What Is Linear Regression?

Linear regression can model the relationship between variables through linear regression equations. These equations quantify the strength and direction of the relationship, allowing for the prediction of future outcomes with a degree of certainty. The precision of these predictions hinges on the regression model's ability to accurately capture the underlying relationship between the variables, which is often refined through techniques like gradient descent to minimize prediction errors.

Among various types of regression analysis, linear regression enjoys widespread popularity due to its simplicity, interpretability, and efficiency. It serves as the most common type of regression analysis, favored for its straightforward application and the clarity of the insights it provides. Whether in academic research, financial forecasting, or technology development, linear regression's role is pivotal, underpinning many of the predictive models that inform data-driven decision-making.

Types of linear regression

Linear regression models are versatile tools in statistical modeling and machine learning, designed to understand and predict the relationships between variables. The most basic form, simple linear regression, focuses on predicting a dependent variable based on a single independent variable. This model assumes a straight-line relationship between the two variables, making it particularly useful for understanding the impact of one variable on another. For example, it could be used to predict sales based on advertising spend, assuming all other factors remain constant.

Multiple linear regression expands on this by incorporating multiple independent variables to predict a dependent variable. This approach allows for a more comprehensive analysis of complex phenomena by considering the simultaneous effects of several predictors on the outcome. Multiple linear regression is particularly useful in scenarios where the outcome is influenced by a combination of factors, such as predicting a home’s price based on its size, location, and age.

Logistic regression, despite its name, is used for classification problems rather than predicting continuous outcomes. It estimates the probability that a given input point belongs to a certain class. For instance, it could be used to predict whether an email is spam or not based on features like the frequency of certain words. Logistic regression is invaluable for binary classification tasks where the outcome is categorical.

Ordinal regression is tailored for situations where the dependent variable is ordinal, meaning it has a natural order. This type of regression is useful when predicting outcomes that fall into naturally ordered categories, such as a rating scale from poor to excellent. It models the probability of the dependent variable falling into one of the categories, taking into account the inherent order of the categories.

Multinomial logistic regression extends logistic regression to handle multiple classes without a natural ordering. It is suitable for classifying instances into three or more categories, such as predicting the type of cuisine of a restaurant based on its menu and location. This model calculates the probabilities of each possible outcome and predicts the category with the highest probability.

Each type of linear regression serves a specific purpose and is applied based on the nature of the variables involved and the question at hand. Understanding the distinctions and appropriate applications of these models is crucial for effectively analyzing data and making predictions.

Simple linear regression (SLR) vs. multiple linear regression (MLR)

Simple linear regression (SLR) and multiple linear regression (MLR) are suited to different types of analysis depending on the number of independent variables involved. SLR explores the relationship between two variables—one independent (or predictor) variable and one dependent (or outcome) variable—fitting a linear regression line that best predicts the dependent variable from the independent variable. MLR extends this concept by incorporating two or more independent variables to predict the dependent variable, allowing for a more nuanced analysis of complex relationships.

The assumptions underlying multiple regression are critical to its application and effectiveness. These include:

• Presumption of a linear relationship between the dependent and independent variables
• Independence of the observations
• Homoscedasticity (constant variance of error terms)
• Normal distribution of error terms

Moreover, MLR assumes that there is no perfect multicollinearity among the independent variables. This means that the independent variables should not be too highly correlated with each other, as this can distort the model's estimates and reduce the reliability of the regression coefficients.

In MLR, the relationship between each independent variable and the dependent variable is quantified by regression coefficients, which represent the weight or importance of each independent variable in predicting the dependent variable. These coefficients are crucial for understanding how changes in the independent variables are expected to influence the dependent variable. For instance, in a model predicting rent prices, the regression coefficients could indicate how factors such as square footage, location, and number of bedrooms affect the rent price. A positive coefficient suggests that as the independent variable increases, the dependent variable also increases. A negative coefficient suggests that as the independent variable increases, the dependent variable decreases. This nuanced analysis provided by MLR allows for more accurate predictions and insights, especially in scenarios where multiple factors influence the outcome variable.

Linear regression in artificial intelligence and machine learning

The integration of linear regression into artificial intelligence and machine learning (AI/ML) projects showcases its simplicity and effectiveness in predictive modeling. Linear regression's straightforward approach, involving the prediction of a dependent variable from one or more independent variables, makes it an accessible and powerful tool for AI applications. This simplicity is particularly advantageous in the early stages of model development, where it provides a clear baseline for performance and interpretability.

The interpretability of linear regression results stands out as a significant advantage in machine learning. Unlike more complex models, the output of a linear regression model is easy to understand and explain, making it an invaluable tool for gaining insights into the relationships between variables. This clarity allows data scientists and stakeholders to make informed decisions based on the model's predictions, fostering trust and transparency in AI applications.

Linear regression's scalability is another key attribute that enhances its utility in machine learning. As datasets grow in size and complexity, linear regression models can efficiently scale to accommodate the increased computational demands. This scalability ensures that linear regression remains a viable option for a wide range of applications, from small-scale projects to large, data-intensive tasks.

As a baseline model, linear regression provides a benchmark against which the performance of more complex models can be measured. This benchmarking is crucial in the model selection process, as it identifies when the additional complexity of a more sophisticated model is justified by a significant improvement in performance.

The versatility and robustness of linear regression further underscore its value in AI applications. Capable of handling both simple and complex relationships between variables, linear regression can be applied across a diverse array of domains, from financial services and healthcare to marketing and environmental science. Its robustness for different types of data and its ability to provide reliable predictions even in the face of uncertainty make it a staple in the machine learning toolkit.

Explore 250+ use cases for advanced in-database analytics

Test-drive the most powerful engine for AI innovation: Get a free hands-on demo environment for ClearScape Analytics™, Teradata VantageCloud’s powerful engine for deploying end-to-end AI/ML pipelines. Explore hundreds of AI/ML use cases across multiple industries and analytic functions—from broken digital journeys and energy consumption forecasting to fraud detection and much more. Get started today.