Spatial Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 163

necessarily exist. Thus, in practice, the aim is to find a design that satisfies Eq. ( 7.6 )
with a certain degree of approximation.
Note that many sampling designs can be viewed as particular cases of balanced
sampling. For example, stratified sampling can also be defined as a design respect-
ing the constraint

X

d k ˆ kh ¼ X

k2U ˆ kh ¼ N h , 8 h ¼ 1,

... , H ð 7 : 7 Þ

k2s

where ˆ kh s are indicator variables equal to 1 if the unit k is in the stratum h, and

0 otherwise. This use of indicator variables to constrain the codes of a qualitative
variable is quite interesting; carefully uses of balanced sampling are valuable to
simplify complex problems, such as stratifying two or more coding variables only
on the marginals of the multi-way table, without necessarily using all the cross-
classified codes. This could be a solution when a large number of strata are needed
to define all the estimation domains, a problem that is often encountered in business
surveys.
The constraint in Eq. ( 7.6 ) is called the balancing equation , and it can be viewed
as a restriction on

. In fact, only samples that satisfy the balancing equations have
a strictly positive probability (Till ´ 2006 , Chap. 8).
This background can be reasonably accepted, even in the design-based approach.
It led to the cube method by Deville and Till ´ ( 2004 ), which was later improved by
Chauvet and Till ´ ( 2006 ).
The name of the algorithm comes from the idea that every sample can be seen as
the coordinates of a vertex of the hypercube,

ʩ

ʛ ¼ [0,1] N , in multi-dimensional

N .
Define the q N matrix

space, ℝ

0
@

1
A

x 11
π 1

x N 1
π N
⋮⋱⋮ ... ⋮

x k 1
π k

...

x 1 j
π 1

x Nj
π N
⋮⋱⋮⋱⋮

x kj
π k

...

A ¼ a 1 a k a N ð Þ ¼ : ð 7 : 8 Þ

x 1 q
π 1

x kq
π k

x Nq
π N

Then, A is a vector of the first-order inclusion probabilities, and t x is

a vector of the known population totals of the auxiliary variables. Moreover, if I s is
a vector with the sample membership indicator (see Eq. ( 1.6 ) of Sect. 1.2 ) as the
random sample s , we also have AI s ¼ t x , wheret x is a vector of HT estimates of the
totals of the auxiliary variables. Geometrically, Eq. ( 7.8 ) defines a subspace

π ¼ t x , where π

ʓ in

N , which is characterized in matrix notation as

ℝ