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 ordered pairing (to show that J(A, B) and J(B, A) are the same) of these sets, including self-pairings.
APL
task←{
jaccard ← (≢∩)÷(≢∪)
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
'.ABCDEF' ⍪ 'ABCDEF' , ∘.jaccard⍨ A B C D E F
}
- Output:
. A B C D E F A 1 0 0 0 0 0 B 0 1 0.4285714286 0.25 0.5 0 C 0 0.4285714286 1 0 0.5 0 D 0 0.25 0 1 0.125 0.2 E 0 0.5 0.5 0.125 1 0 F 0 0 0 0.2 0 1
BQN
Jaccard ← ≡◶⟨∊ ÷○(+´) ∊∘∾, 1⟩
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⟩
Jaccard⌜˜ ⟨a,b,c,d,e,f⟩
- Output:
┌─ ╵ 1 0 0 0 0 0 0 1 0.42857142857142855 0.25 0.5 0 0 0.42857142857142855 1 0 0.5 0 0 0.25 0 1 0.125 0.2 0 0.5 0.5 0.125 1 0 0 0 0 0.2 0 1 ┘
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
- 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
Haskell
import Control.Applicative (liftA2)
import Data.List (genericLength, intersect, nub, union)
import Data.List.Split (chunksOf)
import Data.Ratio (denominator, numerator)
import Text.Tabular (Header(..), Properties(..), Table(..))
import Text.Tabular.AsciiArt (render)
-- The Jaccard index of two sets. If both sets are empty we define the index to
-- be 1.
jaccard :: (Eq a, Fractional b) => [a] -> [a] -> b
jaccard [] [] = 1
jaccard xs ys = let uxs = nub xs -- unique xs
isz = genericLength $ intersect uxs ys
usz = genericLength $ union uxs ys
in isz / usz
-- A table of Jaccard indexes for all pairs of sets given in the argument.
-- Associated with each set is its "name", which is only used for display
-- purposes.
jaccardTable :: Eq a => [(String, [a])] -> String
jaccardTable xs = render id id showRat
$ Table (Group SingleLine $ map Header names)
(Group SingleLine $ map Header names)
$ chunksOf (length xs)
$ map (uncurry jaccard)
$ allPairs sets
where names = map fst xs
sets = map snd xs
-- Show a rational number as numerator/denominator. If the denominator is 1
-- then just show the numerator.
showRat :: Rational -> String
showRat r = case (numerator r, denominator r) of
(n, 1) -> show n
(n, d) -> show n ++ "/" ++ show d
-- All pairs of elements from the list. For example:
--
-- allPairs [1,2] == [(1,1),(1,2),(2,1),(2,2)]
allPairs :: [a] -> [(a,a)]
allPairs xs = liftA2 (,) xs xs
main :: IO ()
main = putStrLn $ jaccardTable [ ("A", [] :: [Int])
, ("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])]
- Output:
+---++---+-----+-----+-----+-----+-----+ | || A | B | C | D | E | F | +===++===+=====+=====+=====+=====+=====+ | A || 1 | 0 | 0 | 0 | 0 | 0 | +---++---+-----+-----+-----+-----+-----+ | B || 0 | 1 | 3/7 | 1/4 | 1/2 | 0 | +---++---+-----+-----+-----+-----+-----+ | C || 0 | 3/7 | 1 | 0 | 1/2 | 0 | +---++---+-----+-----+-----+-----+-----+ | D || 0 | 1/4 | 0 | 1 | 1/8 | 1/5 | +---++---+-----+-----+-----+-----+-----+ | E || 0 | 1/2 | 1/2 | 1/8 | 1 | 0 | +---++---+-----+-----+-----+-----+-----+ | F || 0 | 0 | 0 | 1/5 | 0 | 1 | +---++---+-----+-----+-----+-----+-----+
jq
Works with gojq, the Go implementation of jq In the following:
- the Jaccard index is presented as a string representing a reduced fraction, e.g. "0" or "1/7".
- sets are represented by sorted arrays with distinct elements.
