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@jemoka / Jemoka Knowledge Base / raw/concept/kbhnonlinear_linear_regression.md
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--- title: "Normal Equation" source: https://www.jemoka.com/posts/kbhnonlinear_linear_regression/ --- constituents Let’s also define our entire training examples and stack them in rows: \begin{equation} X = \mqty( - x^{(1)}^{T} - \\ \dots \\ - x^{\qty(n)}^{T} - ) \end{equation} \begin{equation} Y = \mqty(y^{(1)} \\ \dots \\ y^{(n)}) \end{equation} requirements least-squares error becomes: \begin{equation} J\qty(\theta) = \frac{1}{2} \sum_{i=1}^{n} \qty(h\qty(x^{(i)}) - y^{(i)}) ^{2} = \qty(X \theta - y)^{T} \qty(X \theta - y) \end{equation} Solving this exactly by taking the derivative of \(J\) and set it to \(0\) (i.e. for a minima, we obtain) \begin{equation} \theta = \qty(X^{T}X)^{-1} X^{T}y \end{equation} additional information