Standardization of Factors to a Continuous Scale

The standardization procedure for WLC is somewhat more involved than in the Boolean case. Factors are not just reclassified into 0’s and 1’s, but are rescaled to a particular common range (0.0-1.0) according to some function. The original constraints in our example, water bodies and wetlands (WATERCON) and certain land use categories (LANDCON), will remain as Boolean images (i.e., constraining criteria) that will simply act as masks in the last step of the WLC.

Let us reconsider our original factors, standardization guidelines, and decision rules. These decision rules were previously in the form of hard decisions. Our factors were reduced to Boolean constraints using crisp set membership functions, 0’s and 1’s. Now our factors will be considered in terms of fuzzy decision rules where suitable and unsuitable areas are continuous measures between 0.0 and 1.0. The resulting continuous factors to be produced below will be developed using fuzzy set membership functions.¹

Land Use Factor

In our Boolean MCE, we reclassified our land use types available for development into suitable (Forest and Open Undeveloped) and unsuitable (all other land use categories) (LANDBOOL). However, according to developers, there are four land use types that are suitable to some degree (Forested Land, Open Undeveloped Land, Pasture, and Cropland), each with a different level of suitability for residential development. Knowing the relative suitability of each category, we can rescale them into the range 0.0-1.0. While most factors can be automatically rescaled using some mathematical function, rescaling categorical data such as land use simply requires giving a rating to each category based on some knowledge. In this case, the suitability rating is specified by developers.

B

Creating a quantitative factor from a qualitative input image can be done using Edit/ASSIGN or RECLASS in order to give each land use category a suitability value. Display the image called LANDFUZZ. It is a standardized factor map derived from the image MCELANDUSE. On the continuous (i.e., fuzzy) 0.0-1.0 scale we gave a suitability rating of 1.0 to Forested Land, 0.75 to Open Undeveloped land, 0.50 to areas under Pasture, 0.3 to Cropland, and gave all other categories a value of 0.0.

We now need to create the remaining standardized factor maps using the FUZZY module. Standardization is necessary to transform the disparate measurement units of the factor images into comparable suitability values. The selection of parameters for this standardization relies upon the user’s knowledge of how suitability changes for each factor. When using FUZZY, it is important to know the minimum and maximum data values for the input image. A fuzzy membership function shape and type must be specified and control point values are entered based on the input image minimum and maximum data values. Below is a description of the standardization criteria used with each factor image.

1 See the Decision Support chapter in the TerrSet Manual for a detailed discussion of fuzzy set membership functions.

EXERCISE 2-8 MCE: NON-BOOLEAN STANDARDIZATION AND WEIGHTED LINEAR COMBINATION 121

Distance to Town Center Factor

The simplest rescaling function for continuous data takes an original range of data and performs a simple linear stretch. For example, measures of relative distance from the town center, an important determinant of profit for developers, will be rescaled to a range of suitability where the greatest cost distance has the lowest suitability score (0) and the least cost distance has the highest suitability score (1). A simple linear distance decay function is appropriate for this criterion, i.e., as cost distance from the town center increases, its suitability decreases.

C

Display the image TOWNDIST.

The TOWNDIST image was created using the COST module. This module transforms a cost distance surface using roads as the source image and a friction image of road types to derive a relative travel time image to the town center. The assumption is that areas that are more accessible to the amenities of the town center will be more suitable for residential development.

D

Open the module FUZZY. Set the membership function type to linear. Enter TOWNDIST as the input file and enter

TOWNFUZZ as the output name. Set the output format to real and choose the monotonically decreasing linear function. Enter the control points for c of 0 and 582 for control point d. These values are the minimum (0) and maximum (582) distance values found in our cost distance image. Click OK to run.

Distance to Open Water Factor

Other factors, such as our distance from water bodies, do not have a constant decrease or increase in suitability based solely on distance. We know, for example, that town regulations require residential development to be at least 50 meters from open water and wetlands, and

EXERCISE 2-8 MCE: NON-BOOLEAN STANDARDIZATION AND WEIGHTED LINEAR COMBINATION 122

environmentalists prefer to see residential development even further from these water bodies. However, a distance of 800 meters might be just as good as a distance of 1000 meters. Suitability may not linearly increase with distance.

