Jaro–Winkler distance

The Jaro Similarity ${\displaystyle sim_{j}}$ of two given strings ${\displaystyle s_{1}}$ and ${\displaystyle s_{2}}$ is

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

Where:

• ${\displaystyle |s_{i}|}$ is the length of the string ${\displaystyle s_{i}}$;
• ${\displaystyle m}$ is the number of matching characters (see below);
• ${\displaystyle t}$ is half the number of transpositions (see below).

Two characters from ${\displaystyle s_{1}}$ and ${\displaystyle s_{2}}$ respectively, are considered matching only if they are the same and not farther than ${\displaystyle \left\lfloor {\frac {\max(|s_{1}|,|s_{2}|)}{2}}\right\rfloor -1}$.

Each character of ${\displaystyle s_{1}}$ is compared with all its matching characters in ${\displaystyle s_{2}}$. The number of matching (but different sequence order) characters divided by 2 defines the number of transpositions. For example, in comparing CRATE with TRACE, only 'R' 'A' 'E' are the matching characters, i.e. m=3. Although 'C', 'T' appear in both strings, they are farther than 1 (the result of ${\displaystyle \lfloor {\tfrac {5}{2}}\rfloor -1}$). Therefore, t=0 . In DwAyNE versus DuANE the matching letters are already in the same order D-A-N-E, so no transpositions are needed.

Jaro–Winkler similarity uses a prefix scale ${\displaystyle p}$ which gives more favorable ratings to strings that match from the beginning for a set prefix length ${\displaystyle \ell }$. Given two strings ${\displaystyle s_{1}}$ and ${\displaystyle s_{2}}$, their Jaro–Winkler similarity ${\displaystyle sim_{w}}$ is:

${\displaystyle sim_{w}=sim_{j}+\ell p(1-sim_{j}),}$

where:

• ${\displaystyle sim_{j}}$ is the Jaro similarity for strings ${\displaystyle s_{1}}$ and ${\displaystyle s_{2}}$
• ${\displaystyle \ell }$ is the length of common prefix at the start of the string up to a maximum of four characters
• ${\displaystyle p}$ is a constant scaling factor for how much the score is adjusted upwards for having common prefixes. ${\displaystyle p}$ should not exceed 0.25, otherwise the similarity could become larger than 1. The standard value for this constant in Winkler's work is ${\displaystyle p=0.1}$

The Jaro-Winkler distance ${\displaystyle d_{w}}$ is defined as ${\displaystyle d_{w}=1-sim_{w}}$.

Although often referred to as a distance metric, the Jaro–Winkler distance is not a metric in the mathematical sense of that term because it does not obey the triangle inequality.[1] The Jaro–Winkler distance also does not satisfy the identity axiom ${\displaystyle d(x,y)=0\leftrightarrow x=y}$.