Alan Geoffrey Wilson, Centre for Advanced Spatial Analysis, University College London, UK. His research interests have been concerned with many aspects of mathematical modelling and the use of models in planning in relation to all aspects of cities and regions - including demography, economic input-output modelling, transport and locational structures. He was responsible for the introduction of a number of model building techniques which are now in common use internationally. These models have been widely used in areas such as transport planning. He made important contributions through the rigorous deployment of accounts' concepts in demography and economic modelling. In recent years he has been particularly concerned with applications of dynamical systems theory in relation to the task of modelling the evolution of urban structure, initially described in Catastrophe theory and bifurcation: applications to urban and regional systems. His current research, supported by ESRC and EPSRC grants of around ?3M, is on the evolution of cities and the dynamics of global trade and migration.

• Notes on Contributors xvAcknowledgements xxiAbout the Companion Website xxiiiPart I Approaches1 The Toolkit 3Alan G. WilsonPart II Estimating Missing Data: Bi-proportional Fitting and Principal Components Analysis2 The Effects of Economic and Labour Market Inequalities on Interregional Migration in Europe 9Adam Dennett2.1 Introduction 92.2 The Approach 122.3 Data 122.4 Preliminary Analysis 132.5 Multinomial Logit Regression Analysis 152.6 Discussion 222.7 Conclusions 24References 253 Test of Bi-Proportional Fitting Procedure Applied to International Trade 26Simone Caschili and Alan G. Wilson3.1 Introduction 263.2 Model 273.3 Notes of Implementation 283.4 Results 30References 324 Estimating Services Flows 33Robert G. Levy4.1 Introduction 334.2 Estimation Via Iterative Proportional Fitting 344.2.1 The Method 344.2.2 With All Initial Values Equal 354.2.3 Equivalence to Entropy Maximisation 364.2.4 Estimation with Some Known Flows 374.2.5 Drawbacks to Estimating Services Flows with IPF 374.3 Estimating Services Flows Using Commodities Flows 374.3.1 The Gravity Model 374.3.2 Splitting Up Value Added 404.4 A Comparison of The Methods 404.4.1 Unbalanced Row and Column Margins 424.4.2 Iterative Proportional Fitting 424.4.3 Gravity Model 424.4.4 Gravity Model Followed by IPF 444.5 Results 454.5.1 Selecting a Representative Sector 454.5.2 Estimated in-Sample Flows 464.5.3 Estimated Export Totals 474.6 Conclusion 49References 505 A Method for Estimating Unknown National Input–Output Tables Using Limited Data 51Thomas P. Oléron Evans and Robert G. Levy5.1 Motivation and Aims 515.2 Obstacles to The Estimation of National Input–Output Tables 525.3 Vector Representation of Input–Output Tables 535.4 Method 545.4.1 Concept 545.4.2 Estimation Procedure 555.4.3 Cross-Validation 575.5 In-Sample Assessment of The Estimates 585.5.1 Summary Statistics 585.5.2 Visual Comparison 615.6 Out-of-Sample Discussion of The Estimates 635.6.1 Final Demand Closeness 635.6.2 Technical Coefficient Clustering 655.7 Conclusion 67References 68Part III Dynamics in Account-based Models6 A Dynamic Global Trade Model With Four Sectors: Food, Natural Resources, Manufactured Goods and Labour 71Hannah M. Fry, Alan G. Wilson and Frank T. Smith6.1 Introduction 716.2 Definition of Variables for System Description 736.3 The Pricing and Trade Flows Algorithm 736.4 Initial Setup 756.5 The Algorithm to Determine Farming Trade Flows 776.5.1 The Accounts for the Farming Industry 796.5.2 A Final Point on The Farming Flows 796.6 The Algorithm to Determine The Natural Resources Trade Flows 806.6.1 The Accounts for The Natural Resources Sector 806.7 The Algorithm to Determine Manufacturing Trade Flows 816.7.1 The Accounts for The Manufacturing Industry 826.8 The Dynamics 836.9 Experimental Results 846.9.1 Concluding Comments 88References 907 Global Dynamical Input–Output Modelling 91Anthony P. Korte and Alan G. Wilson7.1 Towards a Fully Dynamic Inter-country Input–Output Model 917.2 National Accounts 927.2.1 Definitions 927.2.2 The Production Account 