R: Fuzzy membership functions

fuzzyfuns {sets} | R Documentation

## Fuzzy membership functions

### Description

Fuzzy membership and set creator functions.

### Usage

    charfun_generator(FUN, height = 1)
    fuzzy_tuple(FUN = fuzzy_normal, n = 5, ...,
                     universe = NULL, names = NULL)
    is.charfun_generator(x)

    fuzzy_normal(mean = NULL, sd = 1, log = FALSE, height = 1, chop = 0)
    fuzzy_two_normals(mean = NULL, sd = c(1,1), log = c(FALSE, FALSE),
                      height = 1, chop = 0)
    fuzzy_bell(center = NULL, cross = NULL, slope = 4, height = 1, chop = 0)
    fuzzy_sigmoid(cross = NULL, slope = 0.5, height = 1, chop = 0)
    fuzzy_trapezoid(corners = NULL, height = c(1,1),
                    return_base_corners = TRUE)
    fuzzy_triangular(corners = NULL, height = 1,
                     return_base_corners = TRUE)
    fuzzy_cone(center = NULL, radius = 2, height = 1,
               return_base_corners = TRUE)
    fuzzy_pi3(mid = NULL, min = NULL, max = NULL, height = 1,
              return_base_corners = TRUE)
    fuzzy_pi4(knots, height = 1, return_base_corners = TRUE)

    fuzzy_normal_gset(mean = NULL, sd = 1, log = FALSE, height = 1,
                      chop = 0, universe = NULL)
    fuzzy_two_normals_gset(mean = NULL, sd = c(1,1), log = c(FALSE, FALSE),
                          height = 1, chop = 0, universe = NULL)
    fuzzy_bell_gset(center = NULL, cross = NULL, slope = 4, height = 1,
                    chop = 0, universe = NULL)
    fuzzy_sigmoid_gset(cross = NULL, slope = 0.5, height = 1,
                       chop = 0, universe = NULL)
    fuzzy_trapezoid_gset(corners = NULL, height = c(1,1), universe = NULL,
                         return_base_corners = TRUE)
    fuzzy_triangular_gset(corners = NULL, height = 1, universe = NULL,
                          return_base_corners = TRUE)
    fuzzy_cone_gset(center = NULL, radius = 2, height = 1, universe = NULL,
                    return_base_corners = TRUE)
    fuzzy_pi3_gset(mid = NULL, min = NULL, max = NULL, height = 1,
                  universe = NULL, return_base_corners = TRUE)
    fuzzy_pi4_gset(knots, height = 1,
                   universe = NULL, return_base_corners = TRUE)

### Arguments

`x` | An R object.
`n` | Positive integer indicating the number of sets to be created, or integer vector of location parameters used to create the sets.
`FUN` | Function to be used for creating a membership function. Needs to be vectorized, i.e., is expected to take a vector of set elements and to return a vector of numeric values.
`height` | Numeric value in the unit interval specifying the height of the set resulting from applying the membership function to the universe, i.e., the maximum value to which the values will be scaled to.
`chop` | Threshold value below which the membership function is truncated, i.e., has a value of 0.
`center, mean` | Numeric mean value(s) used for the resulting membership function.
`sd` | Numeric scale factor(s) (standard deviation(s)) used for the resulting membership function.
`radius` | Double added/subtracted from `center` to get the base line corners of the cone.
`log` | Logical (vector), indicating whether normal or log-normal distributions should be used.
`cross` | Double indicating the cross-over point for the sigmoidal distribution.
`slope` | Double indicating the slope at the cross-over point.
`corners` | Double values (vector of length four) indicating the abscissas of the four corners of the resulting trapezoid.
`min, mid, max` | Doubles indicating the abscissas of the three spline knots (\code{min}, 0), (\code{mid}, \code{height}) and (\code{max}, 0) the curve passes through.
`knots` | Vector of doubles of length four indicating the abscissas of the spline knots the curve passes through.
`return_base_corners` | Logical indicating whether membership grades for the base line corner elements should be returned as small values (`.Machine$double.eps`) instead of 0. Otherwise, if a set is created from the memberships, the corner elements would be removed from the set.
`universe` | Universal set used for computing the memberships grades.
`names` | optional character vector of component labels for the return value.
`...` | Further arguments passed to `FUN`.

