AR - Absolute Pearson’s Correlation Coefficient

The Absolute Pearson’s Correlation Coefficient (AR) is a custom evaluation metric introduced natively in permetrics. It modifies the standard Pearson’s Correlation Coefficient by taking the absolute value of the deviations from the mean in the numerator.

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

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

Description

Because AR enforces absolute values in the covariance-like numerator, it behaves very differently from standard correlation:

Advantages:

  • Volatility and Magnitude Tracking: AR strictly measures whether the magnitude of variation in your predictions matches the true data, regardless of phase. This makes it a highly specialized metric for evaluating volatility models (e.g., financial market fluctuations, seismic signal processing) where predicting the size of a movement is the primary goal, even if the exact direction is wrong.

Disadvantages:

  • Directional Blindness (Crucial Flaw): AR completely ignores the phase or direction of the relationship. If your actual data spikes by +10 units, but your model predicts a drop of -10 units, standard Pearson R would penalize it severely. However, AR will score it perfectly (1.0) because the absolute magnitude of the deviation is identical. It should never be used as a standalone accuracy metric for standard regression tasks.

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 Absolute Pearson's Correlation Coefficient
print("AR: ", evaluator.absolute_pearson_correlation_coefficient())

## 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("AR (Multi-output): ", evaluator.AR(multi_output="raw_values"))