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Gravity Models

(http://faculty.washington.edu/krumme/systems/gravity.html)


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Definition

Iij = f (Ri, Aj)
-----------
f (Dij)

Interaction (I) between i and j is a function of repulsive forces (R) at i and attractive forces (A) at j, and an inverse function of the friction/distance (D) between i and j

Iij The interaction volume from i to j
Ria parameter representing (repulsive) factors which are associated with "leaving" i (such as outmigration)
Aja parameter representing (attractive) factors related to going to j (such as inmigration)
Dijthe distance between i and j

Iij = k Pi Pj
-------
Dijb

PPopulation of i & j
k & bconstants

The most basic formulation of the gravity model simply substitutes the populations at i and j for the R and A factors. George Zipf's Pi Pj/Dij hypothesis is probably the most widely accepted form of the gravity model. [Migration between two cities i and j is proportional to the product of the two cities' populations and inversely proportional to the intervening distance.]

Caffrey & Isaacs' Consumption Expenditures Spatial Distribution Model:

The Gravity Model as applied by John Caffrey and Herbert H. Isaacs, Estimating the Impact of a College or University on the Local Economy, [p.46]

eL = proportion of total non-housing expenditures that an individual is likely to make in her local environment
RSL = total period retail sales in the local environment
DL = average distance or travel time for a local individual to make a purchase within her local environment
RSNn = total period retail sales in the nth competing neighboring community
DNn = average distance or travel time for the local individual to make a purchase in the nth competing neighboring community


eL =
RSL
-------
DL2

-------------------------------------------------

RSL
------
DL2
 +
RSN1
------
DN12
 +
RSN2
------
DN22
 +
RSNn
------
DNn2

Lowry Migration Model

Migration from i to j is directly related to high wages at j, low relative unemployment at j and a large civilian labor force at either origin and/or destination. It is inversely related to high wages at i, low unemployment at i and increasing distance between i and j.

Mij = k [ Ui
--------
Uj		 x Wj
--------
Wi		 x Li Lj
----------
Dij		 ] eij

M = number of migrants
L = persons in labor force
U = unemployment in %
W = hourly wage in manufacturing
D = airline distance
k = gravitational constant
e = error term

Source: Ira Lowry, Migration and Metropolitan Growth: Two Analytical Models (1966)

Internet Sites:

Travel Demand Models for the San Francisco Bay Area [June 1997; (BAYCAST-90) Technical Summary Charles L. Purvis, AICP]

describe in detail the new set of travel demand models in the San Francisco Bay Area. These travel demand models were developed by staff of the Metropolitan Transportation Commission (MTC) in Oakland, California. MTC is the metropolitan planning organization (MPO) for the nine-county San Francisco Bay Area.

Calibrating and Adjustment of System Planning Models - December 1990 [National Transportation Library]

The four-step transportation modeling process, as applied at the regional level, has traditionally been dependent upon an extensive and reliable origin-destination (O-D) data base. In the early years of such models, this data base was developed largely through household surveys, a time-consuming and expensive undertaking. Such surveys were instrumental in developing the transportation models that have been used during the past 30 years. Since 1980, over 30 urban areas have conducted new home interview surveys to update their data base and ensure the validity of their modeling process. However, given limited resources, many planning agencies have had to rely on other means to validate their system models.

Calibrating & Testing a GRAVITY MODEL for Any Size Urban Area [U.S. DEPARTMENT OF TRANSPORTATION Federal Highway Administration Reprinted August 1983]

Trip distribution is an important and complex phase of the transportation planning process. It provides the planner with a systematic procedure capable of estimating zonal trip interchanges for alternate plans of both land use and transportation facilities. These zonal interchanges constitute a basic part of the travel information necessary for transportation planning. This manual documents in detail the process of trip distribution utilizing the gravity model as it is now defined.

TRANSPORTATION MODELS AND FORECASTS [The Council of Fresno County Governments (COG) is the State Regional Transportation Planning Agency and the Federal Metropolitan Planning Organization for Fresno County. As a transportation planning agency, COG is responsible for developing and maintaining a microcomputer-based traffic simulation model that represents Fresno County.]

Transportation Modeling, or Travel Demand Forecasting, estimates the amount of travel on the transportation system and provides an abstract view of travel on study facilities. The travel patterns are based on relationships developed from survey data between housing, employment sites and transportation facilities. Future year forecasts assume that human behavior or travel pattern relationships will remain constant for the forecast period and can therefore produce representative traffic flows.

Predictive Gravity Modeling for Marketing

Using GIS and psychographic segmentation, you can evaluate a trade area neighborhood by neighborhood. Gravity modeling, using distance or drive times, can be automated for greatest cost savings.