947.2.3 The Commodity Markets Account 947.2.4 The Household Account 947.2.5 The Capital Markets Account 947.2.6 The Rest of the World (RoW) Account 947.2.7 The Government Account 957.2.8 The Net Worth of an Economy and Revaluations 957.2.9 Overview of the National Accounts 957.2.10 Closing the Model: Making Final Demand Endogenous 967.3 The Dynamical International Model 977.3.1 Supply and Demand 977.3.2 The National Accounts Revisited 997.4 Investment: Modelling Production Capacity: The Capacity Planning Model 1007.4.1 The Multi-region, Multi-sector Capacity Planning Model 1007.5 Modelling Production Capacity: The Investment Growth Approach 1037.5.1 Multi-region, multi-sector Investment Growth Models with Reversibility 1037.5.2 One-country, One-sector Investment Growth Model with Reversibility 1047.5.3 Two-country, Two-sector Investment Growth Model with Reversibility 1067.5.4 A Multi-region, Multi-sector, Investment Growth Model without Reversibility 1087.5.5 A Multi-region, Multi-sector, Investment Growth Model without Reversibility, with Variable Trade Coefficients 1117.5.6 Dynamical Final Demand 1147.5.7 Labour 1157.5.8 The Price Model 1187.6 Conclusions 121References 122Appendix 123A.1 Proof of Linearity of the Static Model and the Equivalence of Two Modelling Approaches 123Part IV Space–Time Statistical Analysis8 Space–Time Analysis of Point Patterns in Crime and Security Events 127Toby P. Davies, Shane D. Johnson, Alex Braithwaite and Elio Marchione8.1 Introduction 1278.1.1 Clustering 1278.1.2 Clustering of Urban Crime 1298.1.3 The Knox Test 1308.2 Application in Novel Areas 1328.2.1 Maritime Piracy 1328.2.2 Space–Time Clustering of Piracy 1348.2.3 Insurgency and Counterinsurgency in Iraq 1368.3 Motif Analysis 1388.3.1 Introduction 1388.3.2 Event Networks 1408.3.3 Network Motifs 1408.3.4 Statistical Analysis 1418.3.5 Random Network Generation 1428.3.6 Results 1438.4 Discussion 147References 148Part V Real-Time Response Models9 The London Riots –1: Epidemiology, Spatial Interaction and Probability of Arrest 153Toby P. Davies, Hannah M. Fry, Alan G. Wilson and Steven R. Bishop9.1 Introduction 1539.2 Characteristics of Disorder 1569.3 The Model 1589.3.1 Outline 1589.3.2 General Concepts 1589.3.3 Riot Participation 1599.3.4 Spatial Assignment 1609.3.5 Interaction between Police and Rioters 1629.4 Demonstration Case 1629.5 Concluding Comments 166References 166Appendix 168A.1 Note on Methods: Data 168A.2 Numerical Simulations 16910 The London Riots –2: A Discrete Choice Model 170Peter Baudains, Alex Braithwaite and Shane D. Johnson10.1 Introduction 17010.2 Model Setup 17010.3 Modelling the Observed Utility 17210.4 Results 17610.5 Simulating the 2011 London Riots: Towards a Policy Tool 18110.6 Modelling Optimal Police Deployment 187References 190Part VI The Mathematics of War11 Richardson Models with Space 195Peter Baudains11.1 Introduction 19511.2 The Richardson Model 19611.3 Empirical Applications of Richardson’s Model 20211.4 A Global Arms Race Model 20411.5 Relationship to a Spatial Conflict Model 20611.6 An Empirical Application 20711.6.1 Two Models of Global Military Expenditure 20711.6.2 The Alliance Measure C ij 20811.6.3 A Spatial Richardson Model of Global Military Expenditure 21011.6.4 Results 21111.7 Conclusion 212References 213Part VII Agent-based Models12 Agent-based Models of Piracy 217Elio Marchione, Shane D. Johnson and Alan G. Wilson12.1 Introduction 21712.2 Data 21912.3 An Agent-based Model 22112.3.1 Defining Maritime Piracy Maps 22112.3.2 Defining Vessel Route Maps 22212.3.3 Defining Pirates’, Naval Units’ and Vessels’ Behaviours 22412.3.4 Comparing Risk Maps 22712.4 Model Calibration 23212.5 Discussion 232References 23513 A Simple Approach for the Prediction of Extinction Events in Multi-agent Models 237Thomas P. Oléron Evans, Steven R. Bishop and Frank T. Smith13.1 Introduction 23713.2 Key Concepts 23813.2.1 Binary Classification 23813.2.2 Measures of Classifier Performance 23813.2.3 Stochastic Processes 24013.3 The NANIA Predator–prey Model 24113.3.1 Background 24113.3.2 An ODD Description of the NANIA Model 24113.3.3 Behaviour of the NANIA