R2s - (Pearson’s Correlation Index)**2

\[\text{R2s}(y, \hat{y}) = \Bigg[ \frac{ \sum_{i=0}^{N - 1} ((y_i - mean(y))*(\hat{y_i} - mean(\hat{y}))) }{ \sqrt{ \sum_{i=0}^{N - 1} (y_i - mean(y))^2}*\sqrt{\sum_{i=0}^{N - 1} (\hat{y_i} - mean(\hat{y}))^2} } \Bigg]^2\]

Latex equation code:

\text{R2s}(y, \hat{y}) = \Bigg[ \frac{ \sum_{i=0}^{N - 1} ((y_i - mean(y))*(\hat{y_i} - mean(\hat{y}))) }{ \sqrt{ \sum_{i=0}^{N - 1} (y_i - mean(y))^2}*\sqrt{\sum_{i=0}^{N - 1} (\hat{y_i} - mean(\hat{y}))^2} } \Bigg]^2

• (Pearson’s Correlation Index)^2 = R^2 = R2s (R square): Best possible score is 1.0, bigger value is better. Range = [0, 1]

• This actually a useless metric that I implemented here just to demonstrate the misunderstanding between R2s and R2 (Coefficient of Determination).

• Most of online tutorials (article, wikipedia,…) or even scikit-learn library are denoted the wrong R2s and R2.

• R^2 = R2s = R squared makes people think it as (Pearson’s Correlation Index)^2

• However, R2 = Coefficient of Determination, link

Example to use R2s metric:

from numpy import array
from permetrics.regression import RegressionMetric

## For 1-D array
y_true = array([3, -0.5, 2, 7])
y_pred = array([2.5, 0.0, 2, 8])

evaluator = RegressionMetric(y_true, y_pred)
print(evaluator.pearson_correlation_coefficient_square())

## For > 1-D array
y_true = array([[0.5, 1], [-1, 1], [7, -6]])
y_pred = array([[0, 2], [-1, 2], [8, -5]])

evaluator = RegressionMetric(y_true, y_pred)
print(evaluator.R2s(multi_output="raw_values"))