### Details

These functions can be used to create sets with certain membership patterns.

The core functions are function generators, taking parameters and returning a corresponding fuzzy function (i.e., with values in the unit interval). All of them are normalized, i.e., scaled to have a maximum value of `height` (default: 1):

`fuzzy_normal`

  is simply based on `dnorm`.

`fuzzy_two_normals`

  returns a function composed of the left and right parts of two normal distributions (each normalized), with possibly different means and standard deviations.

`fuzzy_bell`

  returns a function defined as: 1 / (1 + |(x - c) / w|) ^ 2s with center c, crossover points c +/- w, and slope at the crossover points of s / 2w.

`fuzzy_sigmoid`

  yields a function whose values are computed as 1 / (1 + exp{-s (x - c)}) with slope s at crossover point c.

`fuzzy_trapezoid`

  creates a function with trapezoidal shape, defined by four `corners` elements and two `height` values for the second and third corner (the heights of the first and fourth corner being fixed at 0).

`fuzzy_triangular`

  similar to the above with only three corners.

`fuzzy_cone`

  is a special case of `fuzzy_triangular`, defining an isosceles triangle with corners (element, membership degree): (\code{center} - \code{radius}, 0), (\code{center}, \code{height}), and (\code{center} + \code{radius}, 0).

`fuzzi_pi3`

  constructs a spline based on two quadratic functions, passing through the knot points (\code{min}, 0), (\code{mid}, \code{height}) and (\code{max}, 0).

`fuzzi_pi4`

  constructs a spline based on an S-shaped and a Z-shaped curve forming a π-shaped one, passing through the four knots (\code{knots[1]}, 0), (\code{knots[2]}, \code{height}), (\code{knots[3]}, \code{height}), and (\code{knots[4]}, 0).

`charfun_generator` takes a vectorized function as argument, returning a function normalized by `height`.

The `fuzzy_foo_gset` functions directly generate generalized sets from `fuzzy_foo`, using the values defined by `universe`, `sets_options("universe")`, or `seq(0, 20, by = 0.1)` (in that order, whichever is not `NULL`).

`fuzzy_tuple` generates a sequence of `n` sets based on any of the generating functions (except `fuzzy_trapezoid` and `fuzzy_triangular`). The chosen generating function `FUN` is called with `n` different values chosen along the `universe` passed to the first argument, thus varying the position or the resulting graph.

### Value

For `charfun_generator`, a generating function taking an argument list of parameters, and returning a membership function, mapping elements to membership values (from of the unit interval).

For `fuzzy_tuple`, a tuple of `n` fuzzy sets.

For `is.charfun_generator`, a logical.

For `fuzzy_foo_gset`, a fuzzy set.

For the other functions, a membership function.

### See Also

`set`, `gset`, and `tuple` for the set types, and `plot.gset` for the available plot functions.

### Examples

    ## creating a fuzzy normal function
    N <- fuzzy_normal(mean = 0, sd = 1)
    N(-3:3)

    ## create a fuzzy set with it
    gset(charfun = N, universe = -3:3)

    ## same using wrapper
    fuzzy_normal_gset(universe = -3:3)

    ## creating a user-defined fuzzy function
    fuzzy_poisson <- charfun_generator(dpois)
    gset(charfun = fuzzy_poisson(10), universe = seq(0, 20, 2))

    ## creating a series of fuzzy normal sets
    fuzzy_tuple(fuzzy_normal, 5)

    ## creating a series of fuzzy cones with specific locations
    fuzzy_tuple(fuzzy_cone, c(2,3,7))

[Package sets version 1.0-18 Index]