# Spatial Distribution Analysis: Calculations

Unlike the other models in this book, the gravity model lends itself to a wide variety of data. The model specifies an interaction between bodies based on their size and distance from each other. Any phenomenon that can be assumed to display such behavior can be fitted to the model. One of the beauties of the tool is its adaptability; data for the model can come from anywhere. Rather than specific sources of data, possible measures of the key variables will be discussed.

One of the key variables has been expressed as "Population." Population may be measured, however, as individuals, households, or wage earners. The "population" variable represents the relative size of the locations which are generating "trips." For some purposes it could be measured by housing units or even manufacturing establishments.

"Opportunity" is the variable which expresses the relative strength of attraction of the locations to which trips are directed. Opportunity can be measured as individuals or households. It can also be measured as sales volume, display area, or number of employees. For some purposes, the number of motel rooms or number of trout could be used.

"Total trips" is the variable which expresses the pool of resources which are being transferred. Frequently, the resources are people and "trips" is measured by round-trip journeys to work. There are some problems with this--not all journeys-to-work are round trips; not all trips are journeys-to-work; but when scaled by the constant, it returns a fairly good estimate of travel behavior. Other resources can be moved besides people: "trips" can be measured by household or per capita income, if one is doing a market survey. It can be measured by book circulation if one is doing a study of library usage.

Finally, "distance" is the variable which expresses the resistance to movement which is imposed by space (the model is, after all, a spatial distribution model). Usually it is measured by distance (feet, yards, miles). For some purposes, travel-time may be a better measure; after all, ten minutes on a freeway isn't much more trouble than ten minutes on surface roads.

The spreadsheet model is a constrained gravity model composed of seven worksheets. The first worksheet puts in the "opportunity" measures for the model (Figure 4.1). Throughout the model, "SITE" will designate locations toward which trips are directed. The user specifies a name for each site and a measure of the relative attractiveness of each site. The name for each site is included only for the user's convenience in documenting what is being done; the spreadsheet does not use the names anywhere else.

` `
`                        ATTRACTION BY SITE`
`                                NAME    WEIGHT`
`                        SITE 1  ....   ....               `
`                        SITE 2  ....   ....                                `
`                        SITE 3  ....   .... `
`                        SITE 4  ....   ....               `
`                                Figure 4.1`
`                        Input:  Site Attraction`
` `

The second worksheet puts in the "distance" measures (Figure 4.2). Throughout the model, "NBRHD" (neighborhood) will designate locations from which trips are generated. The user may use any type of measure--time, distance, or any other measure that makes sense. As long as they are expressed as real numbers and all are the same type of measure, the spreadsheet will work.

` `
`                        FRICTION OF DISTANCE MATRIX`
`                        SITE 1  SITE 2  SITE 3  SITE 4`
`                NBRHD 1 ....   ....   ....   ....`
`                NBRHD 2 ....   ....   ....   ....`
`                NBRHD 3 ....   ....   ....   ....`
`                NBRHD 4 ....   ....   ....   ....`
`                                Figure 4.2`
`                        Input:  Friction of Distance Matrix`
` `

The third worksheet puts in the "total trips," or the constraints on the system (Figure 4.3):

` `
`                                CONSTRAINTS`
`                                        IMPACTS`
`                               NBRHD 1      ....    `
`                               NBRHD 2      ....    `
`                               NBRHD 3      ....    `
`                               NBRHD 4      ....    `
`                                Figure 4.3`
`                        Input:  Constraints`
` `
` `

Throughout the model, "IMPACTS" will designate the resources which are being moved around. In this worksheet, the impacts are the total trips (or income or whatever) to be allocated among the various sites--the constraints which give this model its name.

Once these three worksheets are completed, the spreadsheet will calculate the model and produce three tables of results.

The first set of results are called the "relative attractiveness matrix" (Figure 4.4). These values represent the relative attractiveness for each neighborhood to each site. It is the ratio of the attractiveness of the site for that neighborhood, relative to the attractiveness of all sites. The relative attractiveness should total 1.0 for each neighborhood; because of rounding error it often will not. In addition, topographic and other social/spatial characteristics may affect the relative attractiveness of sites in ways not previously included in the model.

` `
`                RELATIVE ATTRACTIVENESS MATRIX`
`                        SITE 1 SITE 2 SITE 3 SITE 4`
`                NBRHD 1 xxxx   xxxx   xxxx   xxxx`
`                NBRHD 2 xxxx   xxxx   xxxx   xxxx`
`                NBRHD 3 xxxx   xxxx   xxxx   xxxx`
`                NBRHD 4 xxxx   xxxx   xxxx   xxxx`
` `
`        IF YOU WANT TO MAKE ADJUSTMENTS TO THIS MATRIX, GO TO H5O. `
`        `
`                                Figure 4.4`
`                        Relative Attractiveness Matrix`
`The spreadsheet includes an "adjustment" matrix to allow for those effects (Figure 4.5):`
` `
`                        ADJUSTMENTS TO MATRIX`
`                        SITE 1 SITE 2 SITE 3 SITE 4`
`                NBRHD 1   1       1      1      1`
`                NBRHD 2   1       1      1      1`
`                NBRHD 3   1       1      1      1`
`                NBRHD 4   1       1      1      1`
` `
`                                Figure 4.5`
`                        Matrix Adjustments Table`
` `

The spreadsheet begins with the value "1" specified for each cell of the matrix. In effect, the default values specify no change in the original relative attractiveness matrix. If a value greater than 1 is specified, the relative attractiveness of that site for that neighborhood will be increased; if less than 1, it will be decreased. Values in the matrix may be decimals or whole numbers.

The second table of results is the "spatial distribution matrix" (Figure 4.6):

` `
`                        SPATIAL DISTRIBUTION MATRIX`
`                        SITE 1 SITE 2 SITE 3 SITE 4`
`                NBRHD 1 xxxx  xxxx    xxxx   xxxx`
`                NBRHD 2 xxxx  xxxx    xxxx   xxxx`
`                NBRHD 3 xxxx  xxxx    xxxx   xxxx`
`                NBRHD 4 xxxx  xxxx    xxxx   xxxx`
`                         xxxx  xxxx    xxxx   xxxx`
`                                Figure 4.6`
`                        Spatial Distribution Matrix`
` `
` `

This table applies the system constraints to the adjusted attractiveness matrix. The result is a table of trips from each neighborhood to each site. The table also provides a summary of the total trips to each site.

` `
`The final worksheet is the "Total Impacts" worksheet (Figure 4.7):`
` `
`                        TOTAL IMPACTS`
`                        NBRHD 1 xxxx `
`                        NBRHD 2 xxxx `
`                        NBRHD 3 xxxx`
`                        NBRHD 4 xxxx`
`                        `
`                        Figure 4.7`
`                Table of Total Impacts`
` `

This table presents the sum of the trips generated by each neighborhood in the spatial distribution matrix. If the relative attractiveness matrix is properly adjusted, the total impacts will be the same as the constraints. Usually it will take several attempts to make the necessary adjustments.

Finally, the user should note the procedure for expanding the worksheets to include more sites. The tables in the worksheets are lined up in such a way that additional columns should be added between site 3 and site 4. This will preserve the other worksheets intact while providing additional columns in the same location for all the tables which include a listing of sites. If the tables are expanded, remember to expand the adjustment matrix as well (it lies outside the range of columns for the rest of the worksheets).