A Comprehensive Guide to Regression Models in Machine Learning
Regression is one of the most important and widely used predictive modeling techniques in machine learning and statistics. At its core, regression involves modeling the relationship between a dependent variable (the outcome we want to predict) and one or more independent variables (the features or predictors). By understanding this relationship, we can make predictions, test hypotheses, and gain insights into how different factors influence an outcome of interest.
While the concept is simple, there are actually many different types of regression models to choose from depending on the characteristics of your data and the assumptions you‘re willing to make. In this post, we‘ll take a comprehensive look at 14 essential regression techniques every data scientist should know. For each, we‘ll discuss what it is conceptually, its strengths and weaknesses, the math behind how it works, and example use cases. Let‘s dive in!
1. Linear Regression
We‘ll start with the grandfather of all regression techniques – ordinary least squares linear regression. Linear regression assumes a linear relationship between the dependent variable y and the independent variable(s) X. Mathematically, we can express this relationship as:
y = β₀ + β₁x₁ + β₂x₂ + … + βₚxₚ + ε
where:
- y is the dependent variable
- x₁, …, xₚ are the independent variables
- β₀, β₁, …, βₚ are the coefficients that quantify the association between each independent variable and the dependent variable
- ε is the error term
The coefficients are estimated by minimizing the sum of squared residuals between the observed values and the predicted values. This results in estimates that are unbiased and have the lowest variance among all linear unbiased estimators (according to the Gauss-Markov theorem).
Linear regression is popular because it‘s simple to understand and implement, and the coefficients have a nice interpretation. However, it makes several key assumptions:
- Linearity – the relationship between X and y is linear
- Independence – the errors are uncorrelated with each other
- Homoscedasticity – the errors have constant variance
- Normality – the errors are normally distributed
When these assumptions are violated, linear regression can perform poorly. It‘s also sensitive to outliers and multicollinearity among the independent variables. Nevertheless, linear regression is a good starting point and baseline for many problems.
2. Logistic Regression
While "regression" is in the name, logistic regression is actually a classification algorithm. It‘s used to model the probability of a binary event occurring as a function of independent variables. The model looks like:
ln(p/(1-p)) = β₀ + β₁x₁ + β₂x₂ + … + βₚxₚ
where:
- p is the probability of the event occurring
- x₁, …, xₚ are the independent variables
- β₀, β₁, …, βₚ are the coefficients
The left side of the equation is the log-odds or logit of the probability. The coefficients are estimated by maximum likelihood. Once we have the estimated coefficients, we can calculate the probability for any given set of independent variable values.
Logistic regression shares some of the same assumptions as linear regression, like independence of errors and lack of multicollinearity. However, it does not assume normality or homoscedasticity of errors. It also tends to be more robust to outliers.
Some example applications of logistic regression include predicting customer churn, credit default, disease diagnosis, and ad click-through rates. By understanding the factors that influence the probability, organizations can take targeted actions to achieve desired outcomes.
3. Polynomial Regression
What if the relationship between the independent and dependent variables is not linear, but can be well-approximated by a polynomial function? In that case, polynomial regression is a good choice.
Polynomial regression extends linear regression by adding polynomial terms of the independent variables as additional predictors in the model. A second-degree polynomial model looks like:
y = β₀ + β₁x + β₂x² + ε
We can extend this to polynomials of higher degree by adding additional higher-order terms. The coefficients are estimated in much the same way as linear regression, by minimizing the sum of squared errors.
Polynomial regression can fit a wide variety of non-linear relationships and provide a better fit than simple linear regression in many cases. However, it‘s important not to overfit by including polynomial terms of too high a degree, as this can lead to highly unstable estimates. Regularization techniques like ridge regression and lasso can help.
Polynomial regression sees common use in fields like economics and ecology for modeling growth curves and other non-linear phenomena. For example, economists might model the relationship between years of education and earnings using a quadratic function.
4. Ridge Regression
Ridge regression is a regularized version of linear regression. It seeks to minimize the sum of squared errors (like OLS), but with an added penalty term on the squared magnitude of the coefficients:
minimize: Σ(y – Xβ)² + λΣβ²
The hyperparameter λ controls the strength of the penalty – larger values of λ result in greater shrinkage of the coefficient estimates towards zero. This biases the estimates, but often reduces their variance.
Ridge regression is effective at mitigating the effects of multicollinearity among the independent variables. When predictors are highly correlated, their coefficient estimates can become very large in magnitude and unstable – ridge regression brings them back down to earth.
