Metrics

Evaluation metrics for quantifying model performance on regression and classification tasks.

Regression Metrics

Mean Squared Error (MSE)

The average of squared differences between predictions and true values. Penalizes large errors quadratically.

$$\text{MSE}(y, \hat{y}) = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2$$

Root Mean Squared Error (RMSE)

The square root of MSE, expressed in the same units as the target variable.

$$\text{RMSE}(y, \hat{y}) = \sqrt{\text{MSE}(y, \hat{y})}$$

Mean Absolute Error (MAE)

The average of absolute differences. Less sensitive to outliers than MSE.

$$\text{MAE}(y, \hat{y}) = \frac{1}{n} \sum_{i=1}^{n} |y_i - \hat{y}_i|$$

Coefficient of Determination ($R^2$)

Measures the proportion of variance in the target explained by the model. A score of $1.0$ indicates perfect prediction; $0.0$ corresponds to always predicting the mean $\bar{y}$.

$$R^2(y, \hat{y}) = 1 - \frac{\sum_{i=1}^{n} (y_i - \hat{y}_i)^2}{\sum_{i=1}^{n} (y_i - \bar{y})^2}$$

The score can be negative if the model is worse than predicting the mean.

#include <Skigen/Metrics>

Skigen::Metrics::mean_squared_error(y_true, y_pred);
Skigen::Metrics::root_mean_squared_error(y_true, y_pred);
Skigen::Metrics::mean_absolute_error(y_true, y_pred);
Skigen::Metrics::r2_score(y_true, y_pred);

Function	Description
mean_squared_error	Mean of squared errors
root_mean_squared_error	Square root of MSE
mean_absolute_error	Mean of absolute errors
r2_score	Coefficient of determination

Classification Metrics

All binary classification metrics below are defined in terms of the confusion matrix entries:

• TP (True Positives): correctly predicted positive
• TN (True Negatives): correctly predicted negative
• FP (False Positives): negative samples incorrectly predicted as positive
• FN (False Negatives): positive samples incorrectly predicted as negative

Accuracy

$$\text{Accuracy} = \frac{\text{TP} + \text{TN}}{n}$$

Precision

Fraction of predicted positives that are truly positive. High precision means few false alarms.

$$\text{Precision} = \frac{\text{TP}}{\text{TP} + \text{FP}}$$

Recall

Fraction of actual positives that are correctly identified. High recall means few missed positives.

$$\text{Recall} = \frac{\text{TP}}{\text{TP} + \text{FN}}$$

F1 Score

The harmonic mean of precision and recall, balancing both concerns equally.

$$F_1 = 2 \cdot \frac{\text{Precision} \cdot \text{Recall}}{\text{Precision} + \text{Recall}} = \frac{2\,\text{TP}}{2\,\text{TP} + \text{FP} + \text{FN}}$$

#include <Skigen/Metrics>

Skigen::Metrics::accuracy_score(y_true, y_pred);
Skigen::Metrics::precision_score(y_true, y_pred);
Skigen::Metrics::recall_score(y_true, y_pred);
Skigen::Metrics::f1_score(y_true, y_pred);
auto cm = Skigen::Metrics::confusion_matrix(y_true, y_pred);

Function	Description
accuracy_score	Fraction of correct predictions
precision_score	Precision (positive predictive value)
recall_score	Recall (sensitivity)
f1_score	Harmonic mean of precision and recall
confusion_matrix	N×N confusion matrix

Example

#include <Skigen/LinearModel>
#include <Skigen/Metrics>
#include <Skigen/ModelSelection>

auto [X_train, X_test, y_train, y_test] = Skigen::train_test_split(X, y);

Skigen::LinearRegression model;
model.fit(X_train, y_train);
auto y_pred = model.predict(X_test);

std::cout << "MSE: " << Skigen::Metrics::mean_squared_error(y_test, y_pred) << "\n";
std::cout << "R²:  " << Skigen::Metrics::r2_score(y_test, y_pred) << "\n";