Re: units as a model term

# Re: units as a model term

```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

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
>

<><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>
Arthur Gilmour PhD                    email: Arthur.Gilmour@agric.nsw.gov.au
Senior Research Scientist (Biometrics)                 fax: <61> 2 6391 3899
NSW Agriculture                                             <61> 2 6391 3922
Orange Agricultural Institute               telephone work: <61> 2 6391 3815
Forest Rd, ORANGE, 2800, AUSTRALIA                    home: <61> 2 6362 0046

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