Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable (Y) and one or more independent variables (X). It assumes that this relationship can be represented as a straight line. The goal is to find the best-fitting line that minimizes the difference between the actual data points and the predicted values.

Types of Linear Regression

1. Simple Linear Regression – Models the relationship between one independent variable and one dependent variable.

   • Equation:
     $Y = mX + b$ where:
     • $Y$ is the dependent variable (output),
     • $X$ is the independent variable (input),
     • $m$ is the slope of the line,
     • $b$ is the y-intercept.

2. Multiple Linear Regression – Extends simple linear regression to multiple independent variables.

   • Equation:
     $Y = b_0 + b_1X_1 + b_2X_2 + ... + b_nX_n$ where:
     • $Y$ is the dependent variable,
     • $X_1, X_2, ..., X_n$ are independent variables,
     • $b_0$ is the intercept,
     • $b_1, b_2, ..., b_n$ are coefficients that represent the impact of each independent variable.

How Linear Regression Works

1. Data Collection – Gather data points with input (X) and output (Y).
2. Model Fitting – Find the line that best fits the data using the Least Squares Method, which minimizes the sum of squared differences between actual and predicted values.
3. Evaluation – Measure the model's accuracy using metrics such as:
   • R² (Coefficient of Determination): Explains how well the model fits the data.
   • Mean Squared Error (MSE): Measures the average squared error of predictions.

Applications of Linear Regression

• Predicting sales based on marketing spend.
• Estimating house prices based on features like area and location.
• Modeling trends in finance and economics.

Would you like a detailed example or a Python implementation?

Related Questions

1. What are the assumptions of linear regression?
2. How is the least squares method used in linear regression?
3. What is multicollinearity, and why is it important in regression?
4. How do you interpret regression coefficients?
5. What is the difference between linear and logistic regression?

Tip:

Linear regression assumes a linear relationship between variables; if the relationship is nonlinear, transformations or alternative models may be needed.