Package {FuzzyRandom}

Type: Package
Title: Level-Wise Statistics for Fuzzy Random Variables
Version: 0.1.0
Date: 2026-10-06
Description: Level-wise statistical methods for fuzzy random variables, implementing the Puri-Ralescu framework. Provides level-wise empirical distribution functions, moments, covariance and correlation, a level-wise Kolmogorov-Smirnov goodness-of-fit test, an approximate fuzzy normal distribution with simulation, and level-wise central limit theorem and law of large numbers inference. The package only depends on base R.
License: GPL-3
Encoding: UTF-8
Imports: stats, graphics, grDevices
Suggests: testthat (≥ 3.0.0), knitr, rmarkdown
VignetteBuilder: knitr
Config/testthat/version: 3
Config/roxygen2/version: 8.1.0
NeedsCompilation: no
Packaged: 2026-10-10 04:42:13 UTC; GUNJAN AGGRWAL
Author: Gunjan Aggrwal ORCID iD [aut, cre], Mukul Bijalwan ORCID iD [aut], Mukul Jain [aut]
Maintainer: Gunjan Aggrwal <gunjanuk1994@gmail.com>
Repository: CRAN
Date/Publication: 2026-10-21 14:20:17 UTC

Statistics of Fuzzy Random Variables: a Puri-Ralescu Framework

Description

FuzzyRandom is a computationally self-contained implementation of the level-wise ("Puri-Ralescu") theory of statistics for fuzzy random variables.

Details

Fuzzy numbers are represented through their alpha-level sets ⁠[X^-(alpha), X^+(alpha)]⁠, and every statistical operator is defined level-wise, so that the expectation, variance, covariance, empirical distribution, law of large numbers and central limit theorem are all handled in a single coherent framework.

The key design choices are:

The package only depends on base R (stats, graphics).

References

Puri, M. L. and Ralescu, D. A. (1986). Fuzzy random variables. Journal of Mathematical Analysis and Applications, 114(2), 409-422.

Klement, E. P., Puri, M. L. and Ralescu, D. A. (1986). Limit theorems for fuzzy random variables. Proceedings of the Royal Society A, 407, 171-182.

Feng, Y., Hu, L. and Shu, H. (2002). The variance and covariance of fuzzy random variables and their applications. Fuzzy Sets and Systems, 128, 335-345.

A fuzzy number given by its alpha-level cuts

Description

Build a FuzzyNumber object from its alpha-level envelopes. When alpha is not supplied it is taken as a regularly spaced grid on [0, 1] with one node per supplied endpoint.

Usage

FuzzyNumber(alpha = NULL, lower, upper)

Arguments

alpha

Numeric vector in [0, 1], strictly increasing.

lower

Numeric; lower envelope, non-decreasing in alpha.

upper

Numeric; upper envelope, non-increasing in alpha.

Value

An object of class "FuzzyNumber".

Examples


fn <- FuzzyNumber(alpha = c(0, 0.5, 1), lower = c(1, 2, 2), upper = c(4, 2, 2))
cuts(fn)

Build a sample of fuzzy random variables

Description

Build a sample of fuzzy random variables

Usage

FuzzySample(x, alpha = NULL)

Arguments

x

Either a list of FuzzyNumber objects, an (n, m, 2) numeric array of cuts (dimension 3 = c("lower", "upper")), or an existing FuzzySample.

alpha

Alpha grid shared by all observations. Required when x is an array with an unlabelled third dimension; otherwise inferred.

Value

An object of class "FuzzySample".

Examples

set.seed(1)
s <- FuzzySample(list(
  triangular_fuzzy(0, 1, 2),
  triangular_fuzzy(1, 2, 3),
  trapezoidal_fuzzy(0.5, 1, 3, 4)))
n_obs(s)

Alpha grid of a fuzzy object

Description

Alpha grid of a fuzzy object

Usage

alpha_levels(x)

Arguments

x

A FuzzyNumber or FuzzySample.

Value

Numeric vector of alpha levels.

Examples

alpha_levels(triangular_fuzzy(1, 2, 4))[1:3]

Coerce to a FuzzySample

Description

Coerces from a list of FuzzyNumber, an array, a data frame (with columns id, alpha, lower, upper) or a single FuzzyNumber (giving a sample of size 1).

