Introduction

Linear regression is one of the most fundamental algorithms in machine learning and statistics used to model the relationship between a dependent variable and one or more independent variables. It attempts to capture a linear relationship between these variables by fitting a line (or hyperplane in multidimensional space) to the data points.

In this article, we focus on the closed-form solution for linear regression, providing technical explanations, examples, and additional insights into the implementation.

Linear Regression Basics

Linear regression aims to find the best-fitting line through a set of points in such a way that the sum of the squared differences (errors) between the observed values and the values predicted by the line is minimized. The equation of the line for univariate linear regression is given as:

• `y` is the dependent variable (target). • `x` is the independent variable (feature). • `β₀` is the y-intercept. • `β₁` is the slope of the line. • `ε` is the error term.

• `$\mathbf{w}$` is the vector of coefficients `[β₀, β₁, ..., βₙ]`. • `$\mathbf{X}$` is the matrix containing the input features. • `$\mathbf{y}$` is the vector of observed outputs or the target variable. 1 & x_{11} & x_{12} \ 1 & x_{21} & x_{22} \ 1 & x_{n1} & x_{n2} • Compute `$\mathbf{X}^T \mathbf{X}$`, which is a square matrix of size `(m+1, m+1)`. • Calculate `$(\mathbf{X}^T \mathbf{X})^{-1}$`, the inverse of the above product. • Compute `$\mathbf{X}^T \mathbf{y}$`, a vector of size `(m+1, 1)`. • Finally, multiply the inverse with this vector to get `$\mathbf{w}$`.

• Analytical Solution: Provides exact coefficients without iterative optimization. • Simplicity: Easier to implement with a few lines of code. • Computationally Intensive: The need to compute a matrix inverse makes it impractical for very large datasets or when the feature space is high-dimensional. • Numerical Stability: Potentially unstable due to the inversion of a potentially ill-conditioned matrix.