Package {IFFuzzyStats}

Type:	Package
Title:	Intuitionistic Fuzzy Statistics
Version:	0.1.0
Maintainer:	Mukul Bijalwan <mukulbijalwan555@gmail.com>
Description:	Provides constructors and arithmetic for intuitionistic fuzzy values (intuitionistic fuzzy value, IFV) for intuitionistic fuzzy sets (Atanassov (1986) <doi:10.1016/S0165-0114(86)80034-3>), including aggregation operators of Xu (2007) <doi:10.1109/TFUZZ.2006.890678> and Xu and Yager (2006) <doi:10.1080/03081070600574353>: intuitionistic fuzzy weighted averaging (IFWA), intuitionistic fuzzy weighted geometric (IFWG), intuitionistic fuzzy ordered weighted averaging (IFOWA), intuitionistic fuzzy ordered weighted geometric (IFOWG), intuitionistic fuzzy hybrid averaging (IFHA), intuitionistic fuzzy hybrid geometric (IFHG), and Einstein operators; entropy (Szmidt and Kacprzyk (2000) <doi:10.1016/S0165-0114(98)00402-3>; Vlachos and Sergiadis (2007) <doi:10.1016/j.patrec.2006.07.004>) and distance and similarity measures, including Ye (2011) <doi:10.1016/j.mcm.2010.07.022> cosine similarity and Hamming, Euclidean and Hausdorff distances; intuitionistic fuzzy c-means and k-means clustering; approximate reasoning via compositional inference; and multi-criteria decision support via intuitionistic fuzzy Technique for Order of Preference by Similarity to Ideal Solution (IF-TOPSIS) (Boran et al. (2009) <doi:10.1016/j.eswa.2009.03.039>) and score ranking. All computations use only base R with fully vectorised implementations.
License:	GPL-3
Encoding:	UTF-8
Suggests:	testthat (≥ 3.0.0), knitr, rmarkdown
VignetteBuilder:	knitr
Config/testthat/edition:	3
URL:	https://github.com/MukulBijalwan/IFFuzzyStats
BugReports:	https://github.com/MukulBijalwan/IFFuzzyStats/issues
NeedsCompilation:	no
Config/roxygen2/version:	8.1.0
Packaged:	2026-10-09 05:37:41 UTC; Admin
Author:	Mukul Bijalwan [aut, cre], Gunjan Aggrwal [aut], Mukul Jain [aut]
Repository:	CRAN
Date/Publication:	2026-10-21 08:40:24 UTC

Intuitionistic Fuzzy Statistics Toolkit

Description

A comprehensive toolkit for intuitionistic fuzzy sets (Atanassov, 1986), where each element carries membership \mu, non-membership \nu and hesitation \pi = 1 - \mu - \nu. Provides constructors and arithmetic, Xu/Yager aggregation operators, entropy and similarity measures, fuzzy clustering, compositional inference, and IF-TOPSIS.

References

Atanassov, K. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87-96. Xu, Z. (2007). Intuitionistic fuzzy aggregation operators. IEEE Trans. Fuzzy Systems, 15(6), 1179-1187. Szmidt, E. & Kacprzyk, J. (2000). Distances between intuitionistic fuzzy sets. Fuzzy Sets and Systems, 114(3), 505-518.

Intuitionistic fuzzy arithmetic (Xu operations)

Description

Addition, multiplication, scalar multiplication and power for IFVs, following Xu (2007): for a=(\mu_a,\nu_a), b=(\mu_b,\nu_b), \lambda>0:

a\oplus b=(\mu_a+\mu_b-\mu_a\mu_b,\ \nu_a\nu_b)

a\otimes b=(\mu_a\mu_b,\ \nu_a+\nu_b-\nu_a\nu_b)

\lambda a=(1-(1-\mu_a)^\lambda,\ \nu_a^\lambda)

a^\lambda=(\mu_a^\lambda,\ 1-(1-\nu_a)^\lambda)

Usage

if_add(a, b)

if_mul(a, b)

if_scalar_mul(lambda, a)

if_pow(a, lambda)

if_einstein_add(a, b)

if_einstein_mul(a, b)

if_einstein_scalar_mul(lambda, a)

if_einstein_pow(a, lambda)

Arguments

Details

Einstein variants use Einstein t-norm / t-conorm:

a\oplus_\epsilon b= \left(\frac{\mu_a+\mu_b}{1+\mu_a\mu_b}, \frac{\nu_a\nu_b}{1+(1-\nu_a)(1-\nu_b)}\right)

a\otimes_\epsilon b= \left(\frac{\mu_a\mu_b}{1+(1-\mu_a)(1-\mu_b)}, \frac{\nu_a+\nu_b}{1+\nu_a\nu_b}\right)

Value

An ifv object.

