Similarity or distance measures/metrics

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What is (and isn't) a metric?

Broadly, a metric is anything that gives you some sort of usable distance between two things - vectors, sequences, but also less-structured things like text.


In that broad view, there are many.

Some handle different types of data,
some assume different things about the data (e.g. their distribution, how related different properties are or are not, etc.),
some are just better at highlighting a specific kind of dissimilarity, etc.
some have mathematica or practical properties that are necessary or useful for specific problems (e.g. machine learning on high dimensional data)


distance measures and metrics and similarity measures and dissimilarity measures and even divergence could all mean the same thing.


In practice people may use these terms more precisely - with more specific formal properties.


For example, in mathematics metrics are a little better defined [1], giving four requirements:

  • non-negativity: d(x, y) ≥ 0
  • identity of indiscernibles: d(x,y) = 0 if and only if x=y
  • symmetry: d(x,y) = d(y,x)
  • triangle inequality: d(x,z) ≤ d(x,y) + d(y,z)

Roughly:

distance can't be negative
often doesn't have sensible meaning, and may be disruptive to the math
things with distance zero are, for these purposes, considered the same thing
the result has to be the same both ways
there are various cases where there being a difference makes sense, e.g. traffic routing when one-way streets are involved
and many where this doesn't make sense, e.g. as-the-crow-flies spatial distance
triangle inequality itself is that the sum of any two sides of a triangle is larger than that of the other. This is probably more intuitive when drawn[2]
as applied to metrics, intuitively sort of a 'does your metric not break basic arithmetic?'
there are metrics that break this but may still be useful, e.g. Lk for k<1 applied to high dimensional data (verify)


These can be relevant in pragmatic ways (e.g. some analyses break, or make no sense, on non-symmetric measures, or with negative values), and in some mathematical ways (e.g. some spaces act differently).


Not all real-world metrics are trying to meet all of that, sometimes by useful design.

Words like distance and metric suggest one of the more mathematical ones

Words like divergence suggest asymmetric comparisons

Words like measure could be anything.

In general, if you care, then read up on the details.

Some assumptions and conventions

Distance measures are often used to shape data for something else.

Often a form of dimensionality reduction where that's relatively easy, e.g. for things that like simple linear(ish) one-dimensional data more than the raw data, such as most data clustering.


A lot of data is close enough to numbers or vectors already.

Or as (often-equivalent) Euclidean coordinates, where each dimension indicates a feature.


...though note that the ability to shove something into a vector doesn't imply all metrics on it are accurate - or useful at all.

For example, it is tempting to treat such vectors as Euclidean coordinates, implied by running basic euclidean distance on it - while city block distance may be more valid because it does not implicitly assume that every feature is entangled with every other one.

But maybe that has validity, yet since you implicitly say feature = linear orthogonal dimension, this runs quickly in to the questions of linear algebra.

Chances are that either will give halfway decent results very easily -- but often no better than halfway, and with distortions that are hard to understand or resolve.




In some cases the things you compare are probability distributions, or sometimes an unordered set of probabilities. For example, in distributional similarity you often use the relative probabilities of a word co-occurence.

Vector/coordinate measures

This article/section is a stub — probably a pile of half-sorted notes, is not well-checked so may have incorrect bits. (Feel free to ignore, fix, or tell me)

Lk Norms

Lk norms are mentioned as a slight mathematical generalisation, that take the form (|a|k + |b|k + ...)(1/k)

Which in practice is an overkill generalization since you mostly just see:

for k=1 this is the city block distance (a.k.a. L1)
for k=2 this is the Euclidian distance (a.k.a. L2)

See also Minkowski distance


City block / Manhattan distance / L1 distance / Taxicab geometry

  • Input: two n-dimensional vectors, q and r
  • Distance: ∑v|q(v)-r(v)|

i.e. in the difference in each dimension, summed up.


The intuition that 'city block' suggests (in 2D) is a city in an entirely regular grid (think US cities), and you can only travel on streets so travel in one direction doesn't affect the distance in the other direction.

