Gravity Models of Spatial Interaction Behavior

TLDR: In this paper, the authors propose a framework for the analysis of spatial interaction processes in the context of gravity models, including the following: 1.1 Measures of Spatial Separation, 1.2 Relaxation of Locational Independence, and 1.3 Structural Independence.

Abstract: 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.