## P

We can then replace b with the maximum sales distance and a with the minimum sales distance from our analysis, and p and k with the estimate of the values of these parameters. We can then use this equation to project total sales across a specified region. A numerical method can also be used to project sales, especially if the distance-sales equation is different from Equation 16.

Edge Effects and the Impracticality of Zero and Infinity

The analysis of spatial data is often hampered by edge effects which are discontinuous, or unusual behavior near the edge of a region. For distance-sales analysis, we often encounter edge effects requiring some adjustment very close to the retailer and far away from the retailer.

The derivation of the distance-sales relationship is based on a demand function that implicitly assumes a maximum sales distance of infinity (®) and a minimum sales distance of zero. In practice, we recommend that you truncate the data to a maximum and a minimum distance.

There are three reasons for this: truncated area will almost always be in line with the actual characteristics of the market area,360 the mathematics of estimating models is much more tractable if we slightly restrict the range, and the data are often aggregated or otherwise deficient.

We consider the rationale for the minimum- and maximum-distance truncations separately.

### Minimum-Distance Truncation

If there are very few residential properties within a mile or more of the retail establishment, a minimum retail distance of zero may result in an overestimate of the total sales of a store with a primarily residential customer base, or allow a handful of observations to bias the estimated parameters of the model.

Recall our derivation of the distance-sales relationship from the fundamental microeconomics of consumer demand. Distance in the demand equation is an indicator of time and other costs of traveling to and from the retailer. Part of that time cost is the same whether the trip is across the street or across the country.361 This de minimis cost is not important when the distances involved are all significant. However, ignoring it can cause problems when a few consumers are within 1mi and others are within 5mi, while the majority is 10mi or more away. The first group is indeed closer than the other groups, but does it really take one fifth the time-cost for a

360 At least until space travel becomes economical, there is an upper limit to the distance a retail customer can travel on earth. In practical applications, truncating the market area to that portion providing 90 to 95% of the sales has worked well, as this eliminates outliers without affecting the fundamental relationship.

361 Consider the actual costs to humans of taking a shopping trip. They include making arrangements for whatever must be watched while the shopper is gone, possibly getting children or others into a vehicle, walking in from the parking lot, and other tasks. These are not cost free and are roughly the same whether the trip is 1mi or 10.

person 1mi away to purchase something at a retailer compared to that for a person 5mi away?

This is further complicated by the imprecision of data. Often the analyst will have aggregate data sorted, for example, by zip-code area. In metropolitan areas, the centroid of the zip-code region may be a good indicator for zip-code areas whose boundaries are more than 5mi away. However, the centroids may not be a good indicator when the retailer is located in the zip-code area or near its boundary. Finally, the mathematical specifications for distance-sales relationships do not work well for distances close to zero.

### Maximum-Distance Truncation

The rationale for a maximum-distance truncation is quite apparent. Any good model, including a polynomial or gravity model, should estimate sales well within the range for which data are available. However, such models will not be accurate at distances where there have been relatively few sales in the past. When only a handful of sales arises from very distant customers, these could be considered outliers that should not be given equal weight with other observations.

In addition, as discussed in "Weaknesses of Polynomial Equations" on page 360, a polynomial specification can produce negative sales projections outside its intended range. With a gravity model specification and reasonably well-behaved data, the expected sales approaches zero as the distance extends outside the range of data. Hence, such models are less susceptible to distorted results because of a handful of large-distance sales observations.

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