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.^[1]

It was first introduced by Georg Bohlmann^[2] (for order 1). E.T. Whittaker independently proposed the same idea in 1923^[3] (for order 3). Robert Henderson contributed to the topic by his two publications in 1924^[4] and 1925.^[5] 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 ${\displaystyle y_{i}}$, ${\displaystyle i=1,\ldots ,n}$, of equidistant steps, e.g. a time series with constant intervals, the Whittaker–Henderson smoothing of order ${\displaystyle p}$ is the solution to the following penalized least squares problem:

${\displaystyle {\hat {x}}=\operatorname {argmin} _{x_{1},\ldots ,x_{n}}\sum _{i=1}^{n}(y_{i}-x_{i})^{2}+\lambda \sum _{i=1}^{n-p}(\Delta ^{p}x_{i})^{2}\,,}$

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

${\displaystyle {\begin{aligned}\Delta x_{i}&=x_{i+1}-x_{i}\\\Delta ^{2}x_{i}&=\Delta (\Delta x_{i})=x_{i+2}-2x_{i+1}+x_{i}\end{aligned}}}$

and so on.

For ${\displaystyle \lambda \rightarrow \infty }$, the solution converges to a polynomial of degree ${\displaystyle p-1}$. For ${\displaystyle \lambda \rightarrow 0}$, the solution converges to the observations ${\displaystyle y}$.

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 ${\displaystyle y}$ just reverses the solution ${\displaystyle {\hat {x}}}$.
• The first ${\displaystyle p}$ moments of the data are preserved, i.e., the j-th momentum ${\displaystyle \sum _{i}y_{i}^{j}=\sum _{i}{\hat {x}}_{i}^{j}}$ for ${\displaystyle j=0\ldots p}$.
• Polynomials of degree ${\displaystyle p-1}$ are unaffected by the smoothing.

Binomial Data

Henderson^[6] formulates the smoothing problem for binomial data, using the logarithm of binomial probabilities in place of the error sum-of-squares,

${\displaystyle \log Q=\sum _{x}\left\{\theta _{x}\log q_{x}+(E_{x}-\theta _{x})\log(1-q_{x})\right\}}$

where ${\displaystyle E_{x}}$ is the number of binary observations made at ${\displaystyle x}$; ${\displaystyle q_{x}}$ is the probability that the event of interest is realized, and ${\displaystyle \theta _{x}}$ is the number of instances in which the event is realized.

Henderson applies the logistic transformation to the probabilities ${\displaystyle q_{x}}$ for the penalty term,

${\displaystyle \lambda _{x}=\log {\frac {q_{x}}{1-q_{x}}};{\mbox{ and }}y=\sum _{x}\left(\Delta ^{3}\lambda _{x}\right)^{2}.}$

Then, Henderson places an a priori probability on a set of graduated values,

${\displaystyle \log P=f(y)}$

for a decreasing function ${\displaystyle f(y)}$ (${\displaystyle f(y)=-y}$ for the usual quadratic penalty). Henderson's penalized criterion is

${\displaystyle \log P+\log Q,}$

which is a modification of the Whittaker-Henderson smoothing criterion for binomial data.

Further reading

• Paul H. C. Eilers (1 July 2003). "A perfect smoother". Analytical Chemistry. 75 (14): 3631–3636. doi:10.1021/AC034173T. ISSN 0003-2700. PMID 14570219. Wikidata Q79189954.
• Frederick Macaulay (1931). "The Whittaker-Henderson Method of Graduation." Chapter VI of The Smoothing of Time Series^[7]
• Weinert, Howard L. (October 15, 2007). "Efficient computation for Whittaker–Henderson smoothing". Computational Statistics & Data Analysis. 52 (2). Elsevier: 959–974. doi:10.1016/j.csda.2006.11.038.