FuzzyStatTraEOO (version 1.0)

TrapezoidalFuzzyNumberList: 'TrapezoidalFuzzyNumberList' is a child class of 'StatList'.

Description

'TrapezoidalFuzzyNumberList' must contain valid 'TrapezoidalFuzzyNumbers'. This class implements a version of the empty 'StatList' methods.

Author

(s) Andrea Garcia Cernuda <[e-mail]>, Asun Lubiano <[e-mail]>, Sara de la Rosa de Saa

Super class

FuzzyStatTraEOO::StatList -> TrapezoidalFuzzyNumberList

Methods

Public methods

• TrapezoidalFuzzyNumberList$new()

• TrapezoidalFuzzyNumberList$add()

• TrapezoidalFuzzyNumberList$dthetaphi()

• TrapezoidalFuzzyNumberList$dwablphi()

• TrapezoidalFuzzyNumberList$gsi()

• TrapezoidalFuzzyNumberList$hyperI()

• TrapezoidalFuzzyNumberList$mEstimator()

• TrapezoidalFuzzyNumberList$mdd()

• TrapezoidalFuzzyNumberList$mean()

• TrapezoidalFuzzyNumberList$median1Norm()

• TrapezoidalFuzzyNumberList$medianWabl()

• TrapezoidalFuzzyNumberList$qn()

• TrapezoidalFuzzyNumberList$rho1()

• TrapezoidalFuzzyNumberList$sn()

• TrapezoidalFuzzyNumberList$tn()

• TrapezoidalFuzzyNumberList$transfTra()

• TrapezoidalFuzzyNumberList$var()

• TrapezoidalFuzzyNumberList$wablphi()

• TrapezoidalFuzzyNumberList$addTrapezoidalFuzzyNumber()

• TrapezoidalFuzzyNumberList$removeTrapezoidalFuzzyNumber()

• TrapezoidalFuzzyNumberList$getDimension()

• TrapezoidalFuzzyNumberList$plot()

• TrapezoidalFuzzyNumberList$getLength()

• TrapezoidalFuzzyNumberList$clone()

Inherited methods

---

Method new()

This method creates a 'TrapezoidalFuzzyNumberList' object with all the attributes set if the 'TrapezoidalFuzzyNumbers' are valid.

Usage

TrapezoidalFuzzyNumberList$new(numbers = NA)

Arguments

numbers

is a list which contains n TrapezoidalFuzzyNumbers.

Details

See examples.

Returns

The TrapezoidalFuzzyNumberList object created with all attributes set if the 'TrapezoidalFuzzyNumbers' are valid.

Examples

TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1, 2, 3, 4),
TrapezoidalFuzzyNumber$new(-8, -6, -4, -2),
TrapezoidalFuzzyNumber$new(-1, -1, 2, 3),
TrapezoidalFuzzyNumber$new(1, 2, 3, 3)))

---

Method add()

This method calculates the scale measure Average Distance Deviation (ADD) of a 'TrapezoidalFuzzyNumberList' with respect to a 'TrapezoidalFuzzyNumberList' or with respect to a 'FuzzyNumberList' containing a unique valid fuzzy number. The employed metric in the calculation can be the 1-norm distance, the mid/spr distance or the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance. See De la Rosa de Saa et al. (2017) [2].

Usage

TrapezoidalFuzzyNumberList$add(s = NA, type = NA, a = 1, b = 1, theta = 1)

Arguments

s

is a TrapezoidalFuzzyNumberList containing a unique valid TrapezoidalFuzzyNumber or it is a FuzzyNumberList containing a unique valid FuzzyNumber.

type

positive integer 1, 2 or 3: if type==1, the 1-norm distance will be considered in the calculation of the measure ADD. If type==2, the mid/spr distance will be considered. By contrast, if type==3, the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance will be used.

a

real number > 0, by default a=1. It is the first parameter of a beta distribution which corresponds to a weighting measure on [0,1] in the mid/spr distance or in the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance.

b

real number > 0, by default b=1. It is the second parameter of a beta distribution which corresponds to a weighting measure on [0,1] in the mid/spr distance or in the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance.

theta

real number > 0, by default theta=1. It is the weight of the spread in the mid/spr distance and the weight of the ldev and rdev in the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance.

Details

See examples.

Returns

the scale measure ADD, which is a real number. If the body's method inner conditions are not met, NA will be returned.

Examples

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(0,0.1,0.2,0.3),
TrapezoidalFuzzyNumber$new(1,2,3,4),TrapezoidalFuzzyNumber$new(2,3,4,5)))$add(
FuzzyNumberList$new(c(FuzzyNumber$new(array(c(0.0, 0.5, 1.0,-1.5,-1.25,-1.0,
3.0, 2.0, 1.0), dim = c(3, 3))))),1L)

# Example 2: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(0,0.1,0.2,0.3), TrapezoidalFuzzyNumber$new(1,2,3,4),TrapezoidalFuzzyNumber$new(2,3,4,5)))$add( FuzzyNumberList$new(c(FuzzyNumber$new(array(c(0.0, 0.5, 1.0,-1.5,-1.25,-1.0, 3.0, 2.0, 1.0), dim = c(3, 3))))),2L,2,1,1)

# Example 3: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(0,0.1,0.2,0.3), TrapezoidalFuzzyNumber$new(1,2,3,4),TrapezoidalFuzzyNumber$new(2,3,4,5)))$add( TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(5,6,7,8))),3L,1,1,1)

# Example 4: F=Simulation$new()$simulCase1(10L) S=F$mean() F$add(S,1L)

# Example 5: F=Simulation$new()$simulCase1(100L) S=F$median1Norm() F$add(S,2L,2,1,1)

# Example 6: F=Simulation$new()$simulCase2(10L) U=Simulation$new()$simulCase2(1L) F$add(U,2L)

# Example 7: F=Simulation$new()$simulCase2(10L) U=F$transfTra() F$add(U,2L)

# Example 8: F=Simulation$new()$simulCase2(10L) U=Simulation$new()$simulCase2(2L) F$add(U,2L)

---

Method dthetaphi()

This method calculates the mid/spr distance between the 'TrapezoidalFuzzyNumbers' contained in the current object and the one passed as parameter. See Lubiano et al. (2016) [5].