The coefficient estimates are found by solving a slightly modified set of linear equations compared to OLS. Efficient implementations are available in most statistical software packages.
Some example applications of ridge regression include genetics (large p, small n problems), forecasting economic time series, and building predictive models where multicollinearity is present and we want to retain all the predictors in the model.
5. Lasso Regression
Lasso (least absolute shrinkage and selection operator) regression is another regularized version of linear regression. The lasso minimizes the sum of squared errors plus an L1 penalty on the absolute values of the coefficients:
minimize: Σ(y – Xβ)² + λΣ|β|
The L1 penalty has the effect of setting some of the coefficient estimates exactly to zero (for sufficiently large λ). Thus, the lasso performs feature selection and produces sparse models, which can be beneficial for both interpretation and avoiding overfitting.
Like ridge, the lasso can help with multicollinearity and high variance in the coefficient estimates. However, it has a few key differences:
- If there are groups of highly correlated predictors, the lasso will tend to pick one and set the others to zero, whereas ridge will shrink them all towards each other.
- If p > n, the lasso can only select at most n predictors, whereas ridge will keep all p in the model (albeit shrunken).
- The lasso is better for sparse problems where we expect many of the predictors to have zero coefficients.
Lasso regression has been widely used in fields like genomics and medicine where there are often a very large number of potential predictors (like gene expression levels) but we believe only a small fraction are truly relevant to the outcome. By setting most of the coefficients to zero, the lasso gives highly interpretable models.
6. Elastic Net Regression
As you might have guessed, elastic net is a compromise between ridge and lasso regression. It combines both the L1 and L2 penalties:
minimize: Σ(y – Xβ)² + λ₁Σ|β| + λ₂Σβ²
The hyperparameters λ₁ and λ₂ control the strength of the L1 and L2 penalties, respectively. When λ₁=0 and λ₂>0, elastic net reduces to ridge regression. When λ₁>0 and λ₂=0, it reduces to lasso. And when both are positive, we get a blend of the two.
Elastic net aims to combine the strengths of both lasso and ridge:
- Like lasso, it can produce sparse models and perform feature selection.
- Like ridge, it can handle multicollinearity and p > n problems.
For very sparse problems, lasso may have an advantage, while for dense problems with many small but non-zero coefficients, ridge may win out. Elastic net is a good choice when you‘re not sure which scenario you‘re in and want to hedge your bets.
7. Bayesian Regression
All of the regression techniques we‘ve discussed so far take a frequentist approach – they yield point estimates of the model parameters. Bayesian regression, on the other hand, embraces uncertainty and seeks to estimate the full posterior distribution of the parameters given the data.
According to Bayes‘ theorem, the posterior distribution is proportional to the likelihood of the data (given the parameters) times the prior distribution of the parameters:
P(β|y,X) ∝ P(y|β,X) · P(β)
To perform Bayesian regression, we need to specify priors for the model parameters (e.g., the regression coefficients). These encode our initial beliefs about what values are more or less likely. Common choices include:
- Flat prior – a uniform distribution, reflecting no prior knowledge
- Gaussian prior – normally distributed with a specified mean and variance
- Laplace prior – zero-centered with exponential tails, encouraging sparsity
Given the priors and the data, we can then compute the posterior distribution either analytically (for certain prior/likelihood combinations) or numerically (via MCMC sampling). The result is a full distribution of plausible values for each model parameter, rather than a single point estimate. This allows us to quantify our uncertainty and make probabilistic predictions.
Bayesian regression can be a powerful tool, particularly when we have prior knowledge we want to incorporate or when we‘re working with small sample sizes where frequentist estimates may be unstable. It‘s commonly used in fields like pharmacology, epidemiology, and marketing research.
Conclusion
We‘ve covered a lot of ground in this post, from the basics of linear regression to more advanced regularized and Bayesian techniques. Of course, we‘ve only scratched the surface – there are many other regression methods out there, each with their own unique strengths and use cases.
The key takeaway is that regression is an incredibly powerful and versatile tool for understanding relationships between variables and making predictions. However, it‘s not a magic bullet – each technique comes with its own set of assumptions and limitations that must be carefully considered.
As a data scientist, your job is to understand your data and choose the right tool for the job. This requires a solid grasp of the different regression methods available, as well as an appreciation for the domain-specific context of your problem. By combining technical know-how with subject matter expertise, you can wield regression to uncover valuable insights and drive better decisions.
I hope this post has given you a solid foundation in regression and a roadmap for further exploration. Happy modeling!
