11/21/2007
1
Advanced Topics in Forest
Biometrics  FOR6934
Introduction to statistical models
for spatial data – Part 1
Lecture outline
Spatial data
Stationarity
Isotropy
Semivariogram analysis
Why spatial statistics?
Spatial statistics first came about in the
cartography and surveying
Significant contributions have been made by
researchers in the mining industry (e.g., D G Krige)
Spatial statistics help us to characterize the
spatial continuity or roughness of a data set
While ordinary (onedimensional) statistics may be
nearly identical for different data sets, their spatial
continuity may
be quite different
A sample size of one?
Consider monitoring 100 seedlings planted in a
1hectare contiguous area
This study will give rise to 100 spatially referenced observations:
Z
(
s
i
)
(
Z
(
s
1
) ,
Z
(
s
2
) , …
Z
(
s
100
) )
where:
s
is a vector representing the location of the observations
Are the observations independent?
Spatial data is a special case of
clustered data
A set of spatial data is one realization of sampling from a
random
field
(a stochastic process in twodimensions)
We expect observations taken close together to be more related
than observations taken far apart
Types of spatial data
Geostatistical data
The domain,
D
, of the data is a fixed, continuous set
Data can be observed at points (
x, y
), where
Lattice data
Data can only be observed at fixed points in
D
, i.e., all possible
sample locations can be enumerated
For example, data by county or city block, biomass harvested in
contiguous 1x1 m
2
areas in 100 m
2
Spatial point patterns
Data consist of locations where a particular attribute is observed
For example, locations of Brazil nut trees in an area
2
)
,
(
y
x
Questions of interest
Whereas the
Z
process is of interest in geostatistical and
lattice data, the
D
process is of interest with spatial point
patterns
Continuous surface maps are often of interest with
geostatistical data
Kriging is used to predict
Z
(
s
0
) at some unsampled location
s
0
The relative randomness or uniformity of pattern is often
of interest with spatial point patterns
Answering these questions often involves making two
assumptions:
stationarity
and
isotropy
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document11/21/2007
2
Assumption of Stationarity
Stationarity is a property of selfreplication of a
stochastic process
Absolute coordinates are not important – only the
distance between points is
The stochastic properties of
Z
(
s
) and
Z
(
s
+
h
) are similar
The random field
looks similar
over the domain,
D
There are several kinds of stationarity
Strong (strict)
Secondorder (weak)
Intrinsic
Types of Stationarity
Strong stationarity
the spatial distribution is invariant under translation (rotating or
stretching of coordinate system)
Secondorder (weak) stationarity
The mean of the random field is constant and the covariance
between observations only depends on the distance between
them
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Staff
 Spatial analysis, Variogram, Geostatistics, Kriging, Random field, H_Al_0_20cm H_Al_0_20cm H_Al_0_20cm

Click to edit the document details