Usage

TrapezoidalFuzzyNumberList$dthetaphi(s = NA, a = 1, b = 1, theta = 1)

Arguments

s

TrapezoidalFuzzyNumberList containing valid TrapezoidalFuzzyNumbers characterized by their four values inf0, inf1, sup1, sup0.

a

real number > 0, by default a=1. It is the first parameter of a beta distribution which corresponds to a weighting measure on [0,1].

b

real number > 0, by default b=1. It is the second parameter of a beta distribution which corresponds to a weighting measure on [0,1].

theta

real number > 0, by default theta=1. It is the weight of the spread in the mid/spr distance.

Details

See examples.

Returns

a matrix containing the mid/spr distances between the two previous mentioned TrapezoidalFuzzyNumberLists.

Examples

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$dthetaphi(
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58))))

# Example 2: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$dthetaphi( TrapezoidalFuzzyNumberList$new( c(TrapezoidalFuzzyNumber$new(1.5,2,3.75,4), TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58))),1,1,1)

# Example 3: F=Simulation$new()$simulCase1(6L) S=Simulation$new()$simulCase1(8L) F$dthetaphi(S,1,5,1)

---

Method dwablphi()

This method calculates the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance between the 'TrapezoidalFuzzyNumbers' contained in two 'TrapezoidalFuzzyNumberLists'. See Sinova et al. (2013) [6] and Sinova et al. (2016) [10].

Usage

TrapezoidalFuzzyNumberList$dwablphi(s = NA, a = 1, b = 1, theta = 1)

Arguments

s

TrapezoidalFuzzyNumberList containing valid TrapezoidalFuzzyNumbers characterized by their four values inf0, inf1, sup1, sup0.

a

real number > 0, by default a=1. It is the first parameter of a beta distribution which corresponds to a weighting measure on [0,1].

b

real number > 0, by default b=1. It is the second parameter of a beta distribution which corresponds to a weighting measure on [0,1].

theta

real number > 0, by default theta=1. It is the weight of the ldev and rdev in the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance.

Details

See examples.

Returns

a matrix containing the (\(\phi\),\(\theta\))-wabl/ldev/rdev distances between the two previous mentioned TrapezoidalFuzzyNumberLists.

Examples

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$dwablphi(
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58))))

# Example 2: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$dwablphi( TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1.5,2,3.75,4), TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58))),5,1,1)

# Example 3: F=Simulation$new()$simulCase1(10L) S=Simulation$new()$simulCase1(20L) F$dwablphi(S)

---

Method gsi()

This method calculates the Gini-Simpson diversity index for a sample of 'TrapezoidalFuzzyNumbers' contained in a 'TrapezoidalFuzzyNumberList'. See De la Rosa de Saa et al. (2015) [1].

Usage

TrapezoidalFuzzyNumberList$gsi()

Details

See examples.

Returns

the Gini-Simpson diversity index, which is a real number.

Examples

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$gsi()

# Example 2: F=Simulation$new()$simulCase1(50L) F$gsi()

---

Method hyperI()

This method calculates the hyperbolic inequality index for a sample of 'TrapezoidalFuzzyNumbers' contained in a 'TrapezoidalFuzzyNumberList'. The method checks if all 'TrapezoidalFuzzyNumbers' are positive. See De la Rosa de Saa et al. (2015) [1] and Lubiano and Gil (2002) [4].

Usage

TrapezoidalFuzzyNumberList$hyperI(c = 0, verbose = TRUE)

Arguments

c

number in [0,0.5]. The c*100 of the hyperbolic inequality index.

verbose

if TRUE the messages are written to the console unless the user actively decides to set verbose=FALSE.

Details

See examples.

Returns

the hyperbolic inequality index, which is a real number. If the body's method inner conditions are not met, NA will be returned.

Examples

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$hyperI()

# Example 2: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$hyperI(0.5)

# Example 3: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(2,4,6,8)))$hyperI()

# Example 4: F=Simulation$new()$simulFRSTra(100L,0.05,0.35,0.6,2,1) F$hyperI()

# Example 5: F=Simulation$new()$simulCase2(10L) F$hyperI(0.5)

---

Method mEstimator()

This method calculates the M-estimator of scale with loss method given in a 'TrapezoidalFuzzyNumberList' containing 'TrapezoidalFuzzyNumbers'. For computing the M-estimator, a method called “iterative reweighting” is used. The employed metric in the M-equation can be the 1-norm distance, the mid/spr distance or the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance.

