Essential Statistical Concepts, Definitions, and Terminology - Sampling Spatial Units for Agricultural Surveys - page 24

fy z ðÞy z j

¼ fy z ðÞy z j , z j 2 NðÞ , j ¼

6 ¼ i

8
<

9
=
; ;

" # 2

y z ðÞʼ i X
i6 ¼ j

1 = 2 exp

c ij y z j ʼ j

1
2

ð 1 : 45 Þ

¼ 2 π˃

i

: ˃

i

where ʼ i ¼ Ey z ðð Þ , and c ij denote spatial dependence parameters that are only

non-zero if z j 2 NðÞ . From these definitions it follows that
Ey z ðÞy z j , z j 2 NðÞ , j

¼ ʼ i þ X
i6 ¼ j

c ij y z j ʼ j

6 ¼ i ð 1 : 46 Þ

and

Var y z ðÞy z j , z j 2 NðÞ , j

¼ ˃

2

6 ¼ i

i :

ð 1 : 47 Þ

To properly perform the estimation and inference, we need to guarantee the
existence of the joint distribution. Under the Hammersley-Clifford conditions,
Besag ( 1974 ) showed that if we assume that the conditional density functions are
normal and the conditional means and variances are defined by Eqs. ( 1.46 ) and
( 1.47 ), these distributions generate a joint multivariate Gaussian distribution with
mean ʼ ¼ ʼ 1 ð ʼ 2 ... ʼ n Þ and variance

Þ 1 ʣ CAR ¼ I C ð ʣ; ð 1 : 48 Þ

, andC ¼ c ij . To ensure that

2 2 n

where ʣ ¼ diag ˃

1 ; ˃ 2 ; ...; ˃

ʣ CAR is symmetric, it

˃

j c ij ¼ ˃ i c ji . For other specifications of the CAR model see

is necessary to set

Besag et al. ( 1991 ).
If a researcher analyzing regional data does not give importance to the Markov
property, one possible alternative to the CAR approach is represented by the
simultaneous approach to random field model specification (i.e., the SAR model).
Let

2 I

(z i ) is the variable associated with site z i . A random
field is said to be Gaussian SAR (Whittle 1954 )if
y z ðÞ ¼ ʼ i þ X
i6 ¼ j

ε / N 0, ð ˃ Þ , where ʵ

b ij y z j ʼ j

þ ʵ

z ðÞ; ð 1 : 49 Þ

where b ii ¼0. In a matrix notation model, Eq. ( 1.49 ) can be written as

ð I B Þ y ʼ ð Þ ¼ ε: ð 1 : 50 Þ

Þ 1 Þ 1 I B t Obviously, E ðÞ ¼ ʼ . From Eq. ( 1.50 ) we can derive ʣ SAR ¼ I B ð ʣ ʵ ð

, assuming that

Þ 1

2 2 2

n

where Var ðÞ ¼ ʣ ʵ ¼ diag ˃

1 ; ˃ 2 ; ...; ˃

ð I B exists. If