An Alternative Formulation for Spatial-Interaction Modeling

1983 ◽  
Vol 15 (5) ◽  
pp. 693-703 ◽  
Author(s):  
W Tobler
Keyword(s):  
Spatial Competition ◽  
Spatial Interaction ◽  
Linear Equations ◽  
Interaction Model ◽  
Neumann Boundary ◽  
The United States ◽  
Alternative Formulation ◽  
Continuous Version ◽  
Movement Rates ◽  
Interaction Modeling

Elementary geometric assumptions are used to derive a spatial-interaction model in which places are related to each other through a set of simultaneous linear equations. The system has simple properties with respect to aggregation and turnover, yet incorporates spatial competition, adjacency, and effects of geographic shadowing. The objective function satisfied by the model reduces congestion and minimizes the per capita work; solves a quadratic transportation-problem and fulfils in-sum and out-sum constraints. The associated Lagrangians can be interpreted as pushes and pulls, or as shadow prices. A spatially continuous version of the model consists of coupled elliptical partial differential equations with Neumann boundary conditions, solvable by numerical methods. With migration data from the United States of America the model yields an amazingly good fit, better than existing models and with fewer free parameters. Inversion of the model yields an estimate of distances between regions. In-movement rates are predicted from out-movement rates in the model.

Tobler's “An Alternative Formulation for Spatial-Interaction Modeling”: An Observation

1983 ◽  
Vol 15 (5) ◽  
pp. 705-706 ◽  
Author(s):  
A G Wilson
Keyword(s):  
Spatial Interaction ◽  
Interaction Model ◽  
Alternative Formulation ◽  
Spatial Interaction Model ◽  
Interaction Modeling ◽  
Kullback Entropy ◽  
Spatial Interaction Modeling

It is shown that there is a possible relationship between the maximand used to produce To bier's spatial interaction model and Kullback entropy.

GEOGRAPHICAL MODELING FOR ARCHAEOLOGY AND HISTORY: TWO CASE STUDIES

2012 ◽  
Vol 15 (01n02) ◽  
pp. 1150008 ◽  
Author(s):  
ALAN WILSON
Keyword(s):  
United States ◽  
Mathematical Models ◽  
Case Studies ◽  
Spatial Interaction ◽  
Interaction Model ◽  
Urban System ◽  
The United States ◽  
Geographical Modeling ◽  
The Impact

This paper explores the role of mathematical models in archaeology and history. Variants of a particular model — an entropy-maximizing spatial interaction model which also functions as a location model — are presented through two case studies. The first is an example from Archaeology which throws light on settlement sizes in the 9th and 8th century BC Greece; the second is from History and explores the evolution of the United States' urban system from 1790–1870 with particular reference to the impact of railways. The approach is essentially interdisciplinary and uses concepts from Geography, Economics, Physics and Ecology.

Spatial Uncertainty and Spatial Dominance in Interaction Modelling: A Theoretical Perspective on Spatial Competition

1992 ◽  
Vol 24 (7) ◽  
pp. 995-1008 ◽  
Author(s):  
J Pooler
Keyword(s):  
Spatial Competition ◽  
Spatial Interaction ◽  
Theoretical Perspective ◽  
Model Performance ◽  
Interaction Model ◽  
Spatial Uncertainty ◽  
Destination Choice ◽  
Relative Location ◽  
Interaction Modelling ◽  
The Empirical Analysis

In this paper a theoretical index of spatial uncertainty is defined, which measures the degree of diversity faced by potential movers in a destination choice set. As a result of variations in attractiveness and distance, destinations are considered to exert varying degrees of spatial dominance on origins and to compete on that basis for the attention of would-be movers. It is hypothesized that at origins where dominances are more uniform, there is ‘less to choose’ among destinations and it is more difficult to predict accurately spatial interaction (and vice versa). The basic concept put forth is that spatial uncertainty and the spatial interaction-model performance are related to the relative location of origins with respect to destinations. The empirical analysis employs data on recreational day travel and population migration.