Preliminaries
def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
def gcd(a; b):
# subfunction expects [a,b] as input
# i.e. a ~ .[0] and b ~ .[1]
def rgcd: if .[1] == 0 then .[0]
else [.[1], .[0] % .[1]] | rgcd
end;
[a,b] | rgcd;
The Task
def rjaccardIndex(x; y):
def i(a;b): a - (a-b);
def u(a;b): a + (b - i(a;b)) | unique;
def idivide($i; $j):
if $i == 0 then "0"
else gcd($i;$j) as $d
| if $j == $d then "\($i/$d)"
else "\($i/$d)/\($j/$d)"
end
end;
if (x|length) == 0 and (y|length) == "0" then "1"
else idivide( i(x;y)|length; u(x;y)|length )
end;
def a : [];
def b : [1, 2, 3, 4, 5];
def c : [1, 3, 5, 7, 9];
def d : [2, 4, 6, 8, 10];
def e : [2, 3, 5, 7];
def f : [8];
def task:
def tidy: map(lpad(4))|join(" ");
[a,b,c,d,e,f] as $sets
| [range(0;$sets|length) | [. + 97] | implode] as $names
| ([""] + $names | tidy),
(range(0; $sets|length) as $i
| ([$i + 97] | implode) as $name
| $sets[$i] as $x
| $sets | map(rjaccardIndex($x; .)) | tidy
| " \($name): \(.)" ) ;
task
- Output:
a b c d e f a: 0 0 0 0 0 0 b: 0 1 3/7 1/4 1/2 0 c: 0 3/7 1 0 1/2 0 d: 0 1/4 0 1 1/8 1/5 e: 0 1/2 1/2 1/8 1 0 f: 0 0 0 1/5 0 1
Julia
J(A, B) = begin i, u = length(A ∩ B), length(A ∪ B); u == 0 ? 1//1 : i // u end
A = Int[]
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]
testsets = [A, B, C, D, E, F]
println("Set A Set B J(A, B)\n", "-"^44)
for a in testsets, b in testsets
println(rpad(isempty(a) ? "[]" : a, 18), rpad(isempty(b) ? "[]" : b, 18),
replace(string(J(a, b)), "//" => "/"))
end
- Output:
Set A Set B J(A, B) -------------------------------------------- [] [] 1/1 [] [1, 2, 3, 4, 5] 0/1 [] [1, 3, 5, 7, 9] 0/1 [] [2, 4, 6, 8, 10] 0/1 [] [2, 3, 5, 7] 0/1 [] [8] 0/1 [1, 2, 3, 4, 5] [] 0/1 [1, 2, 3, 4, 5] [1, 2, 3, 4, 5] 1/1 [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 [1, 3, 5, 7, 9] [] 0/1 [1, 3, 5, 7, 9] [1, 2, 3, 4, 5] 3/7 [1, 3, 5, 7, 9] [1, 3, 5, 7, 9] 1/1 [1, 3, 5, 7, 9] [2, 4, 6, 8, 10] 0/1 [1, 3, 5, 7, 9] [2, 3, 5, 7] 1/2 [1, 3, 5, 7, 9] [8] 0/1 [2, 4, 6, 8, 10] [] 0/1 [2, 4, 6, 8, 10] [1, 2, 3, 4, 5] 1/4 [2, 4, 6, 8, 10] [1, 3, 5, 7, 9] 0/1 [2, 4, 6, 8, 10] [2, 4, 6, 8, 10] 1/1 [2, 4, 6, 8, 10] [2, 3, 5, 7] 1/8 [2, 4, 6, 8, 10] [8] 1/5 [2, 3, 5, 7] [] 0/1 [2, 3, 5, 7] [1, 2, 3, 4, 5] 1/2 [2, 3, 5, 7] [1, 3, 5, 7, 9] 1/2 [2, 3, 5, 7] [2, 4, 6, 8, 10] 1/8 [2, 3, 5, 7] [2, 3, 5, 7] 1/1 [2, 3, 5, 7] [8] 0/1 [8] [] 0/1 [8] [1, 2, 3, 4, 5] 0/1 [8] [1, 3, 5, 7, 9] 0/1 [8] [2, 4, 6, 8, 10] 1/5 [8] [2, 3, 5, 7] 0/1 [8] [8] 1/1
Phix
with javascript_semantics include sets.e function jaccard(sequence a, b) integer i = length(intersection(a,b)), u = length(union(a,b)) return iff(u=0?1:i/u) end function constant tests = {{}, -- A {1, 2, 3, 4, 5}, -- B {1, 3, 5, 7, 9}, -- C {2, 4, 6, 8, 10}, -- D {2, 3, 5, 7}, -- E {8}} -- F for i=1 to length(tests) do for j=i to length(tests) do string s = sprintf("J(%c,%c)",{'A'+i-1,'A'+j-1}) atom jij = jacard(tests[i],tests[j]) if i!=j then atom jji = jacard(tests[j],tests[i]) assert(jji==jij) s &= sprintf(" = J(%c,%c)",{'A'+j-1,'A'+i-1}) end if printf(1,"%s = %g\n",{s,jij}) end for end for