Usage

TrapezoidalFuzzyNumberList$mEstimator(
  f = NA,
  estInitial = NA,
  delta = NA,
  epsilon = NA,
  type = NA,
  a = 1,
  b = 1,
  theta = 1
)

Arguments

f

is the name of the loss function. It can be "Huber", "Tukey" or "Cauchy".

estInitial

real number > 0.

delta

real number in (0,1). It is present in the f-equation.

epsilon

real number > 0. It is the tolerance allowed in the algorithm.

type

positive integer 1, 2 or 3: if type==1, the 1-norm distance will be considered in the calculation of the measure ADD. If type==2, the mid/spr distance will be considered. By contrast, if type==3, the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance will be used.

a

real number > 0, by default a=1. It is the first parameter of a beta distribution which corresponds to a weighting measure on [0,1] in the mid/spr distance or in the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance.

b

real number > 0, by default b=1. It is the second parameter of a beta distribution which corresponds to a weighting measure on [0,1] in the mid/spr distance or in the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance.

theta

real number > 0, by default theta=1. It is the weight of the spread in the mid/spr distance and the weight of the ldev and rdev in the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance.

Details

See examples.

Returns

the value of the M-estimator of scale, which is a real number.

Examples

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$mEstimator("Huber",0.321,0.5,10^(-5),
1L)

# Example 2: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$mEstimator("Tukey",0.123,0.5,10^(-5), 2L,1,1,1)

# Example 3: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$mEstimator("Cauchy",0.123,0.5,10^(-5), 3L,0.75,0.5,1)

# Example 4: F=Simulation$new()$simulCase1(100L) U=F$median1Norm() estInitial=F$mdd(U,1L) delta=0.5 epsilon=10^(-5) F$mEstimator("Huber",estInitial,delta,epsilon,1L)

---

Method mdd()

This method calculates the scale measure Median Distance Deviation (MDD) of a 'TrapezoidalFuzzyNumberList' with respect to a 'TrapezoidalFuzzyNumberList' or with respect to a 'FuzzyNumberList' with a unique valid fuzzy number. The employed metric in the calculation can be the 1-norm distance, the mid/spr distance or the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance. See De la Rosa de Saa et al. (2015) [2] and De la Rosa de Saa et al. (2021) [3].

Usage

TrapezoidalFuzzyNumberList$mdd(s = NA, type = NA, a = 1, b = 1, theta = 1)

Arguments

s

is a TrapezoidalFuzzyNumberList containing a unique TrapezoidalFuzzyNumber or it is a FuzzyNumberList containing a unique FuzzyNumber.

type

positive integer 1, 2 or 3: if type==1, the 1-norm distance will be considered in the calculation of the measure ADD. If type==2, the mid/spr distance will be considered. By contrast, if type==3, the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance will be used.

a

real number > 0, by default a=1. It is the first parameter of a beta distribution which corresponds to a weighting measure on [0,1] in the mid/spr distance or in the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance.

b

real number > 0, by default b=1. It is the second parameter of a beta distribution which corresponds to a weighting measure on [0,1] in the mid/spr distance or in the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance.

theta

real number > 0, by default theta=1. It is the weight of the spread in the mid/spr distance and the weight of the ldev and rdev in the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance.

Details

See examples.

Returns

the scale measure MDD, which is a real number.If the body's method inner conditions are not met, NA will be returned.

Examples

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(0,0.1,0.2,0.3),
TrapezoidalFuzzyNumber$new(1,2,3,4),TrapezoidalFuzzyNumber$new(2,3,4,5)))$mdd(
FuzzyNumberList$new(c(FuzzyNumber$new(array(c(0.0, 0.5, 1.0,-1.5,-1.25,-1.0,
3.0, 2.0, 1.0), dim = c(3, 3))))),1L)

# Example 2: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(0,0.1,0.2,0.3), TrapezoidalFuzzyNumber$new(1,2,3,4),TrapezoidalFuzzyNumber$new(2,3,4,5)))$mdd( FuzzyNumberList$new(c(FuzzyNumber$new(array(c(0.0, 0.5, 1.0,-1.5,-1.25,-1.0, 3.0, 2.0, 1.0), dim = c(3, 3))))),2L,2,1,1)

# Example 3: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(0,0.1,0.2,0.3), TrapezoidalFuzzyNumber$new(1,2,3,4),TrapezoidalFuzzyNumber$new(2,3,4,5)))$mdd( TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(5,6,7,8))),3L,1,1,1)

# Example 4: F=Simulation$new()$simulCase3(10L) U=F$mean() F$mdd(U,3L,1,2,1)

# Example 5: F=Simulation$new()$simulCase2(10L) U=F$median1Norm() F$mdd(U,2L)

# Example 6: F=Simulation$new()$simulCase2(10L) U=Simulation$new()$simulCase2(1L) F$mdd(U,2L)

# Example 7: F=Simulation$new()$simulCase2(10L) U=F$transfTra() F$mdd(U,2L)

# Example 8: F=Simulation$new()$simulCase2(10L) U=Simulation$new()$simulCase2(2L) F$mdd(U,2L)

---

Method mean()

Given a sample of 'TrapezoidalFuzzyNumbers' contained in a 'TrapezoidalFuzzyNumberList', the method calculates the Aumann-type mean of these numbers (which is a 'TrapezoidalFuzzyNumber' too). See Sinova et al. (2015) [7].

Usage

TrapezoidalFuzzyNumberList$mean()

Details

See examples.

Returns

the Aumann-type mean, given as a TrapezoidalFuzzyNumber contained in a TrapezoidalFuzzyNumberList.

