Similarity and Distance Measures in proxyC"

In proxyC: Computes Proximity in Large Sparse Matrices

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

Notation

$$ \vec{x} = [x_i, x_{i + 1}, \dots, x_n] \ \vec{y} = [y_i, y_{i + 1}, \dots, y_n] $$ The length of the vector $n = ||\vec{x}||$, while $|\vec{x}|$ is the absolute values of the elements.

Operations on vectors are element-wise:

$$ \vec{z} = \vec{x}\vec{y} \ n = ||\vec{x}|| = ||\vec{y}|| =||\vec{z}|| $$

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".

$$ e = \sum{\vec{x} \vec{y}} \ w = \text{user-provided weight} \ simil = \frac{e}{\sum{\vec{x} ^ w} + \sum{\vec{y} ^ w} - e} $$

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".

$$ e = \sum{\vec{x} \vec{y}} \ w = \text{user-provided weight} \ simil = \frac{2 e}{\sum{\vec{x} ^ w} + \sum{\vec{y} ^ w}} $$

Hamann similarity ("hamann")

$$ e = \sum{\vec{x} \vec{y}} \ n = ||\vec{x}|| = ||\vec{y}|| \ u = n - e \ simil = \frac{e - u}{e + u} $$

Faith similarity ("faith")

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

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")

$$ p = \text{user-provided parameter} \ dist = \Bigl( \sum{|\vec{x} - \vec{y}| ^ p} \Bigr) ^ \frac{1}{p} $$

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")

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

Kullback--Leibler divergence ("kullback")

$$ \vec{p} = \frac{\vec{x}}{\sum{\vec{x}}} \ \vec{q} = \frac{\vec{y}}{\sum{\vec{y}}} \ dist = \sum{\vec{q} \log_2{\frac{\vec{q}}{\vec{p}}}} $$

Jeffreys divergence ("jeffreys")

$$ \vec{p} = \frac{\vec{x}}{\sum{\vec{x}}} \ \vec{q} = \frac{\vec{y}}{\sum{\vec{y}}} \ dist = \sum{\vec{q} \log_2{\frac{\vec{q}}{\vec{p}}}} + \sum{\vec{p} \log_2{\frac{\vec{p}}{\vec{q}}}} $$

Jensen-Shannon divergence ("jensen")

$$ \vec{p} = \frac{\vec{x}}{\sum{\vec{x}}} \ \vec{q} = \frac{\vec{y}}{\sum{\vec{y}}} \ \vec{m} = \frac{1}{2} (\vec{p} + \vec{q}) \ dist = \frac{1}{2} \sum{\vec{q} \log_2{\frac{\vec{q}}{\vec{m}}}} + \frac{1}{2} \sum{\vec{p} \log_2{\frac{\vec{p}}{\vec{m}}}} $$

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

proxyC documentation built on Oct. 25, 2023, 9:06 a.m.