Usage

as_fuzzy_sample(x, alpha = NULL, ...)

## Default S3 method:
as_fuzzy_sample(x, alpha = NULL, ...)

## S3 method for class 'FuzzyNumber'
as_fuzzy_sample(x, alpha = NULL, ...)

## S3 method for class 'data.frame'
as_fuzzy_sample(x, alpha = NULL, ...)

Arguments

x

Object to coerce.

alpha

Alpha grid (for array/data-frame input).

...

Unused.

Value

A FuzzySample.

Examples

fs <- as_fuzzy_sample(triangular_fuzzy(0, 1, 2))
n_obs(fs)

Central limit theorem for a fuzzy random sample

Description

Uses the level-wise central limit theorem (Puri & Ralescu, 1986): at each level alpha the sample-mean endpoints are approximately normal, so the fuzzy sample mean Xbar is approximately a fuzzy_normal with mean expectation(x) and level- wise covariance variance(x) / n.

Usage

clt(x, alpha = NULL, ...)

Arguments

x

A FuzzySample.

alpha

Numeric levels (default: those of x).

...

Unused.

Value

An object of class c("FuzzyCLT","FuzzyNormal"). Its cov is the covariance of Xbar (i.e. Sigma / n); sigma is the limiting covariance Sigma.

Examples

set.seed(7)
s <- rnorm_fuzzy(500, triangular_fuzzy(0, 1, 2), diag(2) * 0.25)
cl <- clt(s); summary(cl)
simulate(cl, nsim = 4)

Correlation of two fuzzy random variables (sample)

Description

Level-wise Pearson correlation between the lower endpoints (cor(X^-(alpha), Y^-(alpha))) and between the upper endpoints (cor(X^+(alpha), Y^+(alpha))), plus the scalar reduction

\rho_{FHS}(X,Y)=\tfrac12\int_0^1\bigl[r_{--}(a)+r_{++}(a)\bigr]\,da.

Usage

correlation(x, y = NULL, alpha = NULL, ...)

## S3 method for class 'FuzzySample'
correlation(x, y = NULL, alpha = NULL, ...)

Arguments

x, y

Two FuzzySample objects.

alpha

Optional alpha grid.

...

Unused.

Value

An object of class "FuzzyCorrelation" (with a scalar slot).

Examples

set.seed(5)
x <- FuzzySample(replicate(50, triangular_fuzzy(0, 1, 2), simplify = FALSE))
y <- FuzzySample(replicate(50, triangular_fuzzy(0, 1, 2), simplify = FALSE))
correlation(x, y)

Covariance of two fuzzy random variables (sample)

Description

Level-wise covariance. At each level alpha the covariance of X and Y is the 2 x 2 matrix

Cov(X^-, Y^-) Cov(X^-, Y^+)
Cov(X^+, Y^-) Cov(X^+, Y^+).

When y is omitted, x is paired with itself and the result is the variance. A scalar reduction through the Feng–Hu–Shu integral

\mathrm{Cov}_{FHS}(X,Y)=\tfrac12\int_0^1\bigl[\mathrm{Cov}(X^-(a),Y^-(a))+ \mathrm{Cov}(X^+(a),Y^+(a))\bigr]\,da

is also returned.

Usage

covariance(x, y = NULL, alpha = NULL, ...)

## S3 method for class 'FuzzySample'
covariance(x, y = NULL, alpha = NULL, ...)

## S3 method for class 'FuzzySample'
cov(x, y = NULL, ...)

Arguments

x, y

Two FuzzySample objects (or x alone for the variance).

alpha

Optional alpha grid.

...

Unused.

Value

A FuzzyCovariance object.

Examples

set.seed(3)
x <- FuzzySample(replicate(30, triangular_fuzzy(0, 1, 2), simplify = FALSE))
y <- FuzzySample(replicate(30, triangular_fuzzy(1, 2, 3), simplify = FALSE))
covariance(x, y)

Cut of a fuzzy number at a single level

Description

Cut of a fuzzy number at a single level

Usage

cut_at(x, alpha, ...)

## S3 method for class 'FuzzyNumber'
cut_at(x, alpha = 0.5, ...)

Arguments

x

A FuzzyNumber.

alpha

A single level in ⁠[0, 1]⁠.

...

Unused.

Value

A named numeric vector c(lower = ..., upper = ...).