Examples

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$mean()

# Example 2: TrapezoidalFuzzyNumberList$new( c(TrapezoidalFuzzyNumber$new(1.5,2,3.75,4), TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$mean()

# Example 3: F=Simulation$new()$simulCase1(100L) F$mean()

---

Method median1Norm()

Given a sample of 'TrapezoidalFuzzyNumbers' contained in a 'TrapezoidalFuzzyNumberList', the method calculates the 1-norm median of these numbers, characterized by means of nl equidistant \(\alpha\)-levels (by default nl=101), including always the 0 and 1 levels, with their infimum and supremum values. See Sinova et al. (2012) [8].

Usage

TrapezoidalFuzzyNumberList$median1Norm(nl = 101L)

Arguments

nl

integer greater or equal to 2, by default nl=101. It indicates the number of desired \(\alpha\)-levels for characterizing the 1-norm median.

Details

See examples.

Returns

the 1-norm median, given in form of a FuzzyNumber contained in a FuzzyNumberList.

Examples

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$median1Norm()

# Example 2: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4), TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$median1Norm(200L)

# Example 3: F=Simulation$new()$simulCase1(10L) F$median1Norm(200L)

---

Method medianWabl()

Given a sample of 'TrapezoidalFuzzyNumbers' contained in a 'TrapezoidalFuzzyNumberList', the method calculates the \(\phi\)-wabl/ldev/rdev median of these numbers, characterized by means of nl equidistant \(\alpha\)-levels (by default nl=101), including always the 0 and 1 levels, with their infimum and supremum values. See Sinova et al. (2013) [6] and Sinova et al. (2016) [10].

Usage

TrapezoidalFuzzyNumberList$medianWabl(nl = 101L, a = 1, b = 1)

Arguments

nl

integer greater or equal to 2, by default nl=101. It indicates the number of desired \(\alpha\)-levels for characterizing the \(\phi\)-wabl/ldev/rdev median.

a

real number > 0, by default a=1. It is the first parameter of a beta distribution which corresponds to a weighting measure on [0,1].

b

real number > 0, by default b=1. It is the second parameter of a beta distribution which corresponds to a weighting measure on [0,1].

Details

See examples.

Returns

the \(\phi\)-wabl/ldev/rdev median in form of a FuzzyNUmber given in a FuzzyNumberList.

Examples

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$medianWabl()

# Example 2: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4), TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$medianWabl(3L)

# Example 3: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4), TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$medianWabl(3L,2.2,2.8)

# Example 4: F=Simulation$new()$simulCase1(10L) F$medianWabl(3L)

---

Method qn()

This method calculates scale measure Qn for a matrix of 'TrapezoidalFuzzyNumbers' contained in the current 'TrapezoidalFuzzyNumber'. The employed metric in the calculation can be the 1-norm distance, the mid/spr distance or the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance. See De la Rosa de Saa et al. (2021) [3].

Usage

TrapezoidalFuzzyNumberList$qn(type = NA, a = 1, b = 1, theta = 1)

Arguments

type

integer number that can be 1, 2 or 3: if type==1, the 1-norm distance will be considered in the calculation of the measure ADD. If type==2, the mid/spr distance will be considered. By contrast, if type==3, the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance will be used.

a

real number > 0, by default a=1. It is the first parameter of a beta distribution which corresponds to a weighting measure on [0,1] in the mid/spr distance or the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance.

b

real number > 0, by default b=1. It is the second parameter of a beta distribution which corresponds to a weighting measure on [0,1] in the mid/spr distance or the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance.

theta

real number > 0, by default theta=1. It is the weight of the spread in the mid/spr distance and the weight of the ldev and rdev in the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance.

Details

See examples.

Returns

the scale measure Qn, which is a real number.

Examples

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$qn(1L)

# Example 2: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4), TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$qn(2L,5,1,1)

# Example 3: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4), TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$qn(3L,1,1,1)

# Example 4: F=Simulation$new()$simulCase1(10L) F$qn(3L,1,1,1)

---

Method rho1()

This method calculates the 1-norm distance between the 'TrapezoidalFuzzyNumbers' contained in two 'TrapezoidalFuzzyNumberLists'.

Usage

TrapezoidalFuzzyNumberList$rho1(s = NA)

Arguments

s

TrapezoidalFuzzyNumberList containing valid TrapezoidalFuzzyNumbers characterized by their four values inf0, inf1, sup1, sup0.

Details

See examples.

Returns

a matrix containing the 1-norm distances between the two previous mentioned TrapezoidalFuzzyNumberLists.

Examples

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$rho1(TrapezoidalFuzzyNumberList$new(
c(TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58))))

# Example 2: F=Simulation$new()$simulCase1(4L) S=Simulation$new()$simulCase1(5L) F$rho1(S) S$rho1(F)

---

Method sn()

This method calculates scale measure Sn for a matrix of 'TrapezoidalFuzzyNumbers' contained in the current 'TrapezoidalFuzzyNumber'. The employed metric in the calculation can be the 1-norm distance, the mid/spr distance or the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance. See De la Rosa de Saa et al. (2021) [3].

Usage

TrapezoidalFuzzyNumberList$sn(type = NA, a = 1, b = 1, theta = 1)

Arguments

type

integer number that can be 1, 2 or 3: if type==1, the 1-norm distance will be considered in the calculation of the measure ADD. If type==2, the mid/spr distance will be considered. By contrast, if type==3, the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance will be used.

a

real number > 0, by default a=1. It is the first parameter of a beta distribution which corresponds to a weighting measure on [0,1] in the mid/spr distance or the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance.

b

real number > 0, by default b=1. It is the second parameter of a beta distribution which corresponds to a weighting measure on [0,1] in the mid/spr distance or the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance.

theta

real number > 0, by default theta=1. It is the weight of the spread in the mid/spr distance and the weight of the ldev and rdev in the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance.