Parameter Estimation in Spatial Interaction Modeling

1973 ◽  
Vol 5 (4) ◽  
pp. 503-518 ◽  
Author(s):  
F J Cesario
Keyword(s):  
Parameter Estimation ◽  
Growth Factor ◽  
Structural Characteristics ◽  
Spatial Interaction ◽  
Interaction Model ◽  
Solution Properties ◽  
Spatial Interaction Model ◽  
Interaction Modeling ◽  
Constrained Model ◽  
Spatial Interaction Modeling

New ways of estimating the parameters of the so-called ‘doubly-constrained’ spatial interaction model are developed and tested against contemporary procedures. Two variations of the doubly-constrained model—growth-factor and interactance (that is, gravity and entropy) models—are considered: The structural characteristics of these models are specified and several new solution properties are established. Then alternative parameter-estimation strategies are formulated and tested. The proposed techniques (which solve problems that are equivalent to existing models) are shown to be inferior to current methodology with respect to all criteria related to solution speed. However, it is shown that the proposed methods are more flexible than current approaches since a wider range of assumptions can be incorporated into the analysis.

Equilibrium and Nonequilibrium Statistical Thermodynamical Approaches to Modeling Spatial Interaction Dynamics

1992 ◽  
Vol 24 (3) ◽  
pp. 441-446 ◽  
Author(s):  
P B Slater
Keyword(s):  
Nonequilibrium Thermodynamics ◽  
Spatial Interaction ◽  
Interaction Model ◽  
Human Movement ◽  
Evolutionary Processes ◽  
Spatial Interaction Model ◽  
Interaction Dynamics ◽  
Interaction Modeling ◽  
Migratory Population ◽  
Over Time

Spatial interaction modeling can be embedded in the framework of statistical thermodynamics. This has been shown in part by Wilson and by Erlander, in particular. However, to complete the embedding, it is necessary to ask the dynamic question of how a migratory population evolves over time from one (observed) distribution at time t0 to another (observed) distribution at time t1, rather than just the question of how many people who started in one area at t0 ended in another at t1—with no regard to the intermediate locations of the migrants. The manner in which such evolutionary processes are addressed in the context of equilibrium and nonequilibrium thermodynamics serves as a basis for extending the spatial interaction model of human movement.

Effect of Interaction Modeling in AFM-Based Nanomanipulation

2008 ◽  
Author(s):  
Fakhreddine Landolsi ◽  
Fathi H. Ghorbel ◽  
James B. Dabney
Keyword(s):  
Experimental Data ◽  
Numerical Simulations ◽  
Interaction Model ◽  
Collision Process ◽  
Visual Sensing ◽  
Proposed Model ◽  
Interaction Modeling

AFM-based nanomanipulation is very challenging because of the complex mechanics in tip-sample interactions and the limitations in AFM visual sensing capabilities. In the present paper, we investigate the modeling of AFM-based nanomanipulation emphasizing the effects of the relevant interactions at the nanoscale. The major contribution of the present work is the use of a combined DMT-JKR interaction model in order to describe the complete collision process between the AFM tip and the sample. The coupling between the interactions and the friction at the nanoscale is emphasized. The efficacy of the proposed model to reproduce experimental data is demonstrated via numerical simulations.

Comments on the Urban Spatial-Interaction Model Based on the Intervening-Opportunities Principle

1979 ◽  
Vol 11 (5) ◽  
pp. 527-539 ◽  
Author(s):  
Irena Chudzyńska ◽  
Z Słodkowski
Keyword(s):  
Mathematical Model ◽  
Spatial Interaction ◽  
Interaction Model ◽  
Spatial Interaction Model ◽  
Model Based ◽  
Number Of Equilibria ◽  
Intervening Opportunities

A mathematical model of urban spatial interaction based on the intervening-opportunities principle is discussed and its equilibria are studied. It is shown that, under natural assumptions, the number of equilibria is finite, and a mathematical criterion for distinguishing the equilibrium corresponding to reality is given.

Incorporating Additional Tree and Environmental Variables in a Lodgepole Pine Stem Profile Model

1993 ◽  
Vol 8 (3) ◽  
pp. 86-90
Author(s):  
John C. Byrne
Keyword(s):  
Environmental Variables ◽  
Pinus Contorta ◽  
Lodgepole Pine ◽  
Linear Equations ◽  
The United States ◽  
National Forest ◽  
Data Sets ◽  
Tree Form ◽  
Northern Rocky Mountains ◽  
Profile Model

Abstract A new variable-form segmented stem profile model is developed for lodgepole pine (Pinus contorta) trees from the northern Rocky Mountains of the United States. I improved estimates of stem diameter by predicting two of the model coefficients with linear equations using a measure of tree form, defined as a ratio of dbh and total height. Additional improvements were obtained by fitting this model to individual national forest data sets. Other tree and environmental variables tested but found of little use in improving stem profile estimates were crown ratio, habitat series, elevation, slope percent, and aspect. West. J. Appl. For. 8(3):86-90.

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