This vignette explains how proxyC compute the similarity and distance measures.
$$ \vec{x} = [x_i, x_{i + 1}, \dots, x_n] \\ \vec{y} = [y_i, y_{i + 1}, \dots, y_n] $$ The length of the vector n = ||x⃗||, while |x⃗| 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 are available in
proxyC::simil().
$$ simil = \frac{\sum{\vec{x}\vec{y}}}{\sqrt{\sum{\vec{x} ^ 2}} \sqrt{\sum{\vec{y} ^ 2}}} $$
$$ simil = \frac{Cov(\vec{x},\vec{y})}{Var(\vec{x}) Var(\vec{y})} $$
The values of x and y are Boolean for “jaccard”.
$$ e = \sum{\vec{x} \vec{y}} \\ w = \text{user-provided weight} \\ simil = \frac{e}{\sum{\vec{x} ^ w} + \sum{\vec{y} ^ w} - e} $$
The values must be 0 ≤ x ≤ 1.0 and 0 ≤ y ≤ 1.0.
$$ simil = \frac{\sum{min(\vec{x}, \vec{y})}}{\sum{max(\vec{x}, \vec{y})}} $$
The values of x and y are Boolean for “dice”.
$$ e = \sum{\vec{x} \vec{y}} \\ w = \text{user-provided weight} \\ simil = \frac{2 e}{\sum{\vec{x} ^ w} + \sum{\vec{y} ^ w}} $$
$$ e = \sum{\vec{x} \vec{y}} \\ n = ||\vec{x}|| = ||\vec{y}|| \\ u = n - e \\ simil = \frac{e - u}{e + u} $$
$$ t = \sum{[\vec{x} = 1][\vec{y} = 1]} \\ f = \sum{[\vec{x} = 0][\vec{y} = 0]} \\ n = ||\vec{x}|| = ||\vec{y}|| \\ simil = \frac{t + 0.5 f}{n} $$
simil = ∑[x⃗ = y⃗]
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 x⃗ and y⃗.
dist = ∑|x⃗ − y⃗|
$$ dist = \frac{|\vec{x} - \vec{y}|}{|\vec{x}| + |\vec{y}|} $$
$$ dist = \sum{\sqrt{\vec{x}^2 + \vec{y}^2}} $$
$$ p = \text{user-provided parameter} \\ dist = \Bigl( \sum{|\vec{x} - \vec{y}| ^ p} \Bigr) ^ \frac{1}{p} $$
dist = ∑[x⃗ ≠ y⃗]
dist = max x⃗ − y⃗
$$ O_{ij} = \text{augmented matrix from } \vec{x} \text{ and } \vec{y} \\ E_{ij} = \text{matrix of expected count for } O_{ij} \\ dist = \sum{\frac{(O_{ij} - E_{ij}) ^ 2}{ E_{ij}}} \\ $$
$$ \vec{p} = \frac{\vec{x}}{\sum{\vec{x}}} \\ \vec{q} = \frac{\vec{y}}{\sum{\vec{y}}} \\ dist = \sum{\vec{q} \log_2{\frac{\vec{q}}{\vec{p}}}} $$
$$ \vec{p} = \frac{\vec{x}}{\sum{\vec{x}}} \\ \vec{q} = \frac{\vec{y}}{\sum{\vec{y}}} \\ dist = \sum{\vec{q} \log_2{\frac{\vec{q}}{\vec{p}}}} + \sum{\vec{p} \log_2{\frac{\vec{p}}{\vec{q}}}} $$
$$ \vec{p} = \frac{\vec{x}}{\sum{\vec{x}}} \\ \vec{q} = \frac{\vec{y}}{\sum{\vec{y}}} \\ \vec{m} = \frac{1}{2} (\vec{p} + \vec{q}) \\ dist = \frac{1}{2} \sum{\vec{q} \log_2{\frac{\vec{q}}{\vec{m}}}} + \frac{1}{2} \sum{\vec{p} \log_2{\frac{\vec{p}}{\vec{m}}}} $$