- Output:
J(A,A) = 1 J(A,B) = J(B,A) = 0 J(A,C) = J(C,A) = 0 J(A,D) = J(D,A) = 0 J(A,E) = J(E,A) = 0 J(A,F) = J(F,A) = 0 J(B,B) = 1 J(B,C) = J(C,B) = 0.428571 J(B,D) = J(D,B) = 0.25 J(B,E) = J(E,B) = 0.5 J(B,F) = J(F,B) = 0 J(C,C) = 1 J(C,D) = J(D,C) = 0 J(C,E) = J(E,C) = 0.5 J(C,F) = J(F,C) = 0 J(D,D) = 1 J(D,E) = J(E,D) = 0.125 J(D,F) = J(F,D) = 0.2 J(E,E) = 1 J(E,F) = J(F,E) = 0 J(F,F) = 1
Perl
#!/usr/bin/perl
use strict;
use warnings;
my %sets = (
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],
);
use Data::Dump 'dd'; dd \%sets;
for my $left (sort keys %sets )
{
for my $right (sort keys %sets )
{
my %union;
$union{ $_ }++ for @{ $sets{$left} }, @{ $sets{$right} };
print "J($left,$right) = ",
%union ? (grep $_ == 2, values %union) / (keys %union) : 1, "\n";
}
}
- Output:
{ A => [], B => [1 .. 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,A) = 0 J(B,B) = 1 J(B,C) = 0.428571428571429 J(B,D) = 0.25 J(B,E) = 0.5 J(B,F) = 0 J(C,A) = 0 J(C,B) = 0.428571428571429 J(C,C) = 1 J(C,D) = 0 J(C,E) = 0.5 J(C,F) = 0 J(D,A) = 0 J(D,B) = 0.25 J(D,C) = 0 J(D,D) = 1 J(D,E) = 0.125 J(D,F) = 0.2 J(E,A) = 0 J(E,B) = 0.5 J(E,C) = 0.5 J(E,D) = 0.125 J(E,E) = 1 J(E,F) = 0 J(F,A) = 0 J(F,B) = 0 J(F,C) = 0 J(F,D) = 0.2 J(F,E) = 0 J(F,F) = 1
Prolog
show([]).
show([X|Xs]):- write(X), show(Xs).
j(N,M,X):- M > 0 -> X is N/M; X is 1.
task:- L = [[], [1,2,3,4,5], [1,3,5,7,9], [2,4,6,8,10], [2,3,5,7], [8]],
forall((member(A,L), member(B,L)), (
findall(X, (member(X,A), member(X,B)), I), length(I,N),
findall(X, (member(X,B), not(member(X,A))), T), append(A,T,U), length(U,M),
j(N,M,J), show(["A = ",A,", B = ",B,", J = ",J]), nl)).
- Output:
?- task. A = [], B = [], J = 1 A = [], B = [1,2,3,4,5], J = 0 A = [], B = [1,3,5,7,9], J = 0 A = [], B = [2,4,6,8,10], J = 0 A = [], B = [2,3,5,7], J = 0 A = [], B = [8], J = 0 A = [1,2,3,4,5], B = [], J = 0 A = [1,2,3,4,5], B = [1,2,3,4,5], J = 1 A = [1,2,3,4,5], B = [1,3,5,7,9], J = 0.42857142857142855 A = [1,2,3,4,5], B = [2,4,6,8,10], J = 0.25 A = [1,2,3,4,5], B = [2,3,5,7], J = 0.5 A = [1,2,3,4,5], B = [8], J = 0 A = [1,3,5,7,9], B = [], J = 0 A = [1,3,5,7,9], B = [1,2,3,4,5], J = 0.42857142857142855 A = [1,3,5,7,9], B = [1,3,5,7,9], J = 1 A = [1,3,5,7,9], B = [2,4,6,8,10], J = 0 A = [1,3,5,7,9], B = [2,3,5,7], J = 0.5 A = [1,3,5,7,9], B = [8], J = 0 A = [2,4,6,8,10], B = [], J = 0 A = [2,4,6,8,10], B = [1,2,3,4,5], J = 0.25 A = [2,4,6,8,10], B = [1,3,5,7,9], J = 0 A = [2,4,6,8,10], B = [2,4,6,8,10], J = 1 A = [2,4,6,8,10], B = [2,3,5,7], J = 0.125 A = [2,4,6,8,10], B = [8], J = 0.2 A = [2,3,5,7], B = [], J = 0 A = [2,3,5,7], B = [1,2,3,4,5], J = 0.5 A = [2,3,5,7], B = [1,3,5,7,9], J = 0.5 A = [2,3,5,7], B = [2,4,6,8,10], J = 0.125 A = [2,3,5,7], B = [2,3,5,7], J = 1 A = [2,3,5,7], B = [8], J = 0 A = [8], B = [], J = 0 A = [8], B = [1,2,3,4,5], J = 0 A = [8], B = [1,3,5,7,9], J = 0 A = [8], B = [2,4,6,8,10], J = 0.2 A = [8], B = [2,3,5,7], J = 0 A = [8], B = [8], J = 1 true.
