gset: Generalized sets
In sets: Sets, Generalized Sets, Customizable Sets and Intervals
gset R Documentation
Generalized sets
Description
Creation and manipulation of generalized sets.
Usage
gset(support, memberships, charfun, elements, universe, bound,
assume_numeric_memberships)
as.gset(x)
is.gset(x)
gset_support(x)
gset_core(x, na.rm = FALSE)
gset_peak(x, na.rm = FALSE)
gset_height(x, na.rm = FALSE)
gset_universe(x)
gset_bound(x)
gset_memberships(x, filter = NULL)
gset_transform_memberships(x, FUN, ...)
gset_concentrate(x)
gset_dilate(x)
gset_normalize(x, height = 1)
gset_defuzzify(x,
method = c("meanofmax", "smallestofmax",
"largestofmax", "centroid"))
gset_is_empty(x, na.rm = FALSE)
gset_is_subset(x, y, na.rm = FALSE)
gset_is_proper_subset(x, y, na.rm = FALSE)
gset_is_equal(x, y, na.rm = FALSE)
gset_contains_element(x, e)
gset_is_set(x, na.rm = FALSE)
gset_is_multiset(x, na.rm = FALSE)
gset_is_fuzzy_set(x, na.rm = FALSE)
gset_is_set_or_multiset(x, na.rm = FALSE)
gset_is_set_or_fuzzy_set(x, na.rm = FALSE)
gset_is_fuzzy_multiset(x)
gset_is_crisp(x, na.rm = FALSE)
gset_has_missings(x)
gset_cardinality(x, type = c("absolute", "relative"), na.rm = FALSE)
gset_union(...)
gset_sum(...)
gset_difference(...)
gset_product(...)
gset_mean(x, y, type = c("arithmetic", "geometric", "harmonic"))
gset_intersection(...)
gset_symdiff(...)
gset_complement(x, y)
gset_power(x)
gset_cartesian(...)
gset_combn(x, m)
e(x, memberships = 1L)
is_element(e)
## S3 method for class 'gset'
cut(x, level = 1, type = c("alpha", "nu"), strict = FALSE, ...)
## S3 method for class 'gset'
mean(x, ..., na.rm = FALSE)
## S3 method for class 'gset'
## median(x, na.rm = FALSE, ...) [R >= 3.4.0]
## median(x, na.rm) [R < 3.4.0]
## S3 method for class 'gset'
length(x)
## S3 method for class 'gset'
lengths(x, use.names = TRUE)
Arguments
x
For e(), as.gset() and is.gset():
an R object. A (g)set object otherwise. gset_memberships()
also accepts tuple objects.
y
A (g)set object.
e
An object of class element.
filter
Optional vector of elements to be filtered.
m
Number of elements to choose.
support
A set of elements giving the support of the gset
(elements with non-zero memberships). Must be a subset of the
universe, if specified.
memberships
For an (“ordinary”) set: 1L (or simply missing).
For a fuzzy set: a value between 0 and 1. For a multiset: a
positive integer. For a fuzzy multiset: a list of
multisets with elements from the unit interval (or a list of vectors
interpreted as such).
Otherwise, the argument will be transformed using as.gset.
elements
A set (or list) of e objects which are
object/memberships-pairs.
charfun
A function taking an object and returning the
membership.
bound
Integer used to compute the absolute complement for
(fuzzy) multisets. If NULL,
defaults to the value of sets_options("bound").
If the latter is also NULL, the maximum multiplicity
will be used in computations.
assume_numeric_memberships
When applying carfun() to
the universe, should numeric memberships (i.e. fuzzy sets or
multisets) be assumed (default)? If FALSE, fuzzy multisets
will be created.
FUN
A function, to be applied to a membership vector.
type
For gset_cardinality():
cardinality type (either "absolute" or
"relative"). For gset_mean(): mean type
("arithmetic", "geometric", or "harmonic").
For "cut": either "alpha" or "nu".
strict
Logical indicating whether the cut level must be
exceeded strictly (“greater than”) or not (“greater
than or equal”).
height
Double from the unit interval for scaling memberships.
universe
An optional set of elements. If NULL,
defaults to the value of sets_options("universe").
If the latter is also NULL, the support
will be used in computations.
method
"centroid" computes the arithmetic
mean of the set elements, using the membership values as
weights. "smallestofmax" / "meanofmax" /
"largestofmax" returns the minimum/mean/maximum of all
set elements with maximal membership degree.
level
The minimum membership level.
na.rm
logical indicating whether NA values should be
removed.
use.names
logical; should the names of x be used in the result?