Details

See examples.

Returns

the scale measure Sn, which is a real number.

Examples

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$sn(1L)

# Example 2: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4), TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$sn(2L,1,1,1)

# Example 3: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4), TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$sn(3L,5,1,0.5)

# Example 4: F=Simulation$new()$simulCase1(10L) F$sn(2L,5,1,0.5)

---

Method tn()

This method calculates scale measure Tn for a matrix of 'TrapezoidalFuzzyNumbers' contained in the current 'TrapezoidalFuzzyNumber'. The employed metric in the calculation can be the 1-norm distance, the mid/spr distance or the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance. See De la Rosa de Saa et al. (2021) [3].

Usage

TrapezoidalFuzzyNumberList$tn(type = NA, a = 1, b = 1, theta = 1)

Arguments

type

integer number that can be 1, 2 or 3: if type==1, the 1-norm distance will be considered in the calculation of the measure ADD. If type==2, the mid/spr distance will be considered. By contrast, if type==3, the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance will be used.

a

real number > 0, by default a=1. It is the first parameter of a beta distribution which corresponds to a weighting measure on [0,1] in the mid/spr distance or the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance.

b

real number > 0, by default b=1. It is the second parameter of a beta distribution which corresponds to a weighting measure on [0,1] in the mid/spr distance or the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance.

theta

real number > 0, by default theta=1. It is the weight of the spread in the mid/spr distance and the weight of the ldev and rdev in the (\(\phi\),\(\theta\))-wabl/ldev/rdev distance.

Details

See examples.

Returns

the scale measure Tn, which is a real number.

Examples

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$tn(1L)

# Example 2: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4), TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$tn(2L,1,1,1)

# Example 3: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4), TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$tn(3L,5,1,0.5)

# Example 4: F=Simulation$new()$simulCase1(10L) F$tn(1L)

---

Method transfTra()

This method transforms a 'TrapezoidalFuzzyNumberList' containing valid 'TrapezoidalFuzzyNumbers' characterized by their four values inf0, inf1, sup1, sup0 into a 'FuzzyNumberList' containing these same amount of fuzzy numbers, characterized by means of nl equidistant \(\alpha\)-levels each (by default nl=101).

Usage

TrapezoidalFuzzyNumberList$transfTra(nl = 101L)

Arguments

nl

integer greater or equal to 2, by default nl=101. It indicates the number of desired \(\alpha\)-levels for characterizing the trapezoidal fuzzy numbers.

Details

See examples.

Returns

a FuzzyNumberList containing the transformed TrapezoidalFuzzyNumbers into FuzzyNumbers.

Examples

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$transfTra()

# Example 2: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4), TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$transfTra(3L)

# Example 3: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4), TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$transfTra(10L)

# Example 4: F=Simulation$new()$simulCase3(10L) F$transfTra(200L)

---

Method var()

Given a sample of 'TrapezoidalFuzzyNumbers' contained in a 'TrapezoidalFuzzyNumberList', the method calculates the variance of these numbers with respect to the mid/spr distance. See De la Rosa de Saa et al. (2017) [2].

Usage

TrapezoidalFuzzyNumberList$var(a = 1, b = 1, theta = 1)

Arguments

a

real number > 0, by default a=1. It is the first parameter of a beta distribution which corresponds to a weighting measure on [0,1].

b

real number > 0, by default b=1. It is the second parameter of a beta distribution which corresponds to a weighting measure on [0,1].

theta

real number > 0, by default theta=1. It is the weight of the spread in the mid/spr distance.

Details

See examples.

Returns

the variance of the sample with respect to the mid/spr distance, which is a real number.

Examples

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$var()

# Example 2: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4), TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$var(1,1,1)

# Example 3: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4), TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$var(1/3,1/3,1/5)

# Example 4: F=Simulation$new()$simulCase1(10L) F$var(1,1,1)

---

Method wablphi()

Given a sample of 'TrapezoidalFuzzyNumbers' contained in a 'TrapezoidalFuzzyNumberList', the method calculates the \(\phi\)-wabl value for each of these numbers. See Sinova et al. (2014) [9].

Usage

TrapezoidalFuzzyNumberList$wablphi(a = 1, b = 1)

Arguments

a

real number > 0, by default a=1. It is the first parameter of a beta distribution which corresponds to a weighting measure on [0,1].

b

real number > 0, by default b=1. It is the second parameter of a beta distribution which corresponds to a weighting measure on [0,1].

Details

See examples.

Returns

a vector giving the \(\phi\)-wabl values of each TrapezoidalFuzzyNumber.

Examples

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$wablphi()

# Example 2: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4), TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$wablphi(2,1)

# Example 3: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4), TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$wablphi(2.2,1.1)

# Example 4: F=Simulation$new()$simulCase4(60L) F$wablphi(2,1)

---

Method addTrapezoidalFuzzyNumber()

This method adds a 'TrapezoidalFuzzyNumber' to the current collection inside the current 'TrapezoidalFuzzyNumberList'. Therefore, the dimensions' field is increased in a unit.

Usage

TrapezoidalFuzzyNumberList$addTrapezoidalFuzzyNumber(n = NA, verbose = TRUE)

Arguments

n

is the TrapezoidalFuzzyNumber to be added to the current collection inside the current TrapezoidalFuzzyNumberList.

verbose

if TRUE the messages are written to the console unless the user actively decides to set verbose=FALSE.