References

Xu, Z. (2007). Intuitionistic fuzzy aggregation operators. IEEE Trans. Fuzzy Systems, 15(6), 1179-1187.

Examples

a <- ifv(0.6, 0.3); b <- ifv(0.5, 0.3)
if_add(a, b); if_mul(a, b); if_scalar_mul(2, a); if_pow(a, 2)

Aggregate rows of an IF decision matrix

Description

Aggregate rows of an IF decision matrix

Usage

if_aggregate_matrix(mat, w, ncrit, fun = ifwa)

Arguments

Value

ifv vector (one fused value per alternative).

Examples

m <- ifv(c(0.6, 0.5, 0.7, 0.4), c(0.2, 0.3, 0.1, 0.4))
# 2 alternatives x 2 criteria stored row-wise:
if_aggregate_matrix(m, w = c(0.6, 0.4), ncrit = 2, fun = ifwa)

Validate IFVs

Description

Validate IFVs

Usage

if_check(x, tol = 1e-09)

Arguments

Value

invisibly TRUE; errors if invalid.

Complement of an IFV

Description

Complement of an IFV

Usage

if_comp(x)

Arguments

Value

ifv with mu/nu swapped.

Examples

if_comp(ifv(0.6, 0.2))

Intuitionistic fuzzy relations: composition and inference

Description

Max-min / max-product composition of IF relations and the compositional rule of inference (Zadeh, 1973, extended to IFS).

Usage

if_compose(R, S, nx, ny, nz, op = c("maxmin", "maxprod"))

Arguments

Details

An IF relation on X \times Y is a matrix of IFVs. Composition R \circ S on X \times Z is:

(R \circ S)(x,z) = \bigoplus_{y} (R(x,y) \otimes S(y,z))

where \oplus,\otimes are max-min (default) or max-product. Membership composes with t-conorm over t-norm, non-membership with the dual operation (min over max for max-min; product-sum dual for max-product).

Value

An ifv vector of length nx*nz (row-major).

Examples

R <- ifv(c(0.6, 0.3, 0.8, 0.2), c(0.2, 0.5, 0.1, 0.6))  # 2x2
S <- ifv(c(0.5, 0.7, 0.4, 0.9), c(0.3, 0.1, 0.4, 0.0))  # 2x2
if_compose(R, S, nx = 2, ny = 2, nz = 2)

Defuzzify IFVs to crisp scores

Description

Defuzzify IFVs to crisp scores

Usage

if_defuzzify(x, method = c("score", "expected", "accuracy", "mu"))

Arguments

Value

numeric vector.

Examples

if_defuzzify(ifv(c(0.6, 0.5), c(0.3, 0.2)))

Distances between intuitionistic fuzzy sets

Description

For sets A,B of equal cardinality n with elements (\mu^A_i,\nu^A_i,\pi^A_i), (\mu^B_i,\nu^B_i,\pi^B_i):

Usage

if_dist(
  a,
  b,
  method = c("hamming", "euclidean", "hausdorff", "ye", "ye_cosine", "park"),
  w = NULL
)

Arguments

Details

• "hamming" (normalised Szmidt-Kacprzyk): \frac{1}{2n}\sum(|\Delta\mu|+|\Delta\nu|+|\Delta\pi|).

• "euclidean" (normalised): \sqrt{\frac{1}{2n}\sum((\Delta\mu)^2+(\Delta\nu)^2+(\Delta\pi)^2)}.

• "hausdorff" (normalised Hamming-Hausdorff): \frac{1}{n}\sum\max(|\Delta\mu|,|\Delta\nu|,|\Delta\pi|).

• "ye" (Ye's modified Hamming-type distance): \sum w_i \frac{|\Delta\mu_i|+|\Delta\nu_i|} {|\Delta\mu_i|+|\Delta\nu_i| +\min(\mu^A_i,\mu^B_i)+\min(\nu^A_i,\nu^B_i) +\min(\pi^A_i,\pi^B_i)+1}.

• "ye_cosine": 1 - S_C where S_C is Ye's cosine similarity (see if_sim).

• "park": normalised Hamming including hesitation (same numeric form as "hamming" here).

Weighted versions: pass w (sums to 1) to weight elements: D_w=\sum w_i d_i with per-element d_i.

Value

numeric scalar distance.

References

Szmidt, E. & Kacprzyk, J. (2000). Distances between intuitionistic fuzzy sets. Fuzzy Sets and Systems, 114(3), 505-518.

