Linear Regression:

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. The goal is to find a linear equation that best predicts the dependent variable based on the independent variables.

Here’s a step-by-step explanation of simple linear regression (with one independent variable):

Step 1: Understand the Data

You need a dataset with:

Independent variable (X): The input feature(s) (also called the predictor or explanatory variable).
Dependent variable (Y): The output you want to predict (also called the target or response variable).

For example, you might have data about house prices (Y) based on house sizes (X).

Step 2: Visualize the Data

Start by plotting the data points on a scatter plot to visually inspect the relationship between the independent and dependent variables. If the data seems to follow a trend that resembles a straight line, linear regression can be a suitable model.

Step 3: Define the Hypothesis Function

The goal is to find the best-fitting line for the data. The general form of the linear regression equation is:

$Y = mX + b$

m: Slope of the line (the change in Y for a one-unit change in X).
b: Intercept (the value of Y when X = 0).

For multiple independent variables (multiple linear regression), the equation generalizes to:

$Y = b_0 + b_1X_1 + b_2X_2 + ... + b_nX_n$

Where:

$b_0$ $b_{0}$ is the intercept.
$b_1, b_2, ..., b_n$ $b_{1}, b_{2}, ..., b_{n}$ are the coefficients (slopes) of the independent variables.

Step 4: Calculate the Best-Fitting Line

To determine the slope (m) and intercept (b), we minimize the difference between the predicted values (Ŷ) and the actual values (Y). This is done using least squares regression.

The least squares method minimizes the sum of squared residuals:

$\text{Residual} = Y - \hat{Y}$

The formula for the slope $m$ $m$ and intercept $b$ $b$ for simple linear regression can be derived as:

$m = \frac{n \sum (X_i Y_i) - \sum X_i \sum Y_i}{n \sum (X_i^2) - (\sum X_i)^2}$

$b = \frac{\sum Y_i - m \sum X_i}{n}$

Where $n$ $n$ is the number of data points, $X_i$ $X_{i}$ and $Y_i$ $Y_{i}$ are individual data points.

Step 5: Make Predictions

Once the slope (m) and intercept (b) are determined, you can use the equation to make predictions for new values of X.

For example, if $m = 0.5$ $m = 0.5$ and $b = 2$ $b = 2$ , and you want to predict Y for $X = 4$ $X = 4$ , the predicted value would be:

$\hat{Y} = 0.5(4) + 2 = 4$

Step 6: Evaluate the Model

After fitting the model, evaluate its performance using metrics such as:

R-squared (R²): Represents the proportion of the variance in the dependent variable explained by the independent variable(s). R² ranges from 0 to 1, with values closer to 1 indicating a better fit.
Mean Squared Error (MSE): Measures the average squared difference between the actual and predicted values.

$MSE = \frac{1}{n} \sum_{i=1}^n (Y_i - \hat{Y}_i)^2$

Lower MSE indicates a better fit.

Step 7: Assumptions of Linear Regression

Before drawing conclusions, check that your data meets these assumptions for linear regression to be valid:

Linearity: The relationship between X and Y should be linear.
Independence: Observations should be independent of each other.
Homoscedasticity: The variance of residuals (errors) should be constant across all levels of X.
Normality: The residuals should be normally distributed.

Step 8: Improve the Model (Optional)

Feature selection: Choose important independent variables if you're doing multiple linear regression.
Polynomial regression: If the relationship isn’t linear, you might try fitting a polynomial function instead.