Package 'FuzzyNumbers'

Description

FuzzyNumbers is an open source (LGPL 3) package for R. It provides S4 classes and methods to deal with fuzzy numbers. The package may be used by researchers in fuzzy numbers theory (e.g., for testing new algorithms, generating numerical examples, preparing figures).

Details

Fuzzy set theory gives one of many ways (in particular, see Bayesian probabilities) to represent imprecise information. Fuzzy numbers form a particular subclass of fuzzy sets of the real line. The main idea behind this concept is motivated by the observation that people tend to describe their knowledge about objects through vague numbers, e.g., "I'm about 180 cm tall" or "The event happened between 2 and 3 p.m.".

For the formal definition of a fuzzy number please refer to the FuzzyNumber man page. Note that this package also deals with particular types of fuzzy numbers like trapezoidal, piecewise linear, or “parametric” FNs (see TrapezoidalFuzzyNumber PiecewiseLinearFuzzyNumber, PowerFuzzyNumber, and *EXPERIMENTAL* DiscontinuousFuzzyNumber)

The package aims to provide the following functionality:

1. Representation of arbitrary fuzzy numbers (including FNs with discontinuous side functions and/or alpha-cuts), as well as their particular types, e.g. trapezoidal and piecewise linear fuzzy numbers,

2. Defuzzification and approximation by triangular and piecewise linear FNs (see e.g. expectedValue, value, trapezoidalApproximation, piecewiseLinearApproximation),

3. Visualization of FNs (see plot, as.character),

4. Basic operations on FNs (see e.g. fapply and Arithmetic),

5. etc.

For a complete list of classes and methods call help(package="FuzzyNumbers"). Moreover, you will surely be interested in a step-by-step guide to the package usage and features which is available at the project's webpage.

Keywords: Fuzzy Numbers, Fuzzy Sets, Shadowed Sets, Trapezoidal Approximation, Piecewise Linear Approximation, Approximate Reasoning, Imprecision, Vagueness, Randomness.

Acknowledgments: Many thanks to Jan Caha, Przemyslaw Grzegorzewski, Lucian Coroianu, and Pablo Villacorta Iglesias for stimulating discussion.

The development of the package in March-June 2013 was partially supported by the European Union from resources of the European Social Fund, Project PO KL “Information technologies: Research and their interdisciplinary applications”, agreement UDA-POKL.04.01.01-00-051/10-00.

Author(s)

Marek Gagolewski, with contributions from Jan Caha

References

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.

Bodjanova S. (2005), Median value and median interval of a fuzzy number, Information Sciences 172, pp. 73-89.

Chanas S. (2001), On the interval approximation of a fuzzy number, Fuzzy Sets and Systems 122, pp. 353-356.

Coroianu L., Gagolewski M., Grzegorzewski P. (2013), Nearest Piecewise Linear Approximation of Fuzzy Numbers, Fuzzy Sets and Systems 233, pp. 26-51.

Coroianu L., Gagolewski M., Grzegorzewski P., Adabitabar Firozja M., Houlari T. (2014), Piecewise linear approximation of fuzzy numbers preserving the support and core, In: Laurent A. et al. (Eds.), Information Processing and Management of Uncertainty in Knowledge-Based Systems, Part II (CCIS 443), Springer, pp. 244-254.

Delgado M., Vila M.A., Voxman W. (1998), On a canonical representation of a fuzzy number, Fuzzy Sets and Systems 93, pp. 125-135.

Dubois D., Prade H. (1978), Operations on fuzzy numbers, Int. J. Syst. Sci. 9, pp. 613-626.

Dubois D., Prade H. (1987a), The mean value of a fuzzy number, Fuzzy Sets and Systems 24, pp. 279-300.

Dubois D., Prade H. (1987b), Fuzzy numbers: An overview, In: Analysis of Fuzzy Information. Mathematical Logic, vol. I, CRC Press, pp. 3-39.

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. (1998), Metrics and orders in space of fuzzy numbers, Fuzzy Sets and Systems 97, pp. 83-94.

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.

Klir G.J., Yuan B. (1995), Fuzzy sets and fuzzy logic. Theory and applications, Prentice Hall, New Jersey.

Stefanini L., Sorini L. (2009), Fuzzy arithmetic with parametric LR fuzzy numbers, In: Proc. IFSA/EUSFLAT 2009, pp. 600-605.

Yeh C.-T. (2008), Trapezoidal and triangular approximations preserving the expected interval, Fuzzy Sets and Systems 159, pp. 1345-1353.

Description

For fuzzy numbers the equality of X*X == X^2 does not hold.

Usage

## S4 method for signature 'PiecewiseLinearFuzzyNumber,numeric'
e1 ^ e2

Arguments

Details

This function calculates integer power of a PiecewiseLinearFuzzyNumber according to the reference below.

Value

Returns a fuzzy number of the class PiecewiseLinearFuzzyNumber indicating e1^e2.

References

KAUFMANN, A., GUPTA, M. M. (1985) Introduction to Fuzzy Arithmetic. New York : Van Nostrand Reinhold Company. ISBN 044230079.

See Also

Other extension_principle: Arithmetic, fapply()

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, alphaInterval(), arctan2(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval(), fapply(), maximum(), minimum(), necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), plot(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Examples

x = as.PiecewiseLinearFuzzyNumber(TriangularFuzzyNumber(-2, 1, 9), knot.n = 2)
x^2
x^3

Description

If $A$ is a fuzzy number, then its $\alpha$-cuts are always in form of intervals. Moreover, the $\alpha$-cuts form a nonincreasing chain w.r.t. $alpha$.

Usage

## S4 method for signature 'FuzzyNumber,numeric'
alphacut(object, alpha)

Arguments

Value

Returns a matrix with two columns (left and right alha cut bounds). if some elements in alpha are not in [0,1], then NA is set.

See Also

Other FuzzyNumber-method: Arithmetic, Extract, FuzzyNumber-class, FuzzyNumber, alphaInterval(), ambiguity(), as.FuzzyNumber(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), core(), distance(), evaluate(), expectedInterval(), expectedValue(), integrateAlpha(), piecewiseLinearApproximation(), plot(), show(), supp(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Other alpha_cuts: core(), supp()

Examples

A <- TrapezoidalFuzzyNumber(1, 2, 3, 4)
alphacut(A, c(-1, 0.4, 0.2))

Description

We have $\alpha-Int(A) := [\int_0^1 \alpha A_L(\alpha)\,d\alpha, \int_0^1 \alpha A_U(\alpha)\,d\alpha]$.

Usage

## S4 method for signature 'FuzzyNumber'
alphaInterval(object, ...)

## S4 method for signature 'TrapezoidalFuzzyNumber'
alphaInterval(object)

## S4 method for signature 'PiecewiseLinearFuzzyNumber'
alphaInterval(object)

## S4 method for signature 'PowerFuzzyNumber'
alphaInterval(object)

Arguments

Details

Note that if an instance of the FuzzyNumber or DiscontinuousFuzzyNumber class is given, the calculation is performed via numerical integration. Otherwise, the computation is exact.

Value

Returns numeric vector of length 2.

