 ## NAME

r.fuzzy.set - Calculate membership value of any raster map according user's rules.

## KEYWORDS

raster, fuzzy logic

## SYNOPSIS

r.fuzzy.set
r.fuzzy.set --help
r.fuzzy.set input=name output=name points=string[,string,...] [side=string] [boundary=string] [shape=float] [height=float] [--overwrite] [--help] [--verbose] [--quiet] [--ui]

### Flags:

--overwrite
Allow output files to overwrite existing files
--help
Print usage summary
--verbose
Verbose module output
--quiet
Quiet module output
--ui
Force launching GUI dialog

### Parameters:

input=name [required]
Raster map to be fuzzified
output=name [required]
Membership map
points=string[,string,...] [required]
Inflection points: a,b[,c,d]
Default: a,b[,c,d]
side=string
Fuzzy range
Options: both, left, right
Default: both
boundary=string
Type of fuzzy boundaries
Options: Linear, S-shaped, J-shaped, G-shaped
Default: S-shaped
shape=float
Shape modifier: -1 to 1
Options: -1-1
Default: 0.
height=float
Membership height: 0 to 1
Options: 0-1
Default: 1.

## OPTIONS

input
Name of input raster map to be fuzzified. This map may be of any type and may require null values.
points
A list containing 4 (A,B,C,D) or 2 (A,B) points defining set boundaries. Points do not have to be in map range, but this may lead to only 0 o 1 membership for the whole map. For side parameter "both", range between A and D defines base, but range between B and C core of the fuzzy set. Between A and B and C and D are set's boundaries. If side is "both" it require 4 points, else 2 points.

### Fuzzy set definition: side
Option indicates if set is fuzzified of both sides (both), left or right side. See description for details.

### Boundary definition: ## OUTPUTS

output
Map containing membership value of original map. Map is alvays of type FCELLS and contains values from 0 (no membership) to 1 (full membership). Values between 0 and 1 indicate partial membership

## FUZZY SET PARAMETERS

boundary
Parameter defined the shape of the fuzzy boundary. The default and most popular is S-shaped, linear, J-shaped and G-shaped boundaries are also available. The same boundaries are applied to the both sides.
shape
Optional shape modifier. Range from -1 to 1. The default value is 0 and should not be changed in most of the time. The negative values indicate more dilatant set, the positive values more concentrate set. See NOTES for details.

### Impact of shape parameter on shape boundary: height
Optional height modifier. Range from 0 to 1. The default value is 1 and indicates full membership. If height is less than one the maximum membership is equal to height. See image: Fuzzy set definition.

## DESCRIPTION

#### Definition of fuzzy set

Fuzzy sets are sets whose elements have degrees of membership. Zadeh (1965) introduced Fuzzy sets as an extension of the classical notion of sets. Classical membership of elements in a set are binary terms: an element either belongs or does not belong to the set. Fuzzy set theory use the gradual assessment of the membership of elements in a set. A membership function valued in the real unit interval [0, 1]. Classical sets, are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1. Classical sets are in fuzzy set theory usually called crisp sets. Fuzzy set theory can be used in a wide range of domains in which information is imprecise, including many GIS operations.

## NOTES

#### Calculation of boundary shape

Depending on type of the boundary different equation are used to determine its shape:

Linear: the membership is calculated according following equation:

``````
value  <=  A -> x = 0
A < value > B -> x = (value-A)/(B-A)
B <= value >= C -> x = 1
C < value > D -> x = (D-value)/(D-C)
value >= D -> x = 0

where x: membership
``````

S-shaped: it use following equation:

``````
sin(x * Pi/2)^m (for positive shape parameter)
1-cos(x * Pi/2)^m (for negative shape parameter)

where x: membership, and
m = 2^exp(2,shape) (for positive shape parameter)
m = 2^(1+shape) (for negative shape parameter)
where m: shape parameter.
``````
For default shape parameter = 0 m is = 2 which is most common parameter for that equation.

G-shaped and J shaped: it use following equations:

``````
tan(x * Pi/4)^m (for J-shaped)
tan(x * Pi/4)^1/m (for G-shaped)

where x: membership, and
m = 2^exp(2,shape) (for positive shape parameter)
m = 2^(1+shape) (for negative shape parameter)
where m: shape parameter.
``````