simple-linear-regression

Simple Linear Regression

Simple linear regression is a statistical method that allows us to summarize and study relationships between two continuous variables:

• One variable, denoted as X, the variable being used to make the predictions(independent variable)
• The other variable, denoted as Y, the variable that is being predicted(dependent variable)

Concept

The goal of simple linear regression is to find a linear function that predicts the dependent variable (Y) as accurately as possible as a function of the independent variable (X).

Mathematical Representation

The simple linear regression model can be represented by the equation:

Y = mx + c

Y = β₀ + β₁X + ε

Where:

• Y is the dependent variable
• X is the independent variable
• β₀ is the y-intercept (the value of Y when X = 0)
• β₁ is the slope (the change in Y for a unit change in X)
• ε is the error term (the difference between predicted and actual Y values)

Estimating Parameters

To find the best-fitting line, we need to estimate β₀ and β₁. This is typically done using the method of least squares, which minimizes the sum of squared residuals.

The formulas for β₀ and β₁ are:

β₁ = Σ((X - X̄)(Y - Ȳ)) / Σ((X - X̄)²) β₀ = Ȳ - β₁X̄

Where X̄ and Ȳ are the means of X and Y respectively.

Implementation Approach

1. Data Preparation:

   • Load the dataset
   • Split into X (independent) and Y (dependent) variables

2. Calculate Statistics:

   • Compute means of X and Y
   • Compute the sum of squares for X and XY

3. Compute Parameters:

   • Calculate β₁ (slope) using the formula above
   • Calculate β₀ (intercept) using the formula above

4. Make Predictions:

   • Use the equation Y = β₀ + β₁X to predict Y for any given X

5. Evaluate the Model:

   • Calculate residuals (differences between actual and predicted Y values)
   • Compute R-squared (coefficient of determination) to assess model fit

6. Visualize Results:

   • Plot the original data points
   • Plot the regression line