Membership Functions for Fuzzy Logic — membership • mathmodels

A collection of functions to compute membership values for various fuzzy sets, including triangular, trapezoidal, Gaussian, generalized bell, two-parameter Gaussian, sigmoid, difference of sigmoids, product of sigmoids, Z-shaped, PI-shaped, and S-shaped membership functions. Includes a function to visualize membership functions using ggplot2. These are designed for evaluation models in mathematical modeling, compatible with fuzzy_eval in the mathmodels package.

Usage

mf_tri(x, params)

mf_trap(x, params)

mf_gauss(x, params)

mf_gbell(x, params)

mf_gauss2(x, params)

mf_sigmoid(x, params)

mf_dsigmoid(x, params)

mf_psigmoid(x, params)

mf_z(x, params)

mf_pi(x, params)

mf_s(x, params)

plot_mf(mf, xlim = c(0, 10), main = NULL)

Arguments

x

Numeric vector, input values for which to compute membership.

params

Numeric vector, parameters defining the membership function:

For mf_tri: c(a, b, c), where a < b < c (left base, peak, right base).
For mf_trap: c(a, b, c, d), where a < b <= c < d (left base, left top, right top, right base); one-sided shoulders are supported with a = b = -Inf (no rise) or c = d = Inf (no fall).
For mf_gauss: c(sigma, c), where sigma > 0 (spread, center).
For mf_gbell: c(a, b, c), where a > 0, b > 0 (width, shape, center).
For mf_gauss2: c(s1, c1, s2, c2), where s1 > 0, s2 > 0 (left spread, left center, right spread, right center).
For mf_sigmoid: c(a, b), where a > 0 (slope, inflection point).
For mf_dsigmoid: c(a1, c1, a2, c2), where a1 > 0, a2 > 0 (slopes and inflection points for two sigmoids).
For mf_psigmoid: c(a1, c1, a2, c2), where a1 > 0, a2 > 0 (slopes and inflection points for two sigmoids).
For mf_z: c(a, b), where a < b (left base, right base).
For mf_pi: c(a, b, c, d), where a < b <= c < d (left base, left shoulder, right shoulder, right base).
For mf_s: c(a, b), where a < b (left base, right base).

mf

Function, a membership function with fixed parameters (e.g., function(x) mf_tri(x, c(2, 5, 8))).

xlim

Numeric vector of length 2, x-axis limits for plotting (default c(0, 10)).

main

Character, plot title (default NULL, no title).

Value

For membership functions (mf_tri, mf_trap, mf_gauss, mf_gbell, mf_gauss2, mf_sigmoid, mf_psigmoid, mf_z, mf_pi, mf_s): A numeric vector of membership values in [0, 1], same length as x.
For mf_dsigmoid: A numeric vector holding the difference of two sigmoids. Values lie in \([-1, 1]\) and are negative when c1 > c2, so unlike the other membership functions it is not guaranteed to be in [0, 1]. Clip the result to [0, 1], or use mf_psigmoid() instead, before passing it to fuzzy_eval().
For plot_mf: A ggplot2 object, plotting the membership function.

Details

These functions support evaluation models in mathematical modeling:

mf_tri: Triangular membership, linear rise from a to b (peak) and fall to c.
mf_trap: Trapezoidal membership, linear rise from a to b, plateau from b to c, fall to d.
mf_gauss: Gaussian membership, bell-shaped curve centered at c with spread sigma.
mf_gbell: Generalized bell membership, bell-shaped curve with width a, shape b, and center c.
mf_gauss2: Two-parameter Gaussian membership, combining two Gaussians with spreads s1, s2 and centers c1, c2.
mf_sigmoid: Sigmoid membership, S-shaped curve with slope a and inflection point b.
mf_dsigmoid: Difference of two sigmoids, combining slopes a1, a2 and inflection points c1, c2.
mf_psigmoid: Product of two sigmoids, combining slopes a1, a2 and inflection points c1, c2.
mf_z: Z-shaped membership, decreasing from 1 at a to 0 at b.
mf_pi: PI-shaped membership, rising from a to b, plateau from b to c, falling to d.
mf_s: S-shaped membership, increasing from 0 at a to 1 at b.
plot_mf: Plots a membership function over xlim using ggplot2, suitable for tidyverse workflows.

Membership values can be used to construct fuzzy evaluation matrices for fuzzy_eval. Implemented in base R, except plot_mf, which requires ggplot2.

