Help for package FuzzySTs

Title:

Fuzzy Statistical Tools

Description:

The main goal of this package is to present various fuzzy statistical tools. It intends to provide an implementation of the theoretical and empirical approaches presented in the book entitled "The signed distance measure in fuzzy statistical analysis. Some theoretical, empirical and programming advances" <doi:10.1007/978-3-030-76916-1>. For the theoretical approaches, see Berkachy R. and Donze L. (2019) <doi:10.1007/978-3-030-03368-2_1>. For the empirical approaches, see Berkachy R. and Donze L. (2016) <ISBN: 978-989-758-201-1>). Important (non-exhaustive) implementation highlights of this package are as follows: (1) a numerical procedure to estimate the fuzzy difference and the fuzzy square. (2) two numerical methods of fuzzification. (3) a function performing different possibilities of distances, including the signed distance and the generalized signed distance for instance with all its properties. (4) numerical estimations of fuzzy statistical measures such as the variance, the moment, etc. (5) two methods of estimation of the bootstrap distribution of the likelihood ratio in the fuzzy context. (6) an estimation of a fuzzy confidence interval by the likelihood ratio method. (7) testing fuzzy hypotheses and/or fuzzy data by fuzzy confidence intervals in the Kwakernaak - Kruse and Meyer sense. (8) a general method to estimate the fuzzy p-value with fuzzy hypotheses and/or fuzzy data. (9) a method of estimation of global and individual evaluations of linguistic questionnaires. (10) numerical estimations of multi-ways analysis of variance models in the fuzzy context. The unbalance in the considered designs are also foreseen.

Version:

0.3

Author:

Redina Berkachy <redina.berkachy@unifr.ch>, Laurent Donze <laurent.donze@unifr.ch>

Maintainer:

Redina Berkachy <redina.berkachy@hefr.ch>

Depends:

R (≥ 3.0.0), FuzzyNumbers, polynom

License:

MIT + file LICENSE

RoxygenNote:

7.3.1

Encoding:

UTF-8

Suggests:

knitr, rmarkdown

VignetteBuilder:

knitr

NeedsCompilation:

Packaged:

2024-07-05 13:18:28 UTC; berkachr

Repository:

CRAN

Date/Publication:

2024-07-07 10:40:17 UTC

Calculates a distance by the d_Bertoluzza between fuzzy numbers

Description

Calculates a distance by the d_Bertoluzza between fuzzy numbers

Usage

Bertoluzza(X, Y, i = 1, j = 1, theta = 1/3, breakpoints = 100)

Arguments

X

a fuzzy number.

Y

a fuzzy number.

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

a numerical value between 0 and 1, representing a weighting parameter. By default, theta is fixed to 1/3 referring to the Lebesgue space. This measure is used in the calculations of the following distances: d_Bertoluzza, d_mid/spr and d_phi-wabl/ldev/rdev.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

A numerical value.

Calculates a distance by the D2 between fuzzy numbers

Description

Calculates a distance by the D2 between fuzzy numbers

Usage

D2(X, Y, breakpoints = 100)

Arguments

X

a fuzzy number.

Y

a fuzzy number.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

A numerical value.

Calculates a distance by the SGD between fuzzy numbers

Description

Calculates a distance by the SGD between fuzzy numbers

Usage

DSGD(X, Y, i = 1, j = 1, breakpoints = 100, theta = 1/3)

Arguments

X

a fuzzy number.

Y

a fuzzy number.

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

theta

Value

A numerical value.

Calculates a distance by the d_DSGD.G between fuzzy numbers

Description

Calculates a distance by the d_DSGD.G between fuzzy numbers

Usage

DSGD.G(X, Y, i = 1, j = 1, thetas = 1, breakpoints = 100)

Arguments

X

a fuzzy number.

Y

a fuzzy number.

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

thetas

a decimal value between 0 and 1, representing the weight given to the shape of the fuzzy number. By default, thetas is fixed to 1. This parameter is used in the calculations of the d_theta star and the d_GSGD distances.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

A numerical value.

Defuzzify the fuzzy sums of squares calculated by a FANOVA model by an exact calculation or an approximation

Description

Defuzzify the fuzzy sums of squares calculated by a FANOVA model by an exact calculation or an approximation

Usage

Defuzz.FANOVA(
  res,
  distance.type = "DSGD",
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100
)

Arguments

res

a result of a call of the function FANOVA, where method = "distance".

distance.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, p is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

Returns a list of all the arguments of the function, the defuzzified total, treatment and residuals sums of squares, the decision made etc.

Calculates a distance by the d_Delta.pq between fuzzy numbers

Description

Calculates a distance by the d_Delta.pq between fuzzy numbers

Usage

Delta.pq(X, Y, p, q, breakpoints = 100)

Arguments

X

a fuzzy number.

Y

a fuzzy number.

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

A numerical value.

Calculates the factor Delta_jki

Description

Calculates the factor Delta_jki

Usage

Delta_jki(x, i, K)

Arguments

x

a dataset.

i

an observation index.

K

the total number of linguistics in a sub-item.

Value

The response matrix of binary values (0 or 1) related to the answers of a particular dataset for its corresponding sub-items.

Computes a FANOVA model by a convenient metric, an exact calculation or an approximation

Description

Computes a FANOVA model by a convenient metric, an exact calculation or an approximation

Usage

FANOVA(
  formula,
  dataset,
  data.fuzzified,
  sig,
  method,
  distance.type = "DSGD",
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100,
  int.method = "int.simpson",
  plot = TRUE
)

Arguments

formula

a description of the model to be fitted.

dataset

the data frame containing all the variables of the model.

data.fuzzified

the fuzzified data set constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix.

sig

a numerical value representing the significance level of the test.

method

the choices are the following: "distance", "exact", "approximation".

distance.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, p is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

int.method

the method of numerical integration. It is set by default to the Simpson method, i.e. int.method="int.simpson".

plot

fixed by default to "TRUE". plot="FALSE" if a plot of the fuzzy number is not required.

Value

Returns a list of all the arguments of the function, the total, treatment and residuals sums of squares, the coefficients of the model, the test statistics with the corresponding p-values, and the decision made.

Examples

mat <- matrix(c(1,1,1,1,1,1,1,2,2,2,2,3,2,3,4,2,3,3,2,4), ncol = 2)
data <- data.frame(mat)
data$X1 <- factor(data$X1)
MF121 <- TrapezoidalFuzzyNumber(0,1,1,2.2)
MF122 <- TrapezoidalFuzzyNumber(1.8,1.9,2.2,2.8)
MF123 <- TrapezoidalFuzzyNumber(1.9,2.3,3.1,3.3)
MF124 <- TrapezoidalFuzzyNumber(3.1,3.4,4.1,4.2)
PA12 <- c(1,2,3,4)
data.fuzzified <- GFUZZ(data, 1, 2, PA12, "Identical")
formula = X2 ~ X1
res <- FANOVA(formula, dataset = data, method ="distance", data.fuzzified = data.fuzzified, 
sig = 0.05, distance.type = "wabl")

Computes a FANOVA model by an approximation

Description

Computes a FANOVA model by an approximation

Usage

FANOVA.approximation(
  formula,
  dataset,
  data.fuzzified,
  sig,
  breakpoints = 100,
  int.method = "int.simpson",
  plot = TRUE
)

Arguments

formula

a description of the model to be fitted.

dataset

the data frame containing all the variables of the model.

data.fuzzified

the fuzzified data set constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix.

sig

a numerical value representing the significance level of the test.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

int.method

the method of numerical integration. It is set by default to the Simpson method, i.e. int.method="int.simpson".

plot

fixed by default to "TRUE". plot="FALSE" if a plot of the fuzzy number is not required.

Value

Computes a FANOVA model by a convenient metric

Description

Computes a FANOVA model by a convenient metric

Usage

FANOVA.distance(
  formula,
  dataset,
  data.fuzzified,
  sig,
  distance.type,
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100
)

Arguments

formula

a description of the model to be fitted.

dataset

the data frame containing all the variables of the model.

data.fuzzified

the fuzzified data set constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix.

sig

a numerical value representing the significance level of the test.

distance.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, p is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

Computes a FANOVA model by an exact calculation

Description

Computes a FANOVA model by an exact calculation

Usage

FANOVA.exact(
  formula,
  dataset,
  data.fuzzified,
  sig,
  breakpoints = 100,
  int.method = "int.simpson",
  plot = TRUE
)

Arguments

formula

a description of the model to be fitted.

dataset

the data frame containing all the variables of the model.

data.fuzzified

the fuzzified data set constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix.

sig

a numerical value representing the significance level of the test.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

int.method

the method of numerical integration. It is set by default to the Simpson method, i.e. int.method="int.simpson".

plot

fixed by default to "TRUE". plot="FALSE" if a plot of the fuzzy number is not required.

Value

Prints the summary of the estimation of a FANOVA metric-based model

Description

Prints the summary of the estimation of a FANOVA metric-based model

Usage

FANOVA.summary(res)

Arguments

res

a result of a call of the function FANOVA, where method = "distance".

Value

Returns a list of summary statistics of the estimated model given in res, shown in a FANOVA table. In addition, the F-statistics with their p-values, and the decision are given.

Computes a Mult-FANOVA model by a convenient metric, an exact calculation or an approximation

Description

Computes a Mult-FANOVA model by a convenient metric, an exact calculation or an approximation

Usage

FMANOVA(
  formula,
  dataset,
  data.fuzzified,
  sig = 0.05,
  method,
  distance.type = "DSGD",
  index.var = NA,
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100,
  int.method = "int.simpson",
  plot = TRUE
)

Arguments

formula

a description of the model to be fitted.

dataset

the data frame containing all the variables of the model.

data.fuzzified

the fuzzified data set constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix.

sig

a numerical value representing the significance level of the test.

method

the choices are the following: "distance", "exact", "approximation".

distance.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

index.var

the column index of the considered variable for which the output will be printed. It is an argument of the Mult-FANOVA models by the exact and the approximation methods only.

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, p is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

int.method

the method of numerical integration. It is set by default to the Simpson method, i.e. int.method="int.simpson".

plot

fixed by default to "TRUE". plot="FALSE" if a plot of the fuzzy number is not required.

Value

Examples

mat <- matrix(c(2,2,1,1,2,2,2,2,2,2,2,2,2,2,2,1,1,1,1,2,3,4,4,3,1,2,5,4,4,3),ncol=3)
data <- data.frame(mat)
MF131 <- TrapezoidalFuzzyNumber(0,1,1,2)
MF132 <- TrapezoidalFuzzyNumber(1,2,2,3)
MF133 <- TrapezoidalFuzzyNumber(2,3,3,4)
MF134 <- TrapezoidalFuzzyNumber(3,4,4,5)
MF135 <- TrapezoidalFuzzyNumber(4,5,5,6)
PA13 <- c(1,2,3,4,5); mi <- 1; si <- 3
Yfuzz <- FUZZ(data,1,3,PA13)
attach(data)
formula <- X3 ~ X1 + X2
res <- FMANOVA(formula, data, Yfuzz, method = "distance", distance.type = "wabl")
detach(data)

Computes a Mult-FANOVA model by an approximation

Description

Computes a Mult-FANOVA model by an approximation

Usage

FMANOVA.approximation(
  formula,
  dataset,
  data.fuzzified,
  sig = 0.05,
  breakpoints = 100,
  index.var = NA,
  int.method = "int.simpson",
  plot = TRUE
)

Arguments

formula

a description of the model to be fitted.

dataset

the data frame containing all the variables of the model.

data.fuzzified

the fuzzified data set constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix.

sig

a numerical value representing the significance level of the test.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

index.var

the column index of the considered variable for which the output will be printed. It is an argument of the Mult-FANOVA models by the exact and the approximation methods only.

int.method

the method of numerical integration. It is set by default to the Simpson method, i.e. int.method="int.simpson".

plot

fixed by default to "TRUE". plot="FALSE" if a plot of the fuzzy number is not required.

