Smoothing spline

Smoothing splines are function estimates, ${\displaystyle {\hat {f}}(x)}$, obtained from a set of noisy observations ${\displaystyle y_{i}}$ of the target ${\displaystyle f(x_{i})}$, in order to balance a measure of goodness of fit of ${\displaystyle {\hat {f}}(x_{i})}$ to ${\displaystyle y_{i}}$ with a derivative based measure of the smoothness of ${\displaystyle {\hat {f}}(x)}$. They provide a means for smoothing noisy ${\displaystyle x_{i},y_{i}}$ data. The most familiar example is the cubic smoothing spline, but there are many other possibilities, including for the case where ${\displaystyle x}$ is a vector quantity.