See Also

Other FuzzyNumber-method: Arithmetic, Extract, FuzzyNumber-class, FuzzyNumber, alphacut(), ambiguity(), as.FuzzyNumber(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), core(), distance(), evaluate(), expectedInterval(), expectedValue(), integrateAlpha(), piecewiseLinearApproximation(), plot(), show(), supp(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Other TrapezoidalFuzzyNumber-method: Arithmetic, TrapezoidalFuzzyNumber-class, TrapezoidalFuzzyNumber, TriangularFuzzyNumber(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), expectedInterval(), plot()

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, arctan2(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval(), fapply(), maximum(), minimum(), necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), plot(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Other PowerFuzzyNumber-method: Extract, PowerFuzzyNumber-class, PowerFuzzyNumber, as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval()

Description

The ambiguity (Delgado et al, 1998) is a measure of nonspecificity of a fuzzy number.

Usage

## S4 method for signature 'FuzzyNumber'
ambiguity(object, ...)

Arguments

Details

The ambiguity is defined as $amb(A) := \int_0^1 \alpha\left(A_U(\alpha)-A_L(\alpha)\right)\,d\alpha$.

Value

Returns a single numeric value.

References

Delgado M., Vila M.A., Voxman W. (1998), On a canonical representation of a fuzzy number, Fuzzy Sets and Systems 93, pp. 125-135.

See Also

Other FuzzyNumber-method: Arithmetic, Extract, FuzzyNumber-class, FuzzyNumber, alphaInterval(), alphacut(), as.FuzzyNumber(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), core(), distance(), evaluate(), expectedInterval(), expectedValue(), integrateAlpha(), piecewiseLinearApproximation(), plot(), show(), supp(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Other characteristics: expectedValue(), value(), weightedExpectedValue(), width()

Description

The function may be used to create side generating functions from alpha-cut generators and inversely.

Usage

approxInvert(f, method = c("monoH.FC", "linear", "hyman"), n = 500)

Arguments

Details

The function is a wrapper to splinefun and approxfun. Thus, interpolation is used.

Value

Returns a new function, the approximate inverse of the input.

See Also

FuzzyNumber

Other auxiliary: convertAlpha(), convertSide()

Description

The arc-tangent of two arguments arctan2(y, x) returns the angle between the x-axis and the vector from the origin to (x, y) for PiecewiseLinearFuzzyNumbers.

Usage

## S4 method for signature 
## 'PiecewiseLinearFuzzyNumber,PiecewiseLinearFuzzyNumber'
arctan2(y, x)

Arguments

Details

Note that resulting values are no longer from interval [-pi,pi] but [-1.5pi,pi], in order to provide valid fuzzy numbers as result.

Value

Returns a fuzzy number of the class PiecewiseLinearFuzzyNumber indicating the angle specified by the input fuzzy numbers. The range of results is [-1.5pi,pi].

See Also

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval(), fapply(), maximum(), minimum(), necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), plot(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Examples

y = as.PiecewiseLinearFuzzyNumber(TriangularFuzzyNumber(-2, 3, 5), knot.n = 9)
x = as.PiecewiseLinearFuzzyNumber(TriangularFuzzyNumber(-4.8, -4, 1.5), knot.n = 9)
arctan2(y,x)

Description

Applies arithmetic operations using the extension principle and interval-based calculations.

Usage

## S4 method for signature 'numeric,FuzzyNumber'
e1 + e2 # e2 + e1

## S4 method for signature 'TrapezoidalFuzzyNumber,TrapezoidalFuzzyNumber'
e1 + e2

## S4 method for signature 
## 'PiecewiseLinearFuzzyNumber,PiecewiseLinearFuzzyNumber'
e1 + e2

## S4 method for signature 'PiecewiseLinearFuzzyNumber,numeric'
e1 + e2

## S4 method for signature 'PiecewiseLinearFuzzyNumber,FuzzyNumber'
e1 + e2 # calls as.PiecewiseLinearFuzzyNumber()

## S4 method for signature 'numeric,FuzzyNumber'
e1 - e2 # e2*(-1) + e1

## S4 method for signature 'TrapezoidalFuzzyNumber,TrapezoidalFuzzyNumber'
e1 - e2

## S4 method for signature 
## 'PiecewiseLinearFuzzyNumber,PiecewiseLinearFuzzyNumber'
e1 - e2

## S4 method for signature 'PiecewiseLinearFuzzyNumber,numeric'
e1 - e2

## S4 method for signature 'PiecewiseLinearFuzzyNumber,FuzzyNumber'
e1 - e2 # calls as.PiecewiseLinearFuzzyNumber()

## S4 method for signature 'FuzzyNumber,ANY'
e1 - e2 # -e1

## S4 method for signature 'numeric,FuzzyNumber'
e1 * e2 # e2 * e1

## S4 method for signature 'TrapezoidalFuzzyNumber,numeric'
e1 * e2

## S4 method for signature 
## 'PiecewiseLinearFuzzyNumber,PiecewiseLinearFuzzyNumber'
e1 * e2

## S4 method for signature 'PiecewiseLinearFuzzyNumber,FuzzyNumber'
e1 * e2 # calls as.PiecewiseLinearFuzzyNumber()

## S4 method for signature 'PiecewiseLinearFuzzyNumber,numeric'
e1 * e2

## S4 method for signature 'PiecewiseLinearFuzzyNumber,numeric'
e1 / e2

## S4 method for signature 
## 'PiecewiseLinearFuzzyNumber,PiecewiseLinearFuzzyNumber'
e1 / e2

## S4 method for signature 'PiecewiseLinearFuzzyNumber,FuzzyNumber'
e1 / e2 # calls as.PiecewiseLinearFuzzyNumber()

Arguments

Details

Implemented operators: +, -, *, / for piecewise linear fuzzy numbers. Also some versions may be applied on numeric values and trapezoidal fuzzy numbers.

Note that according to the theory the class of PLFNs is not closed under the operations * and /. However, if you operate on a large number of knots, the results should be satisfactory.

Thanks to Jan Caha for suggestions on PLFN operations.

Value

Returns a fuzzy number of the class PiecewiseLinearFuzzyNumber or TrapezoidalFuzzyNumber.

See Also

Other FuzzyNumber-method: 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(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Other PiecewiseLinearFuzzyNumber-method: Extract, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), arctan2(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval(), fapply(), maximum(), minimum(), necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), plot(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Other TrapezoidalFuzzyNumber-method: TrapezoidalFuzzyNumber-class, TrapezoidalFuzzyNumber, TriangularFuzzyNumber(), alphaInterval(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), expectedInterval(), plot()

Other extension_principle: ^,PiecewiseLinearFuzzyNumber,numeric-method, fapply()

Description

This method is especially useful if you would like to generate LaTeX equations defining a fuzzy numbers.

Usage

## S4 method for signature 'FuzzyNumber'
as.character(x, toLaTeX=FALSE, varnameLaTeX="A")

## S4 method for signature 'TrapezoidalFuzzyNumber'
as.character(x, toLaTeX=FALSE, varnameLaTeX="A")

## S4 method for signature 'PiecewiseLinearFuzzyNumber'
as.character(x, toLaTeX=FALSE, varnameLaTeX="A")

## S4 method for signature 'PowerFuzzyNumber'
as.character(x, toLaTeX=FALSE, varnameLaTeX="A")

Arguments

Details

Consider calling the cat function on the resulting string.

Thanks to Jan Caha for suggesting the toLaTeX arg.

Value

Returns a character vector.