Ye, J. (2012). Multicriteria decision-making method using the Dice similarity measure based on the reduct intuitionistic fuzzy sets of interval-valued intuitionistic fuzzy sets. Applied Mathematical Modelling, 36(9), 4466-4472.

Examples

a <- ifv(c(0.6, 0.5), c(0.3, 0.2)); b <- ifv(c(0.5, 0.4), c(0.3, 0.3))
if_dist(a, b, "hamming"); if_dist(a, b, "euclidean"); if_dist(a, b, "hausdorff")

Entropy of intuitionistic fuzzy sets

Description

Fuzziness / intuitionism of a set A=\{a_1,\dots,a_n\}. All measures lie in [0,1] with larger = more fuzzy.

Usage

if_entropy(x, method = c("burillo", "szmidt", "vlachos", "zeng_li"))

Arguments

Details

• "burillo" (Burillo & Bustince, 1996): E = \frac{1}{n}\sum \pi_i.

• "szmidt" (Szmidt & Kacprzyk, 2001): E = \frac{1}{n}\sum \frac{\min(\mu_i,\nu_i)+\pi_i}{\max(\mu_i,\nu_i)+\pi_i} (define 0/0:=1).

• "vlachos" (Vlachos & Sergiadis, 2007; De Luca-Termini extension): E = \frac{-1}{n\ln 2}\sum [\mu_i\ln\mu_i+\nu_i\ln\nu_i-(1-\pi_i)\ln(1-\pi_i)-\pi_i\ln 2], with 0\ln 0:=0.

• "zeng_li" (Zeng & Li, 2006): E = \frac{1}{n}\sum (1-|\mu_i-\nu_i|).

Value

numeric scalar in [0,1].

References

Burillo, P. & Bustince, H. (1996). Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets. Fuzzy Sets and Systems, 78(3), 305-316.

Szmidt, E. & Kacprzyk, J. (2001). Entropy for intuitionistic fuzzy sets. Fuzzy Sets and Systems, 118(3), 467-477.

Vlachos, I. K. & Sergiadis, G. D. (2007). Intuitionistic fuzzy information - applications to pattern recognition. Pattern Recognition Letters, 28(2), 197-206.

Zeng, L. & Li, H. (2006). Normalized distance, similarity measure, intuitionistic fuzzy entropy and application. Proc. World Congress on Intelligent Control and Automation, 3857-3860.

Examples

x <- ifv(c(0.5, 0.6, 0.4), c(0.3, 0.2, 0.4))
if_entropy(x, "burillo"); if_entropy(x, "szmidt")
if_entropy(x, "vlachos"); if_entropy(x, "zeng_li")

Fuzzify numeric data into intuitionistic fuzzy values

Description

Converts a numeric vector into IFVs by normalising into [0,1] to get membership, then deriving non-membership as \nu = (1-\mu)\times \nu\_ratio so hesitation absorbs the remainder. Useful for feeding measurements into the IF toolkit.

Usage

if_fuzzify(x, mu_range = range(x, na.rm = TRUE), nu_ratio = 0.6)

Arguments

Value

ifv vector.

Examples

if_fuzzify(1:5)
if_fuzzify(c(10, 20, 30), nu_ratio = 0.5)

Compositional rule of inference for IFVs

Description

Given a fact (IF set on X, length nx) and an IF implication relation on X \times Y (nx*ny values row-major), infer the consequent on Y via max-min composition.

Usage

if_infer(fact, relation, nx, ny, op = c("maxmin", "maxprod"))

Arguments

Value

ifv of length ny.

Examples

fact <- ifv(c(0.8, 0.2), c(0.1, 0.7))
rel <- ifv(c(0.6, 0.3, 0.8, 0.2), c(0.2, 0.5, 0.1, 0.6))
if_infer(fact, rel, nx = 2, ny = 2)

Hesitation degree

Description

Hesitation degree

Usage

if_pi(x)

Arguments

Value

numeric vector of \pi = 1 - \mu - \nu.

Rank alternatives from aggregated IFVs

Description

Rank alternatives from aggregated IFVs

Usage

if_rank_df(x, labels = NULL)

Arguments

Value

ranked data.frame.

Examples

if_rank_df(ifv(c(0.6, 0.5, 0.7), c(0.3, 0.2, 0.1)))

Mamdani-style IF rule block evaluation

Description

Evaluate nrules rules, each mapping an nx-vector input fact through its own nx*ny relation, then aggregate consequents by element-wise max (mu) / min (nu) union.

