Linear Regression

Linear regression is a statistical technique that models the linear relationship between a dependent variable (target variable) and one or more independent variables (explanatory variables). At its most basic level, it aims to predict the value of one variable based on the known value of another. This technique is widely used in both statistics and machine learning fields in various forms.

Linear regression is the process of estimating an unknown value through a linear mathematical model. This model assumes a linear (straight-line) relationship between variables. The fundamental equation is:

Linear Regression Fundamental Equation

where:

• y: the dependent variable (to be predicted),
• x: the independent variable (input),
• β0: the constant term (intercept),
• β1: the regression coefficient (slope),
• ε: the error term (residual, the difference between observation and prediction).

The primary goal of linear regression is to identify the linear relationship that best predicts the variable y using the variable x.

Linear regression is widely used in economics, biology, social sciences, marketing, finance and engineering like. Example applications include:

• Predicting the relationship between income and expenditure,
• Analyzing the relationship between product price and sales,
• Modeling the relationship between education duration and salary,
• Extracting trends from stock market data,
• Evaluating interactions between variables in scientific experiments.

• Easy to Implement: Due to its simple structure, it is a fast and computationally inexpensive modeling method.
• Interpretability: The model parameters have direct meaning; it is easy to interpret how much each variable influences the outcome.
• Scalability: It can work effectively even with large datasets.
• Real-Time Prediction: Its lightweight nature allows for instantaneous predictions in online systems.

• Simple Linear Regression: Establishes a linear relationship with a single independent variable.
• Multiple Linear Regression: Makes predictions using more than one independent variable.

Simple and Multiple Linear Regression Equations

1. Data collection and preprocessing.
2. Identification of independent and dependent variables.
3. Visualization of the relationship between variables using scatter plots.
4. Fitting the best straight line using the Ordinary Least Squares (OLS) method.
5. Evaluation of the model (R², p-value, mean squared error, etc.).
6. Checking assumptions and improving the model if necessary.

For linear regression to function properly, the following assumptions must be satisfied:

• Linearity: There must be a linear relationship between the independent and dependent variables.
• Homoscedasticity: The residuals (errors) must have constant variance.
• Normality: The errors must follow a normal distribution.
• No Multicollinearity: There should be no high correlation between independent variables.
• Independence: Observations must be independent of each other.

These assumptions are checked using scatter plots, residual analysis, the Durbin-Watson test, and the Variance Inflation Factor (VIF).

• R² (Coefficient of Determination): Indicates the explanatory power of the model.
• MAE / MSE / RMSE: Measure the average error to assess how accurately the model predicts.
• F-Statistic and p-value: Used to evaluate the statistical significance of the model.

• Predicting future periods based on a company’s historical revenue and expense data,
• Analyzing how changes in a product’s price affect sales,
• Predicting an employee’s performance or income using demographic data (age, education, experience).