A linear regression model learns a straight line or hyperplane that best predicts a target from a set of input features. The prediction is a weighted sum of features plus an intercept. Training chooses the weights to minimize the average squared difference between true values and predictions. This is often done with ordinary least squares or with gradient based optimization.
Problem setup
Define the supervised task clearly. Choose the target to predict and the features that are likely to explain it. Confirm that a linear relationship is reasonable, possibly after transformations. Decide whether the problem is simple regression with one feature or multiple regression with several features, because interpretation and diagnostics differ.
Data preparation
Split the data into training and test sets so that model quality can be measured on unseen data. A common split is about 80 percent train and 20 percent test. Handle missing values by imputing or by removing affected rows when appropriate. Convert categorical variables into numeric representations such as one hot encoding. Consider standardizing features if using gradient methods or if scales differ widely. Inspect distributions and outliers, and add transformations or interaction terms if they help approximate linear behavior.
Fitting the model
Fit the model on the training data. Ordinary least squares finds the weight vector that minimizes the sum of squared residuals. In matrix terms, this solution exists when the design matrix has full rank, but in practice numerical solvers are used to avoid explicitly inverting matrices. For large data or many features, use iterative optimization such as batch, mini batch, or stochastic gradient descent to minimize mean squared error efficiently.
Evaluation
Evaluate on the test set with metrics like R squared for explained variance and with error measures like mean absolute error, mean squared error, and root mean squared error. Inspect residuals versus fitted values. A cloud of points without pattern suggests constant error variance and reasonable linearity. Funnel shapes suggest heteroscedasticity, and curved patterns suggest nonlinearity. Review weight signs and magnitudes for domain sense, and use confidence intervals if statistical inference is required.
Assumptions and diagnostics
Linear regression relies on several assumptions. The relationship should be approximately linear. Errors should be independent. The variance of residuals should be constant. Residuals should be approximately normal if performing inference on coefficients. Use residual versus fitted plots, quantile quantile plots, and scale location plots to detect violations. Statistical tests such as Breusch Pagan for heteroscedasticity and Durbin Watson for autocorrelation can help. Address problems with transformations, added interactions, or alternative models.
Multicollinearity
When predictors are highly correlated, coefficient estimates become unstable and difficult to interpret. Detect multicollinearity using correlation matrices and variance inflation factors. Values of variance inflation factor above about 5 to 10 indicate concern. Mitigate by removing redundant features, combining them, or using regularized models that shrink coefficients.
Regularization
When overfitting or multicollinearity is present, consider penalized regression. Ridge regression adds an L2 penalty that shrinks coefficients toward zero without setting many exactly to zero. Lasso adds an L1 penalty that can set some coefficients to zero and perform feature selection. Elastic Net combines both penalties and can be tuned for a balance between shrinkage and sparsity. Choose penalty strengths with cross validation to optimize generalization.
Interpreting coefficients
In multiple regression, each coefficient estimates the expected change in the target for a one unit increase in that feature while holding other features fixed. This conditional interpretation depends on which features are included and how correlated they are. For comparability, consider standardizing features and reporting standardized coefficients. Confidence intervals and p values give uncertainty and significance, but rely on the model assumptions.
Feature engineering
Improve fit by adding interaction terms when two features jointly affect the target. Add polynomial terms to capture gentle curvature while keeping a linear in parameters model. Transform skewed variables using log or other monotonic transforms. Standardize features before polynomial expansion or when using penalties so that coefficients are comparable and regularization treats features fairly.
Validation and deployment
Use k fold cross validation to estimate generalization error and to select hyperparameters for regularized models. After choosing a final model, save the model artifact, expose it through an API or batch pipeline, and monitor performance over time. Track input drift, residual distributions, and error metrics. Retrain when data shifts or errors exceed thresholds.
Common pitfalls
Avoid data leakage. Fit preprocessing steps such as scaling and encoding only on the training data, then apply them to the test data. Be cautious when extrapolating outside the observed feature ranges, because linear models can produce unrealistic predictions. Remember that high R squared does not imply causation. Use domain knowledge and, if needed, causal methods or experiments to draw causal conclusions.
Closing note
A minimal end to end workflow includes careful splitting of data, robust preprocessing, ordinary least squares or regularized fitting, thorough diagnostics on residuals and assumptions, and cross validated tuning. With these steps, linear regression provides a strong, transparent baseline that is easy to deploy and maintain.