...
For gset_foo(): (g)set objects. For
the mean and sort methods: additional parameters internally passed to
mean and order, respectively. For
gset_transform_memberships: further arguments passed to
FUN. For cut: currently not used.
Details
These functions represent basic infrastructure for handling
generalized sets of general (R) objects.
A generalized set (or gset) is set of pairs (e, f), where
e is some set element and f is the characteristic (or
membership) function. For (“ordinary”) sets
f maps to \{0, 1\},
for fuzzy sets into the unit interval, for multisets into the natural
numbers, and for fuzzy multisets f maps to the set of multisets
over the unit interval.
The gset_is_foo() predicates
are vectorized. In addition
to the methods defined, one can use the following operators:
| for the union, & for the
intersection, + for the sum, - for
the difference, %D% for the symmetric difference,
* and ^n for the
(n-fold) cartesian product, 2^ for the power set,
%e% for the element-of predicate,
< and <= for
the (proper) subset predicate, > and >= for
the (proper) superset predicate, and == and != for
(in)equality.
The Summary methods do also work if
defined for the set elements.
The mean and median
methods try to convert the object to a numeric vector before calling
the default methods. set_combn returns the gset of all
subsets of specified length.
gset_support, gset_core, and gset_peak
return the set of elements with memberships greater than zero, equal
to one, and equal to the maximum membership, respectively.
gset_memberships returns the membership
vector(s) of a given (tuple of) gset(s), optionally
restricted to the elements specified by filter.
gset_height returns only
the largest membership degree.
gset_cardinality computes either the absolute or the
relative cardinality, i.e. the memberships sum, or the absolute
cardinality divided by the number of elements, respectively.
The length method for gsets gives the (absolute) cardinality.
The lengths method coerces the set to a list
before applying the length method on its elements.
gset_transform_memberships applies function FOO to
the membership vector of the supplied gset and returns the transformed
gset. The transformed memberships are guaranteed to be in the unit
interval.
gset_concentrate and gset_dilate are convenience
functions, using the square and the square root,
respectively. gset_normalize divides the memberships by their
maximum and scales with height.
gset_product (gset_mean) of some gsets
compute the gset with the corresponding memberships multiplied (averaged).
The cut method provides both \alpha- and \nu-cuts.
\alpha-cuts “filter” all elements with memberships
greater than (or equal to) level—the result, thus, is a crisp
(multi)set. \nu-cuts select those elements with a
multiplicity exceeding level
(only sensible for (fuzzy) multisets).
Because set elements are unordered, it is not allowed to use
positional indexing. However, it is possible to
do indexing using element labels or
simply the elements themselves (useful, e.g., for subassignment).
In addition, it is possible to iterate over
all elements using for and lapply/sapply.
gset_contains_element is vectorized in e, that is, if e
is an atomic vector or list, the is-element operation is performed
element-wise, and a logical vector returned. Note that, however,
objects of class tuple are taken as atomic objects to
correctly handle sets of tuples.
References
D. Meyer and K. Hornik (2009),
Generalized and customizable sets in R,
Journal of Statistical Software 31(2), 1–27.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v031.i02")}.
See Also
set for “ordinary” sets,
gset_outer, and
tuple for tuples (“vectors”).
Examples
## multisets
(A <- gset(letters[1:5], memberships = c(3, 2, 1, 1, 1)))
(B <- gset(c("a", "c", "e", "f"), memberships = c(2, 2, 1, 2)))
rep(B, 2)
gset_memberships(tuple(A, B), c("a","c"))
gset_union(A, B)
gset_intersection(A, B)
gset_complement(A, B)
gset_is_multiset(A)
gset_sum(A, B)
gset_difference(A, B)
## fuzzy sets
(A <- gset(letters[1:5], memberships = c(1, 0.3, 0.8, 0.6, 0.2)))
(B <- gset(c("a", "c", "e", "f"), memberships = c(0.7, 1, 0.4, 0.9)))
cut(B, 0.5)
A * B
A <- gset(3L, memberships = 0.5, universe = 1:5)
!A
## fuzzy multisets
(A <- gset(c("a", "b", "d"),
memberships = list(c(0.3, 1, 0.5), c(0.9, 0.1),
gset(c(0.4, 0.7), c(1, 2)))))
(B <- gset(c("a", "c", "d", "e"),
memberships = list(c(0.6, 0.7), c(1, 0.3), c(0.4, 0.5), 0.9)))
gset_union(A, B)
gset_intersection(A, B)
gset_complement(A, B)
## other operations
mean(gset(1:3, c(0.1,0.5,0.9)))
median(gset(1:3, c(0.1,0.5,0.9)))
## vectorization
list(gset(1, 0.5), gset(2, 2L), gset()) <= gset(1, 2L)
sets documentation built on May 29, 2024, 10:09 a.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
| gset | R Documentation |
Generalized sets
Description
Creation and manipulation of generalized sets.