Value

Computes a Mult-FANOVA model by a convenient metric

Description

Computes a Mult-FANOVA model by a convenient metric

Usage

FMANOVA.distance(
  formula,
  dataset,
  data.fuzzified,
  distance.type,
  sig = 0.05,
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100
)

Arguments

formula

a description of the model to be fitted.

dataset

the data frame containing all the variables of the model.

data.fuzzified

the fuzzified data set constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix.

distance.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

sig

a numerical value representing the significance level of the test.

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, p is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

Computes a Mult-FANOVA model by an exact calculation

Description

Computes a Mult-FANOVA model by an exact calculation

Usage

FMANOVA.exact(
  formula,
  dataset,
  data.fuzzified,
  sig = 0.05,
  breakpoints = 100,
  int.method = "int.simpson",
  index.var = NA,
  plot = TRUE
)

Arguments

formula

a description of the model to be fitted.

dataset

the data frame containing all the variables of the model.

data.fuzzified

the fuzzified data set constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix.

sig

a numerical value representing the significance level of the test.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

int.method

the method of numerical integration. It is set by default to the Simpson method, i.e. int.method="int.simpson".

index.var

the column index of the considered variable for which the output will be printed. It is an argument of the Mult-FANOVA models by the exact and the approximation methods only.

plot

fixed by default to "TRUE". plot="FALSE" if a plot of the fuzzy number is not required.

Value

Prints the summary of the estimation of the interaction in a Mult-FANOVA metric-based model

Description

Prints the summary of the estimation of the interaction in a Mult-FANOVA metric-based model

Usage

FMANOVA.interaction.summary(res)

Arguments

res

a result of a call of the function FMANOVA, where method = "distance".

Value

Returns a list of summary statistics of the estimated model given in res, shown in a FANOVA table. In addition, the F-statistics with their p-values, and the decision are given.

Prints the summary of the estimation of a Mult-FANOVA metric-based model

Description

Prints the summary of the estimation of a Mult-FANOVA metric-based model

Usage

FMANOVA.summary(res)

Arguments

res

a result of a call of the function FMANOVA, where method = "distance".

Value

Returns a list of summary statistics of the estimated model given in res, shown in a FANOVA table. In addition, the F-statistics with their p-values, and the decision are given.

Calculates the Tukey HSD test corresponding to the fuzzy response variable

Description

Calculates the Tukey HSD test corresponding to the fuzzy response variable

Usage

FTukeyHSD(test, variable, cont = c(1, -1), conf.level = 0.95)

Arguments

test

a result of a call of the function FMANOVA.

variable

the name of a variable in the data set.

cont

the contrasts of the model. It is set by default to c(1,-1).

conf.level

the confidence level of the test. It is set by default to 0.95.

Value

Returns a table of comparisons of means of the different levels of a given factor, two by two. The table contains the means of populations, the lower and upper bounds of the confidence intervals, and their p-values.

Examples

mat <- matrix(c(2,2,1,1,2,2,2,2,2,2,2,2,2,2,2,1,1,1,1,2,3,4,4,3,1,2,5,4,4,3),ncol=3)
data <- data.frame(mat)
MF131 <- TrapezoidalFuzzyNumber(0,1,1,2)
MF132 <- TrapezoidalFuzzyNumber(1,2,2,3)
MF133 <- TrapezoidalFuzzyNumber(2,3,3,4)
MF134 <- TrapezoidalFuzzyNumber(3,4,4,5)
MF135 <- TrapezoidalFuzzyNumber(4,5,5,6)
PA13 <- c(1,2,3,4,5); mi <- 1; si <- 3
Yfuzz <- FUZZ(data,1,3,PA13)
attach(data)
formula <- X3 ~ X1 + X2
res <- FMANOVA(formula, data, Yfuzz, method = "distance", distance.type = "wabl")
FTukeyHSD(res, "X1")[[1]]
detach(data)

Fuzzifies a variable modelled by trapezoidal or triangular fuzzy numbers

Description

Fuzzifies a variable modelled by trapezoidal or triangular fuzzy numbers

Usage

FUZZ(data, mi, si, PA)

Arguments

data

a data set.

mi

the index of the main-item containing the concerned variable.

si

the index of the sub-item of a given main-item mi.

PA

a vector of the linguistic terms of the considered variable.

Value

A fuzzification matrix composed by 4 columns c(p,q,r,s), and m lines, i.e. number of observations. No NA is allowed.

Examples

data <- matrix(c(1,2,3,2,2,1,1,3,1,2),ncol=1)
MF111 <- TrapezoidalFuzzyNumber(0,1,1,2)
MF112 <- TrapezoidalFuzzyNumber(1,2,2,3)
MF113 <- TrapezoidalFuzzyNumber(2,3,3,3)
PA11 <- c(1,2,3)
data.fuzzified <- FUZZ(data,mi=1,si=1,PA=PA11)
is.trfuzzification(data.fuzzified)

Calculates multiple tests corresponding to the fuzzy response variable

Description

Calculates multiple tests corresponding to the fuzzy response variable

Usage

Ftests(test)

Arguments

test

a result of a call of the function FMANOVA.

Value

Returns a table of the following different indicators "Wilks","F-Wilks", "Hotelling-Lawley trace" and "Pillai Trace".

Examples

mat <- matrix(c(2,2,1,1,2,2,2,2,2,2,2,2,2,2,2,1,1,1,1,2,3,4,4,3,1,2,5,4,4,3),ncol=3)
data <- data.frame(mat)
MF131 <- TrapezoidalFuzzyNumber(0,1,1,2)
MF132 <- TrapezoidalFuzzyNumber(1,2,2,3)
MF133 <- TrapezoidalFuzzyNumber(2,3,3,4)
MF134 <- TrapezoidalFuzzyNumber(3,4,4,5)
MF135 <- TrapezoidalFuzzyNumber(4,5,5,6)
PA13 <- c(1,2,3,4,5); mi <- 1; si <- 3
Yfuzz <- FUZZ(data,1,3,PA13)
attach(data)
formula <- X3 ~ X1 + X2
res <- FMANOVA(formula, data, Yfuzz, method = "distance", distance.type = "wabl")
Ftests(res)
detach(data)

Computes a fuzzy inference test by the fuzzy confidence intervals method calculated by the Likelihood method

Description

Computes a fuzzy inference test by the fuzzy confidence intervals method calculated by the Likelihood method

Usage

Fuzzy.CI.ML.test(
  data.fuzzified,
  H0,
  H1,
  t,
  mu = NA,
  sigma = NA,
  sig,
  distribution,
  coef.boot,
  distance.type = "DSGD",
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100,
  step = 0.05,
  margin = c(5, 5),
  plot = TRUE
)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

H0

a trapezoidal or a triangular fuzzy number representing the fuzzy null hypothesis.

H1

a trapezoidal or a triangular fuzzy number representing the fuzzy alternative hypothesis.

t

a given numerical or fuzzy type parameter of the distribution.

mu

if the mean of the normal distribution is known, mu should be a numerical value. Otherwise, the argument mu is fixed to NA.

sigma

if the standard deviation of the normal distribution is known, sigma should be a numerical value. Otherwise, the argument sigma is fixed to NA.

sig

a numerical value representing the significance level of the test.

distribution

a distribution chosen between "normal", "poisson", "Student" or "Logistic".

coef.boot

a decimal representing the 1-sig-quantile of the bootstrap distribution of LR.

distance.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, p is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

step

a numerical value fixed to 0.05, defining the step of iterations on the interval [t-5; t+5].

margin

an optional numerical couple of values fixed to [5; 5], representing the range of calculations around the parameter t.

plot

fixed by default to "FALSE". plot="FALSE" if a plot of the fuzzy number is not required.

Value

Returns a list composed by the arguments, the fuzzy confidence intervals, the fuzzy decisions, the defuzzified values and the decision made.

Examples

data <- matrix(c(1,2,3,2,2,1,1,3,1,2),ncol=1)
MF111 <- TrapezoidalFuzzyNumber(0,1,1,2)
MF112 <- TrapezoidalFuzzyNumber(1,2,2,3)
MF113 <- TrapezoidalFuzzyNumber(2,3,3,4)
PA11 <- c(1,2,3)
data.fuzzified <- FUZZ(data,mi=1,si=1,PA=PA11)
Fmean <- Fuzzy.sample.mean(data.fuzzified)
H0 <- TriangularFuzzyNumber(2.2,2.5,3)
H1 <- TriangularFuzzyNumber(2.5,2.5,5)
coef.boot <- 3.494829
(res <- Fuzzy.CI.ML.test(data.fuzzified, H0, H1, t = Fmean, sigma=0.7888,
coef.boot = coef.boot, sig=0.05, distribution="normal", distance.type="GSGD"))
res$decision

Computes a fuzzy inference test by the traditional fuzzy confidence intervals

Description

Computes a fuzzy inference test by the traditional fuzzy confidence intervals

Usage

Fuzzy.CI.test(
  type,
  H0,
  H1,
  t,
  s.d,
  n,
  sig,
  distribution,
  distance.type = "DSGD",
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100,
  plot = TRUE
)

Arguments

type

a category betwenn "0", "1" and "2". The category "0" refers to a bilateral test, the category "1" for a lower unilateral one, and "2" for an upper unilateral test.

H0

a trapezoidal or a triangular fuzzy number representing the fuzzy null hypothesis.

H1

a trapezoidal or a triangular fuzzy number representing the fuzzy alternative hypothesis.

t

a given numerical or fuzzy type parameter of the distribution.

s.d

a numerical value for the standard deviation of the distribution.

n

the total number of observations of the data set.

sig

a numerical value representing the significance level of the test.

distribution

a distribution chosen between "normal", "poisson", "Student" or "Logistic".

distance.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, p is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

plot

a logical rule "TRUE" or "FALSE" for defining whether to plot the corresponding graphs or not.

Value

Returns a list composed by the arguments, the fuzzy confidence intervals, the fuzzy decisions, the defuzzified values and the decision made.

Examples

H0 <- TriangularFuzzyNumber(2.9,3,3.1)
H1 <- TriangularFuzzyNumber(3,3,5)
res <- Fuzzy.CI.test(type = 0, H0, H1, t = TriangularFuzzyNumber(0.8,1.80,2.80), s.d = 0.79, 
n = 10, sig = 0.05, distribution = "normal", distance.type="GSGD")

Calculates the difference between two fuzzy numbers

Description

Calculates the difference between two fuzzy numbers

Usage

Fuzzy.Difference(X, Y, alphacuts = FALSE, breakpoints = 100)

Arguments

X

a fuzzy number of any type.

