Dice-Sørensen coefficient

The Dice-Sørensen coefficient (see below for other names) is a statistic used to gauge the similarity of two samples. It was independently developed by the botanists Lee Raymond Dice^[1] and Thorvald Sørensen,^[2] who published in 1945 and 1948 respectively.

Name

The index is known by several other names, especially Sørensen–Dice index,^[3] Sørensen index and Dice's coefficient. Other variations include the "similarity coefficient" or "index", such as Dice similarity coefficient (DSC). Common alternate spellings for Sørensen are Sorenson, Soerenson and Sörenson, and all three can also be seen with the –sen ending (the Danish letter ø is phonetically equivalent to the German/Swedish ö, which can be written as oe in ASCII).

Other names include:

• F1 score
• Czekanowski's binary (non-quantitative) index^[4]
• Measure of genetic similarity^[5]
• Zijdenbos similarity index,^[6][7] referring to a 1994 paper of Zijdenbos et al.^[8][3]

Formula

Sørensen's original formula was intended to be applied to discrete data. Given two sets, X and Y, it is defined as

${\displaystyle DSC={\frac {2|X\cap Y|}{|X|+|Y|}}}$

where |X| and |Y| are the cardinalities of the two sets (i.e. the number of elements in each set). The Sørensen index equals twice the number of elements common to both sets divided by the sum of the number of elements in each set. Equivalently, the index is the size of the intersection as a fraction of the average size of the two sets.

When applied to Boolean data, using the definition of true positive (TP), false positive (FP), and false negative (FN), it can be written as

${\displaystyle DSC={\frac {2{\mathit {TP}}}{2{\mathit {TP}}+{\mathit {FP}}+{\mathit {FN}}}}}$.

It is different from the Jaccard index which only counts true positives once in both the numerator and denominator. DSC is the quotient of similarity and ranges between 0 and 1.^[9] It can be viewed as a similarity measure over sets.

Similarly to the Jaccard index, the set operations can be expressed in terms of vector operations over binary vectors a and b:

${\displaystyle s_{v}={\frac {2|{\bf {{a}\cdot {\bf {{b}|}}}}}{|{\bf {{a}|^{2}+|{\bf {{b}|^{2}}}}}}}}$

which gives the same outcome over binary vectors and also gives a more general similarity metric over vectors in general terms.

For sets X and Y of keywords used in information retrieval, the coefficient may be defined as twice the shared information (intersection) over the sum of cardinalities :^[10]

When taken as a string similarity measure, the coefficient may be calculated for two strings, x and y using bigrams as follows:^[11]

${\displaystyle s={\frac {2n_{t}}{n_{x}+n_{y}}}}$

where n_t is the number of character bigrams found in both strings, n_x is the number of bigrams in string x and n_y is the number of bigrams in string y. For example, to calculate the similarity between:

night

nacht

We would find the set of bigrams in each word:

{ni,ig,gh,ht}

{na,ac,ch,ht}

Each set has four elements, and the intersection of these two sets has only one element: ht.

Inserting these numbers into the formula, we calculate, s = (2 · 1) / (4 + 4) = 0.25.

Continuous Dice Coefficient

Source:^[12]

For a discrete (binary) ground truth ${\displaystyle A}$ and continuous measures ${\displaystyle B}$ in the interval [0,1], the following formula can be used:

${\displaystyle cDC={\frac {2|A\cap B|}{c*|A|+|B|}}}$

Where ${\displaystyle |A\cap B|=\Sigma _{i}a_{i}b_{i}}$ and ${\displaystyle |B|=\Sigma _{i}b_{i}}$

c can be computed as follows:

${\displaystyle c={\frac {\Sigma _{i}a_{i}b_{i}}{\Sigma _{i}a_{i}\operatorname {sign} {(b_{i})}}}}$

If ${\displaystyle \Sigma _{i}a_{i}\operatorname {sign} {(b_{i})}=0}$ which means no overlap between A and B, c is set to 1 arbitrarily.