Usage

if_rule_block(fact, relations, nx, ny, op = c("maxmin", "maxprod"))

Arguments

Value

ifv length ny.

Score, accuracy and ranking of IFVs

Description

Score S=\mu-\nu \in [-1,1], accuracy H=\mu+\nu \in [0,1] (Chen & Tan, 1994; Hong & Choi, 2000). Ranking is lexicographic: larger score first, ties broken by larger accuracy, then larger hesitation-handling via tie = "pi_min" (smaller hesitation wins).

Usage

if_score(x)

if_accuracy(x)

if_hesitation(x)

if_rank(x)

if_order(x)

if_best(x)

Arguments

Value

if_score()/if_accuracy()/if_hesitation() numeric vectors; if_rank() integer ranks (1 = best); if_order() ordering permutation (best first); if_best() best element(s).

References

Chen, S. M. & Tan, J. M. (1994). Handling multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets and Systems, 67(2), 163-172.

Hong, D. H. & Choi, C.-H. (2000). Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets and Systems, 114(1), 103-113.

Examples

x <- ifv(c(0.6, 0.5, 0.7), c(0.3, 0.2, 0.1))
if_score(x); if_accuracy(x); if_rank(x); if_order(x)

Similarity between intuitionistic fuzzy sets

Description

Similarities in [0,1] (larger = more similar), complementing if_dist.

Usage

if_sim(
  a,
  b,
  method = c("chen", "hong_kim", "ye_cosine", "ye_weighted", "deng", "liang_shi", "dice"),
  w = NULL
)

Arguments

Details

• "chen": 1 - D_H (one minus normalised Hamming).

• "hong_kim": 1 - D_E (one minus normalised Euclidean).

• "ye_cosine" (Ye, 2011): weighted cosine S_C=\sum w_i \frac{\mu_A\mu_B+\nu_A\nu_B+\pi_A\pi_B} {\sqrt{\mu_A^2+\nu_A^2+\pi_A^2}\sqrt{\mu_B^2+\nu_B^2+\pi_B^2}}.

• "ye_weighted" (Ye, 2010 correlation): S_W=\frac{C(A,B)}{\max(C(A,A),C(B,B))} with C(A,B)=\sum w_i(\mu_A\mu_B+\nu_A\nu_B+\pi_A\pi_B).

• "deng" / "liang_shi": Jaccard-style S_J=\sum w_i \frac{\mu_A\mu_B+\nu_A\nu_B+\pi_A\pi_B} {(\mu_A^2+\nu_A^2+\pi_A^2)+(\mu_B^2+\nu_B^2+\pi_B^2) -(\mu_A\mu_B+\nu_A\nu_B+\pi_A\pi_B)}.

• "dice": Dice-style S_D=\sum w_i \frac{2(\mu_A\mu_B+\nu_A\nu_B+\pi_A\pi_B)} {(\mu_A^2+\nu_A^2+\pi_A^2)+(\mu_B^2+\nu_B^2+\pi_B^2)}.

Value

numeric scalar similarity in [0,1].

References

Ye, J. (2011). Cosine similarity measures for intuitionistic fuzzy sets and their applications. Mathematical and Computer Modelling, 53(1-2), 91-97.

Ye, J. (2010). Fuzzy decision-making method based on the weighted correlation coefficient under intuitionistic fuzzy environment. European J. Operational Research, 205(1), 202-204.

Liang, Z. & Shi, P. (2003). Similarity measures on intuitionistic fuzzy sets. Pattern Recognition Letters, 24(15), 2687-2693.

Examples

a <- ifv(c(0.6, 0.5), c(0.3, 0.2)); b <- ifv(c(0.5, 0.4), c(0.3, 0.3))
if_sim(a, b, "chen"); if_sim(a, b, "ye_cosine"); if_sim(a, b, "dice")

Intuitionistic fuzzy TOPSIS and decision helpers

Description

IF-TOPSIS after Boran et al. (2009): aggregate each alternative's criterion IFVs with weights w (via ifwa by default), then measure separation from the positive ideal A^+=(1,0) and negative ideal A^-=(0,1) using normalised Euclidean IF distance. Closeness C_i = D^-_i/(D^+_i+D^-_i); larger is better.

Usage

if_topsis(mat, nalt, ncrit, w, dist = "euclidean", agg = ifwa)

Arguments

Value

data.frame with columns alt, mu, nu, score, d_plus, d_minus, closeness, rank, ordered by rank.

References

Boran, F. E., Genc, S., Kurt, M. & Akay, D. (2009). A multi-criteria intuitionistic fuzzy group decision making for supplier selection with TOPSIS method. Expert Systems with Applications, 36(8), 11363-11368.