Examples

cut_at(triangular_fuzzy(1, 2, 4), 0.5)

Alpha-level cuts of a fuzzy number

Description

Alpha-level cuts of a fuzzy number

Usage

cuts(x, alpha = NULL, ...)

## S3 method for class 'FuzzyNumber'
cuts(x, alpha = NULL, ...)

Arguments

x

A FuzzyNumber (or other compatible object).

alpha

Optional alpha grid at which to evaluate the cuts.

...

Unused.

Value

A numeric matrix with columns lower and upper.

Examples

fn <- triangular_fuzzy(1, 2, 4)
cuts(fn, alpha = c(0, 0.5, 1))

Default alpha grid

Description

A fine, regularly spaced grid on the closed unit interval used whenever a level-wise grid is not supplied explicitly by the caller.

Usage

default_alpha()

Value

A numeric vector of length 51 on [0, 1], containing 0 and 1.

Internal 2D ellipse points

Description

Internal helper used by plot.FuzzyCovariance.

Usage

ellipse_points(Sigma, mu, level = 0.95)

Expectation of a fuzzy random variable (sample)

Description

Level-wise expectation. For a sample the estimator is the level-wise mean of the alpha-cuts:

[E[X]]_{\alpha} = \Bigl[\,\frac1n\sum_i X_i^{-}(\alpha),\; \frac1n\sum_i X_i^{+}(\alpha)\Bigr].

Usage

expectation(x, alpha = NULL, ...)

## S3 method for class 'FuzzySample'
expectation(x, alpha = NULL, ...)

## S3 method for class 'FuzzySample'
mean(x, ...)

Arguments

x

A FuzzySample.

alpha

Optional alpha grid on which to evaluate.

...

Unused.

Value

A FuzzyNumber, the sample expectation.

Examples

set.seed(2)
s <- FuzzySample(replicate(40, triangular_fuzzy(-1, 0, 1), simplify = FALSE))
ex <- expectation(s); plot(ex, main = "sample mean")

Empirical distribution function of a fuzzy random sample

Description

Level-wise empirical cumulative distribution functions of the alpha-cut endpoints.

Usage

fuzzy_ecdf(x, alpha = NULL, at = NULL, ...)

## S3 method for class 'FuzzySample'
fuzzy_ecdf(x, alpha = NULL, at = NULL, ...)

Arguments

x

A FuzzySample.

alpha

Numeric vector of levels (default: the levels stored in x).

at

Numeric grid on which to evaluate the CDFs (default: 201 points over the data range).

...

Unused.

Value

An object of class "FuzzyECDF".

Examples

set.seed(8)
s <- FuzzySample(replicate(40, triangular_fuzzy(0, 1, 2), simplify = FALSE))
fe <- fuzzy_ecdf(s, alpha = c(0, 0.25, 0.5, 0.75, 1))
plot(fe)

A fuzzy number from a membership function evaluated on a grid

Description

Reconstructs the alpha-cuts of a (normal, quasi-concave) fuzzy number from a membership function evaluated on a grid. The level set {x : mu(x) >= alpha} is assumed to be a single interval.

Usage

fuzzy_from_membership(x, mu, alpha = NULL)

Arguments

x

Sorted numeric grid of the support (unsorted grids are sorted).

mu

Membership values in [0, 1] evaluated at x.

alpha

Optional alpha grid.

Value

An object of class "FuzzyNumber".

Examples

x  <- seq(-3, 3, length.out = 201)
mu <- pmax(1 - abs(x), 0)            # triangular membership
fn <- fuzzy_from_membership(x, mu)
plot(fn)

Construct a fuzzy normal distribution

Description

A fuzzy normal distribution is the family, over levels alpha, of bivariate-normal laws on the alpha-cut endpoints with fuzzy mean mean and level-wise covariance cov.

Usage

fuzzy_normal(mean, cov = NULL, n = NULL)

Arguments

mean

A FuzzyNumber (the fuzzy mean).

cov

A 2 x 2 covariance matrix, a length-4 vector, or a FuzzyCovariance whose levels match mean.

n

Optional sample size associated with the mean (used by clt).

Value

An object of class "FuzzyNormal".

