Linear Models

Linear models are foundational in machine learning (ML), particularly for regression tasks. These models assume a linear relationship between the input features (independent variables) and the target variable (Dependent variable). The equation for a linear model in regression is typically represented with a following equation: \begin{equation} y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \cdots + \beta_nx_n + \epsilon, \end{equation} wehre \(y\) is the predicted target, \(x_1,x_2,...,x_n\) are the input features, \(\beta_0\) is the intercept, \(\beta_1, \beta_2,...,\beta_n\) are the coefficients (weights) that determine the contribution of each features, and \(\epsilon\) is the error term. Linear regression is widely used due to its simplicty, interpretability, and ability to provide insights into relationships between features and the target variable.
Linear models are particularly effective when the relationship between the input features nad the target is approximately linear. However, they struggle with the non-linear patterns unless feature engineering or kernel transformations are applied. Despite this limtation, their computational efficiency and robustnes make them a preffered choice from many practical applications.

Linear Models in scikit-learn for Regression

Scikit-learn is a popular machine learning library in Python programming language. It offers a variety of linear models for regression, each tailored to different data scenarios. Sum of the examples of linear models for regression in sckit-learn are:

• Linear REgression - The linear regression class implements ordinary least squares regression. It minimizes the residual sum of squares between observed and predicted values, making it a default choice for simple regression tasks.
• Ridge Regression - or shortly Ridge is a type of linear regression that includes an L2-regulariation term in the cost function. This regularization penalizes large coefficients, which helps prevent overfitting, especially whne working with datasets that have multicollinearity or many features.
• Lasso Regression (Lasso) - adds an L1-regularization term to the cost function. This penalty cna shrink some coefficinets to zero, effectively performing feature selection, which is useful when dealing with sparse dataset or when feature selection is critical.
• ElasticNet - the algorithm combines L1 and L2 regularization terms. It balances the strenghts of Ridge and Lasso regression and is effective when there are many correlated features or when you want both regularization and feature selection.
• Bayesain Rige Regression - this estiamtor incorporates Bayesian principles producing probabilistic predictions, It assumes the regression coefficients follow a Gaussian distribution and estimates their distribution using the training data.
• SGDRegression - is a linear model optimized using stohastic gradient descent (SGD). IT is well-suited for large-scale datasets due to its ability to handle streaming data and incremental updates efficiently.
• HuberRegressor - is the model that is robust to outliers by modifying the loss function, Instead of minimizing squared errors, it uses the Huber loss, which is less snesitive to large deviations.
• Quantile Regressor - the model minimizes the quantile loss function to predict conditional quantiles of the target variable, making it ideal for estimating confidence intervals or distributions.

How to choos the right linear model?

The most common answer to the question above would be that the choice of a linear regression model in scikit-learn depends on the data characteristics and the problem being solved. For example if the overfitting is concern then Rige, Lasso, or ElasticNet are useful since they control the model complexity through regularization. In case the dataset contains the outliers then the HubberRegressor or QuantileRegressor are more apporapriate. BayesianRige is excellent choice for tasks requirining probabilitstic predicitons, hwile SGDRegressor is preferred for large scale dataset due to its computational efficiency.