Author: Kohei Watanabe
Date: 2025-04-25

# Similarity and Distance Measures in proxyC

#### Kohei Watanabe

#### 2025-04-25

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

## Notation

\[ \begin{gather} \vec{x} = [x_i, x_{i + 1}, \dots, x_n] \\ \vec{y} = [y_i, y_{i + 1}, \dots, y_n] \end{gather} \]

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

Operations on vectors are element-wise:

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

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

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

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:

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

## Similarity Measures

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

#### Cosine similarity (“cosine”)

\[ simil = \frac{\sum{\vec{x}\vec{y}}}{\sqrt{\sum{\vec{x} ^ 2}} \sqrt{\sum{\vec{y} ^ 2}}} \]

#### Pearson correlation coefficient (“correlation”)

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

#### Jaccard similarity (“jaccard” and “ejaccard”)

The values of \(x\) and \(y\) are Boolean for “jaccard”.

\[ \begin{gather} e = \sum{\vec{x} \vec{y}} \\ w = \text{user-provided weight} \\ simil = \frac{e}{\sum{\vec{x} ^ w} + \sum{\vec{y} ^ w} - e} \end{gather} \]

#### Fuzzy Jaccard similarity (“fjaccard”)

The values must be \(0 \le x \le 1.0\) and \(0 \le y \le 1.0\).

\[ simil = \frac{\sum{min(\vec{x}, \vec{y})}}{\sum{max(\vec{x}, \vec{y})}} \]

#### Dice similarity (“dice” and “edice”)

The values of \(x\) and \(y\) are Boolean for “dice”.

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

#### Hamann similarity (“hamann”)

\[ \begin{gather} e = \sum{\vec{x} \vec{y}} \\ n = ||\vec{x}|| = ||\vec{y}|| \\ u = n - e \\ simil = \frac{e - u}{e + u} \end{gather} \]

#### Faith similarity (“faith”)

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

#### Simple matching (“matching”)

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

## 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 \(\vec{x}\) and \(\vec{y}\).

#### Manhattan distance (“manhattan”)

\[ dist = \sum{|\vec{x} - \vec{y}|} \]

#### Canberra distance (“canberra”)

\[ dist = \frac{|\vec{x} - \vec{y}|}{|\vec{x}| + |\vec{y}|} \]

#### Euclidian (“euclidian”)

\[ dist = \sum{\sqrt{\vec{x}^2 + \vec{y}^2}} \]

#### Minkowski distance (“minkowski”)

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

#### Hamming distance (“hamming”)

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

#### The largest difference between values (“maximum”)

\[ dist = \max{\vec{x} - \vec{y}} \]

#### Chi-squared divergence (“chisquared”)

\[ \begin{gather} O_{ij} = \text{augmented matrix from } \vec{x} \text{ and } \vec{y} \\ E_{ij} = \text{matrix of expected count for } O_{ij} \\ dist = \sum{\frac{(O_{ij} - E_{ij}) ^ 2}{ E_{ij}}} \end{gather} \]

#### Kullback–Leibler divergence (“kullback”)

\[ \begin{gather} \vec{p} = \frac{\vec{x}}{\sum{\vec{x}}} \\ \vec{q} = \frac{\vec{y}}{\sum{\vec{y}}} \\ dist = \sum{\vec{q} \log_2{\frac{\vec{q}}{\vec{p}}}} \end{gather} \]

#### Jeffreys divergence (“jeffreys”)

\[ \begin{gather} \vec{p} = \frac{\vec{x}}{\sum{\vec{x}}} \\ \vec{q} = \frac{\vec{y}}{\sum{\vec{y}}} \\ dist = \sum{\vec{q} \log_2{\frac{\vec{q}}{\vec{p}}}} + \sum{\vec{p} \log_2{\frac{\vec{p}}{\vec{q}}}} \end{gather} \]

#### Jensen-Shannon divergence (“jensen”)

\[ \begin{gather} \vec{p} = \frac{\vec{x}}{\sum{\vec{x}}} \\ \vec{q} = \frac{\vec{y}}{\sum{\vec{y}}} \\ \vec{m} = \frac{1}{2} (\vec{p} + \vec{q}) \\ dist = \frac{1}{2} \sum{\vec{q} \log_2{\frac{\vec{q}}{\vec{m}}}} + \frac{1}{2} \sum{\vec{p} \log_2{\frac{\vec{p}}{\vec{m}}}} \end{gather} \]

## 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