Examples

fn <- fuzzy_normal(triangular_fuzzy(0, 1, 2), diag(2))
simulate(fn, nsim = 5)

Linearly interpolate the alpha-cuts of a fuzzy number

Description

Returns the lower and upper envelopes evaluated on the grid at.

Usage

interpolate_cuts(alpha, lower, upper, at)

Arguments

alpha

Original grid of the fuzzy number (sorted in [0,1]).

lower

Lower envelope on alpha (non-decreasing).

upper

Upper envelope on alpha (non-increasing).

at

Grid on which to evaluate the cuts.

Interpolate a sample onto a new alpha grid

Description

Interpolate a sample onto a new alpha grid

Usage

interpolate_sample(obj, alpha)

Arguments

obj

A FuzzySample object.

alpha

Target alpha grid.

Value

An (n, m, 2) array of cuts on alpha.

Whether an object is a (valid) fuzzy number

Description

Whether an object is a (valid) fuzzy number

Usage

is_fuzzy_number(x)

Arguments

x

Any R object.

Value

Logical; TRUE if x is a valid FuzzyNumber.

Examples

is_fuzzy_number(triangular_fuzzy(0, 1, 2))
is_fuzzy_number(1:3)

Whether an object is a FuzzySample

Description

Whether an object is a FuzzySample

Usage

is_fuzzy_sample(x)

Arguments

x

Any R object.

Value

Logical.

Examples

is_fuzzy_sample(FuzzySample(list(triangular_fuzzy(0, 1, 2))))

One- and two-sample fuzzy Kolmogorov-Smirnov tests

Description

Performs, at each level alpha, a Kolmogorov-Smirnov test on the lower and upper endpoints of the alpha-cuts. With p0 a one-sample test against the CDF p0 is run; otherwise a two-sample test against the second sample y is run. The result is a family of level-wise tests consistent with the Puri-Ralescu level-wise approach.

Usage

ks_test(x, ...)

## S3 method for class 'FuzzySample'
ks_test(x, y = NULL, p0 = NULL, alpha = NULL, ...)

Arguments

x

A FuzzySample.

...

Additional arguments passed on to ks.test.

y

A second FuzzySample for a two-sample test (default NULL).

p0

Null CDF for the one-sample test: a character string naming a CDF in stats (e.g. "pnorm") or a function p0(q).

alpha

Numeric vector of levels (default: those of x).

Value

An object of class "FuzzyKS".

Examples

set.seed(9)
s <- FuzzySample(replicate(50, triangular_fuzzy(-1, 0, 1), simplify = FALSE))
ks_test(s, p0 = "pnorm")

Level-wise sample-mean convergence (LLN)

Description

Following Puri & Ralescu (1986), statistics on fuzzy random variables are evaluated level-wise on the alpha-cuts. This function computes the fuzzy sample mean for an increasing sequence of sample sizes and measures, at a fixed level, the sup-norm distance between the size-s mean and the full-sample mean. By the level-wise (fuzzy) law of large numbers this distance shrinks to zero with the sample size (at rate 1/sqrt(s) under the level-wise CLT).

Usage

lln(x, sizes = NULL, alpha = NULL, levels = c(0.5, 1), ...)

Arguments

x

A FuzzySample.

sizes

Integer vector of sample sizes (default: a grid from 5 up to x$n).

alpha

Optional alpha grid on which to evaluate the means.

levels

Numeric levels at which to measure convergence, used by the summary, as.data.frame and plot methods (default c(0.5, 1)).

...

Unused.

Value

An object of class "FuzzyLLN" with components sizes, means, full and alpha.

Examples

set.seed(23)
s <- rnorm_fuzzy(400, triangular_fuzzy(0, 1, 2), diag(2) * 0.25)
w <- lln(s)
plot(w)
summary(w)

Sup-norm distance of each mean to the full mean, at the requested levels

Description

Sup-norm distance of each mean to the full mean, at the requested levels

Usage

lln_distances(means, full, alpha, levels)

Arguments

means

List of FuzzyNumber means (size-ascending).

full

The full-sample FuzzyNumber mean.

alpha

Alpha grid shared by means and full.

levels

Levels at which the distance is measured.

Value

A matrix with one column per level.

Sample from a bivariate normal distribution

Description

n independent draws from N_2(mu, Sigma) using the Cholesky factorisation (after repair of Sigma).

