Creation and manipulation of generalized sets.
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 gset
cut(x, level = 1, type = c("alpha", "nu"), strict = FALSE, ...)
# S3 method for 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 gset
length(x)
# S3 method for gset
lengths(x, use.names = TRUE)
For e(), as.gset() and is.gset():
an R object. A (g)set object otherwise. gset_memberships()
also accepts tuple objects.
A (g)set object.
An object of class element.
Optional vector of elements to be filtered.
Number of elements to choose.
A set of elements giving the support of the gset (elements with non-zero memberships). Must be a subset of the universe, if specified.
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.
A set (or list) of e objects which are
object/memberships-pairs.
A function taking an object and returning the membership.
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.
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.
A function, to be applied to a membership vector.
For gset_cardinality():
cardinality type (either "absolute" or
"relative"). For gset_mean(): mean type
("arithmetic", "geometric", or "harmonic").
For "cut": either "alpha" or "nu".
Logical indicating whether the cut level must be exceeded strictly (“greater than”) or not (“greater than or equal”).
Double from the unit interval for scaling memberships.
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.
"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.
The minimum membership level.
logical indicating whether NA values should be
removed.
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.
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.
D. Meyer and K. Hornik (2009), Generalized and customizable sets in R, Journal of Statistical Software 31(2), 1--27. tools:::Rd_expr_doi("10.18637/jss.v031.i02").
set for “ordinary” sets,
gset_outer, and
tuple for tuples (“vectors”).
## 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)
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