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). Robert Henderson contributed to the topic by his two publications in 1924 and 1925. 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 {\displaystyle y_{i}} , i = 1 , … , n {\displaystyle i=1,\ldots ,n} , of equidistant steps, e.g. a time series with constant intervals, the Whittaker–Henderson smoothing of order p {\displaystyle p} is the solution to the following penalized least squares problem:
x ^ = argmin x 1 , … , x n ∑ i = 1 n ( y i − x i ) 2 + λ ∑ i = 1 n − p ( Δ p x i ) 2 , {\displaystyle {\hat {x}}=\operatorname {argmin} _{x_{1},\ldots ,x_{n}}\sum _{i=1}^{n}(y_{i}-x_{i})^{2}+\lambda \sum _{i=1}^{n-p}(\Delta ^{p}x_{i})^{2}\,,}
with penalty parameter λ {\displaystyle \lambda } and difference operator Δ {\displaystyle \Delta } :
Δ x i = x i + 1 − x i Δ 2 x i = Δ ( Δ x i ) = x i + 2 − 2 x i + 1 + x i {\displaystyle {\begin{aligned}\Delta x_{i}&=x_{i+1}-x_{i}\\\Delta ^{2}x_{i}&=\Delta (\Delta x_{i})=x_{i+2}-2x_{i+1}+x_{i}\end{aligned}}}
and so on. For λ → ∞ {\displaystyle \lambda \rightarrow \infty } , the solution converges to a polynomial of degree p − 1 {\displaystyle p-1} . For λ → 0 {\displaystyle \lambda \rightarrow 0} , the solution converges to the observations y {\displaystyle y} . The Whittaker-Henderson method is very similar to modern Smoothing spline methods; the latter use derivatives rather than differences of the smoothed values in the penalty term.
Properties Reversing y {\displaystyle y} just reverses the solution x ^ {\displaystyle {\hat {x}}} . The first p {\displaystyle p} moments of the data are preserved, i.e., the j-th momentum ∑ i y i j = ∑ i x ^ i j {\displaystyle \sum _{i}y_{i}^{j}=\sum _{i}{\hat {x}}_{i}^{j}} for j = 0 … p {\displaystyle j=0\ldots p} . Polynomials of degree p − 1 {\displaystyle p-1} are unaffected by the smoothing.
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