Summary  
This chapter explains how to implement multivariate linear regression by constructing an augmented feature matrix (with a bias column of ones) and solving for the model parameters using the normal equation.  

General domain of usage  
Machine learning regression

## N-Feature Linear Regression Equation
As we have seen, adding the new feature to the linear regression model is as easy as adding it along with the new parameter to the model's equation. We can add much more than two parameters that way. 

Consider **n** to be a whole number greater than two.

Note

$$
y_{\text{pred}} = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \dots + \beta_n x_n
$$

Where:

- $$\beta_0, \beta_1, \beta_2, \dots, \beta_n$$ – are the model's parameters;
- $$y_{\text{pred}}$$ – is the prediction of a target;
- $$x_1$$ – is the first feature value;
- $$x_2$$ – is the second feature value;
- $$\dots$$
- $$x_n$$ – is the n-th feature value.

## Normal Equation
The only problem is the visualization. If we have two parameters, we need to build a 3D plot. But if we have more than two parameters, the plot will be more than three-dimensional. But we live in a 3-dimensional world and cannot imagine higher-dimensional plots. However, it is not necessary to visualize the result. We only need to find the parameters for the model to work. Luckily, it is relatively easy to find them. The good old Normal Equation will help us:














$$
\vec{\beta} = \begin{pmatrix} \beta_0 \\ \beta_1 \\ \dots \\ \beta_n \end{pmatrix} = (\tilde{X}^T \tilde{X})^{-1} \tilde{X}^T y_{\text{true}}
$$

Where:
* $$\beta_0, \beta_1, \dots, \beta_n$$ – are the model's parameters;
* $$\tilde{X}$$ – is a matrix, containing 1s as a first column, and $$X_1 - X_n$$ as other columns: 

$$
\tilde{X} = \begin{pmatrix} | & | & \dots & | \\ 1 & X_1 & \dots & X_n \\ | & | & \dots & | \end{pmatrix}
$$
* $$X_k$$ – is an array of k-th feature values from the training set;
* $$y_{\text{true}}$$ – is an array of target values from the training set.


## X̃ Matrix
Notice that only the **X̃**  matrix has changed. You can think of the columns of this matrix as each responsible for its **β** parameter. The following video explains what I mean.

The first column of 1s is needed to find the **β₀** parameter.

Master the core algorithms of supervised learning and implement them using Scikit-learn. Explore linear and polynomial regression for price prediction, and transition into classification using k-NN, Logistic Regression, and Decision Trees. Learn to evaluate models through cross-validation, manage overfitting with regularization, and optimize hyperparameters. Build robust predictive systems and define complex decision boundaries for multi-class classification tasks.

Linear Regression with N Features

N-Feature Linear Regression Equation

Normal Equation

X̃ Matrix