Spatial Distribution Analysis: Definitions

There are, actually, several forms of the gravity model as it is applied to the spatial distribution of people.

The simplest, or "pure," gravity model expresses the relationship:

Attraction = K {(Pop. 1) * (Pop. 2)} / Distance**2

This formula says that the force of attraction is proportional to the size of two locations (as measured by population) divided by the square of the distance between them, and scaled to some "constant" value (or "fudge factor"), K.

The impact of distance is squared to reflect the perception that movement is discouraged with greater impact as distance increases. In other words, if the store is out of cigarettes and you have to walk an additional block to find some, you might think twice about it but you would probably go. If you had to walk two additional blocks, your resistance to going would more than double; if you had to go three more blocks you might decide to give up smoking for awhile unless you were really addicted. The resistance you feel does not increase in equal increments, but seems to grow by leaps and bounds.

The potential attractiveness of one location is the sum of its attractiveness to all possible locations:

Attraction = K * Sum {Population[n] / Distance[n]**2}
where n = each of all the possible locations

The pure gravity model is reflected in "Reilly's Law of Retail Gravitation" (Reilly, 1929), which is used to calculate market areas. In its simplest form, it assumes two market centers (towns, shopping malls, etc.) which will divide the market between themselves. In this simplified schema, customers will shop only at one center or the other (or, what amounts to the same thing, any crossover from one market will be exactly offset by crossovers from the other). The division of the market can then be expressed as the product of the ratio of the sizes of the centers and the ratio of their distances from the boundary:

Attraction = (Pop.1 / Pop.2 ) * ({Distance2 **2}/ {Distance1**2})

In this form, the equation presents the structure of the model. In actual use, the true distances are usually unknown, in which case the calculating formula takes the form:

Distance to loc.1 = Total Distance / (1 + {Pop.2 / Pop.1}**.5)
where Attraction = 1

The derivation of the calculating formula from the structural formula is given in Krueckeberg & Silvers (1974). It comes from the fact that the total distance between the two centers is divided into two market ranges (or, Total Distance = Distance 1 + Distance 2) and the total attraction of the two centers is 100 per cent (unity, or 1) of the resident population.

The pure gravity model returns surprisingly good predictions, considering the simplifications assumed by the model. Recent work has refined the model, redefining the key variables to allow greater specification. These more recent gravity models approach the problem not as predicting the behavior of population aggregates, but as estimating the probability of an individual's behavior. Rather than measuring attraction, they estimate the attractiveness of a location. This measure of attractiveness ("attraction potential," if you will) can then be applied to a population pool to return an estimate of the attraction which will be realized. The distinction is a subtle one, and holding everything else equal the two families of models would return roughly the same results. The advance was, rather, a conceptual breakthrough: by redefining the problem it opened up new strategies for solving it.

The simplest of the probability models is the "unconstrained" gravity model. It takes the form:

Trips1-2 = K {(Pop.1**a) * (Opportunities2**b)} / Distance**c

The similarity to the pure gravity model should be apparent; the differences are equally instructive. The unconstrained gravity model no longer speaks of "attraction"; attraction is now expressed in behavior--number of trips. Notice that it also specifies that the trips go from location 1 to location 2. The fudge factor (constant) remains, as does population at center 1 and the distance between the two centers.

But notice that there is a variable exponent ("a" and c, respectively) attached to each. The pure gravity model is retained if the population exponent is "1" and the distance exponent is "2"; but the formula allows the exponents to be adjusted to reflect theoretical assumptions or (depending on the use to which the model is being put) to model observed events. This is a slight change, but one that allows the model to be more finely tuned than the pure gravity model. The third term, population at center 2, is transformed into "opportunities at center 2," also with a variable exponent. Again, this is an important conceptual shift. If the opportunities at center 2 are measured by the population at that location, the pure gravity model is approximated. But the model lays the groundwork for mixing the terms of the analysis. It is possible to measure the impact on people of, say, sales volume or floor area or labor pool; with the pure gravity model, both center 1 and center 2 had to be measured in the same terms.

In the unconstrained gravity model, the total number of trips is the sum of the trips generated by all the centers. In other words, the available opportunities will determine the total number of trips. There is some merit to this--if the opportunities do not present themselves, it will be difficult to make a trip to use them. But it is also true that people will not necessarily use every opportunity which presents itself; after all, how many cars will you buy in a year? The unconstrained gravity model has a tendency to overestimate the number of trips which will be generated.

The problem of overestimation was resolved with the "constrained gravity model." This model specifies that the sum of the trips to each center is limited by the pool of total trips possible. Substituting this constraint into the unconstrained model leads, eventually, to the calculating formula:

Trips1-2 = Trips[1] * {(Opportunities[2]**b) / (Distance[1-2]**c)} / Sum {(Opportunities[all]**b) / (Distance[all]**c)}

The complete derivation can be seen in Krueckeberg & Silvers (1974).

The constrained gravity model states that the trips generated from location 1 to location 2 is a proportion of the total trips generated by location 1 ("Trips[1]"). That proportion is the ratio of the attractiveness of location 2 to the total attractiveness of all locations (including location 2). In other words, people will distribute their trips (for shopping, recreation, or any other activity) among the potential locations based on the relative attractiveness of each location.

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