Model 24513.3.4 Extinctions in the NANIA Model 24613.4 Computer Simulation 24713.4.1 Data Generation 24713.4.2 Categorisation of the Data 24913.5 Period Detection 24913.6 A Monte Carlo Approach to Prediction 25213.6.1 Binned Data 25213.6.2 Confidence Intervals 25713.6.3 Predicting Extinctions using Binned Population Data 25713.6.4 ROC and Precision-recall Curves for Monte Carlo Prediction of Predator Extinctions 26013.7 Conclusions 263References 264Part VIII Diffusion Models14 Urban Agglomeration Through the Diffusion of Investment Impacts 269Minette D’Lima, Francesca R. Medda and Alan G. Wilson14.1 Introduction 26914.2 The Model 27014.3 Mathematical Analysis for Agglomeration Conditions 27214.3.1 Introduction 27214.3.2 Case: r < c 27414.3.3 Case: r ≥ c 27414.4 Simulation Results 27514.5 Conclusions 279References 279Part IX Game Theory15 From Colonel Blotto to Field Marshall Blotto 283Peter Baudains, Toby P. Davies, Hannah M. Fry and Alan G. Wilson15.1 Introduction 28315.2 The Colonel Blotto Game and its Extensions 28515.3 Incorporating a Spatial Interaction Model of Threat 28615.4 Two-front Battles 28815.5 Comparing Even and Uneven Allocations in a Scenario with Five Fronts 28915.6 Conclusion 292References 29216 Modelling Strategic Interactions in a Global Context 293Janina Beiser16.1 Introduction 29316.2 The Theoretical Model 29416.3 Strategic Estimation 29516.4 International Sources of Uncertainty in the Context of Repression and Rebellion 29716.4.1 International Sources of Uncertainty Related to Actions 29716.5 International Sources of Uncertainty Related to Outcomes 29916.6 Empirical Analysis 30116.6.1 Data and Operationalisation 30116.7 Results 30316.8 Additional Considerations Related to International Uncertainty 30416.9 Conclusion 304References 30517 A General Framework for Static, Spatially Explicit Games of Search and Concealment 306Thomas P. Oléron Evans, Steven R. Bishop and Frank T. Smith17.1 Introduction 30617.2 Game Theoretic Concepts 30717.3 Games of Search and Security: A Review 31017.3.1 Simple Search Games 31017.3.2 Search Games with Immobile Targets 31117.3.3 Accumulation Games 31117.3.4 Search Games with Mobile Targets 31117.3.5 Allocation Games 31217.3.6 Rendez-vous Games 31217.3.7 Security Games 31317.3.8 Geometric Games 31317.3.9 Motivation for Defining a New Spatial Game 31417.4 The Static Spatial Search Game (SSSG) 31417.4.1 Definition of the SSSG 31417.4.2 The SSSG and other Games 31617.4.3 The SSSG with Finite Strategy Sets 31717.4.4 Dominance and Equivalence in the SSSG 31817.4.5 Iterated Elimination of Dominated Strategies 32317.5 The Graph Search Game (GSG) 32417.5.1 Definition of the GSG 32417.5.2 The GSG with r ≠ 1 32617.5.3 Preliminary Observations 32717.5.4 Bounds on the Value of the GSG 33017.6 Summary and Conclusions 335References 336Part X Networks18 Network Evolution: A Transport Example 343Francesca Pagliara, Alan G. Wilson and Valerio de Martinis18.1 Introduction 34318.2 A Hierarchical Retail Structure Model as a Building Block 34418.3 Extensions to Transport Networks 34518.4 An Application in Transport Planning 34718.5 A Case Study: Bagnoli in Naples 35018.6 Conclusion 360References 36119 The Structure of Global Transportation Networks 363Sean Hanna, Joan Serras and Tasos Varoudis19.1 Introduction 36319.2 Method 36419.3 Analysis of the European Map 36619.4 Towards a Global Spatial Economic Map: Economic Analysis by Country 36819.5 An East-west Divide and Natural Economic Behaviour 37319.6 Conclusion 376References 37720 Trade Networks and Optimal Consumption 378Robert J. Downes and Robert G. Levy20.1 Introduction 37820.2 The Global Economic Model 37920.2.1 Introduction 37920.2.2 Data Sources 38020.2.3 Model Overview 38020.3 Perturbing Final Demand Vectors 38020.3.1 Introduction 38020.3.2 Perturbation Process 38220.4 Analysis 38420.4.1 Introduction 38420.4.2 A Directed Network Representation 38420.4.3 A Weighted Directed Network Representation 38920.4.4 Communities in the Network of Improvements 39020.5 Conclusions 393Acknowledgements 394References 394Appendix 396Part XI Integration21 Research Priorities 399Alan G. WilsonIndex 403