Quackery
[ $ "bigrat.qky" loadfile ] now!
[ over size - space swap of
join echo$ ] is recho$ ( $ n --> $ )
[ dip unbuild recho$ ] is recho ( x n --> $ )
[ 0 swap witheach [ bit | ] ] is set ( [ --> n )
[ & ] is intersection ( n --> n )
[ | ] is union ( n --> n )
[ [] 0 rot
[ dup 0 > while
dup 1 & if
[ dip [ tuck join swap ] ]
dip 1+
1 >> again ]
2drop ] is items ( n --> [ )
[ 2dup = iff [ 2drop 1 1 ] done
2dup union items size
dip [ intersection items size ]
dup 0 = if [ 2drop 0 1 ]
] is jaccard ( n n --> n/d )
[ ' [ ] set ] constant is A ( --> n )
[ ' [ 1 2 3 4 5 ] set ] constant is B ( --> n )
[ ' [ 1 3 5 7 9 ] set ] constant is C ( --> n )
[ ' [ 2 4 6 8 10 ] set ] constant is D ( --> n )
[ ' [ 2 3 5 7 ] set ] constant is E ( --> n )
[ ' [ 8 ] set ] constant is F ( --> n )
' [ A B C D E F ]
dup witheach
[ over witheach
[ over items 15 recho
dup items 15 recho
say "--> "
2dup jaccard
proper$ echo$
cr drop ]
drop
behead drop ]
drop
- Output:
[ ] [ ] --> 1 [ ] [ 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 2 3 4 5 ] --> 1 [ 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 ] [ 1 3 5 7 9 ] --> 1 [ 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 4 6 8 10 ] --> 1 [ 2 4 6 8 10 ] [ 2 3 5 7 ] --> 1/8 [ 2 4 6 8 10 ] [ 8 ] --> 1/5 [ 2 3 5 7 ] [ 2 3 5 7 ] --> 1 [ 2 3 5 7 ] [ 8 ] --> 0 [ 8 ] [ 8 ] --> 1
Raku
sub J(\A, \B) { A ∪ B ?? (A ∩ B) / (A ∪ B) !! A ∪ B == A ∩ B ?? 1 !! 0 }
my %p =
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>,
;
.say for %p.sort;
say '';
say "J({.join: ','}) = ", J |%p{$_} for [X] <A B C D E F> xx 2;
- Output:
A => () B => (1 2 3 4 5) C => (1 3 5 7 9) D => (2 4 6 8 10) E => (2 3 5 7) F => 8 J(A,A) = 1 J(A,B) = 0 J(A,C) = 0 J(A,D) = 0 J(A,E) = 0 J(A,F) = 0 J(B,A) = 0 J(B,B) = 1 J(B,C) = 0.428571 J(B,D) = 0.25 J(B,E) = 0.5 J(B,F) = 0 J(C,A) = 0 J(C,B) = 0.428571 J(C,C) = 1 J(C,D) = 0 J(C,E) = 0.5 J(C,F) = 0 J(D,A) = 0 J(D,B) = 0.25 J(D,C) = 0 J(D,D) = 1 J(D,E) = 0.125 J(D,F) = 0.2 J(E,A) = 0 J(E,B) = 0.5 J(E,C) = 0.5 J(E,D) = 0.125 J(E,E) = 1 J(E,F) = 0 J(F,A) = 0 J(F,B) = 0 J(F,C) = 0 J(F,D) = 0.2 J(F,E) = 0 J(F,F) = 1
Wren
Note that the Set object in the above module is implemented as a Map and consequently the iteration order (and the order in which elements are printed) is undefined.
import "./set" for Set
import "./trait" for Indexed
import "./fmt" for Fmt
var jaccardIndex = Fn.new { |a, b|
if (a.count == 0 && b.count == 0) return 1
return a.intersect(b).count / a.union(b).count
}
var a = Set.new([])
var b = Set.new([1, 2, 3, 4, 5])
var c = Set.new([1, 3, 5, 7, 9])
var d = Set.new([2, 4, 6, 8, 10])
var e = Set.new([2, 3, 5, 7])
var f = Set.new([8])
var isets = Indexed.new([a, b, c, d, e, f])
for (se in isets) {
var i = String.fromByte(se.index + 65)
var v = se.value
v = v.toList.sort() // force original sorted order
Fmt.print("$s = $n", i, v)
}
System.print()
for (se1 in isets) {
var i1 = String.fromByte(se1.index + 65)
var v1 = se1.value
for (se2 in isets) {
var i2 = String.fromByte(se2.index + 65)
var v2 = se2.value
Fmt.print("J($s, $s) = $h", i1, i2, jaccardIndex.call(v1, v2))
}
}
- Output:
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