Smoothing spline

Smoothing splines are function estimates, f ^ ( x ) {displaystyle {hat {f}}(x)} , obtained from a set of noisy observations y i {displaystyle y_{i}} of the target f ( x i ) {displaystyle f(x_{i})} , in order to balance a measure of goodness of fit of f ^ ( x i ) {displaystyle {hat {f}}(x_{i})} to y i {displaystyle y_{i}} with a derivative based measure of the smoothness of f ^ ( x ) {displaystyle {hat {f}}(x)} . They provide a means for smoothing noisy x i , y i {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 x {displaystyle x} is a vector quantity. Let { x i , Y i : i = 1 , … , n } {displaystyle {x_{i},Y_{i}:i=1,dots ,n}} be a set of observations, modeled by the relation Y i = f ( x i ) + ϵ i {displaystyle Y_{i}=f(x_{i})+epsilon _{i}} where the ϵ i {displaystyle epsilon _{i}} are independent, zero mean random variables (usually assumed to have constant variance). The cubic smoothing spline estimate f ^ {displaystyle {hat {f}}} of the function f {displaystyle f} is defined to be the minimizer (over the class of twice differentiable functions) of