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 formula_1, formula_2, of equidistant steps, e.g. a time series with constant intervals, the Whittaker–Henderson smoothing of order formula_3 is the solution to the following penalized least squares problem:
formula_4
with penalty parameter formula_5 and difference operator formula_6:
formula_7
and so on.
For formula_8, the solution converges to a polynomial of degree formula_9. For formula_10, the solution converges to the observations formula_11.
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.
Binomial Data.
Henderson formulates the smoothing problem for binomial data, using the logarithm of binomial probabilities in place of the error sum-of-squares,
formula_18
where formula_19 is the number of binary observations made at formula_20; formula_21 is the probability that the event of interest is realized, and formula_22 is the number of instances in which the event is realized.
Henderson applies the logistic transformation to the probabilities formula_21 for the penalty term,
formula_24
Then, Henderson places an "a priori" probability on a set of graduated values,
formula_25
for a decreasing function formula_26 (formula_27 for the usual quadratic penalty). Henderson's penalized criterion is
formula_28
which is a modification of the Whittaker-Henderson smoothing criterion for binomial data.