Trapezoidal Approximation of a Fuzzy Number
This method finds a trapezoidal approximation T(A)
of a given fuzzy number A by using the algorithm specified by the
method parameter.
## S4 method for signature 'FuzzyNumber'
trapezoidalApproximation(object,
method=c("NearestEuclidean", "ExpectedIntervalPreserving",
"SupportCoreRestricted", "Naive"),
..., verbose=FALSE)object |
a fuzzy number |
... |
further arguments passed to |
method |
character; one of: |
verbose |
logical; should some technical details on the computations being performed be printed? |
method may be one of:
NearestEuclidean: see (Ban, 2009);
uses numerical integration, see integrateAlpha
Naive:
We have core(A)==core(T(A)) and supp(A)==supp(T(A))
ExpectedIntervalPreserving:
L2-nearest trapezoidal approximation preserving the expected interval given in
(Grzegorzewski, 2010; Ban, 2008; Yeh, 2008)
Unfortunately, for highly skewed membership functions
this approximation operator may have
quite unfavourable behavior.
For example, if Val(A) < EV_1/3(A) or Val(A) > EV_2/3(A),
then it may happen that the core of the output
and the core of the original fuzzy number A are disjoint
(cf. Grzegorzewski, Pasternak-Winiarska, 2011)
SupportCoreRestricted:
This method was proposed in (Grzegorzewski, Pasternak-Winiarska, 2011).
L2-nearest trapezoidal approximation with constraints
core(A) SUBSETS core(T(A))
and supp(T(A)) SUBSETS supp(A), i.e.
for which each point that surely belongs to A also belongs to T(A),
and each point that surely does not belong to A also does not belong to T(A).
Returns a TrapezoidalFuzzyNumber object.
Ban A.I. (2008), Approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the expected interval, Fuzzy Sets and Systems 159, pp. 1327-1344.
Ban A.I. (2009), On the nearest parametric approximation of a fuzzy number - Revisited, Fuzzy Sets and Systems 160, pp. 3027-3047.
Grzegorzewski P. (2010), Algorithms for trapezoidal approximations of fuzzy numbers preserving the expected interval, In: Bouchon-Meunier B. et al (Eds.), Foundations of Reasoning Under Uncertainty, Springer, pp. 85-98.
Grzegorzewski P, Pasternak-Winiarska K. (2011), Trapezoidal approximations of fuzzy numbers with restrictions on the support and core, Proc. EUSFLAT/LFA 2011, Atlantis Press, pp. 749-756.
Yeh C.-T. (2008), Trapezoidal and triangular approximations preserving the expected interval, Fuzzy Sets and Systems 159, pp. 1345-1353.
Other approximation: piecewiseLinearApproximation
Other FuzzyNumber-method: Arithmetic,
FuzzyNumber-class,
FuzzyNumber, alphaInterval,
alphacut, ambiguity,
as.FuzzyNumber,
as.PiecewiseLinearFuzzyNumber,
as.PowerFuzzyNumber,
as.TrapezoidalFuzzyNumber,
as.character, core,
distance, evaluate,
expectedInterval,
expectedValue,
integrateAlpha,
piecewiseLinearApproximation,
plot, show,
supp, value,
weightedExpectedValue, width
(A <- FuzzyNumber(-1, 0, 1, 40, lower=function(x) sqrt(x), upper=function(x) 1-sqrt(x))) (TA <- trapezoidalApproximation(A, "ExpectedIntervalPreserving")) # Note that the cores are disjoint! expectedInterval(A) expectedInterval(TA)
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