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.


 
// Calculate Jaro distance of 2 strings. Nigel Galloway: August 7th., 2020
let fG n g=Seq.map2(fun n g->if g=1 then Some n else None) n g |> Seq.choose id
let J (n:string) (g:string)=
  let s1,s2=n.Length,g.Length
  let w,m1,m2=(max s1 s2)/2-1,Array.zeroCreate<int>s1,Array.zeroCreate<int>s2
  g|>Seq.iteri(fun i g->match [max(i-w) 0..min(i+w)(s1-1)]|>Seq.tryFind(fun i->n.[i]=g&&m1.[i]=0) with Some n->m1.[n]<-1;m2.[i]<-1 |_->())
  let t=float(Seq.fold2(fun Σ n g->Σ + (if n<>g then 1 else 0)) 0 (fG n m1) (fG g m2))/2.0
  let m=float(m2|>Array.sum) in if m=0.0 then m else ((m/float s1)+(m/float s2)+((m-t)/m))/3.0
 
printfn "MARTHA MARHTA->%f" (J "MARTHA" "MARHTA")
printfn "DIXON DICKSONX->%f" (J "DIXON" "DICKSONX")
printfn "JELLYFISH SMELLYFISH->%f" (J "JELLYFISH" "SMELLYFISH")
 