Smoothing spline

The smoothing spline is a method of fitting a smooth curve to a set of noisy observations using a spline function.

Let ${\displaystyle (x_{i},Y_{i});x_{1}<x_{2}<\dots <x_{n},i\in \mathbb {Z} }$ be a sequence of observations, modeled by the relation ${\displaystyle Y_{i}=f(x_{i})}$. The smoothing spline estimate ${\displaystyle {\hat {f}}}$ of the function ${\displaystyle f}$ is defined to be the minimizer (over the class of twice differentiable functions) of

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