Package 'pfclust'

Power Fuzzy Clustering

Description

Clusters the rows of Y into K groups using Minkowski or adaptive regularised Mahalanobis distance.

Usage

PFC(
  Y,
  K,
  m = 2,
  q = 2,
  distance = "Minkowski",
  p = 2,
  alpha = 0.5,
  beta = 10^15,
  threshold = 0.01,
  max.iter = 100
)

Arguments

Y	An ⁠n x dy⁠ data frame or matrix of observations.
K	Number of clusters (positive integer).
m	Fuzzifier, must be strictly greater than 1. Default 2.
q	Distance exponent, must be strictly greater than 0. Default 2.
distance	Either "Minkowski" or "Mahalanobis".
p	Minkowski exponent (⁠>= 1⁠). Ignored when distance = "Mahalanobis". Default 2.
alpha	Regularisation weight for the Mahalanobis covariance. Default 0.5.
beta	Eigenvalue ratio bound for the Mahalanobis covariance. Default 1e15.
threshold	Convergence tolerance. Default 0.01.
max.iter	Maximum number of iterations. Default 100.

Value

A list with elements B (or C) for cluster centres, d (distances), p (memberships), JDF (objective history), and l (hard labels). For Mahalanobis, also rho and cov.

References

P.T. Nguyen, C. Tortora, and A. Punzo. (2026) "Power fuzzy clustering: flexible distance metrics and inclusion of covariates". IEEE Transactions on Fuzzy Systems 10.1109/TFUZZ.2026.3683998

D. E. Gustafson and W. C. Kessel, “Fuzzy clustering with a fuzzy covariance matrix,” in 1978 IEEE conference on decision and control including the 17th symposium on adaptive processes. IEEE, 1979, pp. 761–766.

Examples

Y <- iris[, 1:4]
l <- iris[, 5]

# --- 2a. Mahalanobis distance ---
resMahC <- PFC(Y, K = 3, distance = "Mahalanobis")
table(resMahC$l, l)
pairs(Y, col = resMahC$l)

# --- 2b. Minkowski distance (default p=2) ---
resMinC <- PFC(Y, K = 3)
table(resMinC$l, l)
pairs(Y, col = resMinC$l)

# --- 2c. Minkowski distance (p=1, Manhattan) ---
resMin1C <- PFC(Y, K = 3, p = 1)
table(resMin1C$l, l)
pairs(Y, col = resMin1C$l)

---

Power Fuzzy Cluster-wise Regression

Description

Fits K cluster-specific linear models $Y = X B_k + \varepsilon$, selecting an internal solver based on the chosen distance and exponent.

Usage

PFCR(
  X,
  Y,
  K,
  m = 2,
  q = 2,
  distance = "Minkowski",
  p = 2,
  alpha = 0.5,
  beta = 10^15,
  threshold = 0.01,
  max.iter = 100
)

Arguments

X	An ⁠n x dx⁠ data frame or matrix of covariates.
Y	An ⁠n x dy⁠ data frame or matrix of dependent variables.
K	Number of clusters (positive integer).
m	Fuzzifier, must be strictly greater than 1. Default 2.
q	Distance exponent, must be strictly greater than 0. Default 2.
distance	Either "Minkowski" or "Mahalanobis".
p	Minkowski exponent (⁠>= 1⁠). Ignored when distance = "Mahalanobis". Default 2.
alpha	Regularisation weight for the Mahalanobis covariance. Default 0.5.
beta	Eigenvalue ratio bound for the Mahalanobis covariance. Default 1e15.
threshold	Convergence tolerance on successive coefficient updates. Default 0.01.
max.iter	Maximum number of iterations. Default 100.

Value

A list with elements:

B

Array of regression coefficients.

d

Data frame of distances (⁠n x K⁠).

p

Data frame of membership degrees (⁠n x K⁠).

JDF

Vector of objective-function values per iteration.

l

Hard cluster labels (length n).

rho, cov

(Mahalanobis only) cluster proportions and covariance matrices.

References

P.T. Nguyen, C. Tortora, and A. Punzo. (2026) "Power fuzzy clustering: flexible distance metrics and inclusion of covariates". IEEE Transactions on Fuzzy Systems 10.1109/TFUZZ.2026.3683998

Examples

library(flexCWM)
data("students")
Y <- students[, 2:3]
X <- students[, 4]
l <- students[, 1]


# --- 1a. Mahalanobis distance (default m=2, q=2) ---
resMah <- PFCR(X, Y, K = 2, distance = "Mahalanobis")
table(resMah$l, l)

# --- 1b. Minkowski distance (default m=2, q=2, p=2) ---
resMin <- PFCR(X, Y, K = 2)
table(resMin$l, l)

# --- 1c. Mahalanobis with m=3, q=3 ---
resMah33 <- PFCR(X, Y, K = 2, m = 3, q = 3, distance = "Mahalanobis")
table(resMah33$l, l)

# --- Plots for the (m=3, q=3) Mahalanobis result ---
pp    <- apply(resMah33$p, 1, max)
color <- ifelse(resMah33$l == unique(resMah33$l)[1], "blue", "red")

plot(Y, col = color, pch = 16, cex = pp,
     main = "PFCR Mahalanobis (3,3)")

plot(X, Y[, 1], ylab = "HEIGHT", xlab = "HEIGHT.F",
     pch = 16, cex = pp, col = color,
     main = "PFCR Mahalanobis (3,3)")
abline(resMah33$B[1, 1, 1], resMah33$B[2, 1, 1], col = "blue", lwd = 2)
abline(resMah33$B[1, 1, 2], resMah33$B[2, 1, 2], col = "red",  lwd = 2)

plot(X, Y[, 2], ylab = "WEIGHT", xlab = "HEIGHT.F",
     pch = 16, cex = pp, col = color,
     main = "PFCR Mahalanobis (3,3)")
abline(resMah33$B[1, 2, 1], resMah33$B[2, 2, 1], col = "blue", lwd = 2)
abline(resMah33$B[1, 2, 2], resMah33$B[2, 2, 2], col = "red",  lwd = 2)

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