Description Usage Arguments Details Value References See Also Examples
This method finds a trapezoidal approximation T(A)
of a given fuzzy number A by using the algorithm specified by the
method
parameter.
1 2 3 4 5 | ## 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
,
Extract
,
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()
1 2 3 4 5 6 | (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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