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E-raamat: Approaches to Geo-mathematical Modelling: New Tools for Complexity Science

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Geo-mathematical modelling: models from complexity science

 

Sir Alan Wilson, Centre for Advanced Spatial Analysis, University College London

 

Mathematical and computer models for a complexity science tool kit

 

Geographical systems are characterised by locations, activities at locations, interactions between them and the infrastructures that carry these activities and flows. They can be described at a great variety of scales, from individuals and organisations to countries. Our understanding, often partial, of these entities, and in many cases this understanding is represented in theories and associated mathematical models.

 

In this book, the main examples are models that represent elements of the global system covering such topics as trade, migration, security and development aid together with examples at finer scales. This provides an effective toolkit that can not only be applied to global systems, but more widely in the modelling of complex systems. All complex systems involve nonlinearities involving path dependence and the possibility of phase changes and this makes the mathematical aspects particularly interesting. It is through these mechanisms that new structures can be seen to ‘emerge’, and hence the current notion of ‘emergent behaviour’. The range of models demonstrated include account-based models and biproportional fitting, structural dynamics, space-time statistical analysis, real-time response models, Lotka-Volterra models representing ‘war’, agent-based models, epidemiology and reaction-diffusion approaches, game theory, network models and finally, integrated models.

 

Geo-mathematical modelling:

  • Presents mathematical models with spatial dimensions.
  • Provides representations of path dependence and phase changes.
  • Illustrates complexity science using models of trade, migration, security and development aid.
  • Demonstrates how generic models from the complexity science tool kit can each be applied in a variety of situations

 

This book is for practitioners and researchers in applied mathematics, geography, economics, and interdisciplinary fields such as regional science and complexity science. It can also be used as the basis of a modelling course for postgraduate students.

