Similarity Index

[Melissa Russian]

Itwas developed by Grove Karl Gilbert in 1884 as his ratio of verification (v)[1] and now is often called the critical success index in meteorology.[2] It was later developed independently by Paul Jaccard, originally giving the French name coefficient de communaut,[3][4] and independently formulated again by T. Tanimoto.[5] Thus, it is also called Tanimoto index or Tanimoto coefficient in some fields. However, they are identical in generally taking the ratio of size of intersection over union.

The Jaccard distance, which measures dissimilarity between sample sets, is complementary to the Jaccard coefficient and is obtained by subtracting the Jaccard coefficient from 1 or, equivalently, by dividing the difference of the sizes of the union and the intersection of two sets by the size of the union:

There is also a version of the Jaccard distance for measures, including probability measures. If μ \displaystyle \mu is a measure on a measurable space X \displaystyle X , then we define the Jaccard coefficient by

The MinHash min-wise independent permutations locality sensitive hashing scheme may be used to efficiently compute an accurate estimate of the Jaccard similarity coefficient of pairs of sets, where each set is represented by a constant-sized signature derived from the minimum values of a hash function.

Statistical inference can be made based on the Jaccard similarity coefficients, and consequently related metrics.[6] Given two sample sets A and B with n attributes, a statistical test can be conducted to see if an overlap is statistically significant. The exact solution is available, although computation can be costly as n increases.[6] Estimation methods are available either by approximating a multinomial distribution or by bootstrapping.[6]

When used for binary attributes, the Jaccard index is very similar to the simple matching coefficient. The main difference is that the SMC has the term M 00 \displaystyle M_00 in its numerator and denominator, whereas the Jaccard index does not. Thus, the SMC counts both mutual presences (when an attribute is present in both sets) and mutual absence (when an attribute is absent in both sets) as matches and compares it to the total number of attributes in the universe, whereas the Jaccard index only counts mutual presence as matches and compares it to the number of attributes that have been chosen by at least one of the two sets.

In market basket analysis, for example, the basket of two consumers who we wish to compare might only contain a small fraction of all the available products in the store, so the SMC will usually return very high values of similarities even when the baskets bear very little resemblance, thus making the Jaccard index a more appropriate measure of similarity in that context. For example, consider a supermarket with 1000 products and two customers. The basket of the first customer contains salt and pepper and the basket of the second contains salt and sugar. In this scenario, the similarity between the two baskets as measured by the Jaccard index would be 1/3, but the similarity becomes 0.998 using the SMC.

In other contexts, where 0 and 1 carry equivalent information (symmetry), the SMC is a better measure of similarity. For example, vectors of demographic variables stored in dummy variables, such as gender, would be better compared with the SMC than with the Jaccard index since the impact of gender on similarity should be equal, independently of whether male is defined as a 0 and female as a 1 or the other way around. However, when we have symmetric dummy variables, one could replicate the behaviour of the SMC by splitting the dummies into two binary attributes (in this case, male and female), thus transforming them into asymmetric attributes, allowing the use of the Jaccard index without introducing any bias. The SMC remains, however, more computationally efficient in the case of symmetric dummy variables since it does not require adding extra dimensions.

With even more generality, if f \displaystyle f and g \displaystyle g are two non-negative measurable functions on a measurable space X \displaystyle X with measure μ \displaystyle \mu , then we can define

The Probability Jaccard Index has a geometric interpretation as the area of an intersection of simplices. Every point on a unit k \displaystyle k -simplex corresponds to a probability distribution on k + 1 \displaystyle k+1 elements, because the unit k \displaystyle k -simplex is the set of points in k + 1 \displaystyle k+1 dimensions that sum to 1. To derive the Probability Jaccard Index geometrically, represent a probability distribution as the unit simplex divided into sub simplices according to the mass of each item. If you overlay two distributions represented in this way on top of each other, and intersect the simplices corresponding to each item, the area that remains is equal to the Probability Jaccard Index of the distributions.

