robr2: Robust R-squared (RobR2) Metric

The robr2 function computes the Robust R-squared (RobR2) value for evaluating the goodness of fit between observed data and model predictions. Unlike the classical R-squared metric, RobR2 is robust to outliers and incorporates sample weights, making it ideal for noisy or irregular datasets.

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Overview

Robust R-squared (RobR2) measures the proportion of variance in the observed data explained by the fitted data, while reducing sensitivity to outliers. This is achieved by using a weighted formulation, which allows for more reliable model evaluation in real-world scenarios where data may not be perfectly clean.

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Formula

\[ \text{RobR2} = 1 - \frac{\sum_i w_i (e_i - \bar{e})^2}{\sum_i w_i (y_i - \bar{y})^2} \]

Where:

• \(e_i = y_i - \hat{y}_i\) (residuals)
• \(\bar{e}\) = weighted mean of residuals
• \(\bar{y}\) = weighted mean of observed data
• \(w_i\) = weight for each data point

If weights are not provided, equal weights are assumed.

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Parameters

Parameter	Type	Description
y	np.ndarray	Observed data (ground truth). 1D array of numerical values.
y_fit	np.ndarray	Fitted data (model predictions). 1D array, same shape as y.
w	np.ndarray or None	Optional weights for data points. 1D array, same shape as y. If None, equal weights are used.

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Returns

• float The computed Robust R-squared (RobR2) value. Ranges from 0 (no explanatory power) to 1 (perfect fit).

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Raises

• ValueError
• If y and y_fit do not have the same shape.
• If w is provided and does not have the same shape as y.
• If y or y_fit are not 1D arrays.

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Example Usage

import numpy as np
from machinegnostics.metrics import robr2

y = np.array([1.0, 2.0, 3.0, 4.0])
y_fit = np.array([1.1, 1.9, 3.2, 3.8])
w = np.array([1.0, 1.0, 1.0, 1.0])

result = robr2(y, y_fit, w)
print(result)  # Example output: 0.98

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Comparison with Classical R-squared

• Classical R-squared: Assumes equal weights and is sensitive to outliers.
• RobR2: Incorporates weights and is robust to outliers, making it more reliable for datasets with irregularities or noise.

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References

• Kovanic P., Humber M.B (2015) The Economics of Information - Mathematical Gnostics for Data Analysis, Chapter 19

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Notes

• If weights are not provided, the metric defaults to equal weighting for all data points.
• RobR2 is particularly useful for robust regression and model evaluation in the presence of outliers.

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