Re: units as a model term
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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
               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.  
> Date: Tue, 15 Dec 1998 17:28:50 +0800
> X-Sender:
> Mime-Version: 1.0
> To: Arthur Gilmour <>
> From: "N.W. Galwey" <>
> 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:
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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