Linear regression is one of the oldest and most basic machine learning methods, commonly used to understand more advanced algorithms. Despite its simplicity, it is utilized for regression and predictive tasks in economics, healthcare, finance, and marketing. The dependent-independent relationship is modeled using this way. This article discusses linear regression, its types, and its role in machine learning.

What is Linear regression?

Linear regression models the relationship between a dependent variable (goal variable or outcome) and one or more independent variables (features or predictors).Making predictions and comprehending the impact of independent factors on dependent variables are crucial.

Linear regression presupposes that the dependent and independent variables are linearly related, so the target variable changes at a constant rate relative to the predictor variables. Linear regression is employed when the variables are essentially linear because the procedure relies on this assumption.

Supervised linear regression requires labeled data for training. The model must understand the underlying relationship to forecast the target variable’s value for unseen data based on its attributes.

How does Linear Regression Work?

Linear regression is a statistical technique for modeling the connection between a dependent variable (also known as the goal or outcome variable) and one or more independent variables (predictors or characteristics). Simple linear regression consists of one dependent variable and one independent variable. Multiple linear regression involves numerous independent variables.

The general form of a linear regression equation is: y= b​₀+b₁𝑥₁​+b₂𝑥_​2​+⋯+b_n​𝑥_n​+ϵ

Where:

• y is the dependent variable (what you’re trying to predict).
• b_0​ is the intercept (the value of y when all ​x_i​‘s are 0).
• b₁​,b₂​,…,b_n​ are the coefficients (slopes) for each independent variable ​x₁,​x₂,…,x_n.
• ϵ is the error term (the difference between the predicted and actual values, representing randomness or other factors not captured by the model).

Steps in Linear Regression:

• Collect Data: Collect data that includes both the dependent variable (y) and the independent variable (​x).
• Model the Relationship: The objective is to fit a line (or hyperplane in several dimensions) to the data. In simple linear regression, this is the straight line that best fits the data.
• Find the Best Fit Line: The approach for determining the best-fit line is known as least squares. The goal is to minimize the total of the squared discrepancies (residuals) between observed and anticipated values for y. The formula for this is:

Loss Function=∑(𝑦_i​−𝑦_i​’)²

Where y_i are the actual values, and y_i​’ are the predicted values.

• Solve for the Coefficients: After minimizing the loss function, get the coefficients: 𝑏₀ , 𝑏₁ , … , 𝑏_𝑛​. In simple linear regression, this is equivalent to determining the slope and intercept of the line that minimizes the difference between the actual and anticipated values.
• Make predictions: After determining the coefficients, use the equation to predict y for new values of x.

Example (Simple Linear Regression):

Assume you’re trying to guess someone’s weight based on their height. You gather data with height as the independent variable and weight as the dependent variable. The linear regression model could look something like this:

Weight= 50 + 0.5 × Height

In this case:

The intercept b0 is 50 (the anticipated weight when the height is 0, which makes no practical sense but works analytically).
The coefficient b1 is 0.5, indicating that for every 1-unit increase in height, the anticipated weight rises by 0.5 units.

Evaluating the Model:

After fitting the model, you’ll want to evaluate its performance. Common evaluation metrics include the following:

R-squared (R²): It is a measure of a model’s ability to explain data variability. An R² value near to 1 indicates that the model explains the majority of the variability in the dependent variable.
Mean Squared Error (MSE): This is the average of the squared residuals (prediction errors). Lower values indicate improved performance.

Types of Linear Regression

Different types of issues require different versions of linear regression, but the underlying notion remains the same. These differences rely on the number of independent variables and how the model handles overfitting, multicollinearity, and feature selection.

The most popular linear regression types are:

Simple Linear Regression:

Simple linear regression is the simplest. This model predicts a target variable using one independent variable. When the two variables have a linear relationship, the dependent variable changes in proportion to the independent variable.

For instance, simple linear regression may predict weight based on height. According to the model, weight rises linearly with height. This strategy is simple and straightforward to read, but it only handles one predictor.

Multilinear Regression:

many linear regression predicts the dependent variable using many independent variables. This lets the model account for more target variable-influencing factors. When the target and predictors have many features, such as property pricing depending on size, bedrooms, and location, multiple linear regression is utilized.