See Also

Other FuzzyNumber-method: Arithmetic, Extract, FuzzyNumber-class, FuzzyNumber, alphaInterval(), alphacut(), ambiguity(), as.FuzzyNumber(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), core(), distance(), evaluate(), expectedInterval(), expectedValue(), integrateAlpha(), piecewiseLinearApproximation(), plot(), show(), supp(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), arctan2(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), expectedInterval(), fapply(), maximum(), minimum(), necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), plot(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Other PowerFuzzyNumber-method: Extract, PowerFuzzyNumber-class, PowerFuzzyNumber, alphaInterval(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), expectedInterval()

Other conversion: as.FuzzyNumber(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber()

Description

Please note that applying this function on a FuzzyNumber child class causes information loss, as it drops all additional slots defined in the child classes. FuzzyNumber is the base class for all FNs. Note that some functions for TFNs or PLFNs work much faster and are more precise. This function shouldn't be used in normal computations.

Usage

## S4 method for signature 'numeric'
as.FuzzyNumber(object)

## S4 method for signature 'FuzzyNumber'
as.FuzzyNumber(object)

Arguments

Value

Returns an bject of class FuzzyNumber.

See Also

Other FuzzyNumber-method: Arithmetic, Extract, FuzzyNumber-class, FuzzyNumber, alphaInterval(), alphacut(), ambiguity(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), core(), distance(), evaluate(), expectedInterval(), expectedValue(), integrateAlpha(), piecewiseLinearApproximation(), plot(), show(), supp(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Other conversion: as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character()

Description

This method is only for exact conversion. For other cases (e.g. general FNs), use piecewiseLinearApproximation.

Usage

## S4 method for signature 'TrapezoidalFuzzyNumber'
as.PiecewiseLinearFuzzyNumber(object, knot.n=0,
   knot.alpha=seq(0, 1, length.out=knot.n+2)[-c(1,knot.n+2)])

## S4 method for signature 'numeric'
as.PiecewiseLinearFuzzyNumber(object, knot.n=0,
   knot.alpha=seq(0, 1, length.out=knot.n+2)[-c(1,knot.n+2)])

## S4 method for signature 'FuzzyNumber'
as.PiecewiseLinearFuzzyNumber(object, knot.n=0,
   knot.alpha=seq(0, 1, length.out=knot.n+2)[-c(1,knot.n+2)])

## S4 method for signature 'PiecewiseLinearFuzzyNumber'
as.PiecewiseLinearFuzzyNumber(object, knot.n=0,
   knot.alpha=seq(0, 1, length.out=knot.n+2)[-c(1,knot.n+2)])

Arguments

Value

Returns an object of class PiecewiseLinearFuzzyNumber.

See Also

Other TrapezoidalFuzzyNumber-method: Arithmetic, TrapezoidalFuzzyNumber-class, TrapezoidalFuzzyNumber, TriangularFuzzyNumber(), alphaInterval(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), expectedInterval(), plot()

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), arctan2(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval(), fapply(), maximum(), minimum(), necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), plot(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Other FuzzyNumber-method: Arithmetic, Extract, FuzzyNumber-class, FuzzyNumber, alphaInterval(), alphacut(), ambiguity(), as.FuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), core(), distance(), evaluate(), expectedInterval(), expectedValue(), integrateAlpha(), piecewiseLinearApproximation(), plot(), show(), supp(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Other conversion: as.FuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character()

Description

This method is only for exact conversion.

Usage

## S4 method for signature 'numeric'
as.PowerFuzzyNumber(object)

## S4 method for signature 'FuzzyNumber'
as.PowerFuzzyNumber(object)

## S4 method for signature 'PowerFuzzyNumber'
as.PowerFuzzyNumber(object)

## S4 method for signature 'PiecewiseLinearFuzzyNumber'
as.PowerFuzzyNumber(object)

## S4 method for signature 'TrapezoidalFuzzyNumber'
as.PowerFuzzyNumber(object)

Arguments

Value

Returns an object of class PowerFuzzyNumber.

See Also

Other FuzzyNumber-method: Arithmetic, Extract, FuzzyNumber-class, FuzzyNumber, alphaInterval(), alphacut(), ambiguity(), as.FuzzyNumber(), as.PiecewiseLinearFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), core(), distance(), evaluate(), expectedInterval(), expectedValue(), integrateAlpha(), piecewiseLinearApproximation(), plot(), show(), supp(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Other TrapezoidalFuzzyNumber-method: Arithmetic, TrapezoidalFuzzyNumber-class, TrapezoidalFuzzyNumber, TriangularFuzzyNumber(), alphaInterval(), as.PiecewiseLinearFuzzyNumber(), as.TrapezoidalFuzzyNumber(), expectedInterval(), plot()

Other PowerFuzzyNumber-method: Extract, PowerFuzzyNumber-class, PowerFuzzyNumber, alphaInterval(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval()

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), arctan2(), as.PiecewiseLinearFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval(), fapply(), maximum(), minimum(), necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), plot(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Other conversion: as.FuzzyNumber(), as.PiecewiseLinearFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character()

Description

This method is only for exact conversion. For other cases (e.g. general FNs), use trapezoidalApproximation.

Usage

## S4 method for signature 'numeric'
as.TrapezoidalFuzzyNumber(object)

## S4 method for signature 'FuzzyNumber'
as.TrapezoidalFuzzyNumber(object)

## S4 method for signature 'PowerFuzzyNumber'
as.TrapezoidalFuzzyNumber(object)

## S4 method for signature 'PiecewiseLinearFuzzyNumber'
as.TrapezoidalFuzzyNumber(object)

## S4 method for signature 'TrapezoidalFuzzyNumber'
as.TrapezoidalFuzzyNumber(object)

Arguments

Value

Returns an bject of class TrapezoidalFuzzyNumber.

See Also

Other FuzzyNumber-method: Arithmetic, Extract, FuzzyNumber-class, FuzzyNumber, alphaInterval(), alphacut(), ambiguity(), as.FuzzyNumber(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.character(), core(), distance(), evaluate(), expectedInterval(), expectedValue(), integrateAlpha(), piecewiseLinearApproximation(), plot(), show(), supp(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Other TrapezoidalFuzzyNumber-method: Arithmetic, TrapezoidalFuzzyNumber-class, TrapezoidalFuzzyNumber, TriangularFuzzyNumber(), alphaInterval(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), expectedInterval(), plot()

Other PowerFuzzyNumber-method: Extract, PowerFuzzyNumber-class, PowerFuzzyNumber, alphaInterval(), as.PowerFuzzyNumber(), as.character(), expectedInterval()

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), arctan2(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.character(), expectedInterval(), fapply(), maximum(), minimum(), necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), plot(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Other conversion: as.FuzzyNumber(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.character()

Description

The resulting function calls the original function and then linearly scales its output.

Usage

convertAlpha(f, y1, y2)

Arguments

Value

Returns a new function defined on [0,1] (scaled input).

See Also

FuzzyNumber

Other auxiliary: approxInvert(), convertSide()

Description

The resulting function linearly scales the input and passes it to the original function.

Usage

convertSide(f, x1, x2)

Arguments

Details

The function works for x1<x2 and x1>x2.

Value

Returns a new function defined on [0,1] (scaled input).

See Also

FuzzyNumber

Other auxiliary: approxInvert(), convertAlpha()

Description

We have $\mathrm{core}(A) := [a2,a3]$. This gives the values that a fuzzy number necessarily represents.

Usage

## S4 method for signature 'FuzzyNumber'
core(object)

Arguments

Value

Returns a numeric vector of length 2.

