Jaro–Winkler distance

In computer science and statistics, the Jaro–Winkler distance (Winkler, 1990) is a measure of similarity between two strings. It is a variant of the Jaro distance metric (Jaro, 1989, 1995) and mainly^{[citation needed]} used in the area of record linkage (duplicate detection). The higher the Jaro–Winkler distance for two strings is, the more similar the strings are. The Jaro–Winkler distance metric is designed and best suited for short strings such as person names. The score is normalized such that 0 equates to no similarity and 1 is an exact match.

Definition

The Jaro distance d_j of two given strings s₁ and s₂ is

$d_j = \frac{1}{3}\left(\frac{m}{|s_1|} + \frac{m}{|s_2|} + \frac{m-t}{m}\right)$

where:

• m is the number of matching characters (see below);
• t is half the number of transpositions (see below).

Two characters from s₁ and s₂ respectively, are considered matching only if they are not farther than $\left\lfloor\frac{\max(|s_1|,|s_2|)}{2}\right\rfloor-1$.

Each character of s₁ is compared with all its matching characters in s₂. The number of matching (but different sequence order) characters divided by the numeric value '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 1.5, i.e., (5/2)-1=1.5. 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 distance uses a prefix scale p which gives more favourable ratings to strings that match from the beginning for a set prefix length $\ell$. Given two strings s₁ and s₂, their Jaro–Winkler distance d_w is:

$d_w = d_j + (\ell p (1 - d_j))$

where:

• d_j is the Jaro distance for strings s₁ and s₂
• $\ell$ is the length of common prefix at the start of the string up to a maximum of 4 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

Although often referred to as a distance metric, the Jaro–Winkler distance is actually not a metric in the mathematical sense of that term.

Example

Note that Winkler's "reference" C code differs in at least two ways from published accounts of the Jaro–Winkler metric. First is his use of a typo table (adjwt) and also some optional additional tolerance for long strings.

Given the strings s₁ MARTHA and s₂ MARHTA we find:

• m = 6
• | s₁ | = 6
• | s₂ | = 6
• There are mismatched characters T/H and H/T leading to $t = \frac{2}{2} = 1$

We find a Jaro score of:

$d_j = \frac{1}{3}\left(\frac{6}{6} + \frac{6}{6} + \frac{6-1}{6}\right) = 0.944$

To find the Jaro–Winkler score using the standard weight p = 0.1, we continue to find:

• $\ell = 3$

Thus:

d_w = 0.944 + (3 * 0.1(1 − 0.944)) = 0.961

Given the strings s₁ DWAYNE and s₂ DUANE we find:

• m = 4
• | s₁ | = 6
• | s₂ | = 5
• t = 0

We find a Jaro score of:

$d_j = \frac{1}{3}\left(\frac{4}{6} + \frac{4}{5} + \frac{4-0}{4}\right) = 0.822$

To find the Jaro–Winkler score using the standard weight p = 0.1, we continue to find:

• $\ell = 1$

Thus:

d_w = 0.822 + (1 * 0.1(1 − 0.822)) = 0.84

Given the strings s₁ DIXON and s₂ DICKSONX we find:

	D	I	X	O	N
D	1	0	0	0	0
I	0	1	0	0	0
C	0	0	0	0	0
K	0	0	0	0	0
S	0	0	0	0	0
O	0	0	0	1	0
N	0	0	0	0	1
X	0	0	0	0	0

• m = 4 Note that the two Xs are not considered matches because they are outside the match window of 3.
• | s₁ | = 5
• | s₂ | = 8
• t = 0

We find a Jaro score of:

$d_j = \frac{1}{3}\left(\frac{4}{5} + \frac{4}{8} + \frac{4-0}{4}\right) = 0.767$

To find the Jaro–Winkler score using the standard weight p = 0.1, we continue to find:

• $\ell = 2$

Thus:

d_w = 0.767 + (2 * 0.1(1 − 0.767)) = 0.813

See also

• Levenshtein distance
• Record linkage
• Census

References

• Jaro, M. A. (1989). "Advances in record linkage methodology as applied to the 1985 census of Tampa Florida". Journal of the American Statistical Society 84 (406): 414–20. 
• 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". Proceedings of the Section on Survey Research Methods (American Statistical Association): 354–359. http://www.amstat.org/sections/srms/Proceedings/papers/1990_056.pdf. 
• Winkler, W. E. (2006). "Overview of Record Linkage and Current Research Directions". Research Report Series, RRS. http://www.census.gov/srd/papers/pdf/rrs2006-02.pdf. 

External links

• Implementation & documentation in Java LingPipe. Features extensive comparison with the original strcmp.c implementation.
• strcmp.c - Original C Implementation by the author of the algorithm
• Open Source implementation in Java and .NET
• PHP implementation released under GPLv3.0