Linear Regression is a statistical method used to model and analyze the relationship between a dependent variable and one or more independent variables.
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It aims to find the best-fitting linear relationship between the variables.
1. Simple Linear Regression:
- Model: Involves a single independent variable. The relationship is described by the equation:
[
Y = \beta_0 + \beta_1 X + \epsilon
]
Where ( Y ) is the dependent variable, ( X ) is the independent variable, ( \beta_0 ) is the y-intercept, ( \beta_1 ) is the slope of the line, and ( \epsilon ) is the error term. - Objective: The goal is to estimate ( \beta_0 ) and ( \beta_1 ) such that the sum of the squared differences between the observed values and the predicted values is minimized.
2. Multiple Linear Regression:
- Model: Extends simple linear regression to include multiple independent variables. The equation is:
[
Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \cdots + \beta_p X_p + \epsilon
]
Where ( X_1, X_2, \ldots, X_p ) are the independent variables. - Purpose: To understand the impact of several predictors on the dependent variable and to assess the relative importance of each predictor.
3. Assumptions:
- Linearity: The relationship between the dependent and independent variables is linear.
- Independence: Observations are independent of each other.
- Homoscedasticity: The variance of the residuals (errors) is constant across all levels of the independent variable(s).
- Normality of Errors: The residuals should be approximately normally distributed.
4. Evaluation Metrics:
- R-Squared: Measures the proportion of the variance in the dependent variable that is predictable from the independent variables. An ( R^2 ) value closer to 1 indicates a better fit.
- Adjusted R-Squared: Adjusts ( R^2 ) for the number of predictors in the model, providing a more accurate measure when multiple predictors are used.
- Residual Analysis: Examines the residuals to check for violations of regression assumptions and model fit.
5. Applications:
- Predictive Modeling: Used to make predictions based on new data.
- Quantifying Relationships: Helps in understanding how changes in independent variables impact the dependent variable.
- Decision Making: Provides insights for business, economics, and various fields where predicting outcomes based on predictor variables is crucial.
In summary, linear regression is a foundational statistical technique that models relationships between variables. It is widely used for prediction, understanding variable relationships, and decision-making based on data.