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--- title: "[redirect] normal equation" source: https://www.jemoka.com/posts/kbhnormal_equation/ --- See Normal Equation for small equations of Linear Regression, we can solve it using normal equation method. Consider \(d\) dimensional feature and \(n\) samples of data. Remember, including the dummy feature, we have a matrix: \(X \in \mathbb{R}^{n \times \qty(d+1)}\) and a target \(Y \in \mathbb{R}^{n}\). Notice: \begin{equation} J\qty(\theta) = \frac{1}{2} \sum_{i=1}^{n} \qty(h_{\theta} \qty(x^{(i)}) - y^{(i)})^{2} \end{equation} and \(h = X \theta\), we we can write: \begin{equation} J(\theta) = \frac{1}{2} \qty(X \theta - y)^{T} \qty(X \theta - y) \end{equation} We can take a derivative of this Setting this to \(0\), taking the pseudoinverse: \begin{equation} \theta = \qty(X^{T}X)^{-1} X^{T}y \end{equation}