R2S - Squared Pearson’s Correlation Coefficient

Squared Pearson’s Correlation Coefficient (R2S / RSQ) is exactly what the name implies: the mathematical square of Pearson’s Correlation Coefficient (R).

In permetrics, this metric is explicitly implemented to combat the widespread nomenclature confusion across online tutorials and major frameworks. Many libraries incorrectly denote the Coefficient of Determination (COD) as \(R^2\) (R-squared), misleading users into thinking COD is merely the square of the linear correlation. R2S serves to physically separate the true squared correlation from the explained variance (COD/R2).

\[\text{R2S}(y, \hat{y}) = \left[ \frac{\sum_{i=1}^{N} (y_i - \bar{y})(\hat{y}_i - \bar{\hat{y}})}{\sqrt{\sum_{i=1}^{N} (y_i - \bar{y})^2} \sqrt{\sum_{i=1}^{N} (\hat{y}_i - \bar{\hat{y}})^2}} \right]^2\]

Note: \(\bar{y}\) and \(\bar{\hat{y}}\) represent the mean of the actual and predicted values, respectively.

Description

Advantages:

  • Educational clarity: Serves as a definitive benchmark to prove whether a given mathematical framework is calculating the true Coefficient of Determination (explained variance) or merely returning the squared correlation.

Disadvantages:

  • Loss of directional context: Because the value is squared, it strips away the negative sign of inversely correlated data. A perfect negative correlation (\(R = -1\)) and a perfect positive correlation (\(R = 1\)) will both yield an R2S of 1.0.

  • Redundancy: In practical modeling, calculating R2S rarely provides actionable value beyond what the raw Pearson’s R already tells you.

Properties

  • Best possible score: 1.0 (Bigger value is better).

  • Range: [0.0, 1.0].

Example Usage

from numpy import array
from permetrics.regression import RegressionMetric

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

evaluator = RegressionMetric(y_true, y_pred)
# Calculate Squared Pearson's Correlation Coefficient
print("R2S: ", evaluator.pearson_correlation_coefficient_square())

## 2. For > 1-D array (Multi-output)
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)
# Return an array of scores for each column
print("R2S (Multi-output): ", evaluator.R2S(multi_output="raw_values"))