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Similarity and Distance Measures in proxyC

Kohei Watanabe

2025-08-05

Source: vignettes/measures.Rmd
measures.Rmd

This vignette explains how proxyC compute the similarity and distance measures.

Notation

x→=[xi,xi+1,…,xn]y→=[yi,yi+1,…,yn] \begin{gather} \vec{x} = [x_i, x_{i + 1}, \dots, x_n] \\ \vec{y} = [y_i, y_{i + 1}, \dots, y_n] \end{gather}

The length of the vector n=||x→||n = ||\vec{x}||, while |x→||\vec{x}| is the absolute values of the elements.

Operations on vectors are element-wise:

z→=x→y→n=||x→||=||y→||=||z→|| \begin{gather} \vec{z} = \vec{x}\vec{y} \\ n = ||\vec{x}|| = ||\vec{y}|| =||\vec{z}|| \end{gather}

Summation of the elements of vectors is written using sigma without specifying the range:

∑x→=∑i=1nxi \sum{\vec{x}} = \sum_{i=1}^{n}{x_i}

When the elements of the vector is compared with a value in a pair of square brackets, the summation is counting the number of elements that equal (or unequal) to the value:

∑[x→=1]=∑i=1n[xi=1] \sum{[\vec{x} = 1]} = \sum_{i=1}^{n}{[x_i = 1]}

Similarity Measures

Similarity measures are available in proxyC::simil().

Cosine similarity (“cosine”)

simil=∑x→y→∑x→2∑y→2 simil = \frac{\sum{\vec{x}\vec{y}}}{\sqrt{\sum{\vec{x} ^ 2}} \sqrt{\sum{\vec{y} ^ 2}}}

Pearson correlation coefficient (“correlation”)

simil=Cov(x→,y→)Var(x→)Var(y→) simil = \frac{Cov(\vec{x},\vec{y})}{Var(\vec{x}) Var(\vec{y})}

Jaccard similarity (“jaccard” and “ejaccard”)

The values of xx and yy are Boolean for “jaccard”.

e=∑x→y→w=user-provided weightsimil=e∑x→w+∑y→w−e \begin{gather} e = \sum{\vec{x} \vec{y}} \\ w = \text{user-provided weight} \\ simil = \frac{e}{\sum{\vec{x} ^ w} + \sum{\vec{y} ^ w} - e} \end{gather}

Fuzzy Jaccard similarity (“fjaccard”)

The values must be 0≤x≤1.00 \le x \le 1.0 and 0≤y≤1.00 \le y \le 1.0.

simil=∑min(x→,y→)∑max(x→,y→) simil = \frac{\sum{min(\vec{x}, \vec{y})}}{\sum{max(\vec{x}, \vec{y})}}

Dice similarity (“dice” and “edice”)

The values of xx and yy are Boolean for “dice”.

e=∑x→y→w=user-provided weightsimil=2e∑x→w+∑y→w \begin{gather} e = \sum{\vec{x} \vec{y}} \\ w = \text{user-provided weight} \\ simil = \frac{2 e}{\sum{\vec{x} ^ w} + \sum{\vec{y} ^ w}} \end{gather}

Hamann similarity (“hamann”)

e=∑x→y→n=||x→||=||y→||u=n−esimil=e−ue+u \begin{gather} e = \sum{\vec{x} \vec{y}} \\ n = ||\vec{x}|| = ||\vec{y}|| \\ u = n - e \\ simil = \frac{e - u}{e + u} \end{gather}

Faith similarity (“faith”)

t=∑[x→=1][y→=1]f=∑[x→=0][y→=0]n=||x→||=||y→||simil=t+0.5fn \begin{gather} t = \sum{[\vec{x} = 1][\vec{y} = 1]} \\ f = \sum{[\vec{x} = 0][\vec{y} = 0]} \\ n = ||\vec{x}|| = ||\vec{y}|| \\ simil = \frac{t + 0.5 f}{n} \end{gather}

Simple matching (“matching”)

simil=∑[x→=y→] simil = \sum{[\vec{x} = \vec{y}]}

Distance Measures

Similarity measures are available in proxyC::dist(). Smoothing of the vectors can be performed when method is “chisquared”, “kullback”, “jefferys” or “jensen”: the value of smooth will be added to each element of x→\vec{x} and y→\vec{y}.