In our case study, suitability is very low within 100 meters of water. Beyond 100 meters, all parties agree that suitability increases with distance. However, environmentalists point out that the benefits of distance level off to maximum suitability at approximately 800 meters.

Beyond 800 meters, suitability is again equal. This function cannot be described by the simple linear function used in the preceding factor. It is best described by an increasing sigmoidal curve. We will use a monotonically increasing sigmoidal function to rescale the values in the distance-from-water image WATERDIST.

E

Display the image WATERDIST and examine the values.

F

Use FUZZY again. Select sigmoidal as the membership function type. Enter WATERDIST as the input file and enter

WATERFUZZ as the output file name. Select real as the output data format and monotonically increasing as the membership function shape. To accommodate the two thresholds of 100 and 800 meters in our function, the control points are no longer the minimum and maximum of our input values. Rather, they are equivalent to the points of inflection on the Sigmoidal curve. In the case of an increasing function, the first control point (a) is the value at which suitability begins to rise sharply above zero and the second control point (b) is the value at which suitability begins to level off and approaches a maximum of 1.0. Therefore, for this factor, input a value of 100 for control point a and a value of 800 for control point b. See the Help for FUZZY for a complete description of the fuzzy curves and control points. Click OK to run.

Distance to Roads Factor

Similar to our distance from water factor, distance from roads is a continuous factor to be rescaled to 0.0-1.0. In the previous exercise, developers identified only areas within 400 meters of roads as suitable. However, given the ability to determine a range of suitability, they have identified areas within 50 meters of roads as the most suitable and areas beyond 50 meters as having a continuously decreasing suitability that approaches, but never reaches 0. This function is adequately described by a decreasing J-shaped curve.

G

Display the image ROADDIST and examine the values.

H

Use FUZZY again. Select J-shaped as the membership function type. Enter ROADDIST as the input file and enter ROADFUZZ as the output file name. Select real as the output data format. To rescale our distance from roads factor to this J-shaped curve, we chose a monotonically decreasing function. As with the other functions, the first control point is the value at which the suitability begins to decline from maximum suitability. However, because the J-shaped function never reaches 0, the second control point is

EXERCISE 2-8 MCE: NON-BOOLEAN STANDARDIZATION AND WEIGHTED LINEAR COMBINATION 123

set at the value at which suitability is halfway between not suitable and perfectly suitable. Specify 50 for the value of the first control point c and 400 for the value of the second control point d. Click OK to run.

Slopes Factor

We know from our discussion in the previous exercise that slopes below 15% are the most cost effective for development. However, the lowest slopes are the best and any slope above 15% is equally unsuitable. We again use a monotonically decreasing sigmoidal function to rescale our data to the 0.0-1.0 range.

I

Display the image SLOPES and examine the values.

J

Using FUZZY, select sigmoidal as the membership function type. Enter SLOPES as the input file and enter SLOPEFUZZ as the output file name. Select real as the output data format. To rescale our slopes factor to this sigmoidal curve, we chose a

monotonically decreasing function. As with the other functions, the first control point is the value at which the suitability begins to decline from maximum suitability. The second control point is the value at which suitability begins to level off and approaches 0.0. Specify 0.0 for the value of the first control point c and 15 for the value of the second control point d. Click OK to run.

EXERCISE 2-8 MCE: NON-BOOLEAN STANDARDIZATION AND WEIGHTED LINEAR COMBINATION 124

Distance from Developed Land Factor

Finally, our last factor, distance from developed land, is also rescaled using a linear distance decay function. Areas closer to currently developed land are more suitable than areas farther from developed land, i.e., suitability decreases with distance.

K

Display the image DEVELOPDIST and examine the values.

L

Open the module FUZZY. Set the membership function type to linear. Enter DEVELOPDIST as the input file and enter

DEVELOPFUZZ as the output name. Set the output format to real and chose the monotonically decreasing linear function. Enter the control points for c of 0 and for 1325 for control point d. These values are the minimum and maximum distance values found in our distance image. Click OK to run

All factors have now been standardized to the same continuous scale of suitability (0.0-1.0). Standardization makes comparable factors representing different criteria measured in different ways. This will also allow us to combine or aggregate all the factor images.