Whittaker–Henderson smoothing

Whittaker–Henderson smoothing or Whittaker–Henderson graduation is a digital filter that can be applied to a set of digital data points for the purpose of smoothing the data, that is, to increase the precision of the data without distorting the signal tendency. It was first introduced by Georg Bohlmann (for order 1). E.T. Whittaker independently proposed the same idea in 1923 (for order 3). Whittaker–Henderson smoothing can be seen as P-Splines of degree 0. The special case of order 2 also goes under the name Hodrick–Prescott filter.

Mathematical Formulation

For a signal $y_i$, $i=1,\dots ,n$, of equidistant steps, e.g. a time series with constant intervals, the Whittaker–Henderson smoothing of order $p$ is the solution to the following penalized least squares problem:

$x={\mathrm{argmin}}_{x_1,\dots ,x_n}\sum _i^n(y_i-x_i{)}^2+\lambda \sum _i^{n-p}(\mathrm{\Delta }{x}_i{)}^2,$

with penalty parameter $\lambda$ and difference operator $\mathrm{\Delta }$:

$\begin{array}{cc}\mathrm{\Delta }{x}_i& =x_{i+1}-x_i\\ {\mathrm{\Delta }}^2{x}_i& =\mathrm{\Delta }(\mathrm{\Delta }{x}_i)=x_{i+2}-2x_{i+1}+x_i\end{array}$

and so on.

References

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