See Also

Other FuzzyNumber-method: Arithmetic, Extract, FuzzyNumber-class, FuzzyNumber, alphaInterval(), alphacut(), ambiguity(), as.FuzzyNumber(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), distance(), evaluate(), expectedInterval(), expectedValue(), integrateAlpha(), piecewiseLinearApproximation(), plot(), show(), supp(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Other alpha_cuts: alphacut(), supp()

Description

For convenience, objects of class DiscontinuousFuzzyNumber may be created with this function.

Usage

DiscontinuousFuzzyNumber(
  a1,
  a2,
  a3,
  a4,
  lower = function(a) rep(NA_real_, length(a)),
  upper = function(a) rep(NA_real_, length(a)),
  left = function(x) rep(NA_real_, length(x)),
  right = function(x) rep(NA_real_, length(x)),
  discontinuities.left = numeric(0),
  discontinuities.right = numeric(0),
  discontinuities.lower = numeric(0),
  discontinuities.upper = numeric(0)
)

Arguments

Value

Object of class DiscontinuousFuzzyNumber

See Also

Other DiscontinuousFuzzyNumber-method: DiscontinuousFuzzyNumber-class, Extract, distance(), integrateAlpha(), plot()

Description

Discontinuity information increase the precision of some numerical integration-based algorithms, e.g. of piecewiseLinearApproximation. It also allows for making more valid fuzzy number plots.

Slots

a1, a2, a3, a4, lower, upper, left, right:

Inherited from the FuzzyNumber class.

discontinuities.left:

nondecreasingly sorted numeric vector with elements in (0,1); discontinuity points for the left side generating function

discontinuities.right:

nondecreasingly sorted numeric vector with elements in (0,1); discontinuity points for the right side generating function

discontinuities.lower:

nondecreasingly sorted numeric vector with elements in (0,1); discontinuity points for the lower alpha-cut bound generator

discontinuities.upper:

nondecreasingly sorted numeric vector with elements in (0,1); discontinuity points for the upper alpha-cut bound generator

Extends

Class FuzzyNumber, directly.

See Also

DiscontinuousFuzzyNumber for a convenient constructor

Other DiscontinuousFuzzyNumber-method: DiscontinuousFuzzyNumber, Extract, distance(), integrateAlpha(), plot()

Examples

showClass("DiscontinuousFuzzyNumber")
showMethods(classes="DiscontinuousFuzzyNumber")

Description

Currently, only Euclidean distance may be calculated. We have $d_E^2(A,B) := \int_0^1 (A_L(\alpha)-B_L(\alpha))^2\,d\alpha,\int_0^1 + (A_U(\alpha)-B_U(\alpha))^2\,d\alpha$, see (Grzegorzewski, 1988).

Usage

## S4 method for signature 'FuzzyNumber,FuzzyNumber'
distance(e1, e2, type=c("Euclidean", "EuclideanSquared"), ...)

## S4 method for signature 'FuzzyNumber,DiscontinuousFuzzyNumber'
distance(e1, e2, type=c("Euclidean", "EuclideanSquared"), ...)

## S4 method for signature 'DiscontinuousFuzzyNumber,FuzzyNumber'
distance(e1, e2, type=c("Euclidean", "EuclideanSquared"), ...)

## S4 method for signature 'DiscontinuousFuzzyNumber,DiscontinuousFuzzyNumber'
distance(e1, e2, type=c("Euclidean", "EuclideanSquared"), ...)

Arguments

Details

The calculation are done using numerical integration,

Value

Returns the calculated distance, i.e. a single numeric value.

References

Grzegorzewski P., Metrics and orders in space of fuzzy numbers, Fuzzy Sets and Systems 97, 1998, pp. 83-94.

See Also

Other FuzzyNumber-method: Arithmetic, Extract, FuzzyNumber-class, FuzzyNumber, alphaInterval(), alphacut(), ambiguity(), as.FuzzyNumber(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), core(), evaluate(), expectedInterval(), expectedValue(), integrateAlpha(), piecewiseLinearApproximation(), plot(), show(), supp(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Other DiscontinuousFuzzyNumber-method: DiscontinuousFuzzyNumber-class, DiscontinuousFuzzyNumber, Extract, integrateAlpha(), plot()

Description

This function returns the value(s) of the membership function of a fuzzy number at given point(s).

Usage

## S4 method for signature 'FuzzyNumber,numeric'
evaluate(object, x)

Arguments

Value

Returns a numeric vector.

See Also

Other FuzzyNumber-method: Arithmetic, Extract, FuzzyNumber-class, FuzzyNumber, alphaInterval(), alphacut(), ambiguity(), as.FuzzyNumber(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), core(), distance(), expectedInterval(), expectedValue(), integrateAlpha(), piecewiseLinearApproximation(), plot(), show(), supp(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Examples

T <- TrapezoidalFuzzyNumber(1,2,3,4)
evaluate(T, seq(0, 5, by=0.5))

Description

We have $EI(A) := [\int_0^1 A_L(\alpha)\,d\alpha,\int_0^1 A_U(\alpha)\,d\alpha]$, see (Duboid, Prade, 1987).

Usage

## S4 method for signature 'FuzzyNumber'
expectedInterval(object, ...)

## S4 method for signature 'TrapezoidalFuzzyNumber'
expectedInterval(object)

## S4 method for signature 'PiecewiseLinearFuzzyNumber'
expectedInterval(object)

## S4 method for signature 'PowerFuzzyNumber'
expectedInterval(object)

Arguments

Details

Note that if an instance of the FuzzyNumber or DiscontinuousFuzzyNumber class is given, the calculation is performed via numerical integration. Otherwise, the computation is exact.

Value

Returns a numeric vector of length 2.

References

Dubois D., Prade H. (1987), The mean value of a fuzzy number, Fuzzy Sets and Systems 24, pp. 279-300.

See Also

Other FuzzyNumber-method: Arithmetic, Extract, FuzzyNumber-class, FuzzyNumber, alphaInterval(), alphacut(), ambiguity(), as.FuzzyNumber(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), core(), distance(), evaluate(), expectedValue(), integrateAlpha(), piecewiseLinearApproximation(), plot(), show(), supp(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Other TrapezoidalFuzzyNumber-method: Arithmetic, TrapezoidalFuzzyNumber-class, TrapezoidalFuzzyNumber, TriangularFuzzyNumber(), alphaInterval(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), plot()

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), arctan2(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), fapply(), maximum(), minimum(), necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), plot(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Other PowerFuzzyNumber-method: Extract, PowerFuzzyNumber-class, PowerFuzzyNumber, alphaInterval(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character()

Description

The calculation of the so-called expected value is one of possible methods to deffuzify a fuzzy number.

Usage

## S4 method for signature 'FuzzyNumber'
expectedValue(object, ...)

Arguments

Details

The expected value of $A$ is defined as $EV(A) := (EI_U(A) + EI_L(A))/2$, where $EI$ is the expectedInterval.

Value

Returns a single numeric value.

See Also

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(), integrateAlpha(), piecewiseLinearApproximation(), plot(), show(), supp(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Other deffuzification: value(), weightedExpectedValue()

Other characteristics: ambiguity(), value(), weightedExpectedValue(), width()

Description

For possible slot names see man pages for the FuzzyNumber class and its derivatives

Usage

## S4 method for signature 'FuzzyNumber,character'
x[i]

## S4 method for signature 'PiecewiseLinearFuzzyNumber,character'
x[i]

## S4 method for signature 'PowerFuzzyNumber,character'
x[i]

## S4 method for signature 'DiscontinuousFuzzyNumber,character'
x[i]

Arguments

Details

All slot accessors are read-only.