Usage

mvrnorm_2d(n, mu, Sigma)

Arguments

n

Number of draws.

mu

Length-2 numeric mean vector.

Sigma

2x2 covariance matrix.

Value

An nx2 numeric matrix; column 1 is the lower endpoint, column 2 the upper endpoint of a level cut.

Number of alpha levels of a fuzzy object

Description

Number of alpha levels of a fuzzy object

Usage

n_cuts(x)

Arguments

x

A FuzzyNumber or FuzzySample.

Value

Length-one integer.

Examples

n_cuts(triangular_fuzzy(1, 2, 4))

Number of observations in a sample

Description

Number of observations in a sample

Usage

n_obs(x)

Arguments

x

A FuzzySample.

Value

Length-one integer.

Examples

s <- FuzzySample(list(triangular_fuzzy(0, 1, 2), triangular_fuzzy(1, 2, 3)))
n_obs(s)

Internal constructor for a FuzzyCovariance object

Description

Internal constructor for a FuzzyCovariance object

Usage

new_FuzzyCovariance(alpha, cov, scalar, type)

Arguments

alpha

Alpha grid.

cov

Array of dim c(m, 2, 2) (rows/cols: lower, upper).

scalar

Scalar reduction (Feng–Hu–Shu integral).

type

"variance" or "covariance".

Internal constructor for a FuzzyNumber (no validation)

Description

Internal constructor for a FuzzyNumber (no validation)

Usage

new_FuzzyNumber(alpha, lower, upper)

Arguments

alpha

Alpha grid.

lower, upper

Envelopes.

Internal constructor (no validation) for FuzzySample objects

Description

Internal constructor (no validation) for FuzzySample objects

Usage

new_FuzzySample(arr, alpha)

Arguments

arr

An (n, m, 2) array of cuts.

alpha

The shared alpha grid.

Value

A FuzzySample object.

Project per-level endpoints onto the cone of valid nested cuts

Description

Enforces lower <= upper and monotonicity (lower non-decreasing, upper non-increasing in alpha).

Usage

project_to_valid_cuts(lower, upper, alpha)

Arguments

lower, upper

Numeric vectors, one entry per level.

alpha

Sorted grid in [0,1].

Value

A length-2 list with components lower and upper.

Repair a 2x2 covariance matrix for positive (semi)definiteness

Description

Symmetrizes Sigma and, if it is not positive (semi)definite, adds a small diagonal ridge so that a Cholesky factor exists.

Usage

repair_cov_2x2(Sigma, eps = 1e-09)

Arguments

Sigma

A 2x2 numeric matrix.

eps

Smallest eigenvalue threshold.

Value

A symmetric, positive-definite 2x2 numeric matrix.

Sample fuzzy random variables from a fuzzy normal distribution

Description

Draws n fuzzy numbers whose alpha-cut endpoints have the bivariate-normal level-wise distribution N_2(mu_alpha, Sigma_alpha).

Usage

rnorm_fuzzy(n, mean, cov = NULL, valid = TRUE, seed = NULL)

## S3 method for class 'FuzzyNormal'
simulate(object, nsim = 1, ...)

## S3 method for class 'FuzzyCLT'
simulate(object, nsim = 1, ...)

Arguments

n

Sample size.

mean

A FuzzyNumber (the fuzzy mean) or a FuzzyNormal.

cov

A 2 x 2 covariance matrix, a length-4 vector, or a FuzzyCovariance. Ignored if mean is a FuzzyNormal.

valid

Logical; whether to project draws to valid fuzzy numbers.

seed

Optional integer seed (NULL leaves the RNG alone).

object

A FuzzyNormal (or FuzzyCLT) object.

nsim

Number of simulations.

...

Unused.

Details

When valid = TRUE (the default) every draw is projected onto the cone of valid, nested alpha-cuts (lower <= upper; lower non-decreasing and upper non-increasing in alpha) so that it is a proper fuzzy number. When valid = FALSE the raw, unprojected level-wise bivariate-normal cuts are returned – useful for checking the level-wise central limit theorem, where projection would bias the marginals.

Value

A FuzzySample of n fuzzy random variables.