Y

a fuzzy number of any type.

alphacuts

fixed by default to "FALSE". No alpha-cuts are printed in this case.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

If the parameter alphacuts="TRUE", the function returns a matrix composed by 2 vectors representing the left and right alpha-cuts. For this output, is.alphacuts = TRUE. If the parameter alphacuts="FALSE", the function returns a trapezoidal fuzzy number given by the quadruple (p,q,r,s), such that p \le q \le r \le s.

Examples

X <- TrapezoidalFuzzyNumber(5,6,7,8)
Y <- TrapezoidalFuzzyNumber(1,2,3,4)
Fuzzy.Difference(X,Y)

Calculates numerically the square of a fuzzy number

Description

Calculates numerically the square of a fuzzy number

Usage

Fuzzy.Square(F1L, breakpoints = 100, plot = FALSE)

Arguments

F1L

a fuzzy number.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

plot

fixed by default to "FALSE". plot="TRUE" if a plot of the fuzzy number is required.

Value

A matrix composed by 2 vectors representing the numerical left and right alpha-cuts. For this output, is.alphacuts = TRUE.

Examples

 X <- TrapezoidalFuzzyNumber(1,2,3,4)
Fuzzy.Square(X, plot=TRUE)

Gives the polynomial expression of the left alpha-levels of the numerical square of a fuzzy number

Description

Gives the polynomial expression of the left alpha-levels of the numerical square of a fuzzy number

Usage

Fuzzy.Square.poly.left(F1L, breakpoints = 100)

Arguments

F1L

a fuzzy number.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

A table containing print the related polynoms at the corresponding definition domains.

Examples

X <- TrapezoidalFuzzyNumber(1,2,3,4)
Fuzzy.Square.poly.left(X)

Gives the polynomial expression of the right alpha-levels of the numerical square of a fuzzy number

Description

Gives the polynomial expression of the right alpha-levels of the numerical square of a fuzzy number

Usage

Fuzzy.Square.poly.right(F1L, breakpoints = 100)

Arguments

F1L

a fuzzy number.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

A table containing print the related polynoms at the corresponding definition domains.

Examples

X <- TrapezoidalFuzzyNumber(1,2,3,4)
Fuzzy.Square.poly.right(X)

Computes the fuzzy decisions of a fuzzy inference test by the traditional fuzzy confidence intervals

Description

Computes the fuzzy decisions of a fuzzy inference test by the traditional fuzzy confidence intervals

Usage

Fuzzy.decisions(
  type,
  H0,
  H1,
  t,
  s.d,
  n,
  sig,
  distribution,
  distance.type = "DSGD",
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100
)

Arguments

type

a category betwenn "0", "1" and "2". The category "0" refers to a bilateral test, the category "1" for a lower unilateral one, and "2" for an upper unilateral test.

H0

a trapezoidal or a triangular fuzzy number representing the fuzzy null hypothesis.

H1

a trapezoidal or a triangular fuzzy number representing the fuzzy alternative hypothesis.

t

a given numerical or fuzzy type parameter of the distribution.

s.d

a numerical value for the standard deviation of the distribution.

n

the total number of observations of the data set.

sig

a numerical value representing the significance level of the test.

distribution

a distribution chosen between "normal", "poisson", "Student" or "Logistic".

distance.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, p is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

Returns a list composed by the arguments, the fuzzy confidence intervals and their complements, the fuzzy decisions and the defuzzified values.

Examples

H0 <- alphacut(TriangularFuzzyNumber(2.9,3,3.1), seq(0,1, 0.01))
H1 <- alphacut(TriangularFuzzyNumber(3,3,5), seq(0,1,0.01))
t <- alphacut(TriangularFuzzyNumber(0.8,1.80,2.80), seq(0,1,0.01))
res <- Fuzzy.decisions(type = 0, H0, H1, t = t, s.d = 0.79, n = 10, sig = 0.05,
distribution = "normal", distance.type = "GSGD")

Computes the fuzzy decisions of a fuzzy inference test by the fuzzy confidence intervals by the likelihood method

Description

Computes the fuzzy decisions of a fuzzy inference test by the fuzzy confidence intervals by the likelihood method

Usage

Fuzzy.decisions.ML(
  data.fuzzified,
  H0,
  H1,
  t,
  coef.boot,
  mu = NA,
  sigma = NA,
  sig,
  distribution,
  distance.type = "DSGD",
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100,
  step = 0.05,
  margin = c(5, 5),
  plot = FALSE
)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

H0

a trapezoidal or a triangular fuzzy number representing the fuzzy null hypothesis.

H1

a trapezoidal or a triangular fuzzy number representing the fuzzy alternative hypothesis.

t

a given numerical or fuzzy type parameter of the distribution.

coef.boot

a decimal representing the 1-sig-quantile of the bootstrap distribution of LR.

mu

if the mean of the normal distribution is known, mu should be a numerical value. Otherwise, the argument mu is fixed to NA.

sigma

if the standard deviation of the normal distribution is known, sigma should be a numerical value. Otherwise, the argument sigma is fixed to NA.

sig

a numerical value representing the significance level of the test.

distribution

a distribution chosen between "normal", "poisson", "Student" or "Logistic".

distance.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, p is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

step

a numerical value fixed to 0.05, defining the step of iterations on the interval [t-5; t+5].

margin

an optional numerical couple of values fixed to [5; 5], representing the range of calculations around the parameter t.

plot

fixed by default to "FALSE". plot="FALSE" if a plot of the fuzzy number is not required.

Value

Returns a list composed by the arguments, the fuzzy confidence intervals, the fuzzy decisions, the defuzzified values and the decision made.

Examples

data <- matrix(c(1,2,3,2,2,1,1,3,1,2),ncol=1) 
MF111 <- TrapezoidalFuzzyNumber(0,1,1,2)
MF112 <- TrapezoidalFuzzyNumber(1,2,2,3)
MF113 <- TrapezoidalFuzzyNumber(2,3,3,4)
PA11 <- c(1,2,3)
data.fuzzified <- FUZZ(data,mi=1,si=1,PA=PA11)
H0 <- alphacut(TriangularFuzzyNumber(2.9,3,3.1), seq(0,1, 0.01))
H1 <- alphacut(TriangularFuzzyNumber(3,3,5), seq(0,1,0.01))
t <- alphacut(TriangularFuzzyNumber(0.8,1.80,2.80), seq(0,1,0.01))
coef.boot <- 3.470085
res <- Fuzzy.decisions.ML(data.fuzzified, H0, H1, t = t, coef.boot = coef.boot, 
sigma = 0.79, sig = 0.05, distribution = "normal", distance.type = "GSGD")

Calculates the exact variance

Description

Calculates the exact variance

Usage

Fuzzy.exact.variance(data.fuzzified, breakpoints = 100, plot = FALSE)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

plot

fixed by default to "FALSE". plot="TRUE" if a plot of the fuzzy number is required.

Value

The numerical alpha-cuts of the estimated fuzzy variance.

Gives the polynomial forms of the numerical alpha-cuts modelling the exact variance

Description

Gives the polynomial forms of the numerical alpha-cuts modelling the exact variance

Usage

Fuzzy.exact.variance.poly.left(data.fuzzified, breakpoints = 100)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

A table composed by the coefficients of the second order equations of the left side, given at the corresponding definitions domains.

Gives the polynomial forms of the numerical alpha-cuts modelling the exact variance

Description

Gives the polynomial forms of the numerical alpha-cuts modelling the exact variance

Usage

Fuzzy.exact.variance.poly.right(data.fuzzified, breakpoints = 100)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

A table composed by the coefficients of the second order equations of the right side, given at the corresponding definitions domains.

Computes the fuzzy p-value of a given fuzzy hypothesis test

Description

Computes the fuzzy p-value of a given fuzzy hypothesis test

Usage

Fuzzy.p.value(
  type,
  H0,
  H1,
  t,
  s.d = 1,
  n,
  sig,
  distribution,
  distance.type = "DSGD",
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100
)

Arguments

type

a category betwenn "0", "1" and "2". The category "0" refers to a bilateral test, the category "1" for a lower unilateral one, and "2" for an upper unilateral test.

H0

a trapezoidal or a triangular fuzzy number representing the fuzzy null hypothesis.

H1

a trapezoidal or a triangular fuzzy number representing the fuzzy alternative hypothesis.

t

a given numerical or fuzzy type parameter of the distribution.

s.d

a numerical value for the standard deviation of the distribution.

n

the total number of observations of the data set.

sig

a numerical value representing the significance level of the test.

distribution

a distribution chosen between "normal", "poisson", "Student" or "Logistic".

distance.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, p is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

Returns the defuzzified p-value and the decision made.

Examples

H0 <- TriangularFuzzyNumber(2.2,2.5,3) 
H1 <- TriangularFuzzyNumber(2.5,2.5,5)
Fuzzy.p.value(type=1, H0, H1, t=TriangularFuzzyNumber(0.8,1.8,2.8),
s.d=0.7888, n=10, sig=0.05, distribution="normal", distance.type="GSGD")

Computes the fuzzy p-value of a given fuzzy hypothesis test for the mean

Description

Computes the fuzzy p-value of a given fuzzy hypothesis test for the mean

Usage

Fuzzy.p.value.mean(
  data.fuzzified,
  type,
  H0,
  H1,
  s.d = 1,
  sig,
  distribution,
  distance.type = "DSGD",
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100
)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

type

a category betwenn "0", "1" and "2". The category "0" refers to a bilateral test, the category "1" for a lower unilateral one, and "2" for an upper unilateral test.

H0

a trapezoidal or a triangular fuzzy number representing the fuzzy null hypothesis.

H1

a trapezoidal or a triangular fuzzy number representing the fuzzy alternative hypothesis.

s.d

a numerical value for the standard deviation of the distribution.

sig

a numerical value representing the significance level of the test.

distribution

a distribution chosen between "normal", "poisson" or "Student".

distance.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, p is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

Returns the defuzzified p-value and the decision made.

Examples

data <- matrix(c(1,2,3,2,2,1,1,3,1,2),ncol=1)
MF111 <- TrapezoidalFuzzyNumber(0,1,1,2)
MF112 <- TrapezoidalFuzzyNumber(1,2,2,3)
MF113 <- TrapezoidalFuzzyNumber(2,3,3,4)
PA11 <- c(1,2,3)
data.fuzzified <- FUZZ(data,mi=1,si=1,PA=PA11)
H0 <- TriangularFuzzyNumber(2.2,2.5,3) 
H1 <- TriangularFuzzyNumber(2.5,2.5,5)
Fuzzy.p.value.mean(data.fuzzified, type=1, H0, H1, s.d=0.7888, sig=0.05, 
distribution="normal", distance.type="GSGD")

Calculates the fuzzy sample mean

Description

Calculates the fuzzy sample mean

Usage

Fuzzy.sample.mean(data.fuzzified, breakpoints = 100, alphacuts = FALSE)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

alphacuts

fixed by default to "FALSE". No alpha-cuts are printed in this case.

Value

If the parameter alphacuts="TRUE", the function returns a matrix composed by 2 vectors representing the numerical left and right alpha-cuts. For this output, is.alphacuts = TRUE. If the parameter alphacuts="FALSE", the function returns a trapezoidal fuzzy number given by the quadruple (p,q,r,s).

