Gravity models describe, and hence help predict, spatial flows of commuters, air-travelers, migrants, commodities and even messages. They are one of the oldest and most widely used of all social science models. This book presents an up-to-date, consistent and unified approach to the theory, methods and application of the gravity model - which spans from the axiomatic foundations of such models all the way to practical hints for their use. "I have found no better general method for use in applied research dealing with spatial interaction... It is against this background that the present book by Sen and Smith is most welcomed." Walter Isard

Inhalt

I Theoretical Development.- 1 Spatial Interaction Processes: An Overview.- 1.1 Introduction.- 1.2 Theoretical Perspectives.- 1.2.1 Macro versus Micro Theories.- 1.2.2 Static versus Dynamic Theories.- 1.2.3 Probabilistic versus Deterministic Theories.- 1.3 Analytical Framework.- 1.3.1 Measures of Spatial Separation.- 1.3.2 Spatial Aggregation Assumptions.- 1.3.3 Structural Independence Assumptions.- 1.4 Spatial Interaction Processes.- 1.4.1 Interaction Patterns.- 1.4.2 General Interaction Processes.- 1.4.3 Independent Interaction Processes.- 1.5 Relaxations of Independence.- 1.5.1 Relaxations of Frequency Independence.- 1.5.2 Relaxations of Locational Independence.- 1.5.3 More Complex Types of Interdependencies.- 2 Gravity Models: An Overview.- 2.1 Introduction.- 2.2 General Gravity Models.- 2.2.1 Model Specifications.- 2.2.2 Illustrative Examples.- 2.2.3 Behavioral Characterizations.- 2.3 Functional Specifications.- 2.3.1 Origin and Destination Functions.- 2.3.2 Deterrence Functions.- 2.4 Exponential Gravity Models.- 2.4.1 Model Specifications.- 2.4.2 Illustrative Examples.- 2.4.3 Behavioral Characterizations.- 2.5 Generalizations of the Gravity Models.- 2.5.1 Generalized Search Processes.- 2.5.2 Interaction Processes with Hierarchical Destinations.- 2.5.3 Interaction Processes with Random Destination Sets.- 3 Spatial Interaction Processes: Formal Development.- 3.1 Introduction.- 3.2 Analytical Preliminaries.- 3.2.1 Measurable Spaces.- 3.2.2 Measurable Functions.- 3.2.3 Probability Spaces.- 3.3 Interaction Probability Spaces.- 3.3.1 Interaction Patterns.- 3.3.2 Locational Attributes of Interactions.- 3.3.3 Interaction Events.- 3.3.4 Frequency Attributes of Interactions.- 3.4 Interaction Processes.- 3.4.1 Separation Configurations.- 3.4.2 General Interaction Processes.- 3.4.3 Independent Interaction Processes.- 3.5 Frequency Processes.- 3.6 Generated Frequency Processes.- 3.6.1 Poisson Frequency Processes.- 3.6.2 Poisson Characterization Theorem.- 3.7 Threshold Interaction Processes.- 3.7.1 Potential Interactions.- 3.7.2 Independent Threshold Interaction Processes.- 3.7.3 Threshold Frequency Processes.- 3.8 Search Processes.- 3.8.1 Search Events.- 3.8.2 Realized-Interaction Frequencies.- 3.8.3 Independent Search Processes.- 3.9 Relaxations of Independence.- 3.9.1 Relaxations of Frequency Independence.- 3.9.2 Relaxation of Locational Independence.- 3.9.3 More Complex Types of Interdependencies.- 3.10 Notes and References.