What is Linear Regression in Machine Learning?

Machine learning algorithms now guide decision-making at organization in healthcare, education, e-commerce, and countless other industries. A healthcare system used in a clinical setting might predict patient hospital stays after surgery by examining past recovery data and patient profiles. An inventory management system might forecast product demand by analyzing past purchasing patterns and seasonal trends. Both rely on linear regression—a method that finds the relationship between different variables by fitting a straight line through data points.

Linear regression is one of the foundational techniques in machine learning. It’s a supervised learning algorithm where the model analyzes labeled training data (like past patient records or retail receipts) to identify patterns, and then uses those patterns to make predictions on new information. While complex AI models get all the headline attention, linear regression remains one of data science’s most practical and widely used tools. Below, we’ll walk you through everything you need to know about linear regression and how it powers modern machine-learning applications.

What is linear regression?

Linear regression is a statistical method used to model the relationship between one or more dependent variables (target) and one or more independent variables (predictors) by fitting a linear equation to the observed data. Here, the term “linear” refers to the straight-line nature of the relationship being modeled, while “regression” describes the process of estimating how variables move back toward (or “regress to”) their average values.

Linear regression finds patterns in your data by establishing relationships between variables. For example, a real estate company might use property size to predict home prices or a retail business might use historical sales data to forecast future revenue. The method calculates both the slope (how much one variable changes in relation to another) and the intercept (where the line starts) to create the most accurate predictions possible.

Breaking down the math behind linear regression models

Linear regression follows a straightforward mathematical formula: y = mx + b. Here’s how the equation works:

• y = dependent variable (output)

• x = independent variable (input)

• m = slope of the line (how much y changes for a unit change in x)

• b = intercept (value of y when x=0)

While this might remind you of high school algebra, its applications in machine learning are far more powerful.

The model works by finding the best line that minimizes prediction errors across all data points. It does this through a process called ordinary least squares (OLS)—this measures the vertical distance between each data point and the predicted line, squares these distances, and minimizes their sum.

These squared differences between actual and predicted values, known as residuals, are a critical measure of model performance—the smaller the sum of squared residuals, the better the line fits your data. By minimizing this value, OLS provides us with the most statistically efficient estimates of the coefficients in the regression equation, giving us confidence in the relationship we’ve discovered.

It’s like adjusting a ruler’s position over scattered dots on a page. The goal is to position the ruler so that (on average) it’s as close as possible to all points. In machine learning terms, this process is called "training”: the model learns the optimal values for slope and intercept that create the best-fitting line through your data.

Ordinary Least Squares (OLS) regression mathematical explanation

OLS is a measure which is used in regression analysis that helps to find the best fitted line for the data points. It minimizes the sum of the squared differences between the observed and predicted values, ensuring that the best-fit line accurately represents this minimization.

Here’s how this is calculated:

By squaring the residuals, OLS ensures:

• No negative values affects the optimization.

• Larger errors are penalized more than smaller errors.

Few assumptions for OLS calculations

For OLS estimates to be unbiased and efficient for linear regression, there are certain assumptions :

1. Linearity: The relationship between independent and dependent variables is linear.

2. Independence: Observations are independent of each other.

3. Homoscedasticity: The variance of errors is constant across all levels of the independent variables.

4. Normality of Errors: The residuals (errors) are normally distributed.

5. No Perfect Multicollinearity: Independent variables should not be highly correlated.

How linear regression works in machine learning

Linear Regression assumes a linear relationship between input features (X) and the target variable (Y).

Steps in Linear Regression:

1. Define the Relationship: The model assumes a linear relationship: y=β0+β1x+ϵ

2. Fit the Best Line: The algorithm calculates the best-fit line by minimizing the difference between actual and predicted values using the Ordinary Least Squares (OLS) method.

3. Minimize Error: Find the optimal values of β0​ and β1​ to reduce the sum of squared differences between actual and predicted values.

4. Make Predictions: Once trained, the model uses the equation to predict new values based on unseen input data.

Linear regression follows a structured process that transforms raw data into predictive insights. The math behind it might feel complicated, but the workflow follows a logical sequence that any data practitioner can master.

Here’s how a typical linear regression model comes together in a machine-learning pipeline:

1. Data preparation: Clean your dataset by handling missing values, removing outliers, and standardizing features. A housing price prediction model (for example) might standardize square footage and normalize price data to work on a similar scale.

2. Feature engineering: Choose which variables will help predict your target. For example, when predicting software development project timelines, you might select team size, project complexity, and historical completion times as key features.

3. Model training: Feed your prepared data into the regression model. The model iteratively adjusts its parameters (slope and intercept) to find the line that best fits your training data. During this phase, it learns the relationships between input features and the target variable.

4. Model evaluation: Test your model’s performance using metrics like Mean Squared Error (MSE) or R-squared values. A model making a prediction should show strong R-squared values to be considered reliable.

5. Fine-tuning: Adjust your model based on its performance. This might involve adding or removing features, handling non-linear relationships, or addressing issues like multicollinearity (where dependent features are too closely related).

6. Deployment: Put your model into production where it can make real predictions. An e-commerce platform might deploy a regression model to predict shipping times based on distance, package weight, and seasonal factors.

7. Monitoring: Regularly check your model’s performance, as changing data patterns (like evolving user behavior) may require updates to maintain accuracy.

3 types of linear regression in ML

Machine learning practitioners typically work with three variations of linear regression. Each is suited for different types of prediction problems. Here’s a quick look at how these approaches differ and when to use each.