Examples

mat <- matrix(c(1,2,2,3,3,4,4,5), ncol =4) 
Fuzzy.sample.mean(mat)

Fuzzy sample variance (approx) - general

Description

Fuzzy sample variance (approx) - general

Usage

Fuzzy.sample.variance.approximation(data.fuzzified, appro.id)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

appro.id

an integer between 1 and 5 giving the method of approximation chosen.

Value

A numerical value.

Fuzzy sample variance (approx) - method 1

Description

Fuzzy sample variance (approx) - method 1

Usage

Fuzzy.sample.variance.approximation1(data.fuzzified)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

Value

A numerical value.

Fuzzy sample variance (approx) - method 2

Description

Fuzzy sample variance (approx) - method 2

Usage

Fuzzy.sample.variance.approximation2(data.fuzzified)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

Value

A numerical value.

Fuzzy sample variance (approx) - method 3

Description

Fuzzy sample variance (approx) - method 3

Usage

Fuzzy.sample.variance.approximation3(data.fuzzified)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

Value

A numerical value.

Fuzzy sample variance (approx) - method 4

Description

Fuzzy sample variance (approx) - method 4

Usage

Fuzzy.sample.variance.approximation4(data.fuzzified)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

Value

A numerical value.

Fuzzy sample variance (approx) - method 5

Description

Fuzzy sample variance (approx) - method 5

Usage

Fuzzy.sample.variance.approximation5(data.fuzzified)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

Value

A numerical value.

Calculates the variance by a chosen method: distance, exact or approximation

Description

Calculates the variance by a chosen method: distance, exact or approximation

Usage

Fuzzy.variance(
  data.fuzzified,
  method,
  dist.type = "DSGD",
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100,
  int.method = "int.simpson",
  plot = FALSE
)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

method

choices are the following: "distance", "exact", "approximation1", "approxi- mation2", "approximation3", "approximation4", "approximation5".

dist.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, q is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

int.method

the integration method could be one of the following four methods: "int.0", "int.t", "int.ct" and "int.simpson".

plot

fixed by default to "FALSE". plot="TRUE" if a plot of the fuzzy number is required.

Value

If the parameter method = "distance", returns a numerical value. If else, returns the numerical \alpha-cuts of the estimated fuzzy variance.

Examples

data <- matrix(c(1,2,3,2,2,1,1,3,1,2),ncol=1) 
MF111 <- TrapezoidalFuzzyNumber(0,1,1,2) 
MF112 <- TrapezoidalFuzzyNumber(1,2,2,3) 
MF113 <- TrapezoidalFuzzyNumber(2,3,3,3) 
PA11 <- c(1,2,3) 
data.fuzzified <- FUZZ(data,mi=1,si=1,PA=PA11) 
Fuzzy.variance(data.fuzzified, method = "approximation5", plot=TRUE) 
Fuzzy.variance(data.fuzzified, method = "distance")

Fuzzifies a variable modelled by any type of fuzzy numbers

Description

Fuzzifies a variable modelled by any type of fuzzy numbers

Usage

GFUZZ(data, mi, si, PA, spec = "Identical", breakpoints = 100)

Arguments

data

a data set.

mi

the index of the main-item containing the concerned variable.

si

the index of the sub-item of a given main-item mi.

PA

a vector of the linguistic terms of the considered variable.

spec

specification of the fuzzification matrix. The possible values are "Identical" and "Not Identical".

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. breakpoints is fixed to 100 by default.

Value

A numerical fuzzification array of 3 dimensions (m,n,2), with m lines, n columns and no NA.

Examples

data <- matrix(c(1,2,3,2,2,1,1,3,1,2),ncol=1)
MF111 <- TrapezoidalFuzzyNumber(0,1,1,2)
MF112 <- TrapezoidalFuzzyNumber(1,2,2,3)
MF113 <- TrapezoidalFuzzyNumber(2,3,3,3)
PA11 <- c(1,2,3)
data.fuzzified <- GFUZZ(data,mi=1,si=1,PA=PA11)

Calculates the global evaluation of a linguistic questionnaire

Description

Calculates the global evaluation of a linguistic questionnaire

Usage

GLOB.EVAL(
  Full_Database,
  MI,
  bmi,
  SI,
  b_jkt,
  p_ind = rep(1/nrow(Full_Database), nrow(Full_Database)),
  distance.type,
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100
)

Arguments

Full_Database

the data set to evaluate.

MI

a numerical value representing the total number of main-items dividing the linguistic questionnaire.

bmi

an array referring to the initial weights of the main-items.

SI

an array representing the total numbers of sub-items per main-item.

b_jkt

a matrix of MI rows and max(SI) columns expressing the initial weights of each sub-item of a given main-item.

p_ind

a vector of the relative sampling weights of the units, for which length(p_ind) = nrow(data). If the weights are not relative, the following expression should be applied on the vector:

\frac{p_{ind}}{\sum_{i=1}^{n} p_{ind}}.

If no sampling weights are used, the vector of weights is reduced to a vector of values 1, i.e. rep(1, nrow(data)).

distance.type

type of distance chosen from the family of distances, set by default to the signed distance. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

A data set of individual evaluations, for which the number of observations is exactly the same as the initial data set.

Examples

data <- matrix(c(3,4,2,3,3,2,4,3,3,4,3,4,4,2,5,3,4,4,3,3,3,4,4,3,
3,3,4,3,3,3,3,4,4,3,5,3,4,3,3,3), ncol = 4)
data <- as.data.frame(data)
MI <- 2
SI1 <- 2
SI2 <- 2
SI <- c(SI1,SI2)
b_j <- c(1/2,1/2)
b_jk <- matrix(c(0.5,0.5,0.5,0.5),nrow=2) 
PA11 <- c(1,2,3,4,5)
PA12 <- c(1,2,3,4,5)
PA21 <- c(1,2,3,4,5)
PA22 <- c(1,2,3,4,5)
# ------------------
MF111 <- TrapezoidalFuzzyNumber(0,2,2,7)
MF112 <- TrapezoidalFuzzyNumber(2,7,7,15)
MF113 <- TrapezoidalFuzzyNumber(7,15,15,23)
MF114 <- TrapezoidalFuzzyNumber(15,23,23,28)
MF115 <- TrapezoidalFuzzyNumber(23,28,28,30)
MF11 <- GFUZZ(data, 1, 1, PA11, spec="Identical", breakpoints = 100)
# ------------------
MF121 <- TrapezoidalFuzzyNumber(0,2,2,7)
MF122 <- TrapezoidalFuzzyNumber(2,7,7,15)
MF123 <- TrapezoidalFuzzyNumber(7,15,15,23)
MF124 <- TrapezoidalFuzzyNumber(15,23,23,28)
MF125 <- TrapezoidalFuzzyNumber(23,28,28,30)
MF12 <- GFUZZ(data, 1, 2, PA12, spec="Identical", breakpoints = 100)
# ------------------
MF211 <- TrapezoidalFuzzyNumber(0,2,2,7)
MF212 <- TrapezoidalFuzzyNumber(2,7,7,15)
MF213 <- TrapezoidalFuzzyNumber(7,15,15,23)
MF214 <- TrapezoidalFuzzyNumber(15,23,23,28)
MF215 <- TrapezoidalFuzzyNumber(23,28,28,30)
MF21 <- GFUZZ(data, 2, 1, PA21, spec="Identical", breakpoints = 100)
# ------------------
MF221 <- TrapezoidalFuzzyNumber(0,2,2,7)
MF222 <- TrapezoidalFuzzyNumber(2,7,7,15)
MF223 <- TrapezoidalFuzzyNumber(7,15,15,23)
MF224 <- TrapezoidalFuzzyNumber(15,23,23,28)
MF225 <- TrapezoidalFuzzyNumber(23,28,28,30)
MF22 <- GFUZZ(data, 2, 2, PA22, spec="Identical", breakpoints = 100)
# ------------------
range <- matrix(c(0,0,0,0,28,28,28,28), ncol=2)
ind.eval <- IND.EVAL(data,MI,b_j,SI,b_jk, range = range, distance.type ="DSGD.G")
GLOB <- GLOB.EVAL(data, MI, b_j, SI, b_jk, distance.type ="GSGD")

Calculates the weighted mean of the set of individual evaluations

Description

Calculates the weighted mean of the set of individual evaluations

Usage

GLOB.EVAL.mean(ind.eval, weight = rep(1, length(ind.eval)))

Arguments

ind.eval

the set of individual evaluations.

weight

a vector of the relative sampling weights of the units, for which length(weight) = length(ind.eval), set by default to rep(1, length(ind.eval)).

Value

An integer.

Examples

data <- matrix(c(3,4,2,3,3,2,4,3,3,4,3,4,4,2,5,3,4,4,3,3,3,4,4,3,
3,3,4,3,3,3,3,4,4,3,5,3,4,3,3,3), ncol = 4)
data <- as.data.frame(data)
MI <- 2
SI1 <- 2
SI2 <- 2
SI <- c(SI1,SI2)
b_j <- c(1/2,1/2)
b_jk <- matrix(c(0.5,0.5,0.5,0.5),nrow=2) 
PA11 <- c(1,2,3,4,5)
PA12 <- c(1,2,3,4,5)
PA21 <- c(1,2,3,4,5)
PA22 <- c(1,2,3,4,5)
# ------------------
MF111 <- TrapezoidalFuzzyNumber(0,2,2,7)
MF112 <- TrapezoidalFuzzyNumber(2,7,7,15)
MF113 <- TrapezoidalFuzzyNumber(7,15,15,23)
MF114 <- TrapezoidalFuzzyNumber(15,23,23,28)
MF115 <- TrapezoidalFuzzyNumber(23,28,28,30)
MF11 <- GFUZZ(data, 1, 1, PA11, spec="Identical", breakpoints = 100)
# ------------------
MF121 <- TrapezoidalFuzzyNumber(0,2,2,7)
MF122 <- TrapezoidalFuzzyNumber(2,7,7,15)
MF123 <- TrapezoidalFuzzyNumber(7,15,15,23)
MF124 <- TrapezoidalFuzzyNumber(15,23,23,28)
MF125 <- TrapezoidalFuzzyNumber(23,28,28,30)
MF12 <- GFUZZ(data, 1, 2, PA12, spec="Identical", breakpoints = 100)
# ------------------
MF211 <- TrapezoidalFuzzyNumber(0,2,2,7)
MF212 <- TrapezoidalFuzzyNumber(2,7,7,15)
MF213 <- TrapezoidalFuzzyNumber(7,15,15,23)
MF214 <- TrapezoidalFuzzyNumber(15,23,23,28)
MF215 <- TrapezoidalFuzzyNumber(23,28,28,30)
MF21 <- GFUZZ(data, 2, 1, PA21, spec="Identical", breakpoints = 100)
# ------------------
MF221 <- TrapezoidalFuzzyNumber(0,2,2,7)
MF222 <- TrapezoidalFuzzyNumber(2,7,7,15)
MF223 <- TrapezoidalFuzzyNumber(7,15,15,23)
MF224 <- TrapezoidalFuzzyNumber(15,23,23,28)
MF225 <- TrapezoidalFuzzyNumber(23,28,28,30)
MF22 <- GFUZZ(data, 2, 2, PA22, spec="Identical", breakpoints = 100)
# ------------------
range <- matrix(c(0,0,0,0,28,28,28,28), ncol=2)
ind.eval <- IND.EVAL(data,MI,b_j,SI,b_jk, range = range, distance.type ="DSGD.G")
GLOB.mean <- GLOB.EVAL.mean(ind.eval)

Calculates a distance between fuzzy numbers

Description

Calculates a distance between fuzzy numbers

Usage

GSGD(X, Y, i = 1, j = 1, thetas = 1, breakpoints = 100)

Arguments

X

a fuzzy number.

