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.


proc jaro {s1 s2} {
    set l1 [string length $s1]
    set l2 [string length $s2]
    set dmax [expr {max($l1, $l2)/2 - 1}]   ;# window size to scan for matches
    set m1 {}                               ;# match indices
    set m2 {}
    for {set i 0} {$i < $l1} {incr i} {
        set jmin [expr {$i - $dmax}]        ;# don't worry about going out-of-bounds
        set jmax [expr {$i + $dmax}]        ;# because [string index] will return {} safely
        for {set j $jmin} {$j <= $jmax} {incr j} {
            if {$j in $m2} continue   ;# don't double-count matches
            if {[string index $s1 $i] eq [string index $s2 $j]} {
                lappend m1 $i
                lappend m2 $j
                break
            }
        }
    }
    set T 0                 ;# number of transpositions
    set oj -1
    foreach j $m2 {
        if {$j < $oj} {incr T}
        set oj $j
    }
    set T [expr {$T / 2.0}]
    set M [expr {1.0 * [llength $m1]}]  ;# number of matches
    expr { ( ($M / $l1) + ($M / $l2) + (($M - $T) / $M) ) / 3.0 }
}
 
 
foreach {s t} {
    DWAYNE DUANE
    MARTHA MARHTA
    DIXON  DICKSONX
    JELLYFISH SMELLYFISH
} {
    puts "[jaro $s $t]:\t$s / $t"
}