Examples

m <- ifv(c(0.6, 0.5, 0.7, 0.4, 0.5, 0.6), c(0.2, 0.3, 0.1, 0.4, 0.3, 0.2))
if_topsis(m, nalt = 3, ncrit = 2, w = c(0.6, 0.4))

Intuitionistic fuzzy c-means (IFCM) clustering

Description

Clusters ifv data with intuitionistic fuzzy memberships. Distance between an observation a and centre v is the normalised Euclidean IF distance (Szmidt-Kacprzyk). An optional hesitation weight lambda inflates distances by \exp(\lambda \cdot \bar\pi) where \bar\pi is the mean hesitation of the pair, emphasising uncertain points when lambda > 0 (Xu et al. style modification).

Usage

ifcmeans(
  x,
  k,
  m = 2,
  lambda = 0,
  maxit = 100,
  tol = 1e-05,
  nstart = 1L,
  seed = NULL
)

Arguments

Value

A list with centers (ifv, k rows), membership (n x k matrix), cluster (hard labels), within_ss (sum of weighted squared distances), iter, converged.

References

Xu, Z., Chen, J. & Wu, J. (2008). Clustering algorithm for intuitionistic fuzzy sets. Information Sciences, 178(19), 3775-3790.

Examples

set.seed(1)
mu <- runif(30, 0.3, 0.7); nu <- runif(30, 0.05, 0.25)
x <- ifv(mu, nu)
r <- ifcmeans(x, k = 2, nstart = 2)
r$cluster; r$centers

Intuitionistic fuzzy k-means (hard labels, IF centroids)

Description

Lloyd-style k-means with normalised Euclidean IF distance and IFWA centroid updates.

Usage

ifkmeans(x, k, maxit = 50, nstart = 5L, seed = NULL)

Arguments

Value

list with centers (ifv), cluster, within_ss, iter, converged.

Examples

set.seed(2)
mu <- runif(20, 0.2, 0.7); nu <- runif(20, 0.0, 0.25)
x <- ifv(mu, nu)
ifkmeans(x, k = 2, nstart = 2)

Create an intuitionistic fuzzy value (IFV)

Description

An intuitionistic fuzzy value is a pair (\mu, \nu) with \mu \ge 0, \nu \ge 0 and \mu + \nu \le 1. Hesitation is \pi = 1 - \mu - \nu.

Usage

ifv(mu, nu)

Arguments

Value

An object of class "ifv": a data.frame with columns mu, nu, pi.

Examples

ifv(0.6, 0.3)
ifv(c(0.5, 0.7), c(0.3, 0.1))

Intuitionistic fuzzy aggregation operators

Description

Weighted / ordered / hybrid aggregation of IFVs after Xu (2007) and Xu & Yager (2006).

Usage

ifwa(x, w)

ifwg(x, w)

ifowa(x, order_w)

ifowg(x, order_w)

ifha(x, w, order_w)

ifhg(x, w, order_w)

if_einstein_wa(x, w)

if_einstein_wg(x, w)

Arguments

Details

For IFVs a_j=(\mu_j,\nu_j) with weights w_j:

IFWA = \left(1-\prod(1-\mu_j)^{w_j},\ \prod \nu_j^{w_j}\right)

IFWG = \left(\prod \mu_j^{w_j},\ 1-\prod(1-\nu_j)^{w_j}\right)

Ordered variants (IFOWA/IFOWG) first reorder a_j by score S=\mu-\nu (descending) and then apply positional weights. Hybrid variants (IFHA/IFHG) weight each a_j by n w_j (via scalar-mul / power) and then apply ordered weighting.

Terms 0^0 arising from zero weights are treated as 1, and 0^0-type products skip zero-weight factors, so unused inputs do not affect the result.

Value

A single-row ifv.

References

Xu, Z. (2007). Intuitionistic fuzzy aggregation operators. IEEE Trans. Fuzzy Systems, 15(6), 1179-1187.

Xu, Z. & Yager, R. R. (2006). Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. General Systems, 35(4), 417-433.

Examples

x <- ifv(c(0.6, 0.5, 0.7), c(0.3, 0.2, 0.1))
ifwa(x, c(0.5, 0.3, 0.2))
ifwg(x, c(0.5, 0.3, 0.2))
ifowa(x, c(0.5, 0.3, 0.2))

Test / coerce intuitionistic fuzzy values

Description

Test / coerce intuitionistic fuzzy values

Usage

is_ifv(x)

as_ifv(x, ...)

Arguments

Value

is_ifv() logical; as_ifv() an ifv object.