That is, no sampling method can achieve more collisions than J P \displaystyle J_\mathcal P on one pair without achieving fewer collisions than J P \displaystyle J_\mathcal P on another pair, where the reduced pair is more similar under J P \displaystyle J_\mathcal P than the increased pair. This theorem is true for the Jaccard Index of sets (if interpreted as uniform distributions) and the probability Jaccard, but not of the weighted Jaccard. (The theorem uses the word "sampling method" to describe a joint distribution over all distributions on a space, because it derives from the use of weighted minhashing algorithms that achieve this as their collision probability.)

Various forms of functions described as Tanimoto similarity and Tanimoto distance occur in the literature and on the Internet. Most of these are synonyms for Jaccard similarity and Jaccard distance, but some are mathematically different. Many sources[12] cite an IBM Technical Report[5] as the seminal reference. The report is available from several libraries.

In "A Computer Program for Classifying Plants", published in October 1960,[13] a method of classification based on a similarity ratio, and a derived distance function, is given. It seems that this is the most authoritative source for the meaning of the terms "Tanimoto similarity" and "Tanimoto Distance". The similarity ratio is equivalent to Jaccard similarity, but the distance function is not the same as Jaccard distance.

In that paper, a "similarity ratio" is given over bitmaps, where each bit of a fixed-size array represents the presence or absence of a characteristic in the plant being modelled. The definition of the ratio is the number of common bits, divided by the number of bits set (i.e. nonzero) in either sample.

If each sample is modelled instead as a set of attributes, this value is equal to the Jaccard coefficient of the two sets. Jaccard is not cited in the paper, and it seems likely that the authors were not aware of it.[citation needed]

This coefficient is, deliberately, not a distance metric. It is chosen to allow the possibility of two specimens, which are quite different from each other, to both be similar to a third. It is easy to construct an example which disproves the property of triangle inequality.

where the same calculation is expressed in terms of vector scalar product and magnitude. This representation relies on the fact that, for a bit vector (where the value of each dimension is either 0 or 1) then

I am using GitExtensions with Visual Studio and when go to commit my change, it says I have added two new files and has a third file (a .resx file) which it seems to be comparing with another .resx file and it says they have similarity index 75%

Git does not store diffs.1 Instead, each commit stores complete files (as listed in the index-at-the-time-the-commit-is-made), as a sort of stand-alone entity. To retrieve a previous commit, git simply finds the commit ID and extracts the associated files.2

The "similarity index" and any presentation of "a file was renamed" or "a file was copied" are simply git guessing at what happened, in an attempt to make things clearer to the human, or present the shortest way to get from one commit to another, for instance. You are correct that the template match is misleading git at this point, but "this point" is the "presentation to user of how to get from Point A to Point B", not "what was or will be stored".

Note that once you have any two given commits to git diff, you can specify different copy and/or rename thresholds to get "what happened" shown to you in different ways. Git does this on demand, by extracting (mostly in-memory) the two commits, comparing them, computing each similarity index (again) at that time, and making its best guess at copies or renames from there.

1This glosses over git's "pack" files, which do use deltas. However, pack files are generally constructed long after a commit (or series of commits). New commits always make new, stand-alone object files, which may be packed and re-packed in various ways later.

2To speed up operation, git will use the current index (cache) information to figure out a quick way to change from "commit currently checked out" (as noted by the index/cache) to "new commit to be checked out" (given as an argument to git checkout). In particular, as long as you have not modified your work-tree so that the index is current, this allows git checkout to avoid touching or even inspecting most files when switching between similar branches or commits.

You don't need to worry about either of these footnotes, though: it's all handled automatically, behind the scenes. (Footnote two can come into play when you start using --work-tree= arguments, as people do in fancy auto-deployment scripts with bare repositories on servers. However, even there it usually just works, all automatically.)

I am new with git and there is something that isn't clear to me.How does git internally know if a file is new file or modified file?

Since git doesn't track files but tracks blobs. Is this related to the similarity index?

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