That analogy works best for integer coordinates (all coordinates are street corners), while the metric stays valid for more number sets.


http://en.wikipedia.org/wiki/Taxicab_geometry

Euclidean / L2 distance

  • Input: two n-dimensional vectors, q and r
  • Distance: length of represented line segment; sqrt( ∑v (q(v)-r(v))2)


From the intuition of city block distance, you're now taking the shortest diagonal.


http://en.wikipedia.org/wiki/Norm_%28mathematics%29#Euclidean_norm

Canberra

  • Input: two n-dimensional vectors, q and r
  • Distance: ∑v|q(v)-r(v)| / ( |q(v)| + |r(v)| )


Variation of the L1 (city block) distance that weighs each distance, roughly by proximity to (0,0).

...which seems like a weird thing to do unless you know that's what you need.


http://en.wikipedia.org/wiki/Canberra_distance

Cosine

  • Input: two n-dimensional vectors
  • Uses fairly obvious form of the cosine rule (TODO: formula)
  • sensitive to vector direction
  • insensitive to vector length
  • Implicitly uses zero point as reference; using reference example not really meaningful


As cosine determines the angle between two vectors, it puts its output on a given scale regardless of vector magnitude.

Useful over euclidean/Lk distances when when length has no important meaning to a given comparison)

However, it lets you put relatively little weight one each dimension when you want that, particularly for high dimensional data.


On high-dimensional data

(Fuzzy) string comparison

Orthographic (look alike)

Based on specific writing systems; mostly useful for phonetic alphabets.

Orthographic methods can regularly be extended from to have phonetical considerations. For example, since Levenshtein compares characters(/tokens) at a time, that comparison can be weighed, or based on a phonetic model (see e.g. L04), or perhaps applied after metaphone was applied to the input.


Methods/algorithms include:


Edit distance (of words/strings)

Edit distance is the idea of that the amount of modifications (inserts, deletes, replacements) to get from one string to the other is a good indicator of its difference.


Depending on context, the term 'edit distance' may refer to Levenshtein distance, or to a group of (often character-based) algorithms that do something similar, including:

  • Hamming distance (boolean inequality, but applied to characters instead of bits. So, only for strings of the same length, and specifically the amount of characters in the same position being identical[5]
which is cheaper (O(n)) but also not fit for many purposes
  • Jaro-Winkler distance [6]
  • Hirschberg's algorithm [7]
  • Wagner-Fischer edit distance [8]
  • Ukkonen's edit distance [9]
Sequences and n-grams
  • sub-sequences
    • Ratcliff/Obershelp
    • LCS: Longest Common Subsequence
      • LCSR: Longest Common Subsequence Ratio, defined as len(lcs)/max(len(one_string),len(other_string))


You can also use n-grams.

N-grams themselves are unordered features, which contain ordered characters.

  • DICE
amount of shared bigrams
2*|X∩Y| / |X|+|Y|
  • XDICE (extended), XXDICE (adds positions)
C Brew, D McKelvie (1996) "Word-pair extraction for lexicography"
  • Jaccard
|X∩Y| / |X∪Y|
  • TRIGRAM-2B
seems to be the name for the best-performing of the best-performing variant in...
B Lambert et al. (1999) "Similarity as a risk factor in drug-name confusion errors: the look-alike (orthographic) and sound-alike (phonetic) model"
and is like DICE but with n=3 and adding two spaces to the start (verify)
  • BI-SIM (verify) and a mentioned TRI-SIM
G Kondrak, B Dorr (2005) "Automatic identification of confusable drug names"
  • WDICE (weighed)
  • WXDICE




Unsorted


C Brew, D McKelvie (1996), "Word-pair extraction for lexicography"


  • ALINE (phonetic sequences)
G Kondrapok (2000) "A new algorithm for the alignment of phonetic sequences"
S Downey et al. (2008) "Computational feature-sensitive reconstruction of language relationships: Developing the ALINE distance for comparative historical linguistic reconstruction"

https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?referer=&httpsredir=1&article=3388&context=etd

Phonological (sound alike)

Distance correctness often depends on the language something is based on.