Predicting a person’s health score may entail considering age, exercise frequency, food, and sleep patterns. many linear regression describes the relationship between many predictors and the target variable as a linear function, which can predict more accurately than basic linear regression.

Ridge Regression(L2 Regularization ):

In linear regression, model coefficient size is penalized by ridge regression (L2 regularization). Ridge regression aims to prevent overfitting, which occurs when the model grows too complex and performs poorly on new data.

Linear regression can be complicated by overfitting, especially when there are many features or the independent and dependent variables are not linear. By including a penalty term in the loss function, ridge regression resolves this issue. The penalty discourages large coefficient values, which can simplify and generalize the model.

Ridge regression works well when predictors are multicollinear, meaning independent variables are highly connected. Ridge regression reduces variance to stabilize model coefficient estimates.

Lasso Regression(L1 Regularization):

Lasso regression, or L1 regularization, penalizes coefficients differently than linear regression. Unlike ridge regression, which penalizes big coefficients by adding their square to the loss function, lasso regression penalizes their absolute values. This is special because it can decrease some coefficients to zero, removing them from the model.

Lasso regression automatically identifies and removes unimportant characteristics, making it suitable for huge feature sets. This helps when there are many predictors but few are meaningful to the target variable. Lasso regression improves model interpretability and performance by deleting irrelevant features.

Elastic Net Regression:

Ridge regression (L2 regularization) and lasso regression (L1 regularization) penalties are combined in elastic net regression. numerous real-world datasets include numerous associated features and more features than observations, which our hybrid technique handles.

Elastic net regression balances coefficient shrinking (ridge) and irrelevant feature elimination (lasso). It prevents overfitting and provides a robust feature selection method for large datasets with highly correlated variables.

Stepwise Regression:

Stepwise regression selects relevant features for a linear regression model. It automatically adds or removes predictors based on p-values, which evaluate feature importance.

Forward stepwise regression starts with no predictors and adds characteristics one by one. At each phase, the feature that improves the model most (usually by reducing the residual sum of squares or improving explanatory power) is included. Beginning with all factors, backward stepwise regression eliminates the least significant ones at each stage.

Stepwise regression helps find the most significant variables in huge datasets with many characteristics, boosting model performance and interpretability.

Assumptions in Linear Regression

Certain assumptions must be true for linear regression to work:

• Linearity: The independent and dependent variables are considered to be linearly related. If the relationship is non-linear, linear regression may fail.
• Error Independence: Residuals (predicted-to-actual errors) should be independent. Time-series data may break this assumption, hence autoregressive integrated moving average (ARIMA) may be better.
• Homoscedasticity: Residual variance should be constant across independent variable levels. Biased estimates might result from residual variance changes at different predictor levels.
• Normality of Errors: Ideally, errors should be naturally distributed.This assumption is essential for hypothesis testing and confidence intervals.
• No Multicollinearity: Predictors should not be highly correlated. When features are highly linked, model coefficients might become unstable and hard to interpret.

Advantages and Limitations of Linear Regression

Advantages:

• Simple and Interpretable: Linear regression is simple to use. Interpretable coefficients demonstrate the relationship between each predictor and the target variable.
• Efficiency: Linear regression works well with small to moderate datasets and is computationally efficient.
• Basis for More Complex Models: Linear regression underpins many advanced machine learning methods. Logistic regression, support vector machines, and other models require linear regression knowledge.

Limitations:

• Assumption of Linearity: In real-world data, the relationship between predictors and target variables is not always linear, but linear regression assumes it is.
• Outlier Sensitivity: Outliers can disproportionately affect linear regression coefficients and predictions.
• Multicollinearity: Highly correlated independent variables might make multiple linear regression coefficient estimations inaccurate.

Linear Regression Applications

Many fields employ linear regression:

Finance: Predicting stock prices, market trends, or economic indicator relationships.
Healthcare: Predicting patient outcomes, illness progression, and therapy efficacy.
Marketing: Monitoring consumer behavior, sales forecasts, and advertising effectiveness.
Real Estate: Valuing properties by location, size, and amenities.

Conclusion

Various kinds of linear regression are still essential to machine learning. Data scientists must understand linear regression, whether they use simple linear regression for simple problems or advanced methods like ridge, lasso, or elastic net for complicated datasets. While it has limitations, its simplicity, ease of understanding, and ability to model linear relationships make it a strong predictive analysis tool for many applications.