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aickley


Total Posts: 37
Joined: Oct 2008
 
Posted: 2016-09-15 20:40
I have always thought of (and have only heard people talking about) exponential smoothing as an ad hoc approach. It is actually not the case.

Developing on some earlier research Hyndman et al. show exponential smoothing can be seen as filtering via so-called innovations state-space models of the form



It is quite straightforward to show that such models reduce to exponential smoothing. Clearly,



so that



Hence the forecast equals



exponentially-weighted past observations plus a constant times initial state.

The material difference with conventional state-space models is of course that we have essentially the same error both in transition and measurement equations (the immaterial difference is that measurement equations of conventional state-space are usually written in terms of the current state and not the previous one).

Counter-intuitively, Hyndman and co show that the class of innovations state-space models includes the conventional ones. The proof is rather indirect (they first show that a conventional state-space model can be written as ARIMA model and then show that every ARIMA model can be written as an innovations model) and I did not read it yet.

P.S. The book is online if one knows where to look.

ronin


Total Posts: 183
Joined: May 2006
 
Posted: 2016-09-19 14:38
That is precisely the reason why people don't like exponential smoothing - it depends on the initial state.

I.e. two people could be doing exactly the same exponential smoothing on the exact same time series, but have started the smoothing at different points in time - their estimates could be completely different.

Whereas even a simple moving average is invariant with respect to starting time / initial state.


"People say nothing's impossible, but I do nothing every day" --Winnie The Pooh

tbrown122387


Total Posts: 34
Joined: Feb 2010
 
Posted: 2016-10-25 23:45
You can prove it by extending the state space.

Another potential problem: poor fit.

tbrown122387


Total Posts: 34
Joined: Feb 2010
 
Posted: 2017-01-09 18:49
Re-reading this now. You say, "[c]ounter-intuitively, Hyndman and co show that the class of innovations state-space models includes the conventional ones." <- I think you have this backwards. Either that or we are not talking about the same thing when we say "state space models."

katastrofa


Total Posts: 356
Joined: Jul 2008
 
Posted: 2017-01-17 14:30
If you're starting the smoothing at different times you're making different assumptions about your data and the noise, so it's not "exactly the same".

I think you just need to be careful about your assumptions, as always (and also in the case of a simple moving average).
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