Linear regression is a fundamental statistical method used for modeling the relationship between a dependent variable (often denoted as 'y') and one or more independent variables (often denoted as 'x'). It assumes a linear relationship between the variables, aiming to find the best-fitting linear equation that describes the relationship between the input variables and the target variable.

The basic form of a linear regression model for a single independent variable (simple linear regression) can be represented as:

y = β _0 + β _1*x + ϵ

Where:

• y represents the dependent variable or the target variable being predicted.

• x represents the independent variable or predictor variable.

• β _0 is the intercept, which represents the value of y when x is zero.

• β _1 is the slope or coefficient of the independent variable, indicating the change in y for a unit change in x.

• ϵ denotes the error term, representing the variability or noise in the relationship not explained by the model. In multiple linear regression, where there are multiple independent variables, the equation extends to accommodate these variables:

y = β _0 + β _1*x_1 + β _2*x_2 + ... + β _n*x_n + ϵ

The primary goal of linear regression is to estimate the values of the coefficients β _0, β _1, ... , β _n that minimize the sum of squared differences between the predicted values and the actual values (the least squares method). This estimation is usually done using optimization techniques to find the best-fitting line or hyperplane in higher dimensions.

Linear regression is used in various fields, including economics, finance, social sciences, and machine learning, for tasks such as prediction, forecasting, and understanding the relationships between variables. It's a foundational technique that forms the basis for more advanced regression and predictive modeling methods.