Notes on Contributors xv
Acknowledgements xxi
About the Companion Website xxiii
Part I Approaches
1 The Toolkit
3(6)
Alan G. Wilson
Part II Estimating Missing Data: Bi-Proportional Fitting And Principal Components Analysis
2 The Effects of Economic and Labour Market Inequalities on Interregional Migration in Europe
9(17)
Adam Dennett
2.1 Introduction
9(3)
2.2 The Approach
12(1)
2.3 Data
12(1)
2.4 Preliminary Analysis
13(2)
2.5 Multinomial Logit Regression Analysis
15(7)
2.6 Discussion
22(2)
2.7 Conclusions
24(1)
References
25(1)
3 Test of Bi-Proportional Fitting Procedure Applied to International Trade
26(7)
Simone Caschili
Alan G. Wilson
3.1 Introduction
26(1)
3.2 Model
27(1)
3.3 Notes of Implementation
28(2)
3.4 Results
30(2)
References
32(1)
4 Estimating Services Flows
33(18)
Robert G. Levy
4.1 Introduction
33(1)
4.2 Estimation Via Iterative Proportional Fitting
34(3)
4.2.1 The Method
34(1)
4.2.2 With All Initial Values Equal
35(1)
4.2.3 Equivalence to Entropy Maximisation
36(1)
4.2.4 Estimation with Some Known Flows
37(1)
4.2.5 Drawbacks to Estimating Services Flows with IPF
37(1)
4.3 Estimating Services Flows Using Commodities Flows
37(3)
4.3.1 The Gravity Model
37(3)
4.3.2 Splitting Up Value Added
40(1)
4.4 A Comparison of The Methods
40(5)
4.4.1 Unbalanced Row and Column Margins
42(1)
4.4.2 Iterative Proportional Fitting
42(1)
4.4.3 Gravity Model
42(2)
4.4.4 Gravity Model Followed by IPF
44(1)
4.5 Results
45(4)
4.5.1 Selecting a Representative Sector
45(1)
4.5.2 Estimated in-Sample Flows
46(1)
4.5.3 Estimated Export Totals
47(2)
4.6 Conclusion
49(1)
References
50(1)
5 A Method for Estimating Unknown National Input-Output Tables Using Limited Data
51(20)
Thomas P. Oleron Evans
Robert G. Levy
5.1 Motivation and Aims
51(1)
5.2 Obstacles to The Estimation of National Input-Output Tables
52(1)
5.3 Vector Representation of Input-Output Tables
53(1)
5.4 Method
54(4)
5.4.1 Concept
54(1)
5.4.2 Estimation Procedure
55(2)
5.4.3 Cross-Validation
57(1)
5.5 In-Sample Assessment of The Estimates
58(5)
5.5.1 Summary Statistics
58(3)
5.5.2 Visual Comparison
61(2)
5.6 Out-of-Sample Discussion of The Estimates
63(4)
5.6.1 Final Demand Closeness
63(2)
5.6.2 Technical Coefficient Clustering
65(2)
5.7 Conclusion
67(1)
References
68(3)
Part III Dynamics In Account-Based Models
6 A Dynamic Global Trade Model With Four Sectors: Food, Natural Resources, Manufactured Goods and Labour
71(20)
Hannah M. Fry
Alan G. Wilson
Frank T. Smith
6.1 Introduction
71(2)
6.2 Definition of Variables for System Description
73(1)
6.3 The Pricing and Trade Flows Algorithm
73(2)
6.4 Initial Setup
75(2)
6.5 The Algorithm to Determine Farming Trade Flows
77(3)
6.5.1 The Accounts for the Farming Industry
79(1)
6.5.2 A Final Point on The Farming Flows
79(1)
6.6 The Algorithm to Determine The Natural Resources Trade Flows
80(1)
6.6.1 The Accounts for The Natural Resources Sector
80(1)
6.7 The Algorithm to Determine Manufacturing Trade Flows
81(2)
6.7.1 The Accounts for The Manufacturing Industry
82(1)
6.8 The Dynamics
83(1)
6.9 Experimental Results
84(6)
6.9.1 Concluding Comments
88(2)
References
90(1)
7 Global Dynamical Input-Output Modelling
91(36)
Anthony P. Korte
Alan G. Wilson
7.1 Towards a Fully Dynamic Inter-country Input-Output Model
91(1)
7.2 National Accounts
92(5)
7.2.1 Definitions
92(2)
7.2.2 The Production Account
94(1)
7.2.3 The Commodity Markets Account
94(1)
7.2.4 The Household Account
94(1)
7.2.5 The Capital Markets Account
94(1)
7.2.6 The Rest of the World (RoW) Account
94(1)
7.2.7 The Government Account
95(1)
7.2.8 The Net Worth of an Economy and Revaluations
95(1)
7.2.9 Overview of the National Accounts
95(1)
7.2.10 Closing the Model: Making Final Demand Endogenous
96(1)
7.3 The Dynamical International Model
97(3)
7.3.1 Supply and Demand
97(2)
7.3.2 The National Accounts Revisited
99(1)
7.4 Investment: Modelling Production Capacity: The Capacity Planning Model
100(3)
7.4.1 The Multi-region, Multi-sector Capacity Planning Model
100(3)
7.5 Modelling Production Capacity: The Investment Growth Approach
103(18)
7.5.1 Multi-region, multi-sector Investment Growth Models with Reversibility
103(1)
7.5.2 One-country, One-sector Investment Growth Model with Reversibility
