Dear Sara:
Sorry, but the "non ergodic" variogram is an artifact!
If you do not believe to me, see the comments about my paper in NACOG 1996 or in
Geostatistics, Volume 8, No.2.
The Mathematical proof is in my paper titled "Acerca del variograma no ergódico", in
Spanish (difficult to get, if you wish I can
send a copy for you).
I think that is more easy to see an example (in my paper you have more examples):
Let a line with data (the data is very regular and has a trend) sampled at regular
intervals of 1. Tha data are (20 data):
1, 1, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 10, 10, 11, 12
Compute by hand or by your software the non ergodic and the classical variogram (the
classical variogram is also non
ergodic!):
Conclusions:
The behavior of the non ergodic variogram, near the origin is linear, and, for the
classical variogram is parabolic.
The non ergodic variogram has a range and a sill, and, the classical variogram always
grows.
I attach a little Qbasic (or QuickBasic) program you can run.
Best regards,
Marco Alfaro.
Listing of the program:
n = 20
DIM z(n)
m = 0 ' mean
v = 0 ' variance
FOR i = 1 TO n
READ z(i)
m = m + z(i)
v = v + z(i) * z(i)
NEXT i
m = m / n ' mean
' the variance is = the non ergodic variogram in the origin
v = v / n - m * m
CLS : SCREEN 12
WINDOW (-10, -10)-(30, 80)
LINE (0, 0)-(20, 65), 8, B
LOCATE 3, 10: PRINT "In red, classical variogram, in green, non ergodic variogram"
FOR k = 0 TO n - 1
cov = 0 ' the non ergodic variogram.
head = 0
tail = 0
gama = 0 ' the classical variogram.
FOR i = 1 TO n - k
hh = z(i)
tt = z(i + k)
head = head + hh
tail = tail + tt
cov = cov + hh * tt
gama = gama + (hh - tt) * (hh - tt)
NEXT i
head = head / (n - k)
tail = tail / (n - k)
cov = cov / (n - k) - head * tail
gama = .5 * gama / (n - k)
CIRCLE (k, v - cov), .1, 2
PAINT (k, v - cov), 2
CIRCLE (k, gama), .1, 4
PAINT (k, gama), 4
NEXT k
a$ = INPUT$(1)
END
DATA 1,1,2,3,3,3,4,4,5,5,6,6,7,7,8,9,10,10,11,12
Sara Kustron wrote:
I have been using Variowin for my variogram analysis. My
covariograms are much better behaved compared to semivariograms, and
I would like to use the covariance to estimate the semivariance.
I understand that under the assumption of strict stationarity, which
is harder to satisfy than the intrinsic hypothesis, semivariance and
covariance are essentially equivalent. Since I have a non-stationary
process I would have expected that my covariograms to be just as bad
as my semivariograms, but this is not the case.
I've read that Variowin uses "non-ergotic"
covariance. Exactly what is this and how does it differ from regular
old covariance? I just want to make sure I understand the
assumptions of using non-ergotic covariance to estimate model
parameters.
Thanks!
Sara Kustron
Boston University Department of Geography
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Sara Kustron wrote:
> I have been using Variowin for my variogram analysis. My
> covariograms are much better behaved compared to semivariograms, and
> I would like to use the covariance to estimate the semivariance.
>
> I understand that under the assumption of strict stationarity, which
> is harder to satisfy than the intrinsic hypothesis, semivariance and
> covariance are essentially equivalent. Since I have a non-stationary
> process I would have expected that my covariograms to be just as bad
> as my semivariograms, but this is not the case.
>
> I've read that Variowin uses "non-ergotic"
> covariance. Exactly what is this and how does it differ from regular
> old covariance? I just want to make sure I understand the
> assumptions of using non-ergotic covariance to estimate model
> parameters.
>
> Thanks!
> Sara Kustron
> Boston University Department of Geography
>
> --
> * To post a message to the list, send it to [EMAIL PROTECTED]
> * As a general service to the users, please remember to post a summary of any useful
>responses to your questions.
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