- 4 Gravity Models: Formal Development.- 4.1 Introduction.- 4.2 Definition of Gravity Model Classes.- 4.2.1 General Gravity Models.- 4.2.2 Exponential Gravity Models.- 4.2.3 Relationships Among Model Types.- 4.3 Examples of Gravity Model Classes.- 4.3.1 Carroll-Bevis Processes.- 4.3.2 Threshold Interaction Processes.- 4.3.3 Kullback-Leibler Processes.- 4.3.4 Simple Search Processes.- 4.4 Axioms for Interaction Processes.- 4.4.1 Positive Interaction Processes.- 4.4.2 Behavioral Axioms.- 4.4.3 Relations among Axioms.- 4.5 Characterizations of Gravity Models.- 4.5.1 Analytical Preliminaries.- 4.5.2 Characterizations of General Gravity Models.- 4.5.3 Characterizations of Exponential Gravity Models.- 4.6 Generalizations of Gravity Models.- 4.6.1 Interaction Processes with Hierarchical Destinations.- 4.6.2 Interaction Processes with Random Destination Sets.- 4.6.3 Interaction Processes with Prominence Effects.- 4.7 Notes and References.- II Methods.- 5 Maximum Likelihood.- 5.1 Introduction.- 5.1.1 Preliminaries.- 5.1.2 Maximum Likelihood Estimation.- 5.1.3 A Preview of this Chapter.- 5.2 Existence and Uniqueness of ML Estimates.- 5.2.1 Condition ML1.- 5.2.2 Condition ML2.- 5.2.3 Proof of Theorem 5.1.- 5.2.4 ML Estimation for Multinomial Gravity Models.- 5.3 ML Estimation Algorithms: Special Cases.- 5.3.1 The DSF Procedure.- 5.3.2 The Evans-Kirby Procedure.- 5.3.3 The Hyman Procedure.- 5.4 The LDSF Procedure.- 5.4.1 The Procedure.- 5.4.2 An Approximation Useful for ML Estimation Algorithms.- 5.4.3 Application to Short-term Forecasting.- 5.5 General Algorithms for ML Estimates.- 5.5.1 Scoring Methods.- 5.5.2 The Modified Scoring Procedure.- 5.5.3 Gradient Search Procedures.- 5.5.4 Modified Gradient Search Procedures.- 5.5.5 GLIM.- 5.6 Performance of General Algorithms.- 5.6.1 The Data.- 5.6.2 Convergence.- 5.6.3 Speeds of Procedures.- 5.7 Covariance of Estimates.- 5.7.1 Covariance of $${\hat \theta k}$$'s.- 5.7.2 Covariance of $$ {\hat{T}{{ij}}} $$.- 5.7.3 Other Forecasts.- 5.8 Goodness of Fit.- 5.8.1 Global Measures.- 5.8.2 Residuals.- 5.9 Other Properties of ML Estimates.- 5.9.1 Asymptotic Properties.- 5.9.2 Small Sample Properties.- 5.9.3 ML Estimates from Factored Data.- 5.10 Notes and Concluding Remarks.- 5.10.1 Conclusion.- 6 Least Squares.- 6.1 Introduction.- 6.1.1 A Preview of this Chapter.- 6.2 LS Procedures.- 6.2.1 Reduction of Parameters.- 6.2.2 Gauss-Markov Conditions.- 6.2.3 Bias.- 6.2.4 Weighting.- 6.2.5 Procedures.- 6.3 Large Sample Theory.- 6.3.1 Preliminaries.- 6.3.2 The Main Theorem.- 6.3.3 A Projection Matrix.- 6.3.4 Proof of Theorem 6.1.- 6.3.5 Some Practical Details.- 6.4 Alternative Methods.- 6.4.1 Use of Iterative Reweighting in Procedure 1.- 6.4.2 Not Reducing Parameters.- 6.4.3 Use of OLS.- 6.4.4 Use of Generalized Inverses.- 6.5 Small Sample Properties.- 6.5.1 The Procedures.- 6.5.2 The Simulations.- 6.5.3 Results from Simulations.- 6.5.4 Conclusions.- 6.6 Non-linear Least Squares.- 6.7 Notes and Concluding Remarks.- 6.7.1 Conclusions.- Appendix: Skokie Data.- References.- List of Principal Definitions and Results.- Author Index.