Y

a fuzzy number.

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

thetas

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

A numerical value.

Creates a Gaussian two-sided bell fuzzy number

Description

Creates a Gaussian two-sided bell fuzzy number

Usage

GaussianBellFuzzyNumber(
  left.mean,
  left.sigma,
  right.mean,
  right.sigma,
  alphacuts = FALSE,
  margin = c(5, 5),
  step = 0.01,
  breakpoints = 100,
  precision = 4,
  plot = FALSE
)

Arguments

left.mean

a numerical value of the parameter mu of the left Gaussian curve.

left.sigma

a numerical value of the parameter sigma of the left Gaussian curve.

right.mean

a numerical value of the parameter mu of the right Gaussian curve.

right.sigma

a numerical value of the parameter sigma of the right Gaussian curve.

alphacuts

fixed by default to "FALSE". No alpha-cuts are printed in this case.

margin

an optional numerical couple of values representing the range of calculations of the Gaussian curve written as [mean - 3*sigma; mean + 3*sigma] by default.

step

a numerical value fixing the step between two knots dividing the interval [mean - 3*sigma; mean + 3*sigma].

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

precision

an integer specifying the number of decimals for which the calculations are made. These latter are set by default to be at the order of 1/10^4 .

plot

fixed by default to "FALSE". plot="TRUE" if a plot of the fuzzy number is required.

Value

If the parameter alphacuts="TRUE", the function returns a matrix composed by 2 vectors representing the left and right alpha-cuts. For this output, is.alphacuts = TRUE. If the parameter alphacuts="FALSE", the function returns a list composed by the Class, the mean, the sigma, the vectors of the left and right alpha-cuts.

Examples

GBFN <- GaussianBellFuzzyNumber(left.mean = -1, left.sigma = 1, 
right.mean = 2, right.sigma = 1, alphacuts = TRUE, plot=TRUE)
is.alphacuts(GBFN)

Creates a Gaussian fuzzy number

Description

Creates a Gaussian fuzzy number

Usage

GaussianFuzzyNumber(
  mean,
  sigma,
  alphacuts = FALSE,
  margin = c(5, 5),
  step = 0.01,
  breakpoints = 100,
  precision = 4,
  plot = FALSE
)

Arguments

mean

a numerical value of the parameter mu of the Gaussian curve.

sigma

a numerical value of the parameter sigma of the Gaussian curve.

alphacuts

fixed by default to "FALSE". No alpha-cuts are printed in this case.

margin

an optional numerical couple of values representing the range of calculations of the Gaussian curve written as [mean - 3*sigma; mean + 3*sigma] by default.

step

a numerical value fixing the step between two knots dividing the interval [mean - 3*sigma; mean + 3*sigma].

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

precision

an integer specifying the number of decimals for which the calculations are made. These latter are set by default to be at the order of 1/10^4 .

plot

fixed by default to "FALSE". plot="TRUE" if a plot of the fuzzy number is required.

Value

If the parameter alphacuts="TRUE", the function returns a matrix composed by 2 vectors representing the left and right alpha-cuts. For this output, is.alphacuts = TRUE. If the parameter alphacuts="FALSE", the function returns a list composed by the Class, the mean, the sigma, the vectors of the left and right alpha-cuts.

Examples

GFN <- GaussianFuzzyNumber(mean = 0, sigma = 1, alphacuts = TRUE, plot=TRUE)
is.alphacuts(GFN)

Calculates the individual evaluations of a linguistic questionnaire

Description

Calculates the individual evaluations of a linguistic questionnaire

Usage

IND.EVAL(
  Full_Database,
  MI,
  bmi,
  SI,
  b_jkt,
  range,
  distance.type,
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100,
  spec = "Identical"
)

Arguments

Full_Database

the data set to evaluate.

MI

a numerical value representing the total number of main-items dividing the linguistic questionnaire.

bmi

an array referring to the initial weights of the main-items.

SI

an array representing the total numbers of sub-items per main-item.

b_jkt

a matrix of MI rows and max(SI) columns expressing the initial weights of each sub-item of a given main-item.

range

a vector of 2 elements giving the range of definition of the produced individual evaluations. The range is usually chosen in the interval between 0 and the maximum of the support set of all the membership functions modelling the data set.

distance.type

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

spec

specification of the fuzzification matrix. The possible values are "Identical" and "Not Identical".

Value

A data set of individual evaluations, for which the number of observations is exactly the same as the initial data set.

Examples

data <- matrix(c(3,4,2,3,3,2,4,3,3,4,3,4,4,2,5,3,4,4,3,3,3,4,4,3,
3,3,4,3,3,3,3,4,4,3,5,3,4,3,3,3), ncol = 4)
data <- as.data.frame(data)
MI <- 2
SI1 <- 2
SI2 <- 2
SI <- c(SI1,SI2)
b_j <- c(1/2,1/2)
b_jk <- matrix(c(0.5,0.5,0.5,0.5),nrow=2) 
PA11 <- c(1,2,3,4,5)
PA12 <- c(1,2,3,4,5)
PA21 <- c(1,2,3,4,5)
PA22 <- c(1,2,3,4,5)
# ------------------
MF111 <- TrapezoidalFuzzyNumber(0,2,2,7)
MF112 <- TrapezoidalFuzzyNumber(2,7,7,15)
MF113 <- TrapezoidalFuzzyNumber(7,15,15,23)
MF114 <- TrapezoidalFuzzyNumber(15,23,23,28)
MF115 <- TrapezoidalFuzzyNumber(23,28,28,30)
MF11 <- GFUZZ(data, 1, 1, PA11, spec="Identical", breakpoints = 100)
# ------------------
MF121 <- TrapezoidalFuzzyNumber(0,2,2,7)
MF122 <- TrapezoidalFuzzyNumber(2,7,7,15)
MF123 <- TrapezoidalFuzzyNumber(7,15,15,23)
MF124 <- TrapezoidalFuzzyNumber(15,23,23,28)
MF125 <- TrapezoidalFuzzyNumber(23,28,28,30)
MF12 <- GFUZZ(data, 1, 2, PA12, spec="Identical", breakpoints = 100)
# ------------------
MF211 <- TrapezoidalFuzzyNumber(0,2,2,7)
MF212 <- TrapezoidalFuzzyNumber(2,7,7,15)
MF213 <- TrapezoidalFuzzyNumber(7,15,15,23)
MF214 <- TrapezoidalFuzzyNumber(15,23,23,28)
MF215 <- TrapezoidalFuzzyNumber(23,28,28,30)
MF21 <- GFUZZ(data, 2, 1, PA21, spec="Identical", breakpoints = 100)
# ------------------
MF221 <- TrapezoidalFuzzyNumber(0,2,2,7)
MF222 <- TrapezoidalFuzzyNumber(2,7,7,15)
MF223 <- TrapezoidalFuzzyNumber(7,15,15,23)
MF224 <- TrapezoidalFuzzyNumber(15,23,23,28)
MF225 <- TrapezoidalFuzzyNumber(23,28,28,30)
MF22 <- GFUZZ(data, 2, 2, PA22, spec="Identical", breakpoints = 100)
# ------------------
range <- matrix(c(0,0,0,0,28,28,28,28), ncol=2)
ind.eval <- IND.EVAL(data,MI,b_j,SI,b_jk, range = range, distance.type ="DSGD.G")

Calculates the excess of kurtosis of a random fuzzy variable

Description

Calculates the excess of kurtosis of a random fuzzy variable

Usage

Kurtosis(
  data.fuzzified,
  dist.type,
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100
)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

dist.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, q is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

A numerical value.

Examples

mat <- matrix(c(1,2,0.25,1.8,2,2.6,0.5,3,3,2.6,3.8,4,4,4.2,3.9,5), ncol =4)
Kurtosis(mat, dist.type = "GSGD")

Calculates a distance by the d_Mid.Spr between fuzzy numbers

Description

Calculates a distance by the d_Mid.Spr between fuzzy numbers

Usage

Mid.Spr(X, Y, i = 1, j = 1, theta = 1/3, breakpoints = 100)

Arguments

X

a fuzzy number.

Y

a fuzzy number.

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

A numerical value.

Calculates a central sample moment of a random fuzzy variable

Description

Calculates a central sample moment of a random fuzzy variable

Usage

Moment(
  data.fuzzified,
  k,
  dist.type,
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100
)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

k

the order of the moment.

dist.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, q is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

A numerical value.

Examples

mat <- matrix(c(1,2,2,3,3,4,4,5), ncol =4)
Moment(mat, k=4, dist.type = "GSGD")

Calculates the indicator of information's rate of the data base

Description

Calculates the indicator of information's rate of the data base

Usage

R(x, p_ind, b_jk, SI)

Arguments

x

the data set to evaluate.

p_ind

a vector of the relative sampling weights of the units, for which length(p_ind) = nrow(data). If the weights are not relative, the following expression should be applied on the vector:

\frac{p_{ind}}{\sum_{i=1}^{n} p_{ind}}.

If no sampling weights are used, the vector of weights is reduced to a vector of values 1, i.e. rep(1, nrow(data)).

b_jk

a matrix of length(b_j) rows and max(SI) columns expressing the initial weights of each sub-item of a given main-item.

SI

an array representing the total numbers of sub-items per main-item.

Value

A numerical value giving the indicator of information's rate of the complete linguistic questionnaire. Note that the obtained value is interpreted as the more it tends to the value 1, the less the complete questionnaire contains missing values.

Examples

data <- matrix(c(3,4,2,3,3,2,4,3,3,4,3,4,4,2,5,3,4,4,3,3,3,4,4,3,
3,3,4,3,3,3,3,4,4,3,5,3,4,3,3,3), ncol = 4)
data <- as.data.frame(data)
p_ind <- c(0.1,0.05,0.05,0.2,0.1,0.05,0.1,0.1,0.2,0.05)
SI1 <- 2
SI2 <- 2
SI <- c(SI1,SI2)
b_jk <- matrix(c(0.5,0.5,0.5,0.5),nrow=2) 
R(data, p_ind, b_jk, SI)

Calculates a distance by the Rho1 between fuzzy numbers

Description

Calculates a distance by the Rho1 between fuzzy numbers

Usage

Rho1(X, Y, breakpoints = 100)

Arguments

X

a fuzzy number.

Y

a fuzzy number.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

A numerical value.

Calculates a distance by the Rho2 between fuzzy numbers

Description

Calculates a distance by the Rho2 between fuzzy numbers

Usage

Rho2(X, Y, breakpoints = 100)

Arguments

X

a fuzzy number.