Details

See examples.

Returns

nothing.

Examples

TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4))
)$addTrapezoidalFuzzyNumber(TrapezoidalFuzzyNumber$new(3,4,5,6))

---

Method removeTrapezoidalFuzzyNumber()

This method removes a 'TrapezoidalFuzzyNumber' to the current collection inside the current 'TrapezoidalFuzzyNumberList'. Therefore, the dimensions' field is decreased in a unit.

Usage

TrapezoidalFuzzyNumberList$removeTrapezoidalFuzzyNumber(i = NA, verbose = TRUE)

Arguments

i

is the position of the TrapezoidalFuzzyNumber to be removed in the current collection inside the current TrapezoidalFuzzyNumberList.

verbose

if TRUE the messages are written to the console unless the user actively decides to set verbose=FALSE.

Details

See examples.

Returns

nothing.

Examples

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(2,3,4,4)))$removeTrapezoidalFuzzyNumber(1L)

# Example 2: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(2,3,4,4)))$removeTrapezoidalFuzzyNumber(2L)

---

Method getDimension()

This method gives the number contained in the dimension passed as parameter when the dimension is greater than 0 and not greater than the dimensions of the TrapezoidalFuzzyNumberList's numbers array.

Usage

TrapezoidalFuzzyNumberList$getDimension(i = NA)

Arguments

i

is the dimension of the TrapezoidalFuzzyNumber wanted to be retrieved.

Details

See examples.

Returns

The TrapezoidalFuzzyNumber contained in the dimension passed as parameter or an error if the dimension is not valid.

Examples

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(2,3,4,4)))$getDimension(1L)

# Example 2: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4), TrapezoidalFuzzyNumber$new(2,3,4,4)))$getDimension(2L)

---

Method plot()

This method shows in a graph the values of the attribute numbers of the corresponding 'TrapezoidalFuzzyNumberList'.

Usage

TrapezoidalFuzzyNumberList$plot(color = "grey")

Arguments

color

is the color of the lines representing the numbers to be shown in the graph. The default value is grey, other colors can be specified, the option palette() too.

Details

See examples.

Returns

a graph with the values of the attribute numbers of the corresponding 'TrapezoidalFuzzyNumberList'.

Examples

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(2,3,4,5)))$plot()

# Example 2: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(-4,-3,-2,-1), TrapezoidalFuzzyNumber$new(-6,0,1,4)))$plot()

# Example 3: Simulation$new()$simulCase1(8L)$plot(palette())

# Example 4: Simulation$new()$simulCase1(5L)$plot(palette()[2:6])

---

Method getLength()

This method returns the number of dimensions that are equivalent to the number of 'TrapezoidalFuzzyNumbers' in the corresponding 'TrapezoidalFuzzyNumberList'.

Usage

TrapezoidalFuzzyNumberList$getLength()

Details

See examples.

Returns

the number of dimensions that are equivalent to the number of 'TrapezoidalFuzzyNumbers' in the corresponding 'TrapezoidalFuzzyNumberList'.

Examples

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4))
)$getLength()

# Example 2: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(-4,-3,-2,-1), TrapezoidalFuzzyNumber$new(-6,0,1,4)))$getLength()

# Example 3: TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(-4,-3,-2,-1), TrapezoidalFuzzyNumber$new(-6,0,1,4),TrapezoidalFuzzyNumber$new(1,2,3,4)) )$getLength()

---

Method clone()

The objects of this class are cloneable with this method.

Usage

TrapezoidalFuzzyNumberList$clone(deep = FALSE)

Arguments

deep

Whether to make a deep clone.

References

[1] De la Rosa de Saa, S.; Gil, M.A.; Gonzalez-Rodriguez, G.; Lopez, M.T.; Lubiano M.A.: Fuzzy rating scale-based questionnaires and their statistical analysis, IEEE Transactions on Fuzzy Systems 23(1), 111-126 (2015)

[2] De la Rosa de Saa, S.; Lubiano M.A.; Sinova, B.; Filzmoser, P.: Robust scale estimators for fuzzy data, Advances in Data Analysis and Classification 11(4), 731-758 (2017)

[3] De la Rosa de Sáa, S.; Lubiano, M.A.; Sinova, B.; Filzmoser, P.; Gil, M.Á.: Location-free robust scale estimates for fuzzy data, IEEE Transactions on Fuzzy Systems 29(6), 1682-1694 (2021)

[4] Lubiano, M.A.; Gil, M.A.: f-Inequality indices for fuzzy random variables, in Statistical Modeling, Analysis and Management of Fuzzy Data (Bertoluzza, C., Gil, M.A., Ralescu, D.A., Eds.), Physica-Verlag, 43-63 (2002)

[5] Lubiano, M.A.; Montenegro, M.; Sinova, B.; De la Rosa de Saa, S.; Gil, M.A.: Hypothesis testing for means in connection with fuzzy rating scale-based data: algorithms and applications, European Journal of Operational Research 251, 918-929 (2016)

[6] Sinova, B.; De la Rosa de Saa, S.; Gil, M.A.: A generalized L1-type metric between fuzzy numbers for an approach to central tendency of fuzzy data, Information Sciences 242, 22-34 (2013)

[7] Sinova, B.; De la Rosa de Saa, S.; Lubiano, M.A.; Gil, M.A.: An overview on the statistical central tendency for fuzzy datasets, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 23 (Suppl. 1), 105-132 (2015)

