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Regression Metrics

These metrics provide essential tools for evaluating the performance of regression models. Each metric measures error or fit quality differently, offering unique insights into model performance.

meanSquaredError

The meanSquaredError (MSE) metric calculates the average squared difference between predicted and actual values, making it useful for measuring the magnitude of prediction errors. Larger errors are penalized more heavily due to the squaring, making this metric sensitive to outliers.

Parameters

  • yTrue: List of true target values, each representing the correct continuous value for a corresponding sample, type f24.
  • yPred: List of predicted target values, each representing the continuous value predicted by the model for a corresponding sample, type f24.

Returns: The mean squared error as a float.

Formula

\[ \text{MSE} = \frac{1}{n} \sum_{i=1}^{n} (y_{\text{true},i} - y_{\text{pred},i})^2 \]

Example

y_true = [3.0, -0.5, 2.0, 7.0]
y_pred = [2.5, 0.0, 2.0, 8.0]

mse = meanSquaredError(y_true, y_pred)
# Output: mse: 0.375

meanAbsoluteError

The meanAbsoluteError (MAE) metric calculates the average absolute difference between predicted and actual values, providing a straightforward measure of error without penalizing larger errors as much as MSE. It’s useful for models where you want to interpret errors in absolute terms.

Parameters

  • yTrue: List of true target values, type f24.
  • yPred: List of predicted target values, type f24.

Returns: The mean absolute error as a float.

Formula

\[ \text{MAE} = \frac{1}{n} \sum_{i=1}^{n} |y_{\text{true},i} - y_{\text{pred},i}| \]

Example

y_true = [3.0, -0.5, 2.0, 7.0]
y_pred = [2.5, 0.0, 2.0, 8.0]

mae = meanAbsoluteError(y_true, y_pred)
# Output: mae: 0.5

r2_score

The r2_score (R-squared) metric, also known as the coefficient of determination, indicates the proportion of variance in the dependent variable that is predictable from the independent variable(s). It ranges from 0.0 to 1.0, with higher values indicating a better fit.

Parameters

  • yTrue: List of true target values, type f24.
  • yPred: List of predicted target values, type f24.

Returns: The R-squared score as a float, where values closer to 1.0 indicate better model performance.

Formula

\[ R^2 = 1 - \frac{\text{Sum of Squared Errors (SSE)}}{\text{Total Sum of Squares (TSS)}} \]

where:
- SSE is the sum of squared errors, calculated as \(\sum (y_{\text{true}} - y_{\text{pred}})^2\).
- TSS is the total sum of squares, calculated as \(\sum (y_{\text{true}} - \text{mean}(y_{\text{true}}))^2\).

Example

y_true = [3.0, -0.5, 2.0, 7.0]
y_pred = [2.5, 0.0, 2.0, 8.0]

r2 = r2_score(y_true, y_pred)
# Output: r2_score: 0.948

totalSumOfSquares

The totalSumOfSquares (TSS) metric provides a measure of the total variance of the true target values. It serves as a baseline metric to compare against the sum of squared errors (SSE) for computing metrics like r2_score.

Parameters

  • yTrue: List of true target values, type f24.

Returns: The total sum of squares as a float.

Formula

\[ \text{TSS} = \sum_{i=1}^{n} (y_{\text{true},i} - \text{mean}(y_{\text{true}}))^2 \]

Example

y_true = [3.0, -0.5, 2.0, 7.0]

tss = totalSumOfSquares(y_true)
# Output: tss: 29.25

Common Issues and Error Handling

  • Mismatched Lengths: Ensure yTrue and yPred are of the same length for all metrics requiring both.
  • Outliers: For datasets with extreme values, consider using MAE over MSE to reduce sensitivity to outliers.
  • Negative R-squared: In rare cases, R^2 can be negative, indicating that the model performs worse than a simple mean-based prediction.