Essential Statistical Concepts, Definitions, and Terminology - Sampling Spatial Units for Agricultural Surveys - page 16

1.4.1 Types of Spatial Data

Spatial statistics is a vast subject, largely because of the many different types of
observations (and data locations) that cover a huge list of phenomena. The obser-
vations, for example, may be univariate or multivariate, categorical or continuous.
They may be based on an observational study, a well-designed experiment, or a
sample survey. The data locations may be points, regions, line segments, or curves.
They may be regularly or irregularly spaced.
There are many different types of spatial data, so different forms of spatial
statistics are required. Here, we consider three main types of spatial data:
geostatistical, lattice, and spatial point patterns. In this classification, spatial data
are distinguished by the nature of the spatial domain 1 (Cressie 1993 ; Schabenberger
and Gotway 2005 ). First, we denote a spatial process in d dimensions as

;

d

y ðÞ: z 2 D ℝ ð 1 : 35 Þ

where y 2 represents the agricultural variable under investigation, observed at a
location z defined using a ( d x 1) vector of coordinates. Most spatial processes are
defined in two dimensional space, and so d ¼2 represents the usual Cartesian
coordinates.
The geostatistical data are defined using a continuous domain D . In this case, the
phenomena can be observed everywhere in D . Consider two locations, z i ¼ ( x i , y i )
and z j ¼ ( x j , y j ). Theoretically, we can place an infinite number of points between
these two sites. The continuity is a property of the domain, not of the variable being
measured. Note that the points in D are non-stochastic. A domain is said to be
non-stochastic, or fixed, if it does not change from one realization of the spatial
process to the next.
In lattice data, the domain D is fixed and discrete. The number of locations can
be infinite, but they must be countable. Generally, neighbor information is available
for the spatial areas. Spatial locations with lattice data often represent areal regions.
The regions can be regular or irregular. One example of spatial regular lattice data is
remote sensed data that divides an area into a series of small rectangles (i.e., pixels,
see Chap. 4 ) . Examples of irregular lattice data are variables collected by ZIP code,
census tracts, provinces, or administrative regions.
The domain of geostatistical or lattice data is non-stochastic. In spatial point
patterns, the set of points changes with each realization of the random process.
More formally, Diggle ( 2003 ) defined a spatial point pattern as
“a set of locations, irregularly distributed within a designated region, and presumed to have
been generated by some form of stochastic mechanism”.

1 Note that here the term domain has a different meaning than that used in survey sampling
literature (in particular, for domain estimation see Chap. 11 ). In this case, it simple denotes the
set of all possible input values for which the function is defined.
2 Note that y is expressed in lowercase though it is a component of a random process.