104(2)
7.5.3 Two-country, Two-sector Investment Growth Model with Reversibility
106(2)
7.5.4 A Multi-region, Multi-sector; Investment Growth Model without Reversibility
108(3)
7.5.5 A Multi-region, Multi-sector, Investment Growth Model without Reversibility, with Variable Trade Coefficients
111(3)
7.5.6 Dynamical Final Demand
114(1)
7.5.7 Labour
115(3)
7.5.8 The Price Model
118(3)
7.6 Conclusions
121(1)
References
122(1)
Appendix
123(4)
A.1 Proof of Linearity of the Static Model and the Equivalence of Two Modelling Approaches
123(4)
Part IV Space-time Statistical Analysis
8 Space-Time Analysis of Point Patterns in Crime and Security Events
127(26)
Toby P. Davies
Shane D. Johnson
Alex Braithwaite
Elio Marchione
8.1 Introduction
127(5)
8.1.1 Clustering
127(2)
8.1.2 Clustering of Urban Crime
129(1)
8.1.3 The Knox Test
130(2)
8.2 Application in Novel Areas
132(6)
8.2.1 Maritime Piracy
132(2)
8.2.2 Space-Time Clustering of Piracy
134(2)
8.2.3 Insurgency and Counterinsurgency in Iraq
136(2)
8.3 Motif Analysis
138(9)
8.3.1 Introduction
138(2)
8.3.2 Event Networks
140(1)
8.3.3 Network Motifs
140(1)
8.3.4 Statistical Analysis
141(1)
8.3.5 Random Network Generation
142(1)
8.3.6 Results
143(4)
8.4 Discussion
147(1)
References
148(5)
Part V Real-Time Response Models
9 The London Riots -1: Epidemiology, Spatial Interaction and Probability of Arrest
153(17)
Toby P. Davies
Hannah M. Fry
Alan G. Wilson
Steven R. Bishop
9.1 Introduction
153(3)
9.2 Characteristics of Disorder
156(2)
9.3 The Model
158(4)
9.3.1 Outline
158(1)
9.3.2 General Concepts
158(1)
9.3.3 Riot Participation
159(1)
9.3.4 Spatial Assignment
160(2)
9.3.5 Interaction between Police and Rioters
162(1)
9.4 Demonstration Case
162(4)
9.5 Concluding Comments
166(1)
References
166(2)
Appendix
168(2)
A.1 Note on Methods: Data
168(1)
A.2 Numerical Simulations
169(1)
10 The London Riots -2: A Discrete Choice Model
170(25)
Peter Baudains
Alex Braithwaite
Shane D. Johnson
10.1 Introduction
170(1)
10.2 Model Setup
170(2)
10.3 Modelling the Observed Utility
172(4)
10.4 Results
176(5)
10.5 Simulating the 2011 London Riots: Towards a Policy Tool
181(6)
10.6 Modelling Optimal Police Deployment
187(3)
References
190(5)
Part VI The Mathematics Of War
11 Richardson Models with Space
195(22)
Peter Baudains
11.1 Introduction
195(1)
11.2 The Richardson Model
196(6)
11.3 Empirical Applications of Richardson's Model
202(2)
11.4 A Global Arms Race Model
204(2)
11.5 Relationship to a Spatial Conflict Model
206(1)
11.6 An Empirical Application
207(5)
11.6.1 Two Models of Global Military Expenditure
207(1)
11.6.2 The Alliance Measure Cij
208(2)
11.6.3 A Spatial Richardson Model of Global Military Expenditure
210(1)
11.6.4 Results
211(1)
11.7 Conclusion
212(1)
References
213(4)
Part VII Agent-Based Models
12 Agent-based Models of Piracy
217(20)
Elio Marchione
Shane D. Johnson
Alan G. Wilson
12.1 Introduction
217(2)
12.2 Data
219(2)
12.3 An Agent-based Model
221(11)
12.3.1 Defining Maritime Piracy Maps
221(1)
12.3.2 Defining Vessel Route Maps
222(2)
12.3.3 Defining Pirates', Naval Units and Vessels' Behaviours
224(3)
12.3.4 Comparing Risk Maps
227(5)
12.4 Model Calibration
232(1)
12.5 Discussion
232(3)
References
235(2)
13 A Simple Approach for the Prediction of Extinction Events in Multi-agent Models
237(32)
Thomas P. Oleron Evans
Steven R. Bishop
Frank T. Smith
13.1 Introduction
237(1)
13.2 Key Concepts
238(3)
13.2.1 Binary Classification
238(1)
13.2.2 Measures of Classifier Performance
238(2)
13.2.3 Stochastic Processes
240(1)
13.3 The NANIA Predator-prey Model
241(6)
13.3.1 Background
241(1)
13.3.2 An ODD Description of the NANIA Model
241(4)
13.3.3 Behaviour of the NANIA Model
245(1)
13.3.4 Extinctions in the NANIA Model
246(1)
13.4 Computer Simulation
247(2)
13.4.1 Data Generation
247(2)
13.4.2 Categorisation of the Data
249(1)
13.5 Period Detection
249(3)
13.6 A Monte Carlo Approach to Prediction
252(11)
13.6.1 Binned Data
252(5)
13.6.2 Confidence Intervals
257(1)
13.6.3 Predicting Extinctions using Binned Population Data
257(3)
13.6.4 ROC and Precision-recall Curves for Monte Carlo Prediction of Predator Extinctions
260(3)
13.7 Conclusions
263(1)
References
