Types of Linear Regression: Simple, Multiple, Ridge, and Lasso

The main types of linear regression are simple, multiple, polynomial, and regularized (Ridge and Lasso). The choice depends on the number of independent variables, the shape of the relationship between them, and the need to control overfitting in high-dimensional datasets.

• What are the main types of linear regression?
• What is simple linear regression?
• How does multiple linear regression work?
• When to use polynomial linear regression?
• What are Ridge and Lasso regression?
• Ridge vs Lasso: which to choose?
• How to choose the right type of linear regression?

What are the main types of linear regression?

Linear regression has four main variations, which differ by the number of independent variables and the shape of the modeled relationship. Before applying any of them, it is worth reviewing the machine learning fundamentals that underpin these supervised models.

1. Simple Linear Regression — models a dependent variable from a single independent variable.
2. Multiple Linear Regression — uses two or more independent variables at the same time.
3. Polynomial Linear Regression — captures curved relationships with higher-degree terms.
4. Regularized Linear Regression — applies penalties to coefficients (Ridge and Lasso) to reduce overfitting.

The table below summarizes when each type is most indicated:

This is the base of one of the most used algorithms in the field: according to the Kaggle State of Data Science and Machine Learning survey, linear and logistic regression figures among the most adopted methods by data professionals, used by more than 80% of respondents in recent editions.

What is simple linear regression?

Simple linear regression is the most basic form of the technique: it assumes a linear relationship between a dependent variable and a single independent variable, represented by a line on the two-dimensional plane. The goal is to find the line that minimizes the difference between the observed values and those predicted by the model.

The equation of simple linear regression is:

Where:

• $Y$ is the dependent variable to be predicted.
• $X$ is the independent (predictor) variable.
• $\beta_0$ is the intercept, the expected value of $Y$ when $X$ equals zero.
• $\beta_1$ is the coefficient of $X$, which represents the slope of the line.
• $\varepsilon$ is the error term, which captures the discrepancies between observed and predicted values.

Simple linear regression is indicated when there is a single variable believed to influence the outcome. It offers a direct way to quantify and interpret that relationship. For a complete picture of how this model is fitted and evaluated, see the guide on what linear regression is and how to model it.

How does multiple linear regression work?

Multiple linear regression extends the simple version by considering several independent variables simultaneously to predict the dependent variable. Instead of a line, the model fits a plane or hyperplane, allowing analysis of each predictor's effect while keeping the others constant.

The equation of multiple linear regression is:

Where:

• $Y$ is the dependent variable.
• $X_1, X_2, \dots, X_n$ are the independent variables.
• $\beta_0, \beta_1, \dots, \beta_n$ are the coefficients measuring each predictor's influence.
• $\varepsilon$ is the error term.

Multiple regression is particularly useful when multiple factors influence the outcome — for example, predicting a property's price from area, location, and number of bedrooms. The central concern is multicollinearity: when predictors are too correlated with each other, coefficients become unstable and hard to interpret.

When to use polynomial linear regression?

Polynomial linear regression is indicated when the relationship between variables is clearly non-linear and a line does not fit the data well. It models the relationship with a higher-degree polynomial, capturing curves and more complex patterns without abandoning the linear structure in the coefficients.

The polynomial regression equation can be represented as:

Where:

• $Y$ is the dependent variable.
• $X$ is the independent variable raised to different powers.
• $\beta_0, \beta_1, \dots, \beta_n$ are the polynomial coefficients.
• $\varepsilon$ is the error term.

In practice, polynomial regression is implemented by generating higher-degree features — for example, with scikit-learn's PolynomialFeatures from the scikit-learn — and applying linear regression over them. The risk is choosing too high a degree: the model starts memorizing the data noise and oscillates exaggeratedly, a classic case of overfitting.

What are Ridge and Lasso regression?

Ridge and Lasso regression are regularized linear regression techniques: they incorporate a penalty on the model's coefficients to avoid overfitting and improve generalization. Both are especially valuable in high-dimensional datasets, with many independent variables and multicollinearity problems.

Ridge regression adds to the loss function a term proportional to the sum of squared coefficients (L2 penalty). This reduces coefficient magnitude, prevents extreme values, and decreases sensitivity to variations in the training data. The technique was proposed by Arthur Hoerl and Robert Kennard in 1970, in the article "Ridge Regression: Biased Estimation for Nonorthogonal Problems" (Technometrics, vol. 12).

Lasso regression also penalizes the loss function, but uses the sum of the absolute values of coefficients (L1 penalty). Besides reducing magnitude, Lasso zeroes out the least relevant coefficients, performing automatic variable selection. The method was introduced by Robert Tibshirani in 1996, in the classic "Regression Shrinkage and Selection via the Lasso" (Journal of the Royal Statistical Society, Series B, vol. 58). Both are readily available in scikit-learn, in the Ridge and Lasso classes.

Ridge vs Lasso: which to choose?

The choice between Ridge and Lasso depends on what you want from the model. Use Ridge when you want to keep all variables in the model and only control coefficient magnitude — it handles highly correlated predictors well. Use Lasso when you seek a leaner, more interpretable model, letting the algorithm itself discard irrelevant variables by zeroing their coefficients.

As a practical rule: if you suspect only a few variables really matter, Lasso tends to produce a simpler model. If all variables seem to contribute, Ridge is usually more stable. An alternative is Elastic Net, which combines L1 and L2 penalties and captures the best of both worlds. To put any of these approaches into practice, the CodeCrush guide on how to create machine learning projects helps structure the flow from training to evaluation.

How to choose the right type of linear regression?

Choosing the type of linear regression starts with three questions: how many independent variables exist, is the relationship between them linear or curved, and is there a risk of overfitting from excess predictors. A single variable and a linear relationship call for simple regression; multiple factors call for multiple; an evident curve calls for polynomial; and many correlated predictors call for regularization with Ridge or Lasso.

Regardless of the type, always validate the model with metrics like R², RMSE, and residual analysis, plus cross-validation. No equation replaces understanding the problem: linear regression is an interpretation tool, and its usefulness grows when chosen based on the nature of the data, not by convenience.

Conclusion

Mastering the types of linear regression is less about memorizing equations and more about matching the model to the shape of your data. Start simple: if a line explains the relationship well, do not complicate it. Scale to multiple, polynomial, or regularized only when the data justifies it — and always validating the assumptions. In the era of applied artificial intelligence, this discernment remains what separates a precise model from a nice-looking number without meaning.