Usage
gset(support, memberships, charfun, elements, universe, bound,
assume_numeric_memberships)
as.gset(x)
is.gset(x)
gset_support(x)
gset_core(x, na.rm = FALSE)
gset_peak(x, na.rm = FALSE)
gset_height(x, na.rm = FALSE)
gset_universe(x)
gset_bound(x)
gset_memberships(x, filter = NULL)
gset_transform_memberships(x, FUN, ...)
gset_concentrate(x)
gset_dilate(x)
gset_normalize(x, height = 1)
gset_defuzzify(x,
method = c("meanofmax", "smallestofmax",
"largestofmax", "centroid"))
gset_is_empty(x, na.rm = FALSE)
gset_is_subset(x, y, na.rm = FALSE)
gset_is_proper_subset(x, y, na.rm = FALSE)
gset_is_equal(x, y, na.rm = FALSE)
gset_contains_element(x, e)
gset_is_set(x, na.rm = FALSE)
gset_is_multiset(x, na.rm = FALSE)
gset_is_fuzzy_set(x, na.rm = FALSE)
gset_is_set_or_multiset(x, na.rm = FALSE)
gset_is_set_or_fuzzy_set(x, na.rm = FALSE)
gset_is_fuzzy_multiset(x)
gset_is_crisp(x, na.rm = FALSE)
gset_has_missings(x)
gset_cardinality(x, type = c("absolute", "relative"), na.rm = FALSE)
gset_union(...)
gset_sum(...)
gset_difference(...)
gset_product(...)
gset_mean(x, y, type = c("arithmetic", "geometric", "harmonic"))
gset_intersection(...)
gset_symdiff(...)
gset_complement(x, y)
gset_power(x)
gset_cartesian(...)
gset_combn(x, m)
e(x, memberships = 1L)
is_element(e)
## S3 method for class 'gset'
cut(x, level = 1, type = c("alpha", "nu"), strict = FALSE, ...)
## S3 method for class 'gset'
mean(x, ..., na.rm = FALSE)
## S3 method for class 'gset'
## median(x, na.rm = FALSE, ...) [R >= 3.4.0]
## median(x, na.rm) [R < 3.4.0]
## S3 method for class 'gset'
length(x)
## S3 method for class 'gset'
lengths(x, use.names = TRUE)
Arguments
x |
For |
y |
A (g)set object. |
e |
An object of class |
filter |
Optional vector of elements to be filtered. |
m |
Number of elements to choose. |
support |
A set of elements giving the support of the gset (elements with non-zero memberships). Must be a subset of the universe, if specified. |
memberships |
For an (“ordinary”) set: 1L (or simply missing).
For a fuzzy set: a value between 0 and 1. For a multiset: a
positive integer. For a fuzzy multiset: a list of
multisets with elements from the unit interval (or a list of vectors
interpreted as such).
Otherwise, the argument will be transformed using |
elements |
A set (or list) of |
charfun |
A function taking an object and returning the membership. |
bound |
Integer used to compute the absolute complement for
(fuzzy) multisets. If |
assume_numeric_memberships |
When applying |
FUN |
A function, to be applied to a membership vector. |
type |
For |
strict |
Logical indicating whether the cut level must be exceeded strictly (“greater than”) or not (“greater than or equal”). |
height |
Double from the unit interval for scaling memberships. |
universe |
An optional set of elements. If |
method |
|
level |
The minimum membership level. |
na.rm |
logical indicating whether |
use.names |
logical; should the names of |
... |
For |
Details
These functions represent basic infrastructure for handling generalized sets of general (R) objects.