[8] Sinova, B.; Gil, M.A.; Colubi, A.; Van Aelst, S.: The median of a random fuzzy number. The 1-norm distance approach, Fuzzy Sets and Systems 200, 99-115 (2012)

[9] Sinova, B.; Gil, M.A.; Lopez, M.T.; Van Aelst, S.: A parameterized L2 metric between fuzzy numbers and its parameter interpretation, Fuzzy Sets and Systems 245, 101-115 (2014)

[10] Sinova, B.; Gil, M.A.; Van Aelst, S.: M-estimates of location for the robust central tendency of fuzzy data, IEEE Transactions on Fuzzy Systems 24(4), 945-956 (2016)

Examples

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$new`
## ------------------------------------------------

TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1, 2, 3, 4),
TrapezoidalFuzzyNumber$new(-8, -6, -4, -2),
TrapezoidalFuzzyNumber$new(-1, -1, 2, 3),
TrapezoidalFuzzyNumber$new(1, 2, 3, 3)))

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$add`
## ------------------------------------------------

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(0,0.1,0.2,0.3),
TrapezoidalFuzzyNumber$new(1,2,3,4),TrapezoidalFuzzyNumber$new(2,3,4,5)))$add(
FuzzyNumberList$new(c(FuzzyNumber$new(array(c(0.0, 0.5, 1.0,-1.5,-1.25,-1.0,
3.0, 2.0, 1.0), dim = c(3, 3))))),1L)

# Example 2:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(0,0.1,0.2,0.3),
TrapezoidalFuzzyNumber$new(1,2,3,4),TrapezoidalFuzzyNumber$new(2,3,4,5)))$add(
FuzzyNumberList$new(c(FuzzyNumber$new(array(c(0.0, 0.5, 1.0,-1.5,-1.25,-1.0,
3.0, 2.0, 1.0), dim = c(3, 3))))),2L,2,1,1)

# Example 3:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(0,0.1,0.2,0.3),
TrapezoidalFuzzyNumber$new(1,2,3,4),TrapezoidalFuzzyNumber$new(2,3,4,5)))$add(
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(5,6,7,8))),3L,1,1,1)

# Example 4:
F=Simulation$new()$simulCase1(10L)
S=F$mean()
F$add(S,1L)

# Example 5:
F=Simulation$new()$simulCase1(100L)
S=F$median1Norm()
F$add(S,2L,2,1,1)

# Example 6:
F=Simulation$new()$simulCase2(10L)
U=Simulation$new()$simulCase2(1L)
F$add(U,2L)

# Example 7:
F=Simulation$new()$simulCase2(10L)
U=F$transfTra()
F$add(U,2L)

# Example 8:
F=Simulation$new()$simulCase2(10L)
U=Simulation$new()$simulCase2(2L)
F$add(U,2L)

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$dthetaphi`
## ------------------------------------------------

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$dthetaphi(
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58))))

# Example 2:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$dthetaphi(
TrapezoidalFuzzyNumberList$new( c(TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58))),1,1,1)

# Example 3:
F=Simulation$new()$simulCase1(6L)
S=Simulation$new()$simulCase1(8L)
F$dthetaphi(S,1,5,1)

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$dwablphi`
## ------------------------------------------------

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$dwablphi(
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58))))

# Example 2:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$dwablphi(
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58))),5,1,1)

# Example 3:
F=Simulation$new()$simulCase1(10L)
S=Simulation$new()$simulCase1(20L)
F$dwablphi(S)

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$gsi`
## ------------------------------------------------

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$gsi()

# Example 2:
F=Simulation$new()$simulCase1(50L)
F$gsi()

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$hyperI`
## ------------------------------------------------

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$hyperI()

# Example 2:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$hyperI(0.5)

# Example 3:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(2,4,6,8)))$hyperI()

# Example 4:
F=Simulation$new()$simulFRSTra(100L,0.05,0.35,0.6,2,1)
F$hyperI()

# Example 5:
F=Simulation$new()$simulCase2(10L)
F$hyperI(0.5)

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$mEstimator`
## ------------------------------------------------

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$mEstimator("Huber",0.321,0.5,10^(-5),
1L)

# Example 2:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$mEstimator("Tukey",0.123,0.5,10^(-5),
2L,1,1,1)

# Example 3:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$mEstimator("Cauchy",0.123,0.5,10^(-5),
3L,0.75,0.5,1)

# Example 4:
F=Simulation$new()$simulCase1(100L)
U=F$median1Norm()
estInitial=F$mdd(U,1L)
delta=0.5
epsilon=10^(-5)
F$mEstimator("Huber",estInitial,delta,epsilon,1L)

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$mdd`
## ------------------------------------------------

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(0,0.1,0.2,0.3),
TrapezoidalFuzzyNumber$new(1,2,3,4),TrapezoidalFuzzyNumber$new(2,3,4,5)))$mdd(
FuzzyNumberList$new(c(FuzzyNumber$new(array(c(0.0, 0.5, 1.0,-1.5,-1.25,-1.0,
3.0, 2.0, 1.0), dim = c(3, 3))))),1L)

# Example 2:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(0,0.1,0.2,0.3),
TrapezoidalFuzzyNumber$new(1,2,3,4),TrapezoidalFuzzyNumber$new(2,3,4,5)))$mdd(
FuzzyNumberList$new(c(FuzzyNumber$new(array(c(0.0, 0.5, 1.0,-1.5,-1.25,-1.0,
3.0, 2.0, 1.0), dim = c(3, 3))))),2L,2,1,1)

