gset {sets} | R Documentation |
Creation and manipulation of generalized sets.
gset(support, memberships, charfun, elements, universe) as.gset(x) is.gset(x) gset_support(x) gset_core(x) gset_peak(x) gset_height(x) gset_universe(x) gset_memberships(x) 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) gset_is_subset(x, y) gset_is_proper_subset(x, y) gset_is_equal(x, y) gset_contains_element(x, e) gset_is_set(x) gset_is_multiset(x) gset_is_fuzzy_set(x) gset_is_set_or_multiset(x) gset_is_set_or_fuzzy_set(x) gset_is_fuzzy_multiset(x) gset_is_crisp(x) gset_cardinality(x, type = c("absolute", "relative")) 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) gset_similarity(x, y, method = "Jaccard") e(x, memberships = 1L) is_element(e) ## S3 method for class 'gset': cut(x, level = 1, ...) ## S3 method for class 'gset': mean(x, ...) ## S3 method for class 'gset': median(x, na.rm = FALSE) ## S3 method for class 'gset': length(x)
x |
For e() , as.gset() and is.gset() :
an R object. A (g)set object otherwise. |
y |
A (g)set object. |
e |
An object of class element . |
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. |
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" ). |
height |
Double from the unit interval for scaling memberships. |
universe |
An optional set of elements. If NULL , d
efaults to the value of sets_options("universe") .
If the latter is also NULL , the support
will be used in computations. |
method |
For gset_similarity() : Currently, only
"Jaccard" is implemented (cardinality of the intersection,
divided by the cardinality of the union). For
gset_defuzzify() : "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. |
... |
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 specfied 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. 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.
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
“filters” all elements with membership not less then
level
— the result, thus, is a crisp
(multi)set. gset_similarity
computes the simple Jaccard
similarity between two generalized sets A and B,
i.e., the cardinality of the
intersection divided by the cardinality of the union of A and B.
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.
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_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)))