Clippings:

Literature:

Anderson, James E., "A Theoretical Foundation for the Gravity Equation," American Economic Review 69(1979), 106-16.

Caffrey, John and Herbert H. Isaacs, Estimating the Impact of College or University on the Local Economy. American Council on Education: Washington D.C. 1971. [Appendix C: The Gravity Model, pp.46ff.]

Erlander, S. and N.F. Stewart, The Gravity Model in Transportation Analysis -- Theory and Extensions: Topics in Transportation 1990; x+226 pages ISBN 90-6764-089-1 [book series Topics in Transportation]

... deals mainly with the gravity model and the distribution problem in transportation planning.... gives a detailed analysis of the fundamental relationships amongst various formulations of the gravity model and shows, in a mathematically rigorous way, which assumptions are necessary for the derivation of each formulation. The mathematical framework is the theory of optimization, specifically the Kuhn--Tucker and Lagrangian theory.

Gossman, Charles S., Calvin F. Schmid et al., Migration of College and University Students in the United States. Seattle: University of Washington Press, 1968. [Ch.11 (pp.145ff.) "The Distance Factor in Student Migration"; Ch.12 (pp.153ff. "A Gravity Model of Student Migration"]]

Haynes, Kingsley E. and A.S. Fotheringham, Gravity and Spatial Interaction Models, Sage-Publications 1984.

Isard, Walter et al., Methods of Interregional and Regional Analysis, Ashgate 1998, ch.6 ("Gravity and Spatial Interaction Models"), pp. 243ff.

Isard, W., "A simple Rationale for Gravity Model Type Behavior," Papers of the Regional Science Association. 35 (1975), 25-30.

Long, W.H., The Economics of Air Travel Gravity Models, Journ. of Regional Science 10(3), 1970, 353ff.

Lowe, J.M. and A.Sen, Gravity Model Applications in Health Planning: Analyses of an Urban Hospital Market," Journal of Regional Science, 36 (1996), 437-62.

Mathur, V.K., "An Economic Derivation of the 'Gravity Law' of Spatial Interaction: A Comment," Journ. of Regional Science 10(3), 1970, 403ff.

Mikkonen-K.; Luoma-M. The parameters of the gravity model are changing - how and why? Journal-of-Transport-Geography. 1999; 7(4): 277-283

Friction of distance in spatial interaction grows when the distance increases. This is a well-known fact and almost self-evident on the basis of the gravity model analogy and numerous empirical studies. Because this friction should apparently decrease over time, the change of the distance-decay parameter should be monotonic along time. We present empirical and theoretical evidence that this is only partially true.

Melling-J.; Turner-R. The road to the asylum: Institutions, distance and the administration of pauper lunacy in Devon, 1845-1914, Journal-of-Historical-Geography. 1999; 25(3): 298-332

The importance of space and distance in the development of social and political institutions ... important methodological and epistemological questions as well as substantive empirical issues about the impact of distance on the provision of medical services over time. This essay reviews this literature with particular reference to the debate about the capacity of distance-decay models to explain the varied use made of lunatic asylums in the nineteenth century.

O'Kelly, M.E., Trade-area models and choice-based samples: Methods. Environment and Planning A. 1999; 31(4): 613-627

The use of choice-based data to generate parameters of a trade-area gravity model is described in detail in this paper. The importance of being able to estimate parameters with data collected from individuals who have selected a particular alternative is explained. The main advantage of the new method is that it allows the efficient use of easily collected data.

Olsson, Gunnar, Distance and Human Interaction: A Review and Bibliography. Regional Scienc e Research Institute, 1965.

Reilly, W.J., The Law of Retail Gravitation. New York: The Knickerbocker Press, 1931

Sen, A. and T.E. Smith, Gravity Models of Spatial Interaction Behavior. NY: Springer, 1995. [Sen & Smith, Gravity Models of Spatial Interaction Behavior Ashish Sen, University of Illinois at Chicago, Chicago and Tony E. Smith, University of Pennsylvania, Philadelphia, Advances in Spatial and Network Economics Series, Springer-Verlag, Berlin Heidelberg, 1995 [ISBN 3-540-60026-4]. Hardcover US $139.00.]

Stewart, John Q., "The Development of Social Physics," American Journal of Physics, 18, 1950, 239-53.

Thorsen, Inge & Jens Petter Gitlesen, Empirical Evaluation of Alternative Model Specifications to Predict Commuting Flows Journal of Regional Science Volume 38 Issue 2 Page 273 - May 1998

Zipf, George Kingsley. "The P1P2/D Hypothesis: On the Intercity Movement of Persons," American Sociological Review, II(December 1946), 677-686.

Zipf, George Kingsley. Human Behavior and the Principle of Least Effort. Cambridge, Mass., : Addison-Wesley, 1949.

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