Jaccard index
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The Jaccard index, also known as the Jaccard similarity coefficient, is a statistic used for gauging the similarity and diversity of sample sets. It was developed by Paul Jaccard, originally giving the French name coefficient de communauté, and independently formulated again by T. Tanimoto. Thus, the Tanimoto index or Tanimoto coefficient are also used in some fields. However, they are identical in generally taking the ratio of Intersection over Union. The Jaccard coefficient measures similarity between finite sample sets, and is defined as the size of the intersection divided by the size of the union of the sample sets:
- J(A, B) = |A ∩ B|/|A ∪ B|
Define sets as follows, using any linear data structure:
A = {}
B = {1, 2, 3, 4, 5}
C = {1, 3, 5, 7, 9}
D = {2, 4, 6, 8, 10}
E = {2, 3, 5, 7}
F = {8}
Write a program that computes the Jaccard index for every unordered pairing (e.g. only one of J(A, B) and J(B, A) need to be done) of these sets, including self-pairings.
Factor
<lang factor>USING: assocs formatting grouping kernel math math.combinatorics prettyprint sequences sequences.repeating sets ;
- jaccard ( seq1 seq2 -- x )
2dup [ empty? ] both? [ 2drop 1 ] [ [ intersect ] [ union ] 2bi [ length ] bi@ / ] if ;
{ { } { 1 2 3 4 5 } { 1 3 5 7 9 } { 2 4 6 8 10 } { 2 3 5 7 } { 8 } } [ 2 <combinations> ] [ 2 repeat 2 group append ] bi [ 2dup jaccard "%u %u -> %u\n" printf ] assoc-each</lang>
- Output:
{ } { 1 2 3 4 5 } -> 0
{ } { 1 3 5 7 9 } -> 0
{ } { 2 4 6 8 10 } -> 0
{ } { 2 3 5 7 } -> 0
{ } { 8 } -> 0
{ 1 2 3 4 5 } { 1 3 5 7 9 } -> 3/7
{ 1 2 3 4 5 } { 2 4 6 8 10 } -> 1/4
{ 1 2 3 4 5 } { 2 3 5 7 } -> 1/2
{ 1 2 3 4 5 } { 8 } -> 0
{ 1 3 5 7 9 } { 2 4 6 8 10 } -> 0
{ 1 3 5 7 9 } { 2 3 5 7 } -> 1/2
{ 1 3 5 7 9 } { 8 } -> 0
{ 2 4 6 8 10 } { 2 3 5 7 } -> 1/8
{ 2 4 6 8 10 } { 8 } -> 1/5
{ 2 3 5 7 } { 8 } -> 0
{ } { } -> 1
{ 1 2 3 4 5 } { 1 2 3 4 5 } -> 1
{ 1 3 5 7 9 } { 1 3 5 7 9 } -> 1
{ 2 4 6 8 10 } { 2 4 6 8 10 } -> 1
{ 2 3 5 7 } { 2 3 5 7 } -> 1
{ 8 } { 8 } -> 1
Wren
Note that the Set object in the above module is implemented as a Map and consequently the iteration order (and the order in which elements are printed) is undefined. <lang ecmascript>import "./set" for Set import "./trait" for Indexed import "./fmt" for Fmt
var jacardIndex = Fn.new { |a, b|
if (a.count == 0 && b.count == 0) return 1 return a.intersect(b).count / a.union(b).count
}
var a = Set.new([]) var b = Set.new([1, 2, 3, 4, 5]) var c = Set.new([1, 3, 5, 7, 9]) var d = Set.new([2, 4, 6, 8, 10]) var e = Set.new([2, 3, 5, 7]) var f = Set.new([8]) var sets = [a, b, c, d, e, f]
for (se in Indexed.new(sets)) {
var i = se.index
var s = se.value
s = s.toList.sort() // force original sorted order
Fmt.print("$s = $n", String.fromByte(65 + i), s)
}
var pairs = [
[a, a], [a, b], [a, c], [a, d], [a, e], [a, f], [b, b], [b, c], [b, d], [b, e], [b, f], [c, c], [c, d], [c, e], [c, f], [d, d], [d, e], [d, f], [e, e], [e, f], [f, f]
]
var names = [
"AA", "AB", "AC", "AD", "AE", "AF", "BB", "BC", "BD", "BE", "BF", "CC", "CD", "CE", "CF", "DD", "DE", "DF", "EE", "EF", "FF"
]
System.print() for (se in Indexed.new(pairs)) {
var n = names[se.index]
var ss = se.value
Fmt.print("J($s, $s) = $h", n[0], n[1], jacardIndex.call(ss[0], ss[1]))
}</lang>
- Output:
A = [] B = [1, 2, 3, 4, 5] C = [1, 3, 5, 7, 9] D = [2, 4, 6, 8, 10] E = [2, 3, 5, 7] F = [8] J(A, A) = 1 J(A, B) = 0 J(A, C) = 0 J(A, D) = 0 J(A, E) = 0 J(A, F) = 0 J(B, B) = 1 J(B, C) = 0.428571 J(B, D) = 0.25 J(B, E) = 0.5 J(B, F) = 0 J(C, C) = 1 J(C, D) = 0 J(C, E) = 0.5 J(C, F) = 0 J(D, D) = 1 J(D, E) = 0.125 J(D, F) = 0.2 J(E, E) = 1 J(E, F) = 0 J(F, F) = 1