Pronunciation models (and distances based on such):

  • Soundex (back from 1919), primarily for names: uses First few letters in roughly phonetic categories



  • MRA (Match Rating Approach) (from 1977), also primarily for names, is similar to and apparently slightly better than NYSIIS.


  • EDITEX: distance metric comparable to levenshtein with some metaphone-like processing (using character equivalence groups)
    • J Zobel, P Dart, Phonetic string matching: Lessons from information retrieval (1996)


  • Caverphone's first version (from 2002) was specifically for names, the second version (from 2004) is more generally applicable.


  • ALINE distance (See Kondrak 2000, A New Algorithm For The Alignment Of Phonetic Sequences)


  • CORDI (See Kondrak 2002, Determining recurrent sound correspondences by inducing translation models)


  • Metaphone (from 1990) Maps whole string into consonants roughly according to a specific language
primarily for names, though it does decently for other words as well.
accounts for more language-specific irregularities
can return two results when pronunciation can be ambiguous


  • Levenshtein on phonetic tokens, e.g. metaphone or any other that returns such a string
Note: Tapered weighing (weighing distances in the start / first quarter/half of the strong) is sometimes effective for certain languages, e.g. those that mark inflection primarily with suffix morphemes.

Unsorted

http://www.morfoedro.it/doc.php?n=223&lang=en

http://www.dcs.shef.ac.uk/~sam/stringmetrics.html#tanimoto

Sample sets (/paired vectors)

Simple matching coefficient, distance

Given two vectors of boolean (there or not) features, and summarizing variables:

  • p as the number of variables that are positive in both vectors
  • q as the number of variables that are positive in the first and negative in the second
  • r as the number of variables that are negative in the first and positive in the second
  • s as the number of variables that are negative in both
  • t as the total number of variables (which is also p+q+r+s, as those are all exclusive)

Then the simple matching coefficient is the number of agreements (on positive and on negative values)

(p+s) / t

...and the simple matching distance is the number of disagreements:

(q+r) / t


Note that for a number of applications, counting s in the results makes no sense, because it counts absence of anything as hits. See Jaccard.


Jaccard index / Jaccard similarity coefficient (sample sets)

This article/section is a stub — probably a pile of half-sorted notes, is not well-checked so may have incorrect bits. (Feel free to ignore, fix, or tell me)

Intuitively, Jaccard similarity measures the amount of features that both vectors agree is present (/true/positive, whatever), divided by the amount of features one or the other has.


You can see this as a variation on simple "count how much they agree" where you also count the disagreements as cases, and of the agreements you only count those on positive values.

Given the same definitions as in simple matching (above), you can say that s are not counted as existing cases at all, and the Jaccard similarity coefficient (also known as Jaccard index) is:

p/(p+q+r)


When you see the data more directly as paired vectors storing booleans, you can also state this as:

  • the size of the boolean intersection (agreement pairs)
  • ...divided by the size of the boolean union (all non-zero pairs)


You can also define a Jaccard distance as 1-jaccard_similarity, which works out as:

(q+r) / (p+q+r)


  • Distance: |V_{qr}|/|V_{q}|\cup|V_{r}|
  • 'Has' is defined by a non-zero probability (I imagine sometimes a low threshold can be useful)
  • Input: N-dimensional vectors, seen to contain boolean values

Tanimoto coefficient

An extension of cosine similarity that, for boolean data, gives the jaccard coefficient.

See also:

Distribution comparisons

This article/section is a stub — probably a pile of half-sorted notes, is not well-checked so may have incorrect bits. (Feel free to ignore, fix, or tell me)


Mutual information

  • For discrete probability distributions (verify)
  • Symmetric comparison

Intuitively: to what degree two distributions estimate the same values, or the degree to which one distribution tells us something about another.

http://www.scholarpedia.org/article/Mutual_information


  • Introduced in (Shannon 1948)
  • regularly shown in context of error correction and such

Lautum Information

Variation on mutual information (reverses the arguments, therefore the reversed name)

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.140.6549

Kullback–Leibler divergence

Also known as information divergence, information gain, relative entropy, cross entropy, mean information for discrimination, information number, information divergence, and other names.

