The Jaro distance is a measure of edit distance between two strings; its inverse, called the Jaro similarity, is a measure of two strings' similarity: the higher the value, the more similar the strings are. The score is normalized such that   0   equates to no similarities and   1   is an exact match.



Definition
The Jaro similarity   




d

j




{\displaystyle d_{j}}

   of two given strings   




s

1




{\displaystyle s_{1}}

   and   




s

2




{\displaystyle s_{2}}

   is

 




d

j


=

{




0



if 

m
=
0






1
3



(


m


|


s

1



|




+


m


|


s

2



|




+



m
−
t

m


)




otherwise









{\displaystyle d_{j}=\left\{{\begin{array}{l l}0&{\text{if }}m=0\\{\frac {1}{3}}\left({\frac {m}{|s_{1}|}}+{\frac {m}{|s_{2}|}}+{\frac {m-t}{m}}\right)&{\text{otherwise}}\end{array}}\right.}


Where:

 



m


{\displaystyle m}

   is the number of matching characters;
 



t


{\displaystyle t}

   is half the number of transpositions.


Two characters from   




s

1




{\displaystyle s_{1}}

   and   




s

2




{\displaystyle s_{2}}

   respectively, are considered matching only if they are the same and not farther apart than   




⌊



max
(

|


s

1



|

,

|


s

2



|

)

2


⌋

−
1


{\displaystyle \left\lfloor {\frac {\max(|s_{1}|,|s_{2}|)}{2}}\right\rfloor -1}

 characters.

Each character of   




s

1




{\displaystyle s_{1}}

   is compared with all its matching characters in   




s

2




{\displaystyle s_{2}}

. Each difference in position is half a transposition; that is, the number of transpositions is half the number of characters which are common to the two strings but occupy different positions in each one.



Example
Given the strings   




s

1




{\displaystyle s_{1}}

   DWAYNE   and   




s

2




{\displaystyle s_{2}}

   DUANE   we find:

 



m
=
4


{\displaystyle m=4}


 




|


s

1



|

=
6


{\displaystyle |s_{1}|=6}


 




|


s

2



|

=
5


{\displaystyle |s_{2}|=5}


 



t
=
0


{\displaystyle t=0}




We find a Jaro score of:

 




d

j


=


1
3



(


4
6


+


4
5


+



4
−
0

4


)

=
0.822


{\displaystyle d_{j}={\frac {1}{3}}\left({\frac {4}{6}}+{\frac {4}{5}}+{\frac {4-0}{4}}\right)=0.822}




Task
Implement the Jaro algorithm and show the similarity scores for each of the following pairs:

 ("MARTHA", "MARHTA")
 ("DIXON", "DICKSONX")
 ("JELLYFISH", "SMELLYFISH")


 See also
 Jaro–Winkler distance on Wikipedia.


def jaro($s1; $s2):
    ($s1|length) as $le1
    | ($s2|length) as $le2
    | if $le1 == 0 and $le2 == 0 then 1
      elif $le1 == 0  or $le2 == 0 then 0
      else ((((if $le2 > $le1 then $le2 else $le1 end) / 2) | floor) - 1) as $dist
      | {matches: 0, matches2: [], matches2: [], transpos: 0 }
      | reduce range(0; $le1) as $i (.;
            (($i - $dist)     | if . < 0    then 0    else . end) as $start
          | (($i + $dist + 1) | if . > $le2 then $le2 else . end) as $stop
          | .k = $start
	  | until(.k >= $stop;
              if (.matches2[.k] or $s1[$i:$i+1] != $s2[.k:.k+1])|not
              then .matches1[$i] = true
              | .matches2[.k] = true
              | .matches += 1
	      | .k = $stop
	      else .k += 1
	      end) )
      | if .matches == 0 then 0
        else .k = 0
        | reduce range(0; $le1) as $i (.;
            if .matches1[$i]
	    then until(.k >= $le2 or .matches2[.k]; .k += 1)
	    | if .k < $le2 and ($s1[$i:$i+1] != $s2[.k:.k+1]) then .transpos += 1 else . end
      	    | .k += 1
	    else .
	    end )
        | .transpos /= 2
        | (.matches/$le1 + .matches/$le2 + ((.matches - .transpos)/.matches)) / 3
        end
      end ;
 
def task:
  [["MARTHA","MARHTA"],
   ["DIXON", "DICKSONX"],
   ["JELLYFISH","SMELLYFISH"],
   ["ABC","DEF"]][]
  | (jaro(.[0]; .[1]) * 1000 | floor / 1000) as $d
  | "jaro(\(.[0]); \(.[1])) => \($d)";
 
task