Spearman's rho¶

Description¶

Spearman's rank correlation coefficient (Spearman's rho) is a non-parametric measure of the strength and direction of a monotonic relationship between two variables. It evaluates how well the relationship between two sets of rankings can be described using a monotonic function. In dimensionality reduction (DR) contexts, it is used to compare the relative order of distances or similarities before and after projection, helping quantify quality loss due to projection.

Spearman's rho illustration

« A positive (resp. negative) Spearman correlation coefficient corresponds to an increasing (resp. decreasing) monotonic trend between X and Y. » Wikipedia

© Wikimedia

Formulas¶

Given two variables \(X=(x_1,...,x_n)\) and \(Y=(y_1,...,y_n)\), and their respective ranks \(R(x_i)\) and \(R(y_i)\), Spearman's rho is computed as :

\[ \rho=1-\frac{6 \displaystyle\sum_{i=1}^{n} d_{i}^{2}} {n(n^2 -1)} \]

where,

\(d_i=R(x_i)-R(y_i)\) is the diference between the ranks of each pair
\(n\) is the number of observations

Sources¶

Gracia, A. et al. A methodology to compare Dimensionality Reduction algorithms in terms of loss of quality, Tech. Rep. 2014

Wikipedia

“Applying Deep Learning algorithm to perform lung cells annotation”, A. Collin, 2020

Code¶

Scipy

Wikipedia