Jaro–Winkler distance

In computer science and statistics, the Jaro–Winkler distance is a string metric measuring an edit distance between two sequences. It is a variant proposed in 1990 by William E. Winkler of the Jaro distance metric (1989, Matthew A. Jaro). Informally, the Jaro distance between two words is the minimum number of single-character transpositions required to change one word into the other.

The Jaro–Winkler distance uses a prefix scale $p$ which gives more favourable ratings to strings that match from the beginning for a set prefix length $\ell$ .

The lower the Jaro–Winkler distance for two strings is, the more similar the strings are. The score is normalized such that 0 equates to no similarity and 1 is an exact match. The Jaro–Winkler similarity is given by 1 − Jaro–Winkler distance.

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.

Definition

Jaro Similarity

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

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:

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

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

Each character of $s_{1}$ is compared with all its matching characters in $s_{2}$ . 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 $\lfloor {\tfrac {5}{2}}\rfloor -1$ ). 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

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

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

where:

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

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

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 $d(x,y)=0\leftrightarrow x=y$ .

Relationship with other edit distance metrics

There are other popular measures of edit distance, which are calculated using a different set of allowable edit operations. For instance,

the Levenshtein distance allows deletion, insertion and substitution;
the Damerau–Levenshtein distance allows insertion, deletion, substitution, and the transposition of two adjacent characters;
the longest common subsequence (LCS) distance allows only insertion and deletion, not substitution;
the Hamming distance allows only substitution, hence, it only applies to strings of the same length.

Edit distance is usually defined as a parameterizable metric calculated with a specific set of allowed edit operations, and each operation is assigned a cost (possibly infinite). This is further generalized by DNA sequence alignment algorithms such as the Smith–Waterman algorithm, which make an operation's cost depend on where it is applied.

Footnotes

↑ "Jaro-Winkler « Inviting Epiphany". RichardMinerich.com. Retrieved 12 June 2017.

References

Cohen, W. W.; Ravikumar, P.; Fienberg, S. E. (2003). "A comparison of string distance metrics for name-matching tasks" (PDF). KDD Workshop on Data Cleaning and Object Consolidation. 3: 73–8.
Jaro, M. A. (1989). "Advances in record linkage methodology as applied to the 1985 census of Tampa Florida". Journal of the American Statistical Association. 84 (406): 414–20. doi:10.1080/01621459.1989.10478785.
Jaro, M. A. (1995). "Probabilistic linkage of large public health data file". Statistics in Medicine. 14 (5–7): 491–8. doi:10.1002/sim.4780140510. PMID 7792443.
Winkler, W. E. (1990). "String Comparator Metrics and Enhanced Decision Rules in the Fellegi-Sunter Model of Record Linkage" (PDF). Proceedings of the Section on Survey Research Methods. American Statistical Association: 354–359.

Winkler, W. E. (2006). "Overview of Record Linkage and Current Research Directions" (PDF). Research Report Series, RRS.

External links

strcmp.c - Original C implementation by the author of the algorithm

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] "Jaro-Winkler « Inviting Epiphany". RichardMinerich.com. Retrieved 12 June 2017.

Strings
String metric	Approximate string matching Bitap algorithm Damerau–Levenshtein distance Edit distance Hamming distance Jaro–Winkler distance Lee distance Levenshtein automaton Levenshtein distance Wagner–Fischer algorithm
String-searching algorithm	Apostolico–Giancarlo algorithm Boyer–Moore string-search algorithm Boyer–Moore–Horspool algorithm Knuth–Morris–Pratt algorithm Rabin–Karp string search algorithm
Multiple string searching	Aho–Corasick Commentz-Walter algorithm Rabin–Karp
Regular expression	Comparison of regular expression engines Regular tree grammar Thompson's construction Nondeterministic finite automaton
Sequence alignment	Hirschberg's algorithm Needleman–Wunsch algorithm Smith–Waterman algorithm
Data structures	DAFSA Suffix array Suffix automaton Suffix tree Generalized suffix tree Rope Ternary search tree Trie
Other	Parsing Pattern matching Compressed pattern matching Longest common subsequence Longest common substring Sequential pattern mining Sorting