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ESE 502 II.1-1 Tony E. Smith
CONTINUOUS SPATIAL DATA ANALYSIS
1. Overview of Spatial Stochastic Processes
The key difference between continuous spatial data and point patterns is that there is
now assumed to be a meaningful value, ()Ys, at every location, s, in the region of
interest. For example, ()Ys might be the temperature at s or the level of air pollution at
s. We shall consider a number of illustrative examples in the next section. But before
doing so, it is convenient to outline the basic analytical framework to be used throughout
this part of the NOTEBOOK.
If the region of interest is again denoted by
R
, and if the value, ()Ys, at each location,
sR∈ is treated as a random variable, then the collection of random variables
(1.1.1) {(): }Ys s R∈
is designated as a spatial stochastic process on
R
(also called a random field on
R
). It
should be clear from the outset that such (uncountably) infinite collections of random
variables cannot be analyzed in any meaningful way without making a number of strong
assumptions. We shall make these assumptions explicit as we proceed.
Observe next that there is a clear parallel between spatial stochastic processes and
temporal stochastic processes,
(1.1.2) { ( ): }Yt t T∈
where the set, T, is some continuous (possibly unbounded) interval of time. In many
respects, the only substantive difference between (1.1) and (1.2) is the dimension of the
underlying domain. Hence it is not surprising that most of the assumptions and analytical
methods to be employed here have their roots in time series analysis. One key difference
that should be mentioned here is that time is naturally ordered (from “past” to “present”
to “future”) whereas physical space generally has no preferred directions. This will have
a number of important consequences that will be discussed as we proceed.
1.1 Standard Notation
The key to studying the infinite collections of random
variables such as (1.1) is of course to take finite samples
of ( )Ys values, and attempt to draw inferences on the
basis of this information. To do so, we shall employ the
following standard notation. For any given set of sample
locations, { : 1,.., }
i
si n R=⊂ (as in Figure 1.1), let the
random vector: Fig.1.1. Sample Locations
•
•
•
•
1
s
n
s
2
s
n
s
−
⋅ ⋅
⋅
R