1. Simple linear regression

Simple linear regression uses just one input variable to predict an outcome. Think of a streaming service predicting watch time based solely on user age or an app predicting user engagement based only on time spent in the first session. Yes, it’s basic, but this approach works well when you have a clear relationship between two variables.

2. Multiple linear regression

Multiple linear regression handles scenarios where several factors influence the outcome. A cloud hosting provider might predict server costs using CPU usage, storage space, and bandwidth consumption. Or a SaaS company might forecast customer lifetime value using subscription tier, usage patterns, and support tickets.

3. Polynomial regression

Sometimes, relationships aren’t straight lines. Polynomial regression handles curved relationships by adding exponential terms to the equation. Think of predicting app performance—as user load increases, performance might drop off exponentially rather than linearly.

Consider a game development company predicting server load during a launch. The relationship between number of players and server requirements often follows a curved pattern, and that makes polynomial regression the ideal choice.

Implementing linear regression in machine learning

You’ll need attention to detail and a systematic approach to implement linear regression reliably. Here are a few things to think about that can make or break your model’s success.

Best practices for implementation

First, focus on your data foundation (before diving into the code). Start by plotting your data to visualize relationships and potential issues. Python libraries like Matplotlib and Seaborn provide essential functions for creating scatter plots, histograms, and correlation matrices, while Plotly, Bokeh, and Altair add interactive features such as hover details and zoom capabilities that help you thoroughly inspect your data before applying regression techniques. Check for outliers that might skew your results—a single anomalous data point can majorly impact your model’s performance.

When your input variables operate on different scales (like comparing user age with annual revenue), standardize them to prevent larger-scale features from dominating the model. Unfortunately, many data scientists learn this lesson the hard way when their models underperform due to unscaled features.

Here’s a practical approach to implementation:

1. Split your data into training, testing, and validation sets

2. Scale your features using standardization or normalization

3. Train your model on the training data

4. Validate performance on your test set

5. Review residual plots to check model assumptions

6. Deploy your model only after thorough validation

Keep an eye out for common pitfalls like multicollinearity, where input variables are too closely related. For example, if you’re predicting software development time, using both “lines of code” and “number of functions” might create issues since they’re likely highly correlated.

How to get started with linear regression

Fortunately, getting started with linear regression doesn’t require a PhD in statistics. Start small, and scale as you learn and grow. Here are a few practical steps to help you get started:

Choose your tools wisely

Python’s scikit-learn library provides the most straightforward path for beginners. It provides clean, consistent APIs and excellent documentation. For those working with larger datasets, platforms like PyTorch or TensorFlow also have linear regression capabilities (though they might feel like using a sledgehammer to crack a nut).

Prepare your first dataset

Start small with a clean, well-understood dataset. Something like the California Housing dataset or Boston Housing dataset are great starting points. They’re clean, well-documented, and show clear linear relationships. Or you could use your own business data, focusing on numerical variables that you suspect might have strong relationships.

Build your first model

Your initial implementation should focus on understanding the process rather than achieving perfect accuracy. A simple project might predict house prices based on square footage, or customer spending based on engagement metrics. Your goal is to understand the workflow: loading data, splitting it into training and test sets, fitting the model, and evaluating results.

Evaluate and iterate

Don’t expect perfection from your first model. Check your R-squared values to understand how well your model fits the data. Look at your residual plots to spot patterns that might suggest problems. Each iteration teaches valuable lessons about feature selection, data preprocessing, and model tuning.

What is linear regression FAQ

What is linear regression in simple terms?

Linear regression is like drawing a line through scattered data points to make predictions. It helps you understand how one thing affects another—like how studying hours affect test scores or how marketing spend affects sales revenue.

What is the best explanation of linear regression?

Linear regression finds the mathematical relationship between variables by fitting a straight line to your data. Think of it as finding the trend line that best represents your data points to help you make predictions about future values based on that relationship.

What is an example of a linear regression in real life?

Game developers use it to predict player churn based on engagement metrics. And cybersecurity teams apply it to detect anomalies by predicting normal network traffic patterns.

Why do we use regression in ML?

We use regression in machine learning to predict continuous numerical values. It helps businesses make data-driven decisions, forecast future trends, and understand relationships between variables. Unlike classification models that predict categories, regression predicts specific numbers.

Is linear regression supervised or unsupervised?

Linear regression is a supervised learning technique. This means it learns from labeled training data where both the input features (like house size) and the target variable (like house price) are known. The model learns these relationships to make predictions on new, unseen data.

What is linear regression used for?

Linear regression serves multiple purposes: predicting future values, understanding variable relationships, identifying trends in data, and quantifying the impact of changes in business metrics. Companies use it for sales forecasting, resource planning, performance prediction, and risk assessment.

How does linear regression differ from logistic regression?

Linear regression predicts continuous numerical values (like prices or temperatures), while logistic regression predicts categorical outcomes (like yes/no decisions or pass/fail results). For example, linear regression might predict a house’s price, while logistic regression would predict whether a house will sell within 30 days.

What are the key assumptions of linear regression?

Linear regression assumes a linear relationship between variables, independent observations, constant variance in errors (homoscedasticity), and normally distributed residuals. Breaking these assumptions can lead to unreliable predictions and should be a signal you need to consider other modeling approaches.

References

• Ordinary Least Squares Regression (OLS)

• Mean squared error

• R Squared in R - How to Calculate R2 in R?

• Multicollinearity

• The California housing dataset

• Boston housing dataset

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