A generalized set (or gset) is set of pairs (e, f), where
e is some set element and f is the characteristic (or
membership) function. For (“ordinary”) sets
f maps to \{0, 1\},
for fuzzy sets into the unit interval, for multisets into the natural
numbers, and for fuzzy multisets f maps to the set of multisets
over the unit interval.
The gset_is_foo() predicates
are vectorized. In addition
to the methods defined, one can use the following operators:
| for the union, & for the
intersection, + for the sum, - for
the difference, %D% for the symmetric difference,
* and ^n for the
(n-fold) cartesian product, 2^ for the power set,
%e% for the element-of predicate,
< and <= for
the (proper) subset predicate, > and >= for
the (proper) superset predicate, and == and != for
(in)equality.
The Summary methods do also work if
defined for the set elements.
The mean and median
methods try to convert the object to a numeric vector before calling
the default methods. set_combn returns the gset of all
subsets of specified length.
gset_support, gset_core, and gset_peak
return the set of elements with memberships greater than zero, equal
to one, and equal to the maximum membership, respectively.
gset_memberships returns the membership
vector(s) of a given (tuple of) gset(s), optionally
restricted to the elements specified by filter.
gset_height returns only
the largest membership degree.
gset_cardinality computes either the absolute or the
relative cardinality, i.e. the memberships sum, or the absolute
cardinality divided by the number of elements, respectively.
The length method for gsets gives the (absolute) cardinality.
The lengths method coerces the set to a list
before applying the length method on its elements.
gset_transform_memberships applies function FOO to
the membership vector of the supplied gset and returns the transformed
gset. The transformed memberships are guaranteed to be in the unit
interval.
gset_concentrate and gset_dilate are convenience
functions, using the square and the square root,
respectively. gset_normalize divides the memberships by their
maximum and scales with height.
gset_product (gset_mean) of some gsets
compute the gset with the corresponding memberships multiplied (averaged).
The cut method provides both \alpha- and \nu-cuts.
\alpha-cuts “filter” all elements with memberships
greater than (or equal to) level—the result, thus, is a crisp
(multi)set. \nu-cuts select those elements with a
multiplicity exceeding level
(only sensible for (fuzzy) multisets).
Because set elements are unordered, it is not allowed to use
positional indexing. However, it is possible to
do indexing using element labels or
simply the elements themselves (useful, e.g., for subassignment).
In addition, it is possible to iterate over
all elements using for and lapply/sapply.
gset_contains_element is vectorized in e, that is, if e
is an atomic vector or list, the is-element operation is performed
element-wise, and a logical vector returned. Note that, however,
objects of class tuple are taken as atomic objects to
correctly handle sets of tuples.
References
D. Meyer and K. Hornik (2009), Generalized and customizable sets in R, Journal of Statistical Software 31(2), 1–27. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v031.i02")}.
See Also
set for “ordinary” sets,
gset_outer, and
tuple for tuples (“vectors”).
Examples
## multisets
(A <- gset(letters[1:5], memberships = c(3, 2, 1, 1, 1)))
(B <- gset(c("a", "c", "e", "f"), memberships = c(2, 2, 1, 2)))
rep(B, 2)
gset_memberships(tuple(A, B), c("a","c"))
gset_union(A, B)
gset_intersection(A, B)
gset_complement(A, B)
gset_is_multiset(A)
gset_sum(A, B)
gset_difference(A, B)
## fuzzy sets
(A <- gset(letters[1:5], memberships = c(1, 0.3, 0.8, 0.6, 0.2)))
(B <- gset(c("a", "c", "e", "f"), memberships = c(0.7, 1, 0.4, 0.9)))
cut(B, 0.5)
A * B
A <- gset(3L, memberships = 0.5, universe = 1:5)
!A
## fuzzy multisets
(A <- gset(c("a", "b", "d"),
memberships = list(c(0.3, 1, 0.5), c(0.9, 0.1),
gset(c(0.4, 0.7), c(1, 2)))))
(B <- gset(c("a", "c", "d", "e"),
memberships = list(c(0.6, 0.7), c(1, 0.3), c(0.4, 0.5), 0.9)))
gset_union(A, B)
gset_intersection(A, B)
gset_complement(A, B)
## other operations
mean(gset(1:3, c(0.1,0.5,0.9)))
median(gset(1:3, c(0.1,0.5,0.9)))
## vectorization
list(gset(1, 0.5), gset(2, 2L), gset()) <= gset(1, 2L)
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Embedding an R snippet on your website
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.