Robust Estimation of the
Variogram
Noel Cressie & Douglas M. Hawkins
Mathematical Geology, Vol. 12, No. 2, 1980
The Problem
• Published work in geostatistics
underplays the problems in estimating
variograms.
• It assumes that the underlying data are
normal.
• If the data are not normal, it assumes
they have been transformed to normality.
The Question
• What are we trying to achieve with
robust estimation of the variogram?
• We are trying to show that other
estimators whose means do not coincide
with the conventional estimator may not
be biased or undesirable.
Reality of Data
• Data are not normal.
• Data are contaminated by outliers which
results in a heavy-tailed distribution.
Problem Solving Approach
Seek estimators whose efficiency is high when
data are normal. (e = O)
Seek estimators which are reasonably
insensitive to departures from normality. (e > 0).
4 methods for estimating the
variogram were delineated
A. Define the quantities Yt = (Zt+h - Zt) 2. Then 2g(h) =E(Y),
and the estimation of the variogram becomes a problem
of estimating the expectation of the random variables Yt
which, under the normality of the Zt, follow scaled C2
distributions.
B. If the sequence Z t may be assumed to be stationary (or
to have been transformed initially to stationarity) then
E(Zt+h)= E(Zt). Thus 2g(h) = var (Zt+h - Zt) and so the
estimation of the variogram may be approached by
estimating the variance of the symmetric random
variables Zt+h - Zt
C. For many purposes (including the estimation of kriging weights), it
is enough to determine the set 3'(h) for all h up to an unknown
constant multiple. Now, provided var (Zt) is finite:
2g(h) = 2 var (Zt) [1 - r(h)]
where r(h) here refers to the autocorrelation at lag h. Thus an
estimate of the set r(h) also determines the set g(h) up to the
unknown variance.
D. Assuming for the moment that the data form a traverse, there
exists an autoregressive-moving average (ARMA) model
describing them
j(B)Z t = t~ (B)a t
where j and ff me power series in the backshift operator B and
the series a t is white noise. From j, y, and the variance of the at,
one may deduce g(h), and any method of estimating j and y can
yield g(h) up to an unknown constant multiple of the variance of
the at.
Preliminary Transformation
• We have powerful tools, algorithms, and
theory for the robust estimation of symmetry.
• Nonsymmetric distributions are not well
understood.
• To help solve the problem, look for a method
of estimating a center of symmetry.
• The class of power transformations was
chosen Yt = {(Zt+h - Zt)2} l because they are
broad and may be suitable.
Transformation
_
• Assume Y is normally distributed.
• The unbiased estimator for 2g(h) is:
_
•
Y4/(0.457 + 0.494n -1 + 0.045n -2).
• Estimators must be asymptotically normal and have
high asymptotic efficiency.
• This applies to Yt even though they are not
independent.
• For problems of practical interest the interdependence
between Yt and Ys will be negligible so there is
justification in treating the { Yt} as if they were an
independent, random sample.
The Study-Testing the Theory
• To test the theory 10 estimators were used:
–
–
–
–
–
–
–
–
–
–
The mean of the Yt values.
The median of the Yt values.
5% trimmed mean of Yt.
10% trimmed mean of Yt.
25% trimmed mean of Yt.
The Huber M-estimator.
The Tukey bisquare M-estimator.
The Hampel M-estimator.
The Andrews M-estimator
The conventional estimator
The Study
• 500 traverses
• Length of 50 units
• Model used: Zt = 0.6Zt-1 + Ut
and estimating g(1) using each estimator. The Ut were
independent identically distributed random variables
from the following distributions–
–
–
–
–
–
–
A: N(0, 1)
B: Standard Laplace: f(x) = ½ exp (- |x|)
C: N(o, 1) 5% contaminated with N(0, 9)
D: N(0, 1) 10% contaminated with N(0, 9)
E: N(0, 1) 20% contaminated with N(0, 9)
F: N(O, 1) 5% contaminated with N(0,100)
G: Real data from the Hartebeestfontein gold mine
Cases
“If one assumes the process to be normally distributed, then a
normal distribution for Zt and a "classical" geostatistical estimation
and kriging scheme may be deduced.”
“If, however, the process remains normal but the et process (white
noise) is assumed to have some heavier-tailed distribution, then
the Zt are not normal but exhibit occasional outliers. It is here that
the use of good robust estimators of the variogram is desirable.”
“These estimators, by reducing the sampling variability of estimates
of g(h), can provide a more stable estimation procedure for gw(h),
and any downward bias will merely imply that they underestimate
s2e while leaving the structure of gw (h) unaffected.”
“This determination in the nonnormal case may however have to be
done in an iterative way by successively computing estimates W t
and residuals Z t -W t and using these residuals to refine our
estimates of the distribution of the et, and, hence, improve our
estimates W t.”
(page 123)
Results
• The conventional estimator is the most stable
for the normal distribution.
• The conventional estimator is the least stable
for distributions with a heavy tail.
• The M-estimators consistently had the
smallest standard deviations for nonnormal
cases.
• The median performs poorly.
• The mean of Yt performs well.
• Tukey’s bisquare M-estimator was best for the
trimmed means.
Results
• “The M-estimators are the estimators of choice, having
quite good efficiency for normal data coupled with
stability for all the heavy-tailed distributions studied.”
• “Another very simple estimator, Y, the mean of the Yt
performs equally well for the distributions
contaminated with N(0, 9) data and, which is more
significant, for the real data.”
• “Y mean is not robust in that it goes to infinity if any Yt
does, however it does seem to be robust enough to
handle data which, while not normal, have outliers that
deviate by not more than six or seven standard
deviations.”
Conclusions
• The arithmetic mean of the fourth root of
(Zt + h- Zt)2 gives a reasonably robust,
stable estimate of the variogram.
• The orthodox M-estimators are
equally stable, and theoretically more
robust.
• Trimmed means and the median do not
perform well.