Spatial Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 170

Define A¼ (X,Z) and R¼ (X,(I-P X )Z), where P X ¼X(X t X) 1 X t is the projec-

tion matrix, and

X t X

0

M ¼ : ð 7 : 13 Þ

2 Q 1

Z t 0 ð I P X Þ Z þ ʻ

So, we can compute

I 0
0 A 1 DA 2

M 1 R t R ¼

; ð 7 : 14 Þ

where D¼ diag { d 1 , ... , d k }, and A 1 DA t 2 is obtained from the singular value decom-

1 Z t I P ð Þ Z. The degrees of freedom of the
linear mixed model in Eq. ( 7.11 ) are represented by tr (RM 1 R t ). The factors d k can
be interpreted as fractional degrees of freedom. They rapidly decay to zero for
many linear mixed models of interest. As a practical rule, we can retain only the first
r columns that together account for much less than one degree of freedom.
Using all the available q+K variables, the new balancing variables are defined
by the N ( q+K ) matrix

2 Q 1

position of Z t ð I P X Þ Z þ ʻ

I 0
0 A 1 D

B 1 ¼ R ; ð 7 : 15 Þ

where the first q columns come from X.
These balancing variables can be geographical coordinates of units or a trans-
formation (i.e., linear, quadratic or splines). The method considers a fixed G -order
local polynomial between knots, under the constraint that the G 1 derivatives at
the intersections of the polynomials are equivalent. The main feature of penalized
balanced sampling is that it considers only a set of covariates and thus, in a spatial
context, it can only manage the presence of a spatial trend. However, selection
strategies based on some moments of the auxiliaries do not use the concept of
distance, and only use linear or nonlinear spatial trends. Distance is a basic tool that
describes the spatial distribution of the sample units, and leads to the intuitive
criterion that units that are close should seldom simultaneously appear in the
sample. This gap reduces the possibility of balanced sampling considering the
complex nature of the spatial structure, which could lead to an efficiency gain in
the selection procedure.
For this reason, Grafstr¨m and Till ´ ( 2013 ) recently introduced a basic change in
the procedure used in the flight phase that selects units respecting a given vector of
first-order inclusion probabilities. This new criterion was inspired by the local
pivotal method (see Sect. 7.6 ). Thus, the resulting algorithm takes advantage of
both the trend and proximity of the population units.
The following R code is an implementation of the transformation of the matrix
X in the matrix B 1 according to Eq. ( 7.15 ). In the first stage, we select the knots