Correlation Distance

The Correlation Distance is a distance measure that quantifies the dissimilarity between vectors based on their linear correlation patterns.

Operation	Computational methods		Programming Interface
dense	dense	compute(…)	compute_input	compute_result

Mathematical formulation

Computing

Given a set \(X\) of \(n\) feature vectors \(x_1 = (x_{11}, \ldots, x_{1p}), \ldots, x_n = (x_{n1}, \ldots, x_{np})\) of dimension \(p\) and a set \(Y\) of \(m\) feature vectors \(y_1 = (y_{11}, \ldots, y_{1p}), \ldots, y_m = (y_{m1}, \ldots, y_{mp})\), the problem is to compute the correlation distance \(D(x_i, y_j)\) for any pair of input vectors:

\[D(x_i, y_j) = 1 - \frac{\sum_{k=1}^{p}(x_{ik} - \bar{x}_i)(y_{jk} - \bar{y}_j)}{\sqrt{\sum_{k=1}^{p}(x_{ik} - \bar{x}_i)^2} \sqrt{\sum_{k=1}^{p}(y_{jk} - \bar{y}_j)^2}}\]

where \(\bar{x}_i = \frac{1}{p}\sum_{k=1}^{p}x_{ik}\) and \(\bar{y}_j = \frac{1}{p}\sum_{k=1}^{p}y_{jk}\) are the means of vectors \(x_i\) and \(y_j\), respectively.

Computation method: dense

The method computes the correlation distance matrix \(D = D(X, Y), D \in \mathbb{R}^{n \times m}\) for dense \(X\) and \(Y\) matrices.

Programming Interface

Refer to API Reference: Correlation Distance.