Y

a fuzzy number.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

A numerical value.

Calculates a distance by the d_Rhop between fuzzy numbers

Description

Calculates a distance by the d_Rhop between fuzzy numbers

Usage

Rhop(X, Y, p, breakpoints = 100)

Arguments

X

a fuzzy number.

Y

a fuzzy number.

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

A numerical value.

Calculates the indicator of information's rate of the data base for a given unit

Description

Calculates the indicator of information's rate of the data base for a given unit

Usage

Ri(x, i, b_jk, SI)

Arguments

x

the data set to evaluate.

i

an observation index.

b_jk

a matrix of length(b_j) rows and max(SI) columns expressing the initial weights of each sub-item of a given main-item.

SI

an array representing the total numbers of sub-items per main-item.

Value

A numerical value giving the indicator of information's rate of the complete linguistic questionnaire for a particular observation. Note that the obtained value is interpreted as the more it tends to the value 1, the less the observation i contains missing values.

Examples

data <- matrix(c(3,4,2,3,3,2,4,3,3,4,3,4,4,2,5,3,4,4,3,3,3,4,4,3,
3,3,4,3,3,3,3,4,4,3,5,3,4,3,3,3), ncol = 4)
data <- as.data.frame(data)
SI1 <- 2
SI2 <- 2
SI <- c(SI1,SI2)
b_jk <- matrix(c(0.5,0.5,0.5,0.5),nrow=2) 
Ri(data, 7, b_jk, SI)

Calculates the sequential sums of squares by a convenient metric

Description

Calculates the sequential sums of squares by a convenient metric

Usage

SEQ.ORDERING(scope, data, f.response)

Arguments

scope

a description of the complete fitting model.

data

the data frame containing all the variables of the model.

f.response

the vector of distances of the fuzzy response variable to the fuzzy origin.

Value

Returns a list of the new sets of sums of squares, as well as the coefficients, the residuals and the fitted.values.

Calculates the sequential sums of squares by an approximation

Description

Calculates the sequential sums of squares by an approximation

Usage

SEQ.ORDERING.APPROXIMATION(scope, data, f.response)

Arguments

scope

a description of the complete fitting model.

data

the data frame containing all the variables of the model.

f.response

the vector of distances of the fuzzy response variable to the fuzzy origin.

Value

Returns a list of the new sets of sums of squares, as well as the coefficients, the residuals and the fitted.values.

Calculates the sequential sums of squares by an exact calculation

Description

Calculates the sequential sums of squares by an exact calculation

Usage

SEQ.ORDERING.EXACT(scope, data, f.response)

Arguments

scope

a description of the complete fitting model.

data

the data frame containing all the variables of the model.

f.response

the vector of distances of the fuzzy response variable to the fuzzy origin.

Value

Returns a list of the new sets of sums of squares, as well as the coefficients, the residuals and the fitted.values.

Calculates a distance by the SGD between fuzzy numbers

Description

Calculates a distance by the SGD between fuzzy numbers

Usage

SGD(X, i = 1, j = 1, breakpoints = 100)

Arguments

X

a fuzzy number.

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

A numerical value.

Calculates the sample variance by a convenient metric

Description

Calculates the sample variance by a convenient metric

Usage

Sample.variance(
  data.fuzzified,
  dist.type,
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100
)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

dist.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, q is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

A numerical value.

Calculates the skewness of a random fuzzy variable

Description

Calculates the skewness of a random fuzzy variable

Usage

Skewness(
  data.fuzzified,
  dist.type,
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100
)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

dist.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, q is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

A numerical value.

Examples

mat <- matrix(c(1,2,0.25,1.8,2,2.6,0.5,3,3,2.6,3.8,4,4,4.2,3.9,5), ncol =4)
Skewness(mat, dist.type = "GSGD")

Calculates the weighted fuzzy sample mean

Description

Calculates the weighted fuzzy sample mean

Usage

Weighted.fuzzy.mean(
  data.fuzzified,
  weight,
  breakpoints = 100,
  alphacuts = FALSE
)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

weight

a weighting vector of the same length of the fuzzification matrix. No NA allowed.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

alphacuts

fixed by default to "FALSE". No alpha-cuts are printed in this case.

Value

Examples

mat <- matrix(c(1,2,2,3,3,4,4,5), ncol =4)
w <- c(1,3)
Weighted.fuzzy.mean(mat, w)

Calculates the adjusted weight for a given main-item of a linguistic questionnaire

Description

Calculates the adjusted weight for a given main-item of a linguistic questionnaire

Usage

adjusted.weight.MI(x, i, j, b_j, b_jk, SI)

Arguments

x

the data set to evaluate.

i

an observation index.

j

a main-item index.

b_j

an array referring to the initial weights given to each main-item of the considered main-item. This array will be afterwards re-calculated.

b_jk

a matrix of length(b_j) rows and max(SI) columns expressing the initial weights of each sub-item of a given main-item.

SI

an array representing the total numbers of sub-items per main-item.

Value

A numerical value giving the readjusted weight of the main-item j for the observation i.

Examples

data <- matrix(c(3,4,2,3,3,2,4,3,3,4,3,4,4,2,5,3,4,4,3,3,3,4,4,3,
3,3,4,3,3,3,3,4,4,3,5,3,4,3,3,3), ncol = 4)
data <- as.data.frame(data)
MI <- 2
SI1 <- 2
SI2 <- 2
SI <- c(SI1,SI2)
b_j <- c(1/2,1/2)
b_jk <- matrix(c(0.5,0.5,0.5,0.5),nrow=2) 
PA11 <- c(1,2,3,4,5)
PA12 <- c(1,2,3,4,5)
PA21 <- c(1,2,3,4,5)
PA22 <- c(1,2,3,4,5)
# ------------------
MF111 <- TrapezoidalFuzzyNumber(0,2,2,7)
MF112 <- TrapezoidalFuzzyNumber(2,7,7,15)
MF113 <- TrapezoidalFuzzyNumber(7,15,15,23)
MF114 <- TrapezoidalFuzzyNumber(15,23,23,28)
MF115 <- TrapezoidalFuzzyNumber(23,28,28,30)
MF11 <- GFUZZ(data, 1, 1, PA11, spec="Identical", breakpoints = 100)
# ------------------
MF121 <- TrapezoidalFuzzyNumber(0,2,2,7)
MF122 <- TrapezoidalFuzzyNumber(2,7,7,15)
MF123 <- TrapezoidalFuzzyNumber(7,15,15,23)
MF124 <- TrapezoidalFuzzyNumber(15,23,23,28)
MF125 <- TrapezoidalFuzzyNumber(23,28,28,30)
MF12 <- GFUZZ(data, 1, 2, PA12, spec="Identical", breakpoints = 100)
# ------------------
MF211 <- TrapezoidalFuzzyNumber(0,2,2,7)
MF212 <- TrapezoidalFuzzyNumber(2,7,7,15)
MF213 <- TrapezoidalFuzzyNumber(7,15,15,23)
MF214 <- TrapezoidalFuzzyNumber(15,23,23,28)
MF215 <- TrapezoidalFuzzyNumber(23,28,28,30)
MF21 <- GFUZZ(data, 2, 1, PA21, spec="Identical", breakpoints = 100)
# ------------------
MF221 <- TrapezoidalFuzzyNumber(0,2,2,7)
MF222 <- TrapezoidalFuzzyNumber(2,7,7,15)
MF223 <- TrapezoidalFuzzyNumber(7,15,15,23)
MF224 <- TrapezoidalFuzzyNumber(15,23,23,28)
MF225 <- TrapezoidalFuzzyNumber(23,28,28,30)
MF22 <- GFUZZ(data, 2, 2, PA22, spec="Identical", breakpoints = 100)
# ------------------
range <- matrix(c(0,0,0,0,28,28,28,28), ncol=2)
adjusted.weight.MI(data, 9, 1, b_j, b_jk, SI)

Calculates the adjusted weight for a given sub-item of a linguistic questionnaire

Description

Calculates the adjusted weight for a given sub-item of a linguistic questionnaire

Usage

adjusted.weight.SI(x, i, k, b_jk)

Arguments

x

the data set to evaluate.

i

an observation index.

k

a sub-item index.

b_jk

an array referring to the initial weights given to each sub-item of the considered main-item. This array will be afterwards re-calculated.

Value

A numerical value giving the readjusted weight of the sub-item k of the considered main-item for the observation i.

Examples

data <- matrix(c(3,4,2,3,3,2,4,3,3,4,3,4,4,2,5,3,4,4,3,3,3,4,4,3,
3,3,4,3,3,3,3,4,4,3,5,3,4,3,3,3), ncol = 4)
adjusted.weight.SI(data, 7, 1, c(0.5,0.5))

Estimates the bootstrap distribution of the likelihood ratio LR by the Algorithm 1 using the mean

Description

Estimates the bootstrap distribution of the likelihood ratio LR by the Algorithm 1 using the mean

Usage

boot.mean.algo1(
  data.fuzzified,
  distribution,
  sig,
  nsim = 100,
  mu = NA,
  sigma = NA,
  step = 0.1,
  margin = c(5, 5),
  breakpoints = 100,
  plot = TRUE
)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

distribution

a distribution chosen between "normal", "poisson", "Student" or "Logistic".

sig

a numerical value representing the significance level of the test.

nsim

an integer giving the number of replications needed in the bootstrap procedure. It is set to 100 by default.

mu

if the mean of the normal distribution is known, mu should be a numerical value. Otherwise, the argument mu is fixed to NA.

sigma

if the standard deviation of the normal distribution is known, sigma should be a numerical value. Otherwise, the argument sigma is fixed to NA.

step

a numerical value fixed to 0.1, defining the step of iterations on the interval [t-5; t+5].

margin

an optional numerical couple of values fixed to [5; 5], representing the range of calculations around the parameter t.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

plot

fixed by default to "FALSE". plot="FALSE" if a plot of the fuzzy number is not required.

Value

Returns a vector of decimals representing the bootstrap distribution of LR.

Estimates the bootstrap distribution of the likelihood ratio LR by the Algorithm 2 using the mean

Description

Estimates the bootstrap distribution of the likelihood ratio LR by the Algorithm 2 using the mean

Usage

boot.mean.algo2(
  data.fuzzified,
  distribution,
  sig,
  nsim = 100,
  mu = NA,
  sigma = NA,
  step = 0.1,
  margin = c(5, 5),
  breakpoints = 100,
  plot = TRUE
)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

distribution

a distribution chosen between "normal", "poisson", "Student" or "Logistic".

sig

a numerical value representing the significance level of the test.

nsim

an integer giving the number of replications needed in the bootstrap procedure. It is set to 100 by default.

mu

if the mean of the normal distribution is known, mu should be a numerical value. Otherwise, the argument mu is fixed to NA.

sigma

if the standard deviation of the normal distribution is known, sigma should be a numerical value. Otherwise, the argument sigma is fixed to NA.

step

a numerical value fixed to 0.1, defining the step of iterations on the interval [t-5; t+5].

margin

an optional numerical couple of values fixed to [5; 5], representing the range of calculations around the parameter t.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

plot

fixed by default to "FALSE". plot="FALSE" if a plot of the fuzzy number is not required.

Value

Returns a vector of decimals representing the bootstrap distribution of LR.

