piecewiseLinearApproximation {FuzzyNumbers}  R Documentation 
Piecewise Linear Approximation of a Fuzzy Number
Description
This method finds a piecewise linear approximation P(A)
of a given fuzzy number A by using the algorithm specified by the
method
parameter.
Usage
## S4 method for signature 'FuzzyNumber' piecewiseLinearApproximation(object, method=c("NearestEuclidean", "SupportCorePreserving", "Naive"), knot.n=1, knot.alpha=seq(0, 1, length.out=knot.n+2)[c(1,knot.n+2)], ..., verbose=FALSE)
Arguments
object 
a fuzzy number 
... 
further arguments passed to 
method 
character; one of: 
knot.n 
desired number of knots (if missing, then calculated from given

knot.alpha 
alphacuts at which knots will be positioned (defaults to equally distributed knots) 
verbose 
logical; should some technical details on the computations being performed be printed?
[only 
Details
'method
' may be one of:

NearestEuclidean
: see (Coroianu, Gagolewski, Grzegorzewski, 2013 and 2014a); uses numerical integration, seeintegrateAlpha
. Slow for largeknot.n
. 
SupportCorePreserving
: This method was proposed in (Coroianu et al., 2014b) and is currently only available forknot.n==1
. It is the L2nearest piecewise linear approximation with constraints core(A)==core(P(A)) and supp(A)==supp(P(A)); uses numerical integration. 
Naive
: We have core(A)==core(P(A)) and supp(A)==supp(P(A)) and the knots are taken directly from the specified alpha cuts (linear interpolation).
Value
Returns a PiecewiseLinearFuzzyNumber
object.
References
Coroianu L., Gagolewski M., Grzegorzewski P. (2013), Nearest Piecewise Linear Approximation of Fuzzy Numbers, Fuzzy Sets and Systems 233, pp. 2651.
Coroianu L., Gagolewski M., Grzegorzewski P., Adabitabar Firozja M., Houlari T. (2014a), Piecewise linear approximation of fuzzy numbers preserving the support and core, In: Laurent A. et al. (Eds.), Information Processing and Management of Uncertainty in KnowledgeBased Systems, Part II (CCIS 443), Springer, pp. 244254.
Coroianu L., Gagolewski M., Grzegorzewski P. (2014b), Nearest Piecewise Linear Approximation of Fuzzy Numbers  General Case, submitted for publication.
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
, plot
,
show
, supp
,
trapezoidalApproximation
,
value
, weightedExpectedValue
,
width
Other approximation: trapezoidalApproximation
Examples
(A < FuzzyNumber(1, 0, 1, 3, lower=function(x) sqrt(x),upper=function(x) 1sqrt(x))) (PA < piecewiseLinearApproximation(A, "NearestEuclidean", knot.n=1, knot.alpha=0.2))