Value

Returns the slot value.

See Also

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(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), arctan2(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval(), fapply(), maximum(), minimum(), necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), plot(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Other PowerFuzzyNumber-method: PowerFuzzyNumber-class, PowerFuzzyNumber, alphaInterval(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval()

Other DiscontinuousFuzzyNumber-method: DiscontinuousFuzzyNumber-class, DiscontinuousFuzzyNumber, distance(), integrateAlpha(), plot()

Examples

A <- FuzzyNumber(1,2,3,4)
A["a1"]
A["right"]

Description

Applies a given monotonic function using the extension principle (i.e. the function is applied on alpha-cuts).

Usage

## S4 method for signature 'PiecewiseLinearFuzzyNumber,function'
fapply(object, fun, ...)

Arguments

Details

Currently only a method for the PiecewiseLinearFuzzyNumber class has been defined. The computations are exact (up to a numeric error) at knots. So, make sure you have a sufficient number of knots if you want good approximation.

For other types of fuzzy numbers, consider using piecewiseLinearApproximation.

Value

Returns a PiecewiseLinearFuzzyNumber.

See Also

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), arctan2(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval(), maximum(), minimum(), necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), plot(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Other extension_principle: Arithmetic, ^,PiecewiseLinearFuzzyNumber,numeric-method

Description

For convenience, objects of class FuzzyNumber may be created with this function.

Usage

FuzzyNumber(
  a1,
  a2,
  a3,
  a4,
  lower = function(a) rep(NA_real_, length(a)),
  upper = function(a) rep(NA_real_, length(a)),
  left = function(x) rep(NA_real_, length(x)),
  right = function(x) rep(NA_real_, length(x))
)

Arguments

Value

Object of class FuzzyNumber

See Also

Other FuzzyNumber-method: Arithmetic, Extract, FuzzyNumber-class, alphaInterval(), alphacut(), ambiguity(), as.FuzzyNumber(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), core(), distance(), evaluate(), expectedInterval(), expectedValue(), integrateAlpha(), piecewiseLinearApproximation(), plot(), show(), supp(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Description

Formally, a fuzzy number $A$ (Dubois, Prade, 1987) is a fuzzy subset of the real line $R$ with membership function $\mu$ given by:

where $a1,a2,a3,a4\in R$, $a1 \le a2 \le a3 \le a4$, $left: [0,1]\to[0,1]$ is a nondecreasing function called the left side generator of $A$, and $right: [0,1]\to[0,1]$ is a nonincreasing function called the right side generator of $A$. Note that this is a so-called L-R representation of a fuzzy number.

Alternatively, it may be shown that each fuzzy number $A$ may be uniquely determined by specifying its $\alpha$-cuts, $A(\alpha)$. We have $A(0)=[a1,a4]$ and

$A(\alpha)=[a1+(a2-a1)*lower(\alpha), a3+(a4-a3)*upper(\alpha)]$

for $0<\alpha\le 1$, where $lower: [0,1]\to[0,1]$ and $upper: [0,1]\to[0,1]$ are, respectively, strictly increasing and decreasing functions satisfying $lower(\alpha)=\inf\{x: \mu(x)\ge\alpha\}$ and $upper(\alpha)=\sup\{x: \mu(x)\ge\alpha\}$.

Details

Please note that many algorithms that deal with fuzzy numbers often use $\alpha$-cuts rather than side functions.

Note that the FuzzyNumbers package also deals with particular types of fuzzy numbers like trapezoidal, piecewise linear, or “parametric” FNs.

Slots

a1:

Single numeric value specifying the left bound for the support.

a2:

Single numeric value specifying the left bound for the core.

a3:

Single numeric value specifying the right bound for the core.

a4:

Single numeric value specifying the right bound for the support.

lower:

A nondecreasing function [0,1]->[0,1] that gives the lower alpha-cut bound.

upper:

A nonincreasing function [0,1]->[1,0] that gives the upper alpha-cut bound.

left:

A nondecreasing function [0,1]->[0,1] that gives the left side function.

right:

A nonincreasing function [0,1]->[1,0] that gives the right side function.

Child/sub classes

TrapezoidalFuzzyNumber, PiecewiseLinearFuzzyNumber, PowerFuzzyNumber, and DiscontinuousFuzzyNumber

References

Dubois D., Prade H. (1987), Fuzzy numbers: An overview, In: Analysis of Fuzzy Information. Mathematical Logic, vol. I, CRC Press, pp. 3-39.

See Also

FuzzyNumber for a convenient constructor, and as.FuzzyNumber for conversion of objects to this class. Also, see convertSide for creating side functions generators, convertAlpha for creating alpha-cut bounds generators, approxInvert for inverting side functions/alpha-cuts numerically.

Other FuzzyNumber-method: Arithmetic, Extract, FuzzyNumber, alphaInterval(), alphacut(), ambiguity(), as.FuzzyNumber(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), core(), distance(), evaluate(), expectedInterval(), expectedValue(), integrateAlpha(), piecewiseLinearApproximation(), plot(), show(), supp(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Examples

showClass("FuzzyNumber")
showMethods(classes="FuzzyNumber")

Description

The function uses multiple calls to integrate.

Usage

integrate_discont_val(f, from, to, discontinuities = numeric(0), ...)

Arguments

Value

Returns the estimate of the integral.

Description

Integrates numerically a transformed or weighted lower or upper alpha-cut bound of a fuzzy number.

Usage

## S4 method for signature 'FuzzyNumber,character,numeric,numeric'
integrateAlpha(object, which=c("lower", "upper"),
   from=0, to=1, weight=NULL, transform=NULL, ...)

## S4 method for signature 'DiscontinuousFuzzyNumber,character,numeric,numeric'
integrateAlpha(object, which=c("lower", "upper"),
   from=0, to=1, weight=NULL, transform=NULL, ...)

Arguments

Value

Returns a single numeric value.

See Also

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(), piecewiseLinearApproximation(), plot(), show(), supp(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Other DiscontinuousFuzzyNumber-method: DiscontinuousFuzzyNumber-class, DiscontinuousFuzzyNumber, Extract, distance(), plot()

Description

Determines maximum fuzzy number based on two inputs.

Usage

## S4 method for signature 
## 'PiecewiseLinearFuzzyNumber,PiecewiseLinearFuzzyNumber'
maximum(e1, e2)

Arguments

Details

The resulting PiecewiseLinearFuzzyNumber is only an approximation of the result it might not be very precise for small number of knots (see examples).

Value

Returns a PiecewiseLinearFuzzyNumber representing maximal value of the inputs.

References

KAUFMANN, A., GUPTA, M. M. (1985) Introduction to Fuzzy Arithmetic. New York : Van Nostrand Reinhold Company. ISBN 044230079.

