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Coefficient of Determination (R²)#

The Coefficient of Determination, commonly denoted as R², is a statistical measure used to assess the goodness of fit of a regression model. It represents the proportion of the variance in the model inferences that is explainable by the model itself.

R² provides a scale that is generally from 0 to 1, where higher values indicate a better fit and imply that the model can better explain the variation of the inferences. It is particularly useful for comparing the explanatory power of different models and their overall fit. Negative values are possible and indicate that the mean of the data itself is a better fit to the data than the regressor/model itself.

Implementation Details#

R² is calculated as the proportion of the total variation of outcomes explained by the model. Mathematically, it can be represented as:

\[ R² = 1 - \frac{\sum_{i=1}^{N}(y_i - \hat{y}_i)^2}{\sum_{i=1}^{N}(y_i - \bar{y})^2} \]

where \(y_i\) is the actual value, \(\hat{y}_i\) is the predicted value from the model, \(\bar{y}\) is the mean of the actual values, and \(N\) is the total number of observations.


Temperature Estimation:

Ground Truth Temperature (°C) Predicted Temperature (°C)
25 27
35 30
\[ \begin{align} R² &= 1 - \frac{(25 - 27)^2 + (35 - 30)^2}{(25 - 30)^2 + (35 - 30)^2} \\ &= 0.42 \end{align} \]

Age Estimation:

Ground Truth Age (Years) Predicted Age (Years)
60 70
40 20
\[ \begin{align} R² &= 1 - \frac{(60 - 70)^2 + (40 - 20)^2}{(60 - 50)^2 + (40 - 50)^2} \\ &= -1.5 \end{align} \]

Limitations and Biases#

When assessing regression models on unseen data, R² quantifies how well model predictions align with observed outcomes indicating the variance explained by the model. However, its utility is nuanced:

  • R² might be misleading if the unseen data's distribution diverges from the training data, hinting at potential data characteristic shifts in addition to model overfitting.
  • It might not capture the full spectrum of model accuracy in complex, non-linear scenarios, where direct error metrics offer clearer insights.
  • High R² values do not imply that changes in predictors causally affect the outcome variable; R² measures correlation strength, not causation.

When evaluating the model's performance under varying conditions, it is important to consider the impact of these conditions on the mean of the ground truth, as this influences the R² value. This is why if you were to stratify by age in an age estimation problem, you may even get negative values depending on the granularity of the strata/test-cases.

Error metrics like Mean Absolute Error (MAE) and Mean Squared Error (MSE) enrich model evaluation. MAE provides the average prediction error magnitude, while MSE emphasizes larger errors, offering a balance to R²'s variance explanation by underscoring prediction accuracy and error severity.