# Example 3:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(0,0.1,0.2,0.3),
TrapezoidalFuzzyNumber$new(1,2,3,4),TrapezoidalFuzzyNumber$new(2,3,4,5)))$mdd(
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(5,6,7,8))),3L,1,1,1)

# Example 4:
F=Simulation$new()$simulCase3(10L)
U=F$mean()
F$mdd(U,3L,1,2,1)

# Example 5:
F=Simulation$new()$simulCase2(10L)
U=F$median1Norm()
F$mdd(U,2L)

# Example 6:
F=Simulation$new()$simulCase2(10L)
U=Simulation$new()$simulCase2(1L)
F$mdd(U,2L)

# Example 7:
F=Simulation$new()$simulCase2(10L)
U=F$transfTra()
F$mdd(U,2L)

# Example 8:
F=Simulation$new()$simulCase2(10L)
U=Simulation$new()$simulCase2(2L)
F$mdd(U,2L)

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$mean`
## ------------------------------------------------

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$mean()

# Example 2:
TrapezoidalFuzzyNumberList$new(
c(TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$mean()

# Example 3:
F=Simulation$new()$simulCase1(100L)
F$mean()

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$median1Norm`
## ------------------------------------------------

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$median1Norm()

# Example 2:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$median1Norm(200L)

# Example 3:
F=Simulation$new()$simulCase1(10L)
F$median1Norm(200L)

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$medianWabl`
## ------------------------------------------------

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$medianWabl()

# Example 2:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$medianWabl(3L)

# Example 3:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$medianWabl(3L,2.2,2.8)

# Example 4:
F=Simulation$new()$simulCase1(10L)
F$medianWabl(3L)

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$qn`
## ------------------------------------------------

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$qn(1L)

# Example 2:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$qn(2L,5,1,1)

# Example 3:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$qn(3L,1,1,1)

# Example 4:
F=Simulation$new()$simulCase1(10L)
F$qn(3L,1,1,1)

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$rho1`
## ------------------------------------------------

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1)))$rho1(TrapezoidalFuzzyNumberList$new(
c(TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58))))

# Example 2:
F=Simulation$new()$simulCase1(4L)
S=Simulation$new()$simulCase1(5L)
F$rho1(S)
S$rho1(F)

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$sn`
## ------------------------------------------------

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$sn(1L)

# Example 2:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$sn(2L,1,1,1)

# Example 3:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$sn(3L,5,1,0.5)

# Example 4:
F=Simulation$new()$simulCase1(10L)
F$sn(2L,5,1,0.5)

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$tn`
## ------------------------------------------------

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$tn(1L)

# Example 2:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$tn(2L,1,1,1)

# Example 3:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$tn(3L,5,1,0.5)

# Example 4:
F=Simulation$new()$simulCase1(10L)
F$tn(1L)

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$transfTra`
## ------------------------------------------------

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$transfTra()

# Example 2:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$transfTra(3L)

# Example 3:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$transfTra(10L)

# Example 4:
F=Simulation$new()$simulCase3(10L)
F$transfTra(200L)

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$var`
## ------------------------------------------------

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$var()

# Example 2:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$var(1,1,1)

# Example 3:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$var(1/3,1/3,1/5)

# Example 4:
F=Simulation$new()$simulCase1(10L)
F$var(1,1,1)

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$wablphi`
## ------------------------------------------------

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$wablphi()

# Example 2:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$wablphi(2,1)

# Example 3:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),TrapezoidalFuzzyNumber$new(1.5,2,3.75,4),
TrapezoidalFuzzyNumber$new(-4.2,-3.6,-2,-1.58)))$wablphi(2.2,1.1)

# Example 4:
F=Simulation$new()$simulCase4(60L)
F$wablphi(2,1)

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$addTrapezoidalFuzzyNumber`
## ------------------------------------------------

TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4))
)$addTrapezoidalFuzzyNumber(TrapezoidalFuzzyNumber$new(3,4,5,6))

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$removeTrapezoidalFuzzyNumber`
## ------------------------------------------------

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(2,3,4,4)))$removeTrapezoidalFuzzyNumber(1L)

# Example 2:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(2,3,4,4)))$removeTrapezoidalFuzzyNumber(2L)

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$getDimension`
## ------------------------------------------------

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(2,3,4,4)))$getDimension(1L)

# Example 2:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(2,3,4,4)))$getDimension(2L)

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$plot`
## ------------------------------------------------

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4),
TrapezoidalFuzzyNumber$new(2,3,4,5)))$plot()

# Example 2:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),
TrapezoidalFuzzyNumber$new(-6,0,1,4)))$plot()

# Example 3:
Simulation$new()$simulCase1(8L)$plot(palette())

# Example 4:
Simulation$new()$simulCase1(5L)$plot(palette()[2:6])

## ------------------------------------------------
## Method `TrapezoidalFuzzyNumberList$getLength`
## ------------------------------------------------

# Example 1:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(1,2,3,4))
)$getLength()

# Example 2:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),
TrapezoidalFuzzyNumber$new(-6,0,1,4)))$getLength()

# Example 3:
TrapezoidalFuzzyNumberList$new(c(TrapezoidalFuzzyNumber$new(-4,-3,-2,-1),
TrapezoidalFuzzyNumber$new(-6,0,1,4),TrapezoidalFuzzyNumber$new(1,2,3,4))
)$getLength()