pfclust: Power Fuzzy Clustering and Cluster-wise Regression

Description

Implements Power Fuzzy Clustering (PFC) and Power Fuzzy Cluster-wise Regression (PFCR) with Minkowski and adaptive regularised Mahalanobis distances.

Details

The two main user-facing functions are:

• PFC() — fuzzy clustering (no covariates)

• PFCR() — fuzzy cluster-wise linear regression

Author(s)

Maintainer: Phuc Thinh Nguyen phucthinh010603@yahoo.com

Authors:

• Cristina Tortora cristina.tortora@sjsu.edu [thesis advisor, degree supervisor]

• Antonio Punzo antonio.punzo@unict.it [thesis advisor, degree supervisor]

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

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

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)