Difference from Jaccard

This coefficient is not very different in form from the Jaccard index. In fact, both are equivalent in the sense that given a value for the Sørensen–Dice coefficient ${\displaystyle S}$, one can calculate the respective Jaccard index value ${\displaystyle J}$ and vice versa, using the equations ${\displaystyle J=S/(2-S)}$ and ${\displaystyle S=2J/(1+J)}$.

Since the Sørensen–Dice coefficient does not satisfy the triangle inequality, it can be considered a semimetric version of the Jaccard index.^[4]

The function ranges between zero and one, like Jaccard. Unlike Jaccard, the corresponding difference function

${\displaystyle d(X,Y)=1-{\frac {2|X\cap Y|}{|X|+|Y|}}}$

is not a proper distance metric as it does not satisfy the triangle inequality.^[4] The simplest counterexample of this is given by the three sets ${\displaystyle X=\{a\}}$, ${\displaystyle Y=\{b\}}$ and ${\displaystyle Z=X\cup Y=\{a,b\}}$. We have ${\displaystyle d(X,Y)=1}$ and ${\displaystyle d(X,Z)=d(Y,Z)=1/3}$. To satisfy the triangle inequality, the sum of any two sides must be greater than or equal to that of the remaining side. However, ${\displaystyle d(X,Z)+d(Y,Z)=2/3<1=d(X,Y)}$.

Applications

The Sørensen–Dice coefficient is useful for ecological community data (e.g. Looman & Campbell, 1960^[13]). Justification for its use is primarily empirical rather than theoretical (although it can be justified theoretically as the intersection of two fuzzy sets^[14]). As compared to Euclidean distance, the Sørensen distance retains sensitivity in more heterogeneous data sets and gives less weight to outliers.^[15] Recently the Dice score (and its variations, e.g. logDice taking a logarithm of it) has become popular in computer lexicography for measuring the lexical association score of two given words.^[16] logDice is also used as part of the Mash Distance for genome and metagenome distance estimation^[17] Finally, Dice is used in image segmentation, in particular for comparing algorithm output against reference masks in medical applications.^[8]

Abundance version

The expression is easily extended to abundance instead of presence/absence of species. This quantitative version is known by several names:

• Quantitative Sørensen–Dice index^[4]
• Quantitative Sørensen index^[4]
• Quantitative Dice index^[4]
• Bray–Curtis similarity (1 minus the Bray-Curtis dissimilarity)^[4]
• Czekanowski's quantitative index^[4]
• Steinhaus index^[4]
• Pielou's percentage similarity^[4]
• 1 minus the Hellinger distance^[18]
• Proportion of specific agreement^[19] or positive agreement^[20]

See also

• Correlation
• F1 score
• Jaccard index
• Hamming distance
• Mantel test
• Morisita's overlap index
• Overlap coefficient
• Renkonen similarity index
• Tversky index
• Universal adaptive strategy theory (UAST)