See Also

Other min_max-operators: minimum()

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), arctan2(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval(), fapply(), minimum(), necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), plot(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Examples

# example with low number of knots, showing the approximate nature
# of the result
x = as.PiecewiseLinearFuzzyNumber(TriangularFuzzyNumber(-4.8, -3 , -1.5))
y = as.PiecewiseLinearFuzzyNumber(TriangularFuzzyNumber(-5.5, -2.5, -1.1))
maxFN = maximum(x,y)
min = min(alphacut(x,0)[1],alphacut(y,0)[1])
max = max(alphacut(x,0)[2],alphacut(y,0)[2])
plot(x, col="red", xlim=c(min,max))
plot(y, col="blue", add=TRUE)
plot(maxFN, col="green", add=TRUE)

# example with high number of knots, that does not suffer 
# from the approximate nature of the result
x = as.PiecewiseLinearFuzzyNumber(TriangularFuzzyNumber(-4.8, -3 , -1.5), knot.n = 9)
y = as.PiecewiseLinearFuzzyNumber(TriangularFuzzyNumber(-5.5, -2.5, -1.1), knot.n = 9)
maxFN = maximum(x,y)
min = min(alphacut(x,0)[1],alphacut(y,0)[1])
max = max(alphacut(x,0)[2],alphacut(y,0)[2])
plot(x, col="red", xlim=c(min,max))
plot(y, col="blue", add=TRUE)
plot(maxFN, col="green", add=TRUE)

Description

Determines minimum fuzzy number based on two inputs.

Usage

## S4 method for signature 
## 'PiecewiseLinearFuzzyNumber,PiecewiseLinearFuzzyNumber'
minimum(e1, e2)

Arguments

Details

The resulting PiecewiseLinearFuzzyNumber is only an approximation of the result it might not be very precise for small number of knots (see examples).

Value

Returns a PiecewiseLinearFuzzyNumber representing maximal value of the inputs.

References

KAUFMANN, A., GUPTA, M. M. (1985) Introduction to Fuzzy Arithmetic. New York : Van Nostrand Reinhold Company. ISBN 044230079.

See Also

Other min_max-operators: maximum()

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), arctan2(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval(), fapply(), maximum(), necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), plot(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Examples

# example with low number of knots, showing the approximate nature
# of the result
x = as.PiecewiseLinearFuzzyNumber(TriangularFuzzyNumber(-4.8, -3 , -1.5))
y = as.PiecewiseLinearFuzzyNumber(TriangularFuzzyNumber(-5.5, -2.5, -1.1))
minFN = minimum(x,y)
min = min(alphacut(x,0)[1],alphacut(y,0)[1])
max = max(alphacut(x,0)[2],alphacut(y,0)[2])
plot(x, col="red", xlim=c(min,max))
plot(y, col="blue", add=TRUE)
plot(minFN, col="green", add=TRUE)

# example with high number of knots, that does not suffer 
# from the approximate nature of the result
x = as.PiecewiseLinearFuzzyNumber(TriangularFuzzyNumber(-4.8, -3 , -1.5), knot.n = 9)
y = as.PiecewiseLinearFuzzyNumber(TriangularFuzzyNumber(-5.5, -2.5, -1.1), knot.n = 9)
minFN = minimum(x,y)
min = min(alphacut(x,0)[1],alphacut(y,0)[1])
max = max(alphacut(x,0)[2],alphacut(y,0)[2])
plot(x, col="red", xlim=c(min,max))
plot(y, col="blue", add=TRUE)
plot(minFN, col="green", add=TRUE)

Description

Determines value of necessity of $e1>=e2$, the result is from range [0,1]. Value 0 indicates no fulfilment of the operator and 1 indicates complete fulfilment.

Usage

## S4 method for signature 
## 'PiecewiseLinearFuzzyNumber,PiecewiseLinearFuzzyNumber'
necessityExceedance(e1, e2)

Arguments

Value

Returns a value from range [0,1] indicating the necessity of exceedance of e2 by e1.

References

DUBOIS, Didier and PRADE, Henri, 1983, Ranking Fuzzy Numbers in the Setting of Possibility Theory. Information Sciences. 1983. Vol. 30, no. 3, p. 183–224.

See Also

Other comparison-operators: necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), arctan2(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval(), fapply(), maximum(), minimum(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), plot(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Examples

a <- TriangularFuzzyNumber(2, 3, 5)
b <- TriangularFuzzyNumber(1.5, 4, 4.8)
a <- as.PiecewiseLinearFuzzyNumber(a, knot.n = 9)
b <- as.PiecewiseLinearFuzzyNumber(b, knot.n = 9)
necessityExceedance(a,b)

Description

Determines value of necessity of $e1>e2$, the result is from range [0,1]. Value 0 indicates no fulfilment of the operator and 1 indicates complete fulfilment.

Usage

## S4 method for signature 
## 'PiecewiseLinearFuzzyNumber,PiecewiseLinearFuzzyNumber'
necessityStrictExceedance(e1, e2)

Arguments

Value

Returns a value from range [0,1] indicating the strict necessity of exceedance of e2 by e1.

References

DUBOIS, Didier and PRADE, Henri, 1983, Ranking Fuzzy Numbers in the Setting of Possibility Theory. Information Sciences. 1983. Vol. 30, no. 3, p. 183–224.

See Also

Other comparison-operators: necessityExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), arctan2(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval(), fapply(), maximum(), minimum(), necessityExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), plot(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Examples

a <- TriangularFuzzyNumber(2, 3, 5)
b <- TriangularFuzzyNumber(1.5, 4, 4.8)
a <- as.PiecewiseLinearFuzzyNumber(a, knot.n = 9)
b <- as.PiecewiseLinearFuzzyNumber(b, knot.n = 9)
necessityStrictExceedance(a,b)

Description

Determines value of necessity of $e1<e2$, the result is from range [0,1]. Value 0 indicates no fulfilment of the operator and 1 indicates complete fulfilment.

Usage

## S4 method for signature 
## 'PiecewiseLinearFuzzyNumber,PiecewiseLinearFuzzyNumber'
necessityStrictUndervaluation(e1, e2)

Arguments

Value

Returns a value from range [0,1] indicating the necessity of exceedance of e2 by e1.

References

DUBOIS, Didier and PRADE, Henri, 1983, Ranking Fuzzy Numbers in the Setting of Possibility Theory. Information Sciences. 1983. Vol. 30, no. 3, p. 183–224.

See Also

Other comparison-operators: necessityExceedance(), necessityStrictExceedance(), necessityUndervaluation(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), arctan2(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval(), fapply(), maximum(), minimum(), necessityExceedance(), necessityStrictExceedance(), necessityUndervaluation(), plot(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Examples

a <- TriangularFuzzyNumber(0.2, 1.0, 2.8)
b <- TriangularFuzzyNumber(0, 1.8, 2.2)
a <- as.PiecewiseLinearFuzzyNumber(a, knot.n = 9)
b <- as.PiecewiseLinearFuzzyNumber(b, knot.n = 9)
necessityStrictUndervaluation(a,b)

Description

Determines value of necessity of $e1<=e2$, the result is from range [0,1]. Value 0 indicates no fulfilment of the operator and 1 indicates complete fulfilment.

Usage

## S4 method for signature 
## 'PiecewiseLinearFuzzyNumber,PiecewiseLinearFuzzyNumber'
necessityUndervaluation(e1, e2)

Arguments

Value

Returns a value from range [0,1] indicating the necessity of exceedance of e2 by e1.

References

DUBOIS, Didier and PRADE, Henri, 1983, Ranking Fuzzy Numbers in the Setting of Possibility Theory. Information Sciences. 1983. Vol. 30, no. 3, p. 183–224.