Examples

set.seed(123)
s <- rnorm_fuzzy(80, triangular_fuzzy(0, 1, 2), diag(2) * 0.25)
plot(s, main = "fuzzy normal sample")

Scalar covariance (Feng–Hu–Shu integral)

Description

spectral_covariance returns the scalar covariance

\mathrm{Cov}_{FHS}(X,Y)=\tfrac12\int_0^1\bigl[\mathrm{Cov}(X^-(a),Y^-(a))+ \mathrm{Cov}(X^+(a),Y^+(a))\bigr]\,da.

Usage

spectral_covariance(x, y = NULL, alpha = NULL, ...)

## S3 method for class 'FuzzySample'
spectral_covariance(x, y = NULL, alpha = NULL, ...)

Arguments

x, y

Two FuzzySample objects (or x alone).

alpha

Optional alpha grid.

...

Unused.

Value

A length-one numeric.

Examples

set.seed(6)
x <- FuzzySample(replicate(30, triangular_fuzzy(0, 1, 2), simplify = FALSE))
spectral_covariance(x)

Scalar variance (Feng–Hu–Shu integral)

Description

spectral_variance returns spectral_covariance(x, x).

Usage

spectral_variance(x, alpha = NULL, ...)

Arguments

x

A FuzzySample.

alpha

Optional alpha grid.

...

Unused.

Value

A length-one numeric.

Examples

set.seed(7)
x <- FuzzySample(replicate(30, triangular_fuzzy(0, 1, 2), simplify = FALSE))
spectral_variance(x)

Standardise an alpha grid

Description

Ensures a strictly increasing grid on [0, 1] containing 0 and 1.

Usage

standardise_alpha(alpha = NULL)

Arguments

alpha

User-supplied alpha grid (may be NULL).

Trapezoidal fuzzy number

Description

The trapezoidal fuzzy number with support [a, d] and core [b, c] (a <= b <= c <= d).

Usage

trapezoidal_fuzzy(a, b, c, d, alpha = NULL)

Arguments

a, b, c, d

Scalars with a <= b <= c <= d.

alpha

Optional alpha grid.

Value

An object of class "FuzzyNumber".

Examples

plot(trapezoidal_fuzzy(1, 2, 3, 5))

Trapezoidal quadrature over an alpha grid

Description

Approximates \int_0^1 f(\alpha)\,d\alpha by the composite trapezoidal rule applied to the values of f on the grid alpha.

Usage

trapz_alpha(f, alpha = default_alpha())

Arguments

f

Numeric vector of function values aligned with alpha.

alpha

Numeric vector of grid points, sorted in [0,1].

Value

A length-one numeric, the approximated integral.

Triangular fuzzy number

Description

The triangular fuzzy number with support [a, c] and core b (a <= b <= c). Its alpha-cuts are [a + (b-a)*alpha, c - (c-b)*alpha].

Usage

triangular_fuzzy(a, b, c, alpha = NULL)

Arguments

a, b, c

Scalars with a <= b <= c.

alpha

Optional alpha grid (see FuzzyNumber).

Value

An object of class "FuzzyNumber".

Examples

plot(triangular_fuzzy(1, 2, 4))

Validate the structure of fuzzy-number cuts

Description

Validate the structure of fuzzy-number cuts

Usage

validate_cuts(alpha, lower, upper)

Arguments

alpha

Sorted grid in [0,1].

lower

Non-decreasing lower envelope.

upper

Non-increasing upper envelope, with lower <= upper.

Variance of a fuzzy random variable (sample)

Description

Level-wise variance, i.e. covariance of a sample with itself. Returns the level-wise 2 x 2 matrix [[Var(X^-), Cov(X^-,X^+)], [Cov(X^+,X^-), Var(X^+)] at each level, together with the scalar Feng–Hu–Shu variance

\mathrm{Var}_{FHS}(X)=\tfrac12\int_0^1\bigl[\mathrm{Var}(X^-(a))+ \mathrm{Var}(X^+(a))\bigr]\,da.

Usage

variance(x, alpha = NULL, ...)

## S3 method for class 'FuzzySample'
variance(x, alpha = NULL, ...)

## S3 method for class 'FuzzySample'
var(x, ...)

Arguments

x

A FuzzySample.

alpha

Optional alpha grid.

...

Unused.

Value

A FuzzyCovariance object (with type = "variance").

Examples

set.seed(4)
s <- FuzzySample(replicate(30, triangular_fuzzy(0, 1, 2), simplify = FALSE))
variance(s)