Estimates the bootstrap distribution of the likelihood ratio LR by the Algorithm 1 or 2 using the mean

Description

Estimates the bootstrap distribution of the likelihood ratio LR by the Algorithm 1 or 2 using the mean

Usage

boot.mean.ml(
  data.fuzzified,
  algorithm,
  distribution,
  sig,
  nsim = 100,
  mu = NA,
  sigma = NA,
  step = 0.1,
  margin = c(5, 5),
  breakpoints = 100,
  plot = TRUE
)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

algorithm

an algorithm chosen between "algo1" or "algo2".

distribution

a distribution chosen between "normal", "poisson", "Student" or "Logistic".

sig

a numerical value representing the significance level of the test.

nsim

an integer giving the number of replications needed in the bootstrap procedure. It is set to 100 by default.

mu

if the mean of the normal distribution is known, mu should be a numerical value. Otherwise, the argument mu is fixed to NA.

sigma

if the standard deviation of the normal distribution is known, sigma should be a numerical value. Otherwise, the argument sigma is fixed to NA.

step

a numerical value fixed to 0.1, defining the step of iterations on the interval [t-5; t+5].

margin

an optional numerical couple of values fixed to [5; 5], representing the range of calculations around the parameter t.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

plot

fixed by default to "FALSE". plot="FALSE" if a plot of the fuzzy number is not required.

Value

Returns a vector of decimals representing the bootstrap distribution of LR.

Examples

mat <- matrix(c(1,2,2,2,2,1),ncol=1)
MF111 <- TrapezoidalFuzzyNumber(0,1,1,2)
MF112 <- TrapezoidalFuzzyNumber(1,2,2,3)
PA11 <- c(1,2)
data.fuzzified <- FUZZ(mat,mi=1,si=1,PA=PA11) 
emp.dist <- boot.mean.ml(data.fuzzified, algorithm = "algo1", distribution = "normal",
 sig = 0.05, nsim = 5, sigma = 1)
eta.boot <- quantile(emp.dist,  probs = 95/100)

Cube a number

Description

Cube a number

Usage

cube(x)

Arguments

x

Number to be cubed

Value

The cube of the input

Calculates a distance between fuzzy numbers

Description

Calculates a distance between fuzzy numbers

Usage

distance(
  X,
  Y,
  type,
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100
)

Arguments

X

a fuzzy number.

Y

a fuzzy number.

type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

A numerical value.

Examples

X <- TrapezoidalFuzzyNumber(1,2,3,4) 
Y <- TrapezoidalFuzzyNumber(4,5,6,7) 
distance(X, Y, type = "DSGD.G")
distance(X, Y, type = "GSGD")

Estimates a fuzzy confidence interval by the Likelihood method

Description

Estimates a fuzzy confidence interval by the Likelihood method

Usage

fci.ml(
  data.fuzzified,
  t,
  distribution,
  sig,
  mu = NA,
  sigma = NA,
  step = 0.05,
  margin = c(5, 5),
  breakpoints = 100,
  plot = TRUE
)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

t

a given numerical or fuzzy type parameter of the distribution.

distribution

a distribution chosen between "normal", "poisson", "Student" or "Logistic".

sig

a numerical value representing the significance level of the test.

mu

if the mean of the normal distribution is known, mu should be a numerical value. Otherwise, the argument mu is fixed to NA.

sigma

if the standard deviation of the normal distribution is known, sigma should be a numerical value. Otherwise, the argument sigma is fixed to NA.

step

a numerical value fixed to 0.05, defining the step of iterations on the interval [t-5; t+5].

margin

an optional numerical couple of values fixed to [5; 5], representing the range of calculations around the parameter t.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

plot

fixed by default to "FALSE". plot="FALSE" if a plot of the fuzzy number is not required.

Value

Returns a matrix composed by 2 vectors representing the numerical left and right alpha-cuts. For this output, is.alphacuts = TRUE.

Examples

data <- matrix(c(1,2,3,2,2,1,1,3,1,2),ncol=1) 
MF111 <- TrapezoidalFuzzyNumber(0,1,1,2)
MF112 <- TrapezoidalFuzzyNumber(1,2,2,3)
MF113 <- TrapezoidalFuzzyNumber(2,3,3,4)
PA11 <- c(1,2,3)
data.fuzzified <- FUZZ(data,mi=1,si=1,PA=PA11)
Fmean <- Fuzzy.sample.mean(data.fuzzified)
fci.ml(data.fuzzified, t = Fmean, distribution = "normal", sig= 0.05, sigma = 0.62)

Estimates a fuzzy confidence interval by the Likelihood method

Description

Estimates a fuzzy confidence interval by the Likelihood method

Usage

fci.ml.boot(
  data.fuzzified,
  t,
  distribution,
  sig,
  coef.boot,
  mu = NA,
  sigma = NA,
  step = 0.05,
  margin = c(5, 5),
  breakpoints = 100,
  plot = TRUE
)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

t

a given numerical or fuzzy type parameter of the distribution.

distribution

a distribution chosen between "normal", "poisson", "Student" or "Logistic".

sig

a numerical value representing the significance level of the test.

coef.boot

a decimal representing the 1-sig-quantile of the bootstrap distribution of LR.

mu

if the mean of the normal distribution is known, mu should be a numerical value. Otherwise, the argument mu is fixed to NA.

sigma

if the standard deviation of the normal distribution is known, sigma should be a numerical value. Otherwise, the argument sigma is fixed to NA.

step

a numerical value fixed to 0.05, defining the step of iterations on the interval [t-5; t+5].

margin

an optional numerical couple of values fixed to [5; 5], representing the range of calculations around the parameter t.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

plot

fixed by default to "FALSE". plot="FALSE" if a plot of the fuzzy number is not required.

Value

Returns a matrix composed by 2 vectors representing the numerical left and right alpha-cuts. For this output, is.alphacuts = TRUE.

Examples

data <- matrix(c(1,2,3,2,2,1,1,3,1,2),ncol=1) 
MF111 <- TrapezoidalFuzzyNumber(0,1,1,2)
MF112 <- TrapezoidalFuzzyNumber(1,2,2,3)
MF113 <- TrapezoidalFuzzyNumber(2,3,3,4)
PA11 <- c(1,2,3)
data.fuzzified <- FUZZ(data,mi=1,si=1,PA=PA11)
Fmean <- Fuzzy.sample.mean(data.fuzzified)
fci.ml.boot(data.fuzzified, t = Fmean, distribution = "normal", sig= 0.05, sigma = 0.62,
coef.boot = 1.8225)

Calculates the fuzzy predicted values

Description

Calculates the fuzzy predicted values

Usage

fuzzy.predicted.values(dataset, coef.model)

Arguments

dataset

the data frame containing all the variables of the model.

coef.model

the coefficients of the model.

Value

Returns a matrix containing the alpha-cuts of the fuzzy prediced values.

Calculates the fuzzy residuals

Description

Calculates the fuzzy residuals

Usage

fuzzy.residuals(data.fuzzified, predicted.values)

Arguments

data.fuzzified

the fuzzified data set constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix.

predicted.values

the fuzzy predicted values constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix.

Value

Returns a matrix containing the alpha-cuts of the fuzzy residuals.

Numerical integration by the trivial method - method 1

Description

Numerical integration by the trivial method - method 1

Usage

int.0(cut, a = 0, b = 1)

Arguments

cut

a vector.

a

fixed by default to 0.

b

fixed by default to 1.

Value

An integer.

Numerical integration by the composite trapezoidal method - method 3

Description

Numerical integration by the composite trapezoidal method - method 3

Usage

int.ct(cut, a = 0, b = 1)

Arguments

cut

a vector.

a

fixed by default to 0.

b

fixed by default to 1.

Value

An integer.

Numerical integration by the Simpson method - method 4

Description

Numerical integration by the Simpson method - method 4

Usage

int.simpson(alpha, cut, a = 0, b = 1)

Arguments

alpha

a vector of alpha values between 0 and 1.

cut

a vector.

a

fixed by default to 0.

b

fixed by default to 1.

Value

An integer.

Numerical integration - method 2

Description

Numerical integration - method 2

Usage

int.t(alpha, cut, a = 0, b = 1)

Arguments

alpha

a vector of alpha values between 0 and 1.

cut

a vector.

a

fixed by default to 0.

b

fixed by default to 1.

Value

An integer.

Numerical integration by a particular method

Description

Numerical integration by a particular method

Usage

integrate.num(alpha, cut, method, a = 0, b = 1)

Arguments

alpha

a vector of alpha values between 0 and 1.

cut

a vector.

method

the integration method could be one of the following four methods: "int.0", "int.t", "int.ct" and "int.simpson".

a

fixed by default to 0.

b

fixed by default to 1.

Value

An integer.

Verifies if a matrix is set of left and right alpha-cuts

Description

Verifies if a matrix is set of left and right alpha-cuts

Usage

is.alphacuts(data)

Arguments

data

a matrix of 2 equal length columns with no NA.

Value

A value TRUE if the concerned object can be a set of numerical left and right alpha-cuts, FALSE otherwise.

Examples

mat <- matrix(c(1,2,3,7,6,5), ncol = 2) 
is.alphacuts(mat)

Verifies if a design is balanced

Description

Verifies if a design is balanced

Usage

is.balanced(ni)

Arguments

ni

a line array given by the contingency table related to the considered variable. Often written as a result of a call of the function table.

Value

Returns a logical decision TRUE or FALSE, to indicate if a given design is respectively balanced or not.

Examples

data <- matrix(c(1,2,3,2,2,1,1,3,1,2),ncol=1)
ni <- t(table(data))
is.balanced(ni)

Verifies if a matrix is a fuzzification matrix

Description

Verifies if a matrix is a fuzzification matrix

Usage

is.fuzzification(data)

Arguments

data

an array of 3 dimensions c(m,n,2), with m lines, n columns. No NA are allowed.

Value

A value TRUE if the concerned object is a numerical fuzzification matrix, FALSE otherwise.

Examples

mat <- array(c(1,1,2,2,3,3,5,5,6,6,7,7),dim=c(2,3,2))
is.fuzzification(mat)

Verifies if a matrix is a fuzzification matrix of trapezoidal fuzzy numbers

Description

Verifies if a matrix is a fuzzification matrix of trapezoidal fuzzy numbers

Usage

is.trfuzzification(data)

Arguments

data

a matrix of 4 columns (p,q,r,s), where p \le q \le r \le s. No NA are allowed.

Value

A value TRUE if the concerned object is a trapezoidal or triangular fuzzification matrix, FALSE otherwise.

Examples

mat <- matrix(c(1,1,2,2,3,3,4,4),ncol=4)
is.trfuzzification(mat)

Calculates the number of answers by a specific sub-item

Description

Calculates the number of answers by a specific sub-item

Usage

n_jk..(x, varindex, PA, p_ind = rep(1, nrow(x)))

Arguments

x

the data set to evaluate.

varindex

index of a particular sub-item.

PA

set of possible linguistic terms.

p_ind

a vector of the relative sampling weights of the units, for which length(p_ind) = nrow(data). If the weights are not relative, the following expression should be applied on the vector:

\frac{p_{ind}}{\sum_{i=1}^{n} p_{ind}}.

