Geoscience Reference
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stochastically independent. This is because element concentration values and reali-
zations from other spatial variables generally are autocorrelated in that values for
samples that are close together resemble one another more closely than values that are
from farther apart. Random variables with autocorrelation properties have a history of
being studied in time series analysis (Bloomfield
2000
) of which the statistical theory
had become well established in the 1920s and 1930s. Filtering in statistical theory of
communication (Lee
1960
) can be regarded as a form of kriging. There is also a
close connection between splines (Eubank
1988
) and kriging. Kriging differs from
time series analysis in that the observation points are located in three-dimensional
space instead of along a line. Nevertheless, autocorrelation commonly is studied
using variograms or correlograms for sampling points with regular spacing along
lines. Although Matheron (
1965
) was most prominent in developing spatial statistics
(or “geostatistics” as he preferred to call it), others such as Mat´rn (
1981
)inforestry
and Gandin (
1965
) in meteorology, to some extent independently, had advanced the
idea of “regionalized random variables” in the 1950s and 1960s as well.
Cressie (
1991
) reasons as follows to explain why spatial autocorrelation must
be considered: Suppose
Z
(1),
,
Z
(
n
) are independent and identically distributed
(i.i.d.) observations drawn from a Gaussian distribution with unknown mean
...
μ
and
2
, then the minimum-variance unbiased estimator of
known variance
˃
μ
is equal to
the sum of the
Z
(
i
) values (
i
¼
1,
...
,
n
) divided by
n
,or
M
¼
{
Σ
Z(
i
)}/
n
. The
2
/
n
. It can be used to construct
estimator
M
is Gaussian with mean
μ
and variance
˃
)/
n
½
}. To-day
many people are familiar with two-sided 95 % confidence intervals on sample
means that have been estimated by means of this method. In practice,
a two-sided 95 % confidence interval for
μ
, which is {
M
(1.96•
˃
2
also is
unknown and is estimated by taking the sum of squares of the differences between
the
Z
(
i
) and their average, and dividing this sum by (
n
˃
1). However, this standard
statistical approach loses its validity when data are not independent but positively
correlated. Normally, the extent of positive correlation decreases with distance
between locations of points in 2-D or 3-D at which two
Z
(
i
) values were measured.
Suppose that this distance between observation points is kept the same and all pairs
of values
Z
(
i
) and
Z
(
i
1) are positively correlated with correlation coefficient
ˁ>
0. Then the variance of
M
is larger than
˃
2
/
n
. This estimator must be multiplied
by a factor
c
that can be estimated although some assumption on the nature of
the theoretical autocorrelation function is required. Consequently, the two-sided
95 % confidence interval for
(
c
½
·1.96·
)/
n
½
} and this is wider than
μ
,is{
M
˃
{
M
(1.96 ·
˃
n
/
c
, which is less than
n
, as the “equivalent number of
independent observations”. To provide an illustrative example: suppose chemical
element concentration values for
n
20 successive drill-core samples (equally
spaced along a straight line) are positively correlated with
¼
0.5 and that the
space series is equivalent to a time series with the first-order Markov property.
Then,
c
ˁ
¼
2.95 and
n
0
¼
6.78. It means that the 20 values are equivalent to about
7 independent observations and that a 95 % confidence interval neglecting the
positive spatial correlation would be 0.58 times too narrow. Obviously, in this
situation it would be misleading to set number of degrees of freedom equal to 19.
The concept of degrees of freedom then has lost its meaning entirely unless one
would base it on the 7 (instead of 20) equivalent “independent” values.
¼
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