OPTIONS
xmap
Name of input x membership operand. This map must be of type FCELL with
range 0 :1 and may require null values. Otherwise program will print error
message and stops.
xmap
Name of input y membership operand. This map must be of type FCELL with
range 0 :1 and may require null values. Otherwise program will print error
message and stops. This map is optional bit is required for all operation except
NOT
operator
A fuzzy set operators are generalization of crisp operators. There is more
than one possible generalization of every opeartor. There are three operations:
fuzzy complements, fuzzy intersections, and fuzzy unions. Addational implication
operator is also provided.
- fuzzy intersection (AND) use T-norm of given family for calculation;
- fuzzy union (OR) use T-conorm of given family for calculation;
- fuzzy complement (NOT) fuzzy negation ussualy 1-x;
- fuzzy implication (IMP) use residuum of given family if available;
family
T-norms, T-conorms and residuals are a generalization of the two-valued
logical conjunction, disjunction and implication used by boolean logic, for
fuzzy logics. Because there is more than one possible generalisation of logial
operations, r.fuzzy.logic provides 6 most popular families for fuzzy operations:
- Zadeh with minimum (Godel) t-norm and maximum T-conorm;
- product with product T-norm and probabilistic sum as T-conorm;
- drastic with drastic T-norm and drastic T-conorm;
- Łukasiewicz with Łukasiewicz T-norm and bounded sum as a T-conorm;
- Fodor with nilpotent minimum as T-norm and nilpotent maximum as
T-conorm;
- Hamacher (simplified) with Hamacher product as T-norm and Einstein
sum as T-conorm;
There is no residuum for drastic and Hamacher families.
For more details see Meyer D, Hornik
K (2009); T-norms;
OUTPUTS
output
Map containing result of two-values operations. Multivalued operations will
be availabel in the future. Map is always of type FCELLS and contains values
from 0 (no membership) to 1 (full membership). Values between 0 and 1 indicate
partial membership
SEE ALSO
r.fuzzy,
r.mapcalc,
REFERENCES
Zadeh, L.A. (1965). "Fuzzy sets". Information and Control 8 (3): 338–353.
doi:10.1016/S0019-9958(65)90241-X. ISSN 0019-9958.
Novák, Vilém (1989). Fuzzy Sets and Their Applications. Bristol: Adam Hilger.
ISBN 0-85274-583-4.
Klir, George J.; Yuan, Bo (1995). Fuzzy sets and fuzzy logic: theory and
applications. Upper Saddle River, NJ: Prentice Hall PTR. ISBN 0-13-101171-5.
Klir, George J.; St Clair, Ute H.; Yuan, Bo (1997). Fuzzy set theory:
foundations and applications. Englewood Cliffs, NJ: Prentice Hall. ISBN
0133410587.
Meyer D, Hornik K (2009a). \Generalized and Customizable Sets in R." Journal of
Statistical Software, 31(2), 1{27. URL http://www.jstatsoft.org/v31/i02/.
Meyer D, Hornik K (2009b). sets: Sets, Generalized Sets, and Customizable Sets.
R~package version~1.0, URL http://CRAN.R-project.org/package=sets.
AUTHOR
Jarek Jasiewicz
Last changed: $Date$