If no sampling weights are used, the vector of weights is reduced to a vector of values 1, i.e. rep(1, nrow(data)).

Value

A positive integer.

Calculates the number of answers by a specific linguistic of a sub-item

Description

Calculates the number of answers by a specific linguistic of a sub-item

Usage

n_jkq.(x, varindex, q, p_ind = rep(1, nrow(x)))

Arguments

x

the data set to evaluate.

varindex

index of a particular sub-item.

q

index of a particular linguistic term.

p_ind

a vector of the relative sampling weights of the units, for which length(p_ind) = nrow(data). If the weights are not relative, the following expression should be applied on the vector:

\frac{p_{ind}}{\sum_{i=1}^{n} p_{ind}}.

If no sampling weights are used, the vector of weights is reduced to a vector of values 1, i.e. rep(1, nrow(data)).

Value

A positive integer.

Calculates the number of breakpoints of a numerical matrix of alpha-cuts

Description

Calculates the number of breakpoints of a numerical matrix of alpha-cuts

Usage

nbreakpoints(data)

Arguments

data

a matrix of numerical alpha-cuts or a 3-dimensional array. No NA are allowed.

Value

A numerical positive integer.

Examples

X <- TrapezoidalFuzzyNumber(1,2,3,4)
alpha.X <- alphacut(X, seq(0,1,0.01)) 
nbreakpoints(alpha.X)

Calculates the p-value of fuzzy observations taken from a Student distribution

Description

Calculates the p-value of fuzzy observations taken from a Student distribution

Usage

p.value.Student(
  type,
  H0,
  H1,
  t,
  n,
  sig,
  dist.type,
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100,
  s.d = 1
)

Arguments

type

a category betwenn "0", "1" and "2". The category "0" refers to a bilateral test, the category "1" for a lower unilateral one, and "2" for an upper unilateral test.

H0

a trapezoidal or a triangular fuzzy number representing the fuzzy null hypothesis.

H1

a trapezoidal or a triangular fuzzy number representing the fuzzy alternative hypothesis.

t

a given numerical or fuzzy type parameter of the distribution.

n

the total number of observations of the data set.

sig

a numerical value representing the significance level of the test.

dist.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, p is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

s.d

a numerical value for the standard deviation of the distribution.

Value

Returns the defuzzified p-value and the decision made.

Calculates the p-value of fuzzy observations taken from a Fisher distribution

Description

Calculates the p-value of fuzzy observations taken from a Fisher distribution

Usage

p.value.fisher(
  type,
  H0,
  H1,
  t,
  n,
  r,
  s.d,
  sig,
  dist.type,
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100
)

Arguments

type

a category betwenn "0", "1" and "2". The category "0" refers to a bilateral test, the category "1" for a lower unilateral one, and "2" for an upper unilateral test.

H0

a trapezoidal or a triangular fuzzy number representing the fuzzy null hypothesis.

H1

a trapezoidal or a triangular fuzzy number representing the fuzzy alternative hypothesis.

t

a given numerical or fuzzy type parameter of the distribution.

n

first degree of freedom.

r

second degree of freedom.

s.d

a numerical value for the standard deviation of the distribution.

sig

a numerical value representing the significance level of the test.

dist.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, p is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

Returns the defuzzified p-value and the decision made.

Calculates the p-value of fuzzy observations taken from a Logistic distribution

Description

Calculates the p-value of fuzzy observations taken from a Logistic distribution

Usage

p.value.log(
  type,
  H0,
  H1,
  t,
  n,
  s.d,
  sig,
  dist.type,
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100
)

Arguments

type

a category betwenn "0", "1" and "2". The category "0" refers to a bilateral test, the category "1" for a lower unilateral one, and "2" for an upper unilateral test.

H0

a trapezoidal or a triangular fuzzy number representing the fuzzy null hypothesis.

H1

a trapezoidal or a triangular fuzzy number representing the fuzzy alternative hypothesis.

t

a given numerical or fuzzy type parameter of the distribution.

n

the total number of observations of the data set.

s.d

a numerical value for the standard deviation of the distribution.

sig

a numerical value representing the significance level of the test.

dist.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, p is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

Returns the defuzzified p-value and the decision made.

Calculates the p-value of fuzzy observations taken from a Student distribution for the mean

Description

Calculates the p-value of fuzzy observations taken from a Student distribution for the mean

Usage

p.value.mean.Student(
  data.fuzzified,
  type,
  H0,
  H1,
  sig,
  dist.type,
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100
)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

type

a category betwenn "0", "1" and "2". The category "0" refers to a bilateral test, the category "1" for a lower unilateral one, and "2" for an upper unilateral test.

H0

a trapezoidal or a triangular fuzzy number representing the fuzzy null hypothesis.

H1

a trapezoidal or a triangular fuzzy number representing the fuzzy alternative hypothesis.

sig

a numerical value representing the significance level of the test.

dist.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, p is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

Returns the defuzzified p-value and the decision made.

Calculates the p-value of fuzzy observations taken from a logistic distribution for the mean

Description

Calculates the p-value of fuzzy observations taken from a logistic distribution for the mean

Usage

p.value.mean.log(
  data.fuzzified,
  type,
  H0,
  H1,
  s.d,
  sig,
  dist.type,
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100
)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

type

a category betwenn "0", "1" and "2". The category "0" refers to a bilateral test, the category "1" for a lower unilateral one, and "2" for an upper unilateral test.

H0

a trapezoidal or a triangular fuzzy number representing the fuzzy null hypothesis.

H1

a trapezoidal or a triangular fuzzy number representing the fuzzy alternative hypothesis.

s.d

a numerical value for the standard deviation of the distribution.

sig

a numerical value representing the significance level of the test.

dist.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, p is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

Returns the defuzzified p-value and the decision made.

Calculates the p-value of fuzzy observations taken from a normal distribution for the mean

Description

Calculates the p-value of fuzzy observations taken from a normal distribution for the mean

Usage

p.value.mean.normal(
  data.fuzzified,
  type,
  H0,
  H1,
  s.d,
  sig,
  dist.type,
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100
)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

type

a category betwenn "0", "1" and "2". The category "0" refers to a bilateral test, the category "1" for a lower unilateral one, and "2" for an upper unilateral test.

H0

a trapezoidal or a triangular fuzzy number representing the fuzzy null hypothesis.

H1

a trapezoidal or a triangular fuzzy number representing the fuzzy alternative hypothesis.

s.d

a numerical value for the standard deviation of the distribution.

sig

a numerical value representing the significance level of the test.

dist.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, p is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

Returns the defuzzified p-value and the decision made.

Calculates the p-value of fuzzy observations taken from a Poisson distribution for the mean

Description

Calculates the p-value of fuzzy observations taken from a Poisson distribution for the mean

Usage

p.value.mean.poisson(
  data.fuzzified,
  type,
  H0,
  H1,
  sig,
  dist.type,
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100
)

Arguments

data.fuzzified

a fuzzification matrix constructed by a call to the function FUZZ or the function GFUZZ, or a similar matrix. No NA are allowed.

type

a category betwenn "0", "1" and "2". The category "0" refers to a bilateral test, the category "1" for a lower unilateral one, and "2" for an upper unilateral test.

H0

a trapezoidal or a triangular fuzzy number representing the fuzzy null hypothesis.

H1

a trapezoidal or a triangular fuzzy number representing the fuzzy alternative hypothesis.

sig

a numerical value representing the significance level of the test.

dist.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, p is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

Returns the defuzzified p-value and the decision made.

Calculates the p-value of fuzzy observations taken from a normal distribution

Description

Calculates the p-value of fuzzy observations taken from a normal distribution

Usage

p.value.normal(
  type,
  H0,
  H1,
  t,
  n,
  s.d,
  sig,
  dist.type,
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100
)

Arguments

type

a category betwenn "0", "1" and "2". The category "0" refers to a bilateral test, the category "1" for a lower unilateral one, and "2" for an upper unilateral test.

H0

a trapezoidal or a triangular fuzzy number representing the fuzzy null hypothesis.

H1

a trapezoidal or a triangular fuzzy number representing the fuzzy alternative hypothesis.

t

a given numerical or fuzzy type parameter of the distribution.

n

the total number of observations of the data set.

s.d

a numerical value for the standard deviation of the distribution.

sig

a numerical value representing the significance level of the test.

dist.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, p is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

Returns the defuzzified p-value and the decision made.

Calculates the p-value of fuzzy observations taken from a Poisson distribution

Description

Calculates the p-value of fuzzy observations taken from a Poisson distribution

Usage

p.value.poisson(
  type,
  H0,
  H1,
  t,
  n,
  sig,
  dist.type,
  i = 1,
  j = 1,
  theta = 1/3,
  thetas = 1,
  p = 2,
  q = 0.5,
  breakpoints = 100,
  s.d = 1
)

Arguments

type

a category betwenn "0", "1" and "2". The category "0" refers to a bilateral test, the category "1" for a lower unilateral one, and "2" for an upper unilateral test.

H0

a trapezoidal or a triangular fuzzy number representing the fuzzy null hypothesis.

H1

a trapezoidal or a triangular fuzzy number representing the fuzzy alternative hypothesis.

t

a given numerical or fuzzy type parameter of the distribution.

n

the total number of observations of the data set.

sig

a numerical value representing the significance level of the test.

dist.type

type of distance chosen from the family of distances. The different choices are given by: "Rho1", "Rho2", "Bertoluzza", "Rhop", "Delta.pq", "Mid/Spr", "wabl", "DSGD", "DSGD.G", "GSGD".

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

thetas

p

a positive integer such that 1 \le p < infinity, referring to the parameter of the Rho_p and Delta_pq. By default, p is fixed to 2.

q

a decimal value between 0 and 1, referring to the parameter of the metric Delta_pq. By default, p is fixed to 0.5.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

s.d

a numerical value for the standard deviation of the distribution.

Value

Returns the defuzzified p-value and the decision made.

Square a number

Description

Takes any numerical value and squares it.

Usage

square(x)

Arguments

x

A numeric value to be squared

Value

The square of the input

Fuzzifies a variable modelled by trapezoidal or triangular fuzzy numbers

Description

Fuzzifies a variable modelled by trapezoidal or triangular fuzzy numbers

Usage

tr.gfuzz(data, breakpoints = 100)

Arguments

data

a matrix of 4 columns (p,q,r,s), where p \le q \le r \le s. No NA are allowed.

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. breakpoints is fixed to 100 by default.

Value

A 3-dimensional array with dimensions (m,n,2), i.e. m lines, n columns, with no NA.

Examples

data <- matrix(c(1,1,2,2,3,3,4,4),ncol=4)
data.tr <- tr.gfuzz(data)

Calculates a distance by the d_wabl between fuzzy numbers

Description

Calculates a distance by the d_wabl between fuzzy numbers

Usage

wabl(X, Y, i = 1, j = 1, theta = 1/3, breakpoints = 100)

Arguments

X

a fuzzy number.

Y

a fuzzy number.

i

parameter of the density function of the Beta distribution, fixed by default to i = 1.

j

parameter of the density function of the Beta distribution, fixed by default to j = 1.

theta

breakpoints

a positive arbitrary integer representing the number of breaks chosen to build the numerical alpha-cuts. It is fixed to 100 by default.

Value

A numerical value.