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.{{citeQ|Q134457353}}

It was first introduced by Georg BohlmannBohlmann, G., 1899. Ein ausgleichungsproblem. Nachrichten Gesellschaft Wissenschaften Gottingen, Math.-Phys. Klasse 260–271. (for order 1). E.T. Whittaker independently proposed the same idea in 1923{{citeQ|Q127739868}} (for order 3).

Robert Henderson contributed to the topic by his two publications in 1924Henderson, R., 1924. A new method of graduation, Trans. Actuarial Soc. Amer. 25, 29–40. and 1925.Henderson, R., 1925. Further remarks on graduation, Trans. Actuarial Soc. Amer. 26, 52–57.

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 $y_i$ , $i=1, \ldots, n$ , 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:

: $\hat{x} = \operatorname{argmin}_{x_1, \ldots, x_n} \sum_{i=1}^n (y_i - x_i)^2 + \lambda \sum_{i=1}^{n-p} (\Delta^p x_i)^2 \,,$

with penalty parameter $\lambda$ and difference operator $\Delta$ :

: $\begin{align}
\Delta x_i &= x_{i+1} - x_i \\
\Delta^2 x_i &= \Delta(\Delta x_i) = x_{i+2} - 2 x_{i+1} + x_i
\end{align}$

and so on.

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

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

Binomial Data

Henderson{{cite conference|last=Henderson|first=R.|date=1924|title=Some points in the general theory of graduation|book-title=Proceedings of the International Mathematical Congress held in Toronto, August 11-16, 1924|volume=2|pages=815-820|url=https://archive.org/details/proceedings-of-the-international-mathematical-congress-1924-vol-2/page/815/mode/2up}} formulates the smoothing problem for binomial data, using the logarithm of binomial probabilities in place of the error sum-of-squares,

: $\log Q = \sum_x \left\{ \theta_x \log q_x + (E_x-\theta_x) \log ( 1-q_x ) \right\}$

where $E_x$ is the number of binary observations made at $x$ ; $q_x$ is the probability that the event of interest is realized, and $\theta_x$ is the number of instances in which the event is realized.

Henderson applies the logistic transformation to the probabilities $q_x$ for the penalty term,

: $\lambda_x = \log \frac{q_x}{1-q_x}; \mbox{ and } y=\sum_x \left ( \Delta^3 \lambda_x \right )^2.$

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

: $\log P = f(y)$

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

: $\log P + \log Q,$

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

References

Category:Signal processing filter

Whittaker–Henderson smoothing

Mathematical Formulation

Properties

Binomial Data

Further reading

References