# Smoothing splines via local linear trends

I was looking over these slides, and on slide 24 he has the following "neat fact" The posterior mean of the local linear trend model is a smoothing spline.

I haven't been able to find any other information on this statement. Does anyone have a formal proof of this fact?

I would also be interested in knowing:

• What order (degree) is the spline?
• Can one use this method to produce higher order splines?
• While the result is unfortunately not shown or even outlined in the answer, Kalman filter vs. smoothing splines at least covers that it's cubic. Aug 29, 2017 at 0:53
• @Glen_b thanks, this seems relevant. Unfortunately I don't have the referenced book.
– Paul
Aug 29, 2017 at 15:33
• I have the book but can't access it (it's packed away in a box I can't get to for some time while my house is repaired). Checking google books (which showed me several pages -- enough to see the details -- quite happily) it is detailed in Section 3.9 of the second edition and looking at the Amazon link of the other post that is 3.11 of 1st edition (though the preceding section would be relevant). You may be able to get the info via google books. The main reference would be Wecker and Ansley 1983 ... ctd Aug 30, 2017 at 0:12
• ctd... (Wecker, W.E. and C.F. Ansley (1983) "The signal extraction approach to non- linear regression and spline smoothing," J. Amer. Statist. Assoc.,78, 81–89.) though there are a bunch of others. You may want to take a look at Hyndman et al 2004 "Local linear forecasts using cubic smoothing splines" which is easily accessible and has a number of the Durbin&Koopman references Aug 30, 2017 at 0:14

I'm the author of the slides.

It turns out the local linear trend model is closely related to a cubic spline, but the actual cubic spline result comes from a slightly different model (so there was an error in my slides, apologies).

The local linear trend model states \begin{align} y[t] &= \mu[t] + \epsilon[t] \\ \mu[t+1] &= \mu[t] + \delta[t] + \eta_0[t] \\ \delta[t+1] &= \delta[t] + \eta_1[t] \\ \end{align} The spline result comes from a model \begin{align} y[t] &= \mu[t] + \epsilon[t] \\ \Delta^2 \mu[t] &= \eta[t] \end{align} where $\Delta$ is the first difference operator. The last line is equivalent to saying that

\begin{align} \mu[t+1] = \mu[t] + \Delta \mu[t] + \eta[t] \end{align} This is awfully close but not the same as the local linear trend. The result is the discrete time equivalent of a result by Kohn and Ansley. You can find it in section 3.11 of Durbin and Koopman (first edition) or section 3.9 of their second edition. I believe that section is available in Google books previews. The reference in DK was Kohn, Ansley, Wong (1992), which you can grab if you have access to JSTOR.

For applied work, you might consider the "integrated random walk model" which is the local linear trend with the variance of eta0 set to zero. It produces smoother posterior means than the local linear trend model.

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– Sycorax
Apr 3, 2018 at 18:18