Manhattan distance (“manhattan”)

dist=∑|x→−y→| dist = \sum{|\vec{x} - \vec{y}|}

Canberra distance (“canberra”)

dist=|x→−y→||x→|+|y→| dist = \frac{|\vec{x} - \vec{y}|}{|\vec{x}| + |\vec{y}|}

Euclidian (“euclidian”)

dist=∑x→2+y→2 dist = \sum{\sqrt{\vec{x}^2 + \vec{y}^2}}

Minkowski distance (“minkowski”)

p=user-provided parameterdist=(∑|x→−y→|p)1p \begin{gather} p = \text{user-provided parameter} \\ dist = \left( \sum{|\vec{x} - \vec{y}| ^ p} \right) ^ \frac{1}{p} \end{gather}

Hamming distance (“hamming”)

dist=∑[x→≠y→] dist = \sum{[\vec{x} \ne \vec{y}]}

The largest difference between values (“maximum”)

dist=maxx→−y→ dist = \max{\vec{x} - \vec{y}}

Chi-squared divergence (“chisquared”)

Oij=augmented matrix from x→ and y→Eij=matrix of expected count for Oijdist=∑(Oij−Eij)2Eij \begin{gather} O_{ij} = \text{augmented matrix from } \vec{x} \text{ and } \vec{y} \\ E_{ij} = \text{matrix of expected count for } O_{ij} \\ dist = \sum{\frac{(O_{ij} - E_{ij}) ^ 2}{ E_{ij}}} \end{gather}

Kullback–Leibler divergence (“kullback”)

p→=x→∑x→q→=y→∑y→dist=∑q→log2q→p→ \begin{gather} \vec{p} = \frac{\vec{x}}{\sum{\vec{x}}} \\ \vec{q} = \frac{\vec{y}}{\sum{\vec{y}}} \\ dist = \sum{\vec{q} \log_2{\frac{\vec{q}}{\vec{p}}}} \end{gather}

Jeffreys divergence (“jeffreys”)

p→=x→∑x→q→=y→∑y→dist=∑q→log2q→p→+∑p→log2p→q→ \begin{gather} \vec{p} = \frac{\vec{x}}{\sum{\vec{x}}} \\ \vec{q} = \frac{\vec{y}}{\sum{\vec{y}}} \\ dist = \sum{\vec{q} \log_2{\frac{\vec{q}}{\vec{p}}}} + \sum{\vec{p} \log_2{\frac{\vec{p}}{\vec{q}}}} \end{gather}

Jensen-Shannon divergence (“jensen”)

p→=x→∑x→q→=y→∑y→m→=12(p→+q→)dist=12∑q→log2q→m→+12∑p→log2p→m→ \begin{gather} \vec{p} = \frac{\vec{x}}{\sum{\vec{x}}} \\ \vec{q} = \frac{\vec{y}}{\sum{\vec{y}}} \\ \vec{m} = \frac{1}{2} (\vec{p} + \vec{q}) \\ dist = \frac{1}{2} \sum{\vec{q} \log_2{\frac{\vec{q}}{\vec{m}}}} + \frac{1}{2} \sum{\vec{p} \log_2{\frac{\vec{p}}{\vec{m}}}} \end{gather}

References

  • Choi, S., Cha, S., & Tappert, C. C. (2010). A survey of binary similarity and distance measures. Journal of Systemics, Cybernetics and Informatics, 8(1), 43–48.
  • Nielsen, F. (2019). On the Jensen–Shannon Symmetrization of Distances Relying on Abstract Means. Entropy, 21(5), 485. https://doi.org/10.3390/e21050485
  • Jain, G., Mahara, T., & Tripathi, K. N. (2020). A Survey of Similarity Measures for Collaborative Filtering-Based Recommender System. In M. Pant, T. K. Sharma, O. P. Verma, R. Singla, & A. Sikander (Eds.), Soft Computing: Theories and Applications (pp. 343–352). Springer. https://doi.org/10.1007/978-981-15-0751-9_32
  • Miyamoto, S. (1990). Hierarchical Cluster Analysis and Fuzzy Sets. In S. Miyamoto (Ed.), Fuzzy Sets in Information Retrieval and Cluster Analysis (pp. 125–188). Springer Netherlands. https://doi.org/10.1007/978-94-015-7887-5_6