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Q: For which data is it appropriate to use state-space modeling and Kalman filtering instead of smoothing splines and vice versa? Is there some equivalence relationship between the two?

I'm trying to get some high-level understanding of how these methods fit together. I browsed through Johnstone's new Gaussian Estimation: Sequence and Multiresolution Models. It surprised be that there was not one mention of state-space models and Kalman filtering. Why wouldn't that be in there? Isn't that the most standard tool for these sort of problems? The focus, instead, was on smoothing splines and wavelet thresholding. I'm now very confused.

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Regarding your question on the equivalence, fitting a univariate local linear trend model using a Kalman filter is equivalent to fitting a cubic spline; see Time Series Analysis by State Space Methods, Section 3.11 for instance.

I think you are right in pointing that the Kalman filter and smoother are sometimes neglected when they could be put to good use. In particular, I find that the Kalman smoother is much more convenient with irregularly spaced and/or missing data.

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  • $\begingroup$ @Tusell. Thx for the reply. I'm going to have to check out the book you pointed out. Not easy to find books that put it all together like that. $\endgroup$
    – lowndrul
    Sep 5, 2011 at 0:09
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    $\begingroup$ State-space based algorithms are very powerful to cope with splines with a scalar argument, or even with tensor product splines. An example is in my answer to this question on smoothing with derivatives. Due to the non-stationary processes involved - known as "Intrinsic Random Functions"- a diffuse initial state is generally required, as now implemented in several toolboxes or packages devoted to SS and Kalman. $\endgroup$
    – Yves
    Oct 10, 2019 at 6:37
  • $\begingroup$ @Yves, a very thorough answer indeed, that I saw a couple of days ago and quickly bookmarked. Thank you anyway for bringing this to my and everyone else's attention. $\endgroup$
    – F. Tusell
    Oct 10, 2019 at 9:13

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