Dear Nick,
I have lost the connection but does this help?
The standard AR1 process is defined as
E_i = r.E_{i-1} + c.Ri
the ith error is r times the previous error + c times
a new independent error. The term 'nugget' refers to
an extra independent error term - additional to the above.
So if V = Var(E) = variance of the process
it equals r^2 V + c^2 (for R_i independent standard normal)
rearranging V = c^2 / (1-r^2)
where c^2 is the innovation variance and r is the autocorrelation.
Check this with a time series book but I think I have it correct.
It seems you are calling c^2 the 'nugget' variance but the nugget
variance is an extra (additional) variance term.
So, without a nugget variance, the covariance of a series of points
is given by V[ 1 r r^2 r^3 r^4 ...]
If we add another Nugget variance say N
the covariance becomes
V+N Vr Vr^2 Vr^3 Vr^4 ...
So the lag 1 correlation actually becomes
Vr/(V+N)
the lag2 correlation becomes Vr^2 /(V+N)
So if V=2, r=.5, N=3
the Covariance sequence is
5 1 .5 .25 .125 ...
This can only approximately be equated to a simple AR
sequence with V=5 and r=0.2 giving
5 1 .2 .04 .008
What usually happens in fitting models in ASREML is that we
start without a nugget variance. Then we might get V=5, r=.2
Then we add a nugget variance and get maybe N=3, V=2, r=0.5
This typically changes all variance parameters because they are expressed
relative to V, not V+N, even though the difference in the model
is largely in the covariance at lags 2 and 3.
I hope this makes it a little clearer.
Arthur
> Date: Tue, 15 Dec 1998 17:28:50 +0800
> X-Sender: ngalwey@cyllene.uwa.edu.au
> Mime-Version: 1.0
> To: Arthur Gilmour <gilmoua@ornsun.agric.nsw.gov.au>
> From: "N.W. Galwey" <ngalwey@cyllene.uwa.edu.au>
> Subject: Re: units as a model term
>
> Dear Arthur,
>
> I think we're going to have to discuss this some more face to face
> before I understand it fully. I want the units term to give me the "new
> bit" of the error for each observation (i.e. the bit that isn't correlated
> with observations earlier in the series), which I'm identifying, perhaps
> incorrectly, with the nugget. Yes, I have a feeling that that is incorrect.
>
> Hmm.
>
> Best wishes,
>
> Nick
> _____________________________________________________________________
> N.W. Galwey,
> Faculty of Agriculture,
> University of Western Australia,
> Nedlands, WA 6709, Australia.
>
> Tel.: +61 9 380 1959 (direct line)
> +61 9 380 2554 (switchboard)
> Fax: +61 9 380 1108
>
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Arthur Gilmour PhD email: Arthur.Gilmour@agric.nsw.gov.au
Senior Research Scientist (Biometrics) fax: <61> 2 6391 3899
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