Examples

# Define input values
x = 0:10

# Triangular membership
mf_tri(x, params = c(3, 6, 8))
#>  [1] 0.0000000 0.0000000 0.0000000 0.0000000 0.3333333 0.6666667 1.0000000
#>  [8] 0.5000000 0.0000000 0.0000000 0.0000000

# Trapezoidal membership
mf_trap(x, params = c(1, 5, 7, 8))
#>  [1] 0.00 0.00 0.25 0.50 0.75 1.00 1.00 1.00 0.00 0.00 0.00

# Gaussian membership
mf_gauss(x, params = c(2, 5))
#>  [1] 0.04393693 0.13533528 0.32465247 0.60653066 0.88249690 1.00000000
#>  [7] 0.88249690 0.60653066 0.32465247 0.13533528 0.04393693

# Generalized bell membership
mf_gbell(x, params = c(2, 4, 6))
#>  [1] 0.0001523926 0.0006549308 0.0038910506 0.0375531759 0.5000000000
#>  [6] 0.9961089494 1.0000000000 0.9961089494 0.5000000000 0.0375531759
#> [11] 0.0038910506

# Two-parameter Gaussian membership
mf_gauss2(x, params = c(1, 3, 3, 4))
#>  [1] 0.0111090 0.1353353 0.6065307 1.0000000 1.0000000 0.9459595 0.8007374
#>  [8] 0.6065307 0.4111123 0.2493522 0.1353353

# Sigmoid membership
mf_sigmoid(x, params = c(2, 4))
#>  [1] 0.0003353501 0.0024726232 0.0179862100 0.1192029220 0.5000000000
#>  [6] 0.8807970780 0.9820137900 0.9975273768 0.9996646499 0.9999546021
#> [11] 0.9999938558

# Difference of sigmoids membership
dsigmoid_y = mf_dsigmoid(x, params = c(5, 2, 5, 7))

# Product of sigmoids membership
mf_psigmoid(x, params = c(2, 3, -5, 8))
#>  [1] 2.472623e-03 1.798621e-02 1.192029e-01 5.000000e-01 8.807971e-01
#>  [6] 9.820135e-01 9.974821e-01 9.929740e-01 4.999773e-01 6.692810e-03
#> [11] 4.539783e-05

# Z-shaped membership
mf_z(x, params = c(3, 7))
#>  [1] 1.000 1.000 1.000 1.000 0.875 0.500 0.125 0.000 0.000 0.000 0.000

# PI-shaped membership
mf_pi(x, params = c(1, 4, 5, 10))
#>  [1] 0.0000000 0.0000000 0.2222222 0.7777778 1.0000000 1.0000000 0.9200000
#>  [8] 0.6800000 0.3200000 0.0800000 0.0000000

# S-shaped membership
mf_s(x, params = c(1, 8))
#>  [1] 0.00000000 0.00000000 0.04081633 0.16326531 0.36734694 0.63265306
#>  [7] 0.83673469 0.95918367 1.00000000 1.00000000 1.00000000

if (FALSE) { # \dontrun{
# Visualize membership functions
plot_mf(\(x) mf_tri(x, c(3, 6, 8)), main = "Triangular MF")
plot_mf(\(x) mf_trap(x, c(1, 5, 7, 8)), main = "Trapezoidal MF")
plot_mf(\(x) mf_gauss(x, c(2, 5)), main = "Gaussian MF")
plot_mf(\(x) mf_gbell(x, c(2, 4, 6)), main = "Generalized Bell MF")
plot_mf(\(x) mf_gauss2(x, c(1, 3, 3, 4)), main = "Two-Parameter Gaussian MF")
plot_mf(\(x) mf_sigmoid(x, c(2, 4)), main = "Sigmoid MF")
plot_mf(\(x) mf_dsigmoid(x, c(5, 2, 5, 7)), main = "Difference of Sigmoids MF")
plot_mf(\(x) mf_psigmoid(x, c(2, 3, -5, 8)), main = "Product of Sigmoids MF")
plot_mf(\(x) mf_z(x, c(3, 7)), main = "Z-Shaped MF")
plot_mf(\(x) mf_pi(x, c(1, 4, 5, 10)), main = "PI-Shaped MF")
plot_mf(\(x) mf_s(x, c(1, 8)), main = "S-Shaped MF")
} # }