Named for the 1951 paper by Kullback and Leibler, "On information and sufficiency", which built on and abstracted earlier work by Shannon and others.


  • Sees vector input as probability distributions
  • Measures the inefficiency of using one distribution for another (entropy idea)
  • Non-symmetric - so not technically a metric, hence 'divergence'

Jensen-Shannon divergence

Also known as information radius(verify)


  • (based on Kullback–Leibler)
  • (mentioned in) Linden, Piitulainen, Discovering Synonyms and Other Related Words [12]

Jeffreys divergence

A symmetrized version of KL divergence


Hellinger distance

http://en.wikipedia.org/wiki/Hellinger_distance


Wasserstein metric, Earth mover's distance, Kantorovich-Rubinstein metric

The second name comes from the informal explanation: when interpreted as two different ways of piling up the same amount of dirt, this is the minimum cost of turning one pile into the other (where cost is amount*distance).

This is only valid for the same amount of volume, so this may require some normalization, though it is always valid on probability density functions, normalized histograms, and some similar things.


See also:


Lévy–Prokhorov metric

Comparing rankings

This article/section is a stub — probably a pile of half-sorted notes, is not well-checked so may have incorrect bits. (Feel free to ignore, fix, or tell me)


What is alpha-skew? something KL based?


Kendall tau rank correlation coefficient, Kendall Tau distance

Kendall's tau is also known as tau coefficient, Kendall's rank

(Correlation on ordinal data)

Kendall's tau is based on the intuition that if q predicts


The Kendall Tau distance gives a distance between two lists. It is also known as the bubble sort distance as it yields the number of swaps that bubble sort would do.


See also:

Unsorted

This article/section is a stub — probably a pile of half-sorted notes, is not well-checked so may have incorrect bits. (Feel free to ignore, fix, or tell me)

Various meaures

  • Cosine correlation
  • Back-off method (Katz 1987) (?)
  • Distance weighted averaging
  • Confusion probability (approximation of KL?)

...and more.


Bhattacharyya



Alpha skew (Sα) divergence

See also:


Chi-squared distance

Confusion probability (τα)

CY dissimilarity

Gower dissimilarity

Jiang's & Conranth similarity, Jiang's & Conranth distance

See also:


Jensen-Shannon divergence

(a.k.a. Information Radius ?)


Kulczynski dissimilarity/distance

a.k.a. Kulsinski ?


Lin's Similarity Measure

See also:

  • Lin, D. (1998) An information-theoretic definition of similarity. Proceedings of the 15th International Conference on Machine Learning, San Francisco, CA, pp. 296–304.


Mahalanobis distance

McArdle-Gower dissimilarity

Orloci's chord distance

Resnik's similarity measure

See also:

  • P. Resnik's (1995) Using Information Content to Evaluate Semantic Similarity in a Taxonomy

Rogers-Tanimoto dissimilarity

Russell-Rao dissimilarity

Sokal-Michener dissimilarity

Sokal-Sneath dissimilarity

Sørensen, Dice, Bray-Curtis, etc.

Names:

  • Sørensen similarity index
  • Dice's coefficient
  • Czekanowski index
  • Hellinger distance
  • Bray-Curtis dissimilarity

...are all related, and some are identical under certain restrictions or when applied to certain types of data.

See also


Yule dissimilarity

Combinations

Back-off (e.g. Katz') vs. averaging

Media

General comparison

  • MAE - Mean Absolute Error
  • MSE - Mean Squared Error
  • PSE -
  • PSNR - Peak Signal-to-Noise Ratio
  • RMSE - Root Mean Squared Error



See also

  • Lee, Measures of Distributional Similarity (1999), [13]
  • Weeds, Weir, McCarthy, Characterising Measures of Lexical Distributional Similarity, [14]


  • CE Shannon (1948) A Mathematical Theory of Communication. [15]



Various:

  • Kruskal - An overview of sequence comparison: time warps, string edits, and macromolecules