Welcome to our exciting second class in the Regression and Gradient Descent series! In the previous lesson, we covered Simple Linear Regression. Now, we're transitioning toward Multiple Linear Regression, a powerful tool for examining the relationship between a dependent variable and several independent variables.

Consider a case where we need to predict house prices, which undoubtedly depend on multiple factors, such as location, size, and the number of rooms. Multiple Linear Regression accounts for these simultaneous predictors. In today's lesson, you'll learn how to implement this concept in C++!

Multiple Linear Regression builds upon the concept of Simple Linear Regression, accounting for more than one independent variable.

Let's recall the Simple Linear Regression equation:

$y = \beta_0 + \beta_1x$

For Multiple Linear Regression, we add multiple independent variables, :

Suppose we had n data points (equations), each with m features (x values) Then X would look like:

Now, for any set of features ${x_{1}}$ through ${x_{m}}$, we can predict the $\hat{y}$ value as:

To implement Multiple Linear Regression, we'll leverage some Linear Algebra concepts. Using the Normal Equation, we can calculate the coefficients for our regression equation:

$\beta = (X^T X)^{-1} X^T y$

Let's roll up our sleeves and start coding! We'll primarily rely on Eigen to handle numerical operations and matrices.

First, we set up our dataset:

Next, we calculate our matrix of coefficients, $\boldsymbol{\beta}$, using the Normal Equation:

• Enhance our feature matrix, $X$, with an extra column of ones to account for the intercept.

• Compute the coefficients $\boldsymbol{\beta}$ using the Normal Equation.

After completing our model, we need to evaluate its performance. We use the coefficient of determination ($R^2$ score) for this purpose. It indicates how well our model fits the data. The formula is:

Congratulations on mastering Multiple Linear Regression! You've effectively bridged the gap from concept to implementation, designing a regression model in C++ from scratch.

Prepare for the upcoming lesson to delve more deeply into Regression Analysis. Meanwhile, make sure to practice and refine your newly acquired skills!