Hi all,
A few more references to the Covariance Vs. Semi-
variograme discussion:
To support Semi-variograme: Cressie N.A.C. (1993)
Statistics for spatial data. New York Wiley. ( Page 70-
73) I believe that the original discussion appears in:
Cressie A.C. Noel. and Grondona O. Martin (1992); A
comparison of Variogram Estimation with Covariogram
Estimation, In The art of statistical Science Edited by K.V.
Mardia Jhon pp:191-208, Wiley & Sons Ltd.
Cressie proves that semi- variogram estimation is to be
preferred over covariogram estimation; the main reasons
for that are:
1.In the Kriging process where we estimate the mean of the
process and then predict the random process both the
variogram estimator and covariogram estimator are biased.
However the variogram bias is of smaller order.
2. If our data has trend contamination then it has
"disastrous effect" on attempts to estimate the covariogram
while on the variogram it has a "small upward shift".
There is more to that; check on the book..
To support Covariance: Barry and Pace (1997) "Kriging
with large data sets using sparse matrix techniques"
Communications in statistics simulation and computation
Vol 26 (2) pp 619-629 exploit the sparseness of covariance
matrix - with stationary models we have zeros for points
outside the range - and they were able to dramatically lower
the time and storage cost of kriging.
Since the covariance matrix is a symmetric positive definite
matrix, we can use the Cholesky factorization for its
inversion. If A is n-by-n, the computational complexity of
Cholesky(A) is O(n^3), but the complexity of the
subsequent inversion solutions is only O(n^2).
With Marco's suggestion of Matheron equations it seems
that one can use Cholesky factorization even with semi-
variogram matrix.
Thanks for interesting discussions.
Yaron Felus
The Ohio-State University
http://felus.mps.ohio-state.edu/yf/
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