similarity {sets}

Similarity and Dissimilarity Functions

Description

Similarities and dissimilarities for (generalized) sets.

Usage

set_similarity(x, y, method = "Jaccard")
gset_similarity(x, y, method = "Jaccard")
cset_similarity(x, y, method = "Jaccard")

set_dissimilarity(x, y,
                  method = c("Jaccard", "Manhattan", "Euclidean",
                             "L1", "L2"))
gset_dissimilarity(x, y,
                   method = c("Jaccard", "Manhattan", "Euclidean",
                              "L1", "L2"))
cset_dissimilarity(x, y,
                   method = c("Jaccard", "Manhattan", "Euclidean",
                              "L1", "L2"))

Arguments

x, y

Two (generalized/customizable) sets.

method

Character string specifying the proximity method (see below).

Details

For two generalized sets X and Y, the Jaccard similarity is |X \cap Y| / |X \cup Y| where |\cdot| denotes the cardinality for generalized sets (sum of memberships). The Jaccard dissimilarity is 1 minus the similarity.

The L1 (or Manhattan) and L2 (or Euclidean) dissimilarities are defined as follows. For two fuzzy multisets A and B on a given universe X with elements x, let M_A(x) and M_B(x) be functions returning the memberships of an element x in sets A and B, respectively. The memberships are returned in standard form, i.e. as an infinite vector of decreasing membership values, e.g. (1, 0.3, 0, 0, \dots). Let M_A(x)_i and M_B(x)_i denote the ith components of these membership vectors. Then the L1 distance is defined as:

d_1(A, B) = \sum_{x \in X}\sum_{i=1}{\infty}|M_A(x)_i - M_B(x)_i|

and the L2 distance as:

d_2(A, B) = \sqrt{\sum_{x \in X}\sum_{i=1}{\infty}|M_A(x)_i - M_B(x)_i|^2}

Value

A numeric value (similarity or dissimilarity, as specified).

Source

T. Matthe, R. De Caluwe, G. de Tre, A. Hallez, J. Verstraete, M. Leman, O. Cornelis, D. Moelants, and J. Gansemans (2006), Similarity Between Multi-valued Thesaurus Attributes: Theory and Application in Multimedia Systems, Flexible Query Answering Systems, Lecture Notes in Computer Science, Springer, 331–342.

K. Mizutani, R. Inokuchi, and S. Miyamoto (2008), Algorithms of Nonlinear Document Clustering Based on Fuzzy Multiset Model, International Journal of Intelligent Systems, 23, 176–198.

See Also

set.

Examples

A <- set("a", "b", "c")
B <- set("c", "d", "e")
set_similarity(A, B)
set_dissimilarity(A, B)

A <- gset(c("a", "b", "c"), c(0.3, 0.7, 0.9))
B <- gset(c("c", "d", "e"), c(0.2, 0.4, 0.5))
gset_similarity(A, B, "Jaccard")
gset_dissimilarity(A, B, "Jaccard")
gset_dissimilarity(A, B, "L1")
gset_dissimilarity(A, B, "L2")

A <- gset(c("a", "b", "c"), list(c(0.3, 0.7), 0.1, 0.9))
B <- gset(c("c", "d", "e"), list(0.2, c(0.4, 0.5), 0.8))
gset_similarity(A, B, "Jaccard")
gset_dissimilarity(A, B, "Jaccard")
gset_dissimilarity(A, B, "L1")
gset_dissimilarity(A, B, "L2")