References

1. Dice, Lee R. (1945). "Measures of the Amount of Ecologic Association Between Species". Ecology. 26 (3): 297–302. doi:10.2307/1932409. JSTOR 1932409. S2CID 53335638.
2. Sørensen, T. (1948). "A method of establishing groups of equal amplitude in plant sociology based on similarity of species and its application to analyses of the vegetation on Danish commons". Kongelige Danske Videnskabernes Selskab. 5 (4): 1–34.
3. Carass, A.; Roy, S.; Gherman, A.; Reinhold, J.C.; Jesson, A.; et al. (2020). "Evaluating White Matter Lesion Segmentations with Refined Sørensen-Dice Analysis". Scientific Reports. 10 (1): 8242. Bibcode:2020NatSR..10.8242C. doi:10.1038/s41598-020-64803-w. ISSN 2045-2322. PMC 7237671. PMID 32427874.
4. Gallagher, E.D., 1999. COMPAH Documentation, University of Massachusetts, Boston
5. Nei, M.; Li, W.H. (1979). "Mathematical model for studying genetic variation in terms of restriction endonucleases". PNAS. 76 (10): 5269–5273. Bibcode:1979PNAS...76.5269N. doi:10.1073/pnas.76.10.5269. PMC 413122. PMID 291943.
6. Prescott, J.W.; Pennell, M.; Best, T.M.; Swanson, M.S.; Haq, F.; Jackson, R.; Gurcan, M.N. (2009). "An automated method to segment the femur for osteoarthritis research". 2009 Annual International Conference of the IEEE Engineering in Medicine and Biology Society. IEEE. pp. 6364–6367. doi:10.1109/iembs.2009.5333257. PMC 2826829.
7. Swanson, M.S.; Prescott, J.W.; Best, T.M.; Powell, K.; Jackson, R.D.; Haq, F.; Gurcan, M.N. (2010). "Semi-automated segmentation to assess the lateral meniscus in normal and osteoarthritic knees". Osteoarthritis and Cartilage. 18 (3): 344–353. doi:10.1016/j.joca.2009.10.004. ISSN 1063-4584. PMC 2826568. PMID 19857510.
8. Zijdenbos, A.P.; Dawant, B.M.; Margolin, R.A.; Palmer, A.C. (1994). "Morphometric analysis of white matter lesions in MR images: method and validation". IEEE Transactions on Medical Imaging. 13 (4): 716–724. doi:10.1109/42.363096. ISSN 0278-0062. PMID 18218550.
9. Murguía, Miguel; Luis Villaseñor, José (2003). "Estimating the effect of the similarity coefficient and the cluster algorithm on biogeographic classifications" (PDF). Annales Botanici Fennici. 40: 415–421. ISSN 0003-3847.
10. van Rijsbergen, Cornelis Joost (1979). Information Retrieval. London: Butterworths. ISBN 3-642-12274-4.
11. Kondrak, Grzegorz; Marcu, Daniel; Knight, Kevin (2003). "Cognates Can Improve Statistical Translation Models" (PDF). Proceedings of HLT-NAACL 2003: Human Language Technology Conference of the North American Chapter of the Association for Computational Linguistics. pp. 46–48.
12. Shamir, Reuben R.; Duchin, Yuval; Kim, Jinyoung; Sapiro, Guillermo; Harel, Noam (2018-04-25). "Continuous Dice Coefficient: a Method for Evaluating Probabilistic Segmentations": 306977. arXiv:1906.11031. doi:10.1101/306977. S2CID 90993940. {{cite journal}}: Cite journal requires |journal= (help)
13. Looman, J.; Campbell, J.B. (1960). "Adaptation of Sorensen's K (1948) for estimating unit affinities in prairie vegetation". Ecology. 41 (3): 409–416. doi:10.2307/1933315. JSTOR 1933315.
14. Roberts, D.W. (1986). "Ordination on the basis of fuzzy set theory". Vegetatio. 66 (3): 123–131. doi:10.1007/BF00039905. S2CID 12573576.
15. McCune, Bruce & Grace, James (2002) Analysis of Ecological Communities. Mjm Software Design; ISBN 0-9721290-0-6.
16. Rychlý, P. (2008) A lexicographer-friendly association score. Proceedings of the Second Workshop on Recent Advances in Slavonic Natural Language Processing RASLAN 2008: 6–9
17. Ondov, Brian D., et al. "Mash: fast genome and metagenome distance estimation using MinHash." Genome biology 17.1 (2016): 1-14.
18. Bray, J. Roger; Curtis, J. T. (1957). "An Ordination of the Upland Forest Communities of Southern Wisconsin". Ecological Monographs. 27 (4): 326–349. doi:10.2307/1942268. JSTOR 1942268.
19. Ayappa, Indu; Norman, Robert G (2000). "Non-Invasive Detection of Respiratory Effort-Related Arousals (RERAs) by a Nasal Cannula/Pressure Transducer System". Sleep. 23 (6): 763–771. doi:10.1093/sleep/23.6.763. PMID 11007443.
20. John Uebersax. "Raw Agreement Indices".

External links