Regression

Regression encompasses supervised learning tasks where the goal is to predict continuous numerical values rather than discrete categorical labels. Unlike classification, which assigns instances to predefined classes, regression models estimate functional relationships between input features and continuous target variables. This distinction is fundamental: regression addresses problems where outputs form a continuous spectrum (e.g., house prices, temperature, stock prices, patient survival times) rather than discrete categories.

The simplest and most foundational form of regression is Linear Regression, which models relationships as straight lines. Ordinary Least Squares (OLS) estimation, the standard approach to linear regression, minimizes the sum of squared differences between predicted and actual values. The Gauss-Markov Theorem (1809), foundational to statistical learning theory, proves that under standard assumptions (linearity, zero-mean errors, constant variance, independence), OLS estimators are the Best Linear Unbiased Estimators (BLUE)—optimal among all unbiased linear estimators.

While linear regression assumes straight-line relationships, real-world data often exhibits non-linear patterns. Several approaches address this:

Polynomial Regression extends linear regression by including polynomial terms (powers of variables), enabling fitting of curved relationships while remaining theoretically linear in parameters. Higher-degree polynomials capture increasingly complex curves but risk overfitting without proper regularization.

Support Vector Regression (SVR) applies the support vector machine framework to regression by finding the optimal hyperplane that minimizes prediction errors while maintaining a specified margin of tolerance around predictions. SVR with kernel functions can efficiently model highly non-linear relationships in high-dimensional spaces.

Gradient Boosting and XGBoost represent ensemble methods that iteratively build decision trees, with each new tree correcting residual errors from previous trees. XGBoost has achieved remarkable success on diverse regression problems, balancing model complexity with predictive accuracy.

Neural Network-Based Regression employs deep learning architectures to automatically discover hierarchical non-linear relationships from data. These approaches excel on high-dimensional complex problems but require substantial data and computational resources.

A critical challenge in regression is balancing model complexity against generalization. Ridge Regression and Lasso Regression add penalties for model complexity, controlling feature importance to reduce variance and prevent overfitting. Elastic Net combines ridge and lasso penalties, offering flexibility for various problem structures.

Assessing regression model quality requires metrics measuring prediction accuracy:

Mean Squared Error (MSE) averages squared differences between predictions and actual values. It emphasizes larger errors through squaring, making it sensitive to outliers but useful when large errors are particularly costly. MSE is measured in squared units of the target variable, making interpretation less intuitive.

Root Mean Squared Error (RMSE) is the square root of MSE, returning errors to the original units for interpretability. RMSE is widely used but shares MSE's sensitivity to outliers.

Mean Absolute Error (MAE) averages absolute differences between predictions and actual values, providing more robust assessment in the presence of outliers while maintaining interpretability. MAE is often more appropriate for practical applications where error magnitude matters equally regardless of direction.

R-Squared (R²), the Coefficient of Determination, represents the proportion of target variable variance explained by the model, ranging from 0 to 1 (with 1 indicating perfect prediction). R-squared enables comparing models across different datasets and scales. Research demonstrates R-squared is often more informative than MSE, MAE, or RMSE for assessing model adequacy.

Root Mean Squared Percentage Error (RMSPE) and Mean Absolute Percentage Error (MAPE)express errors as percentages, enabling comparison across datasets with different scales.

Effective regression modeling requires careful attention to several factors:

Data Scaling: When features have different magnitudes, normalization or standardization can improve algorithm convergence and prevent high-magnitude features from dominating learning.

Feature Engineering: Selecting appropriate features, transforming variables, and creating meaningful feature interactions dramatically impact model performance.

Metric Selection: Choosing appropriate evaluation metrics depends on application requirements—MSE/RMSE when large errors are costly, MAE when errors are equally important, and R-squared for assessing overall model adequacy.

Regression finds widespread application across domains:

The choice between linear, polynomial, kernel-based, and ensemble regression methods should align with data characteristics, problem complexity, available training data size, and computational constraints. Modern practice often employs ensemble methods like gradient boosting and neural networks for complex problems while maintaining linear regression's interpretability advantages in applications requiring model transparency.