trapezoidalApproximation {FuzzyNumbers}  R Documentation 
Trapezoidal Approximation of a Fuzzy Number
Description
This method finds a trapezoidal approximation T(A)
of a given fuzzy number A by using the algorithm specified by the
method
parameter.
Usage
## S4 method for signature 'FuzzyNumber' trapezoidalApproximation(object, method=c("NearestEuclidean", "ExpectedIntervalPreserving", "SupportCoreRestricted", "Naive"), ..., verbose=FALSE)
Arguments
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? 
Details
method
may be one of:

NearestEuclidean
: see (Ban, 2009); uses numerical integration, seeintegrateAlpha

Naive
: We have core(A)==core(T(A)) and supp(A)==supp(T(A)) 
ExpectedIntervalPreserving
: L2nearest 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, PasternakWiniarska, 2011) 
SupportCoreRestricted
: This method was proposed in (Grzegorzewski, PasternakWiniarska, 2011). L2nearest 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).
Value
Returns a TrapezoidalFuzzyNumber
object.
References
Ban A.I. (2008), Approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the expected interval, Fuzzy Sets and Systems 159, pp. 13271344.
Ban A.I. (2009), On the nearest parametric approximation of a fuzzy number  Revisited, Fuzzy Sets and Systems 160, pp. 30273047.
Grzegorzewski P. (2010), Algorithms for trapezoidal approximations of fuzzy numbers preserving the expected interval, In: BouchonMeunier B. et al (Eds.), Foundations of Reasoning Under Uncertainty, Springer, pp. 8598.
Grzegorzewski P, PasternakWiniarska K. (2011), Trapezoidal approximations of fuzzy numbers with restrictions on the support and core, Proc. EUSFLAT/LFA 2011, Atlantic Press, pp. 749756.
Yeh C.T. (2008), Trapezoidal and triangular approximations preserving the expected interval, Fuzzy Sets and Systems 159, pp. 13451353.
See Also
Other FuzzyNumbermethod: Arithmetic
,
FuzzyNumberclass
,
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
Other approximation: piecewiseLinearApproximation
Examples
(A < FuzzyNumber(1, 0, 1, 40, lower=function(x) sqrt(x), upper=function(x) 1sqrt(x))) (TA < trapezoidalApproximation(A, "ExpectedIntervalPreserving")) # Note that the cores are disjoint! expectedInterval(A) expectedInterval(TA)