💎 Algorithms
🗣 Overview
scv supports simpson, jaccard, dice, cosine, pearson, euclidean, manhattan, and chebyshev algorithms.
Those algorithms separated into two categories, similarities, and distances.
The followings describes each algorithms and how to calculate similarity/distance from two vectors.
Before calculating the similarities and/or distance of the following algorithms, we assume that two vectors were given (\(v = { (a_1, b_1), (a_2, b_2), …, (a_n, b_n) }\) and \(w = { (c_1, d_1), (c_2, d_2), …, (c_m, d_m) }\)).
Similarities
Calculate similarities among the given vectors.
For this, scv computes similarity between two vectors from the given vectors of their combinations.
To describe each algorithm, Each element of the vector has key and value.
Simpson Index
$$S = \frac{|\mathrm{intersect}(v, w)|}{\min(|v_1|, |v_2|)}$$
\(\mathrm{intersect}\) function returns the new vector by common keys and sum of thier values (\(a_i = c_j (1 \leq i \leq n, 1 \leq j \leq m)\).
Jaccard index
$$J=\frac{|\mathrm{intersect}(v, w)|}{|\mathrm{union}(v, w)|}$$
\(\mathrm{intersect}\) function returns the new vector which contains every keys of \(v\) and \(w\).
Dice index
$$D=\frac{2\times|\mathrm{intersect}(v, w)|}{|v| + |w|}$$
Cosine similarity
$$C=\cos\theta=\frac{v\cdot w}{\sqrt{\sum_{i=0}^{n}b_i^2}\sqrt{\sum_{j=0}^{m}d_j^2}}$$