Package 'IFFuzzyStats'

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

a, b	objects coercible to ifv (recycled to common length).
lambda	positive numeric scalar or vector.

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

mat	ifv with one row per alternative-criterion cell, or a list of per-alternative ifv vectors; see examples.
w	criterion weights.
ncrit	number of criteria (columns); rows of mat are nalt blocks of ncrit in row-major order.
fun	aggregation function, e.g. ifwa, ifwg.

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

x	object coercible to ifv.
tol	numeric tolerance.

Value

invisibly TRUE; errors if invalid.

---

Complement of an IFV

Description

Complement of an IFV

Usage

if_comp(x)

Arguments

x	object coercible to ifv.

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

R, S	matrices (or data.frames) of IFVs stored as ifv objects with nrow = nx*ny in row-major order, or numeric matrices of membership values with matching non-membership matrices R_nu, S_nu. Simpler: pass ifv vectors plus dimensions.
nx, ny, nz	dimensions.
op	"maxmin" or "maxprod".

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

x	object coercible to ifv.
method	"score" ($\mu-\nu$), "accuracy" ($\mu+\nu$), "expected" ($\mu + \pi/2$), or "mu" (membership only).

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

a, b	objects coercible to ifv of common length.
method	distance name (partial matching allowed).
w	optional element weights.

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

x	object coercible to ifv (the whole set).
method	one of "burillo", "szmidt", "vlachos", "zeng_li" (partial matching allowed).

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

x	numeric vector.
mu_range	range used to scale x into membership (default range(x)).
nu_ratio	share of the non-membership part in $(1-\mu)$ (scalar or vector in $[0,1]$, default 0.6).

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

fact	ifv of length nx.
relation	ifv of length nx*ny.
nx, ny	dimensions.
op	composition operator.

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

x	an ifv object (or coercible).

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

x	object coercible to ifv (one value per alternative).
labels	optional alternative labels.

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

fact	ifv length nx.
relations	list of nrules ifv relations (each length nx*ny).
nx, ny	dimensions.
op	composition operator.

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

x	object coercible to ifv.

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

a, b	objects coercible to ifv.
method	similarity name (partial matching allowed).
w	optional element weights.

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

mat	ifv of length nalt*ncrit in row-major order (alternative 1's criteria first), or an nalt x ncrit list-matrix.
nalt, ncrit	dimensions (guessed from square if missing is impossible, so both are required).
w	criterion weights (length ncrit).
dist	distance method passed to if_dist (default "euclidean").
agg	aggregation function (default ifwa).

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

x	object coercible to ifv (n observations).
k	number of clusters.
m	fuzzifier > 1 (default 2).
lambda	hesitation penalty >= 0 (default 0 = plain IFCM).
maxit	maximum iterations.
tol	convergence tolerance on membership change.
nstart	number of random starts (best kept).
seed	random seed.

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

x	object coercible to ifv.
k	number of clusters.
maxit	maximum iterations.
nstart	random starts.
seed	seed.

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

mu	numeric vector of membership degrees in $[0,1]$.
nu	numeric vector of non-membership degrees in $[0,1]$.

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

x	object coercible to ifv (vector of values to fuse).
w	numeric weight vector (non-negative, sums to 1; normalised if not).
order_w	positional weights for ordered/hybrid operators.

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

x	object to test or coerce.
...	ignored.

Value

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

---