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 , , of equidistant steps, e.g. a time series with constant intervals, the Whittaker–Henderson smoothing of order is the solution to the following penalized least squares problem:

with penalty parameter and difference operator :

and so on.

For , the solution converges to a polynomial of degree . For , the solution converges to the observations .

Properties

  • Reversing just reverses the solution .
  • The first moments of the data are preserved, i.e., the j-th momentum for .
  • Polynomials of degree are unaffected by the smoothing.

Further reading

References

  1. ^ https://www.jstor.org/stable/41139599
  2. ^ Bohlmann, G., 1899. Ein ausgleichungsproblem. Nachrichten Gesellschaft Wissenschaften Gottingen, Math.-Phys. Klasse 260–271.
  3. ^ https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/on-a-new-method-of-graduation/744E6CBD93804DA4DF7CAC50507FA7BB
  4. ^ Henderson, R., 1924. A new method of graduation, Trans. Actuarial Soc. Amer. 25, 29–40.
  5. ^ Henderson, R., 1925. Further remarks on graduation, Trans. Actuarial Soc. Amer. 26, 52–57.


Allikas: https://en.wikipedia.org/wiki/Whittaker%E2%80%93Henderson_smoothing
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