See Also

Other comparison-operators: necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), arctan2(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval(), fapply(), maximum(), minimum(), necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), plot(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Examples

a <- TriangularFuzzyNumber(0.2, 1.0, 2.8)
b <- TriangularFuzzyNumber(0, 1.8, 2.2)
a <- as.PiecewiseLinearFuzzyNumber(a, knot.n = 9)
b <- as.PiecewiseLinearFuzzyNumber(b, knot.n = 9)
necessityUndervaluation(a,b)

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

Details

'method' may be one of:

1. NearestEuclidean: see (Coroianu, Gagolewski, Grzegorzewski, 2013 and 2014a); uses numerical integration, see integrateAlpha. Slow for large knot.n.

2. SupportCorePreserving: This method was proposed in (Coroianu et al., 2014b) and is currently only available for knot.n==1. It is the L2-nearest piecewise linear approximation with constraints core(A)==core(P(A)) and supp(A)==supp(P(A)); uses numerical integration.

3. 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. 26-51.

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 Knowledge-Based Systems, Part II (CCIS 443), Springer, pp. 244-254.

Coroianu L., Gagolewski M., Grzegorzewski P. (2014b), Nearest Piecewise Linear Approximation of Fuzzy Numbers - General Case, submitted for publication.

See Also

Other approximation: trapezoidalApproximation()

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(), plot(), show(), supp(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Examples

(A <- FuzzyNumber(-1, 0, 1, 3,
   lower=function(x) sqrt(x),upper=function(x) 1-sqrt(x)))
(PA <- piecewiseLinearApproximation(A, "NearestEuclidean",
   knot.n=1, knot.alpha=0.2))

Description

For convenience, objects of class PiecewiseLinearFuzzyNumber may be created with this function.

Usage

PiecewiseLinearFuzzyNumber(
  a1,
  a2,
  a3,
  a4,
  knot.n = 0,
  knot.alpha = numeric(0),
  knot.left = numeric(0),
  knot.right = numeric(0)
)

Arguments

Details

If a1, a2, a3, and a4 are missing, then knot.left and knot.right may be of length knot.n+2.

If knot.n is not given, then it guessed from length(knot.left). If knot.alpha is missing, then the knots will be equally distributed on the interval [0,1].

Value

An object of class PiecewiseLinearFuzzyNumber.

See Also

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber-class, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), arctan2(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval(), fapply(), maximum(), minimum(), necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), plot(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Description

A piecewise linear fuzzy number (PLFN) has side functions and alpha-cut bounds that linearly interpolate a given set of points (at fixed alpha-cuts).

Details

If knot.n is equal to 0 or all left and right knots lie on common lines, then a Piecewise Linear Fuzzy Number reduces to a TrapezoidalFuzzyNumber. Note that, however, the TrapezoidalFuzzyNumber does not inherit from PiecewiseLinearFuzzyNumber for efficiency reasons. To convert the former to the latter, call as.PiecewiseLinearFuzzyNumber.

Slots

a1, a2, a3, a4, lower, upper, left, right:

Inherited from the FuzzyNumber class.

knot.n:

number of knots, a single integer value, 0 for a trapezoidal fuzzy number

knot.alpha:

alpha-cuts, increasingly sorted vector of length knot.n with elements in [0,1]

knot.left:

nondecreasingly sorted vector of length knot.n; defines left alpha-cut bounds at knots

knot.right:

nondecreasingly sorted vector of length knot.n; defines right alpha-cut bounds at knots

Extends

Class FuzzyNumber, directly.

See Also

PiecewiseLinearFuzzyNumber for a convenient constructor, as.PiecewiseLinearFuzzyNumber for conversion of objects to this class, and piecewiseLinearApproximation for approximation routines.

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), arctan2(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval(), fapply(), maximum(), minimum(), necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), plot(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Examples

showClass("PiecewiseLinearFuzzyNumber")
showMethods(classes="PiecewiseLinearFuzzyNumber")

Description

The function aims to provide a similar look-and-feel to the built-in plot.default and curve function.

Usage

## S4 method for signature 'FuzzyNumber,missing'
plot(x, y, from=NULL, to=NULL, n=101, at.alpha=NULL,
draw.membership.function=TRUE, draw.alphacuts=!draw.membership.function,
xlab=NULL, ylab=NULL, xlim=NULL, ylim=NULL,
type="l", col=1, lty=1, pch=1, lwd=1,
shadowdensity=15, shadowangle=45, shadowcol=col, shadowborder=NULL,
add=FALSE, ...)

## S4 method for signature 'TrapezoidalFuzzyNumber,missing'
plot(x, y, from=NULL, to=NULL,
draw.membership.function=TRUE, draw.alphacuts=!draw.membership.function,
xlab=NULL, ylab=NULL, xlim=NULL, ylim=NULL,
type="l", col=1, lty=1, pch=1, lwd=1, add=FALSE, ...)

## S4 method for signature 'PiecewiseLinearFuzzyNumber,missing'
plot(x, y, from=NULL, to=NULL,
draw.membership.function=TRUE, draw.alphacuts=!draw.membership.function,
xlab=NULL, ylab=NULL, xlim=NULL, ylim=NULL,
type="l", col=1, lty=1, pch=1, lwd=1, add=FALSE, ...)

## S4 method for signature 'DiscontinuousFuzzyNumber,missing'
plot(x, y, from=NULL, to=NULL,
n=101, draw.membership.function=TRUE, draw.alphacuts=!draw.membership.function,
xlab=NULL, ylab=NULL, xlim=NULL, ylim=NULL,
type="l", col=1, lty=1, pch=1, lwd=1,
add=FALSE, ...)

Arguments

Details

Note that if from > a1 then it is set to a1.

Value

Returns nothing really interesting.

See Also

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(), show(), supp(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), arctan2(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval(), fapply(), maximum(), minimum(), necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Other TrapezoidalFuzzyNumber-method: Arithmetic, TrapezoidalFuzzyNumber-class, TrapezoidalFuzzyNumber, TriangularFuzzyNumber(), alphaInterval(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), expectedInterval()

Other DiscontinuousFuzzyNumber-method: DiscontinuousFuzzyNumber-class, DiscontinuousFuzzyNumber, Extract, distance(), integrateAlpha()

Examples

plot(FuzzyNumber(0,1,2,3), col="gray")
plot(FuzzyNumber(0,1,2,3, left=function(x) x^2, right=function(x) 1-x^3), add=TRUE)
plot(FuzzyNumber(0,1,2,3, lower=function(x) x, upper=function(x) 1-x), add=TRUE, col=2)

Description

Determines value of possibility of $e1>=e2$, the result is from range [0,1]. Value 0 indicates no fulfilment of the operator and 1 indicates complete fulfilment.

Usage

## S4 method for signature 
## 'PiecewiseLinearFuzzyNumber,PiecewiseLinearFuzzyNumber'
possibilityExceedance(e1, e2)

Arguments

Value

Returns a value from range [0,1] indicating the possibility of exceedance of e2 by e1.

References

DUBOIS, Didier and PRADE, Henri, 1983, Ranking Fuzzy Numbers in the Setting of Possibility Theory. Information Sciences. 1983. Vol. 30, no. 3, p. 183–224.

