Geoscience Reference
In-Depth Information
where
h
is the lag distance that separates pairs of points;
Z(x)
is bird diversity at lo-
cation
x
, and
Z(x + h)
is bird diversity at location
x + h
;
n(h)
is the number of pairs
separated by lag distance h.
Kriging is estimated using weighted sums of adjacent sampled concentrations. The
weights depend on the correlation structure exhibited. The weights are determined
by minimizing estimated variance. In this context, kriging estimates (BLUE) are the
most accurate of all linear estimators. Accordingly, kriging estimates the value of the
random variable at unsampled location X
0
based on measured values in a linear form:
N
∑
(3)
()
=
()
Z
∗
x
0
λ
i
0
Zx
i
i
=
1
where Z*(
x
0
) is the estimated value at location
x
0
, λ
i0
is the estimation weight of
Z(x
i
),
x
i
is the location of sampling point for variable Z, and N is the number of the variable
Z involved in the estimation. Based on non-biased constraints and minimizing estima-
tion variance, estimated kriging variance can be presented as:
N
∑
(
)
(4)
2
kriging
σ
=
λ
i
0
γ
zz
x
i
−
x
0
+ μ
i
=
1
where μ is the Lagrange multiplier.
Conditional Latin Hypercube
The cLHS, which is based on the empirical distribution of original data, provides a
full coverage of range each variable by maximally stratifying the marginal distribution
and ensuring a good spread of sampling points (Minasny and McBratney, 2006). This
sampling procedure represents an optimization problem: given N sites with ancillary
data (Z), select n sample sites (n << N) such that the sampled sites z form a Latin hy-
percube. For continuous variables, each component of X (size, N × k) is divided into n
(sample size) equally probable strata based on their distributions, and x (size n × k) is
a sub-sample of X. The procedures of the cLHS algorithm (Minasny and McBratney,
2006) are follows.
1. Divide the quantile distribution of X into n strata, and calculate the quantile
distribution for each variable, q
i
j
,..., q
n+1
j
. Calculate the correlation matrix for Z
(C).
2. Pick n random samples from N; z (i=1,…, n) are the sampled sites. Calculate
the correlation matrix of x (T).
3. Calculate the objective function. The overall objective function is O = w
1
O
1
+
w
2
O
2
+ w
3
O
3
, where w is the weight given to each component of the objective
function. For general applications, w is set to 1 for all components of the objec-
tive function.
