Re: AI-GEOSTATS: Intrinsic Random Functions -- what it mean for lambda to annihilate a polynomial?

[Isobel Clark]

Olumide
   
  I would think what they mean is that each order of polynomial has to be 
balanced between the 'drift' at the actual estimated point and the weighted 
average of the samples which proovides the estimator. For this you have to 
introduce an extra lamda and an extra equation on the kriging system which 
guarantees the unbiassedness of the estimate.
   
  At least, that is what happens in Universal Kriging. What is annihilated is 
any possible bias due to the order k. 
   
  I do not know why lamda is referred to as a discrete measure.
   
  Isobel
  http://www.kriging.com/courses

Olumide [EMAIL PROTECTED] wrote:
  Hello -

I've made some progress understanding what intrinsic random functions 
are, and what increments are in that regard. The next question that's 
still puzzling me is the question of what the discrete measure lambda 
and the annihilation of polynomials.

Quote from Geostatistics Modeling Uncertainty by Chiles and Delfiner 
page 238:

Definition: a discrete measure lambda is allowable at the order k if it 
annihilates polynomials of degree less than or equal to k

Questions:
1. what does it mean for lambda to annihilate a polynomial
2. why the need to annihilate those poor polynomials (what have they 
done wrong? ;-) )

Thanks,

- Olumide
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Re: AI-GEOSTATS: Intrinsic Random Functions -- what it mean for lambda to annihilate a polynomial?

[Isobel Clark]

Olumide
   
  I recommend you work through our free tutorial on kriging with trend. It 
discusses Universal Kriging rather the IRF-K but I think it will answer your 
question better than I can do in a short email.
   
  Yes you can annihilate the trend by making the weighted average of the trend 
equal zero but it makes more sense to make the trend from the samples honour 
the trend at the point being estimated.
   
  Isobel
  http://www.kriging.com

Olumide [EMAIL PROTECTED] wrote:
  Isobel Clark wrote:
 I would think what they mean is that each order of polynomial has to be 
 balanced between the 'drift' at the actual estimated point and the 
 weighted average of the samples which proovides the estimator. For this 
 you have to introduce an extra lamda and an extra equation on the 
 kriging system which guarantees the unbiassedness of the estimate.

Sorry but I don't understand what you mean by this.

I've been doing some more thinking and reading and here's my GUESS -- 
please correct me if I'm wrong:

Suppose a RF Z(x) can be modeled as:

Z(x) = m(x) + Y(x)

where m(x) is the drift which is modeled as weighted sum of 
polynomials of order up to k (e.g. if k = 2, drift is w[0] + w[1].x + 
w[2].y + w[3].xy + w[4].x² + w[5].y²) and Y(x) a fluctuation or residual 
about this drift. Removing this drift would require somehow finding 
values for the weights such that the weighted sum *somehow* becomes zero 
thus annihilating the *effect* of the polynomials.

???