264(5)
Part VIII Diffusion Models
14 Urban Agglomeration Through the Diffusion of Investment Impacts
269(14)
Minette D'Lima
Francesca R. Medda
Alan G. Wilson
14.1 Introduction
269(1)
14.2 The Model
270(2)
14.3 Mathematical Analysis for Agglomeration Conditions
272(3)
14.3.1 Introduction
272(2)
14.3.2 Case: r < c
274(1)
14.3.3 Case: r > or = to c
274(1)
14.4 Simulation Results
275(4)
14.5 Conclusions
279(1)
References
279(4)
Part IX Game Theory
15 From Colonel Blotto to Field Marshall Blotto
283(10)
Peter Baudains
Toby P. Davies
Hannah M. Fry
Alan G. Wilson
15.1 Introduction
283(2)
15.2 The Colonel Blotto Game and its Extensions
285(1)
15.3 Incorporating a Spatial Interaction Model of Threat
286(2)
15.4 Two-front Battles
288(1)
15.5 Comparing Even and Uneven Allocations in a Scenario with Five Fronts
289(3)
15.6 Conclusion
292(1)
References
292(1)
16 Modelling Strategic Interactions in a Global Context
293(13)
Janina Beiser
16.1 Introduction
293(1)
16.2 The Theoretical Model
294(1)
16.3 Strategic Estimation
295(2)
16.4 International Sources of Uncertainty in the Context of Repression and Rebellion
297(2)
16.4.1 International Sources of Uncertainty Related to Actions
297(2)
16.5 International Sources of Uncertainty Related to Outcomes
299(2)
16.6 Empirical Analysis
301(2)
16.6.1 Data and Operationalisation
301(2)
16.7 Results
303(1)
16.8 Additional Considerations Related to International Uncertainty
304(1)
16.9 Conclusion
304(1)
References
305(1)
17 A General Framework for Static, Spatially Explicit Games of Search and Concealment
306(37)
Thomas P. Oleron Evans
Steven R. Bishop
Frank T. Smith
17.1 Introduction
306(1)
17.2 Game Theoretic Concepts
307(3)
17.3 Games of Search and Security: A Review
310(4)
17.3.1 Simple Search Games
310(1)
17.3.2 Search Games with Immobile Targets
311(1)
17.3.3 Accumulation Games
311(1)
17.3.4 Search Games with Mobile Targets
311(1)
17.3.5 Allocation Games
312(1)
17.3.6 Rendez-vous Games
312(1)
17.3.7 Security Games
313(1)
17.3.8 Geometric Games
313(1)
17.3.9 Motivation for Defining a New Spatial Game
314(1)
17.4 The Static Spatial Search Game (SSSG)
314(10)
17.4.1 Definition of the SSSG
314(2)
17.4.2 The SSSG and other Games
316(1)
17.4.3 The SSSG with Finite Strategy Sets
317(1)
17.4.4 Dominance and Equivalence in the SSSG
318(5)
17.4.5 Iterated Elimination of Dominated Strategies
323(1)
17.5 The Graph Search Game (GSG)
324(11)
17.5.1 Definition of the GSG
324(2)
17.5.2 The GSG with r not = to 1
326(1)
17.5.3 Preliminary Observations
327(3)
17.5.4 Bounds on the Value of the GSG
330(5)
17.6 Summary and Conclusions
335(1)
References
336(7)
Part X Networks
18 Network Evolution: A Transport Example
343(20)
Francesca Pagliara
Alan G. Wilson
Valerio de Martinis
18.1 Introduction
343(1)
18.2 A Hierarchical Retail Structure Model as a Building Block
344(1)
18.3 Extensions to Transport Networks
345(2)
18.4 An Application in Transport Planning
347(3)
18.5 A Case Study: Bagnoli in Naples
350(10)
18.6 Conclusion
360(1)
References
361(2)
19 The Structure of Global Transportation Networks
363(15)
Sean Hanna
Joan Serras
Tasos Varoudis
19.1 Introduction
363(1)
19.2 Method
364(2)
19.3 Analysis of the European Map
366(2)
19.4 Towards a Global Spatial Economic Map: Economic Analysis by Country
368(5)
19.5 An East-west Divide and Natural Economic Behaviour
373(3)
19.6 Conclusion
376(1)
References
377(1)
20 Trade Networks and Optimal Consumption
378(21)
Robert J. Downes
Robert G. Levy
20.1 Introduction
378(1)
20.2 The Global Economic Model
379(1)
20.2.1 Introduction
379(1)
20.2.2 Data Sources
380(1)
20.2.3 Model Overview
380(1)
20.3 Perturbing Final Demand Vectors
380(4)
20.3.1 Introduction
380(2)
20.3.2 Perturbation Process
382(2)
20.4 Analysis
384(9)
20.4.1 Introduction
384(1)
20.4.2 A Directed Network Representation
384(5)
20.4.3 A Weighted Directed Network Representation
389(1)
20.4.4 Communities in the Network of Improvements
390(3)
20.5 Conclusions
393(1)
Acknowledgements
394(1)
References
394(2)
Appendix
396
Part XI Integration
21 Research Priorities
399(4)
Alan G. Wilson
Index 403
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.
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