See Also

Other comparison-operators: necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), arctan2(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval(), fapply(), maximum(), minimum(), necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), plot(), possibilityStrictExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Examples

a <- TriangularFuzzyNumber(2, 3, 5)
b <- TriangularFuzzyNumber(1.5, 4, 4.8)
a <- as.PiecewiseLinearFuzzyNumber(a, knot.n = 9)
b <- as.PiecewiseLinearFuzzyNumber(b, knot.n = 9)
possibilityExceedance(a,b)

Description

Determines value of possibility of $e1>e2$, the result is from range [0,1]. Value 0 indicates no fulfilment of the operator and 1 indicates complete fulfilment.

Usage

## S4 method for signature 
## 'PiecewiseLinearFuzzyNumber,PiecewiseLinearFuzzyNumber'
possibilityStrictExceedance(e1, e2)

Arguments

Value

Returns a value from range [0,1] indicating the strict possibility of exceedance of e2 by e1.

References

DUBOIS, Didier and PRADE, Henri, 1983, Ranking Fuzzy Numbers in the Setting of Possibility Theory. Information Sciences. 1983. Vol. 30, no. 3, p. 183–224.

See Also

Other comparison-operators: necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), possibilityExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), arctan2(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval(), fapply(), maximum(), minimum(), necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), plot(), possibilityExceedance(), possibilityStrictUndervaluation(), possibilityUndervaluation()

Examples

a <- TriangularFuzzyNumber(2, 3, 5)
b <- TriangularFuzzyNumber(1.5, 4, 4.8)
a <- as.PiecewiseLinearFuzzyNumber(a, knot.n = 9)
b <- as.PiecewiseLinearFuzzyNumber(b, knot.n = 9)
possibilityStrictExceedance(a,b)

Description

Determines value of possibility of $e1<e2$, the result is from range [0,1]. Value 0 indicates no fulfilment of the operator and 1 indicates complete fulfilment.

Usage

## S4 method for signature 
## 'PiecewiseLinearFuzzyNumber,PiecewiseLinearFuzzyNumber'
possibilityStrictUndervaluation(e1, e2)

Arguments

Value

Returns a value from range [0,1] indicating the necessity of exceedance of e2 by e1.

References

DUBOIS, Didier and PRADE, Henri, 1983, Ranking Fuzzy Numbers in the Setting of Possibility Theory. Information Sciences. 1983. Vol. 30, no. 3, p. 183–224.

See Also

Other comparison-operators: necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), possibilityExceedance(), possibilityStrictExceedance(), possibilityUndervaluation()

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), arctan2(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval(), fapply(), maximum(), minimum(), necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), plot(), possibilityExceedance(), possibilityStrictExceedance(), possibilityUndervaluation()

Examples

a <- TriangularFuzzyNumber(0.2, 1.0, 2.8)
b <- TriangularFuzzyNumber(0, 1.8, 2.2)
a <- as.PiecewiseLinearFuzzyNumber(a, knot.n = 9)
b <- as.PiecewiseLinearFuzzyNumber(b, knot.n = 9)
possibilityStrictUndervaluation(a,b)

Description

Determines value of possibility of $e1<=e2$, the result is from range [0,1]. Value 0 indicates no fulfilment of the operator and 1 indicates complete fulfilment.

Usage

## S4 method for signature 
## 'PiecewiseLinearFuzzyNumber,PiecewiseLinearFuzzyNumber'
possibilityUndervaluation(e1, e2)

Arguments

Value

Returns a value from range [0,1] indicating the possibility of exceedance of e2 by e1.

References

DUBOIS, Didier and PRADE, Henri, 1983, Ranking Fuzzy Numbers in the Setting of Possibility Theory. Information Sciences. 1983. Vol. 30, no. 3, p. 183–224.

See Also

Other comparison-operators: necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation()

Other PiecewiseLinearFuzzyNumber-method: Arithmetic, Extract, PiecewiseLinearFuzzyNumber-class, PiecewiseLinearFuzzyNumber, ^,PiecewiseLinearFuzzyNumber,numeric-method, alphaInterval(), arctan2(), as.PiecewiseLinearFuzzyNumber(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval(), fapply(), maximum(), minimum(), necessityExceedance(), necessityStrictExceedance(), necessityStrictUndervaluation(), necessityUndervaluation(), plot(), possibilityExceedance(), possibilityStrictExceedance(), possibilityStrictUndervaluation()

Examples

a <- TriangularFuzzyNumber(0.2, 1.0, 2.8)
b <- TriangularFuzzyNumber(0, 1.8, 2.2)
a <- as.PiecewiseLinearFuzzyNumber(a, knot.n = 9)
b <- as.PiecewiseLinearFuzzyNumber(b, knot.n = 9)
possibilityUndervaluation(a,b)

Description

For convenience, objects of class PowerFuzzyNumber may be created with this function.

Usage

PowerFuzzyNumber(a1, a2, a3, a4, p.left = 1, p.right = 1)

Arguments

Value

Object of class PowerFuzzyNumber

See Also

Other PowerFuzzyNumber-method: Extract, PowerFuzzyNumber-class, alphaInterval(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval()

Description

Bodjanova-type fuzzy numbers which sides are given by power functions are defined using four coefficients a1 <= a2 <= a3 <= a4, and parameters p.left, p.right>0, which determine exponents for the side functions.

Details

We have $\mathtt{left}(x)=x^{\mathtt{p.left}}$, and $\mathtt{right}(x)=(1-x)^{\mathtt{p.right}}$.

This class is a natural generalization of trapezoidal FNs. For other see PiecewiseLinearFuzzyNumber.

Slots

a1, a2, a3, a4, lower, upper, left, right:

Inherited from the FuzzyNumber class.

p.left:

single numeric value; 1.0 for a trapezoidal FN.

p.right:

single numeric value; 1.0 for a trapezoidal FN.

Extends

Class FuzzyNumber, directly.

References

Bodjanova S. (2005), Median value and median interval of a fuzzy number, Information Sciences 172, pp. 73-89.

See Also

PowerFuzzyNumber for a convenient constructor, as.PowerFuzzyNumber for conversion of objects to this class.

PowerFuzzyNumber for a convenient constructor

Other PowerFuzzyNumber-method: Extract, PowerFuzzyNumber, alphaInterval(), as.PowerFuzzyNumber(), as.TrapezoidalFuzzyNumber(), as.character(), expectedInterval()

Examples

showClass("PowerFuzzyNumber")
showMethods(classes="PowerFuzzyNumber")

Description

See as.character for more details.

Usage

## S4 method for signature 'FuzzyNumber'
show(object)

Arguments

Details

The method as.character is called on given fuzzy number object with default arguments. The results are printed on stdout.

Value

Does not return anything interesting.

See Also

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(), supp(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Description

We have $\mathrm{supp}(A) := [a1,a4]$. This gives the values that a fuzzy number possibly may represent.

Usage

## S4 method for signature 'FuzzyNumber'
supp(object)

Arguments

Value

Returns a numeric vector of length 2.

See Also

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(), trapezoidalApproximation(), value(), weightedExpectedValue(), width()

Other alpha_cuts: alphacut(), core()

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

Details

method may be one of:

1. NearestEuclidean: see (Ban, 2009); uses numerical integration, see integrateAlpha

2. Naive: We have core(A)==core(T(A)) and supp(A)==supp(T(A))

3. 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)

4. SupportCoreRestricted: This method was proposed in (Grzegorzewski, Pasternak-Winiarska, 2011). L2-nearest trapezoidal approximation with constraints core(A) $\subseteq$ core(T(A)) and supp(T(A)) $\subseteq$ 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. 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.

See Also

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()