GINI - Regression Gini

In regression analysis, the term “Gini” refers to two fundamentally different statistical paradigms depending on whether one evaluates the ranking capability of the predictions or the dispersion of the prediction errors.

To prevent statistical misinterpretation, permetrics explicitly separates these into two distinct metrics:

1. Normalized Gini Coefficient (Ranking Power)

The Normalized Gini Coefficient [17] measures the actuarial ranking capability of a regression model. Inherited from economics (the Lorenz curve) and heavily utilized in insurance pricing, credit scoring, and algorithmic trading, it quantifies how well the predicted values \(y_{\text{pred}}\) can rank the actual continuous targets \(y_{\text{true}}\).

\[G_{\text{norm}} = \frac{\text{Gini}(y_{\text{true}}, y_{\text{pred}})}{\text{Gini}(y_{\text{true}}, y_{\text{true}})}\]

where the numerator is the raw covariance Gini of the model, and the denominator is the raw Gini of an optimal model (the ground truth sorted by itself).

Properties

  • Best possible score: 1.0 (Perfect ranking: the model sorts targets in the exact correct order).

  • Worst possible score: 0.0 (Random ranking) or -1.0 (Perfectly inverted ranking).

  • Range: [-1, 1]

  • Function call: evaluator.normalized_gini_coefficient()

2. Residual Gini Index (Error Dispersion)

The Residual Gini Index [18] applies the classic economic Gini index of inequality to the absolute regression residuals \(E = \lvert y_{\text{true}} - y_{\text{pred}} \rvert\).

Instead of measuring ranking, it answers an econometric question: “Is the model’s total error distributed equally across all samples, or is 90% of the total error caused by 3 extreme outliers?”

\[G_{\text{residual}} = \frac{2 \sum_{i=1}^{n} i \cdot e_{(i)}}{n \sum_{i=1}^{n} e_i} - \frac{n+1}{n}\]

where \(e_{(i)}\) represents the absolute errors sorted in non-decreasing order (\(e_{(1)} \le e_{(2)} \le \dots \le e_{(n)}\)).

Properties

  • Best possible score: 0.0 (Complete equality: every single sample in the dataset experiences the exact same magnitude of error).

  • Worst possible score: approaching 1.0 (Extreme disparity: the model predicts perfectly for almost all samples, but fails catastrophically on a tiny fraction).

  • Range: [0, 1]

  • Function call: evaluator.residual_gini_index()

Example Usage

import numpy as np
from permetrics.regression import RegressionMetric

y_true = np.array([10, 20, 30, 40, 50])
y_pred = np.array([12, 18, 33, 39, 55])

evaluator = RegressionMetric(y_true, y_pred)

# 1. Evaluate Ranking capability
gini_rank = evaluator.normalized_gini_coefficient()
print(f"Ranking Gini: {gini_rank:.4f}")  # Expected: ~1.0 (Very good ranker)

# 2. Evaluate Error sparsity
gini_err = evaluator.residual_gini_index()
print(f"Residual Gini: {gini_err:.4f}")  # Expected: closer to 0.0