similarity: 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"))

Value

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

Arguments

x, y: Two (generalized/customizable) sets.
method: Character string specifying the proximity method (see below).

Details

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 $i$ th 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: $d_{2} (A, B) = \sqrt{\sum_{x \in X} \sum_{i = 1} \infty | M_{A} (x)_{i} - M_{B} (x)_{i} |^{2}}$

Examples

Run this code

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")

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