Ecosystems are: Hierarchically structured, Metastable, Far from equilibrium

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Spatial Relationships
Ecosystems are Hierarchically structured,
Metastable, Far from equilibrium
Theoretical Framework
An Introduction to Applied Geostatistics, E.
Isaaks and R. Srivastava, (1989). Factorial
Analysis, C. J. Adcock, (1954) Spatial
Analysis A guide for ecologists, M. Fortin and
M. Dale, (2005)
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Time
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Basic paradigm Ecosystem processes (change) are
constrained and controlled by the pattern of
hierarchical scales Things closer together
(in both space and time) are more alike then
things far apart Toblers Law (1970, Economic
Geography) Everything is related to everything,
but near things are more related then distant
things Ecological scale is the space and time
distance apart (lag) at which significant
variation is NO LONGER correlated with distance
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Applied Geostatistics
Spatial Structure
Regionalized Variable
Spatial Autocorrelation
Moran I (1950)
GEARY C (1954)
Semivariance
Stationarity
Anistotropy
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Applied Geostatistics
Notes on Introduction to Spatial Autocorrelation
Geostatistical methods were developed for
interpreting data that varies continuously over a
predefined, fixed spatial region. The study of
geostatistics assumes that at least some of the
spatial variation observed for natural phenomena
can be modeled by random processes with spatial
autocorrelation.
Geostatistics is based on the theory of
regionalized variables, variable distributed in
space (or time). Geostatiscal theory supports
that any measurement of regionalozed variables
can be viewed as a realization of a random
function (or random process, or random field, or
stochastic process)

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Spatial Structure
Geostatistical techniques are designed to
evaluate the spatial structure of a variable, or
the relationship between a value measured at a
point in one place, versus a value from another
point measured a certain distance
away. Describing spatial structure is useful
for

• Indicating intensity of pattern and the scale
  at which that pattern is exposed
• Interpolating to predict values at unmeasured
  points across the domain (e.g. kriging)
• Assessing independence of variables before
  applying parametric tests of significance

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Regionalized Variable
Regionalized Variables take on values according
to spatial location. Given a variable z, measured
at a location i , the variability in z can be
broken down into three components
Usually removed by detrending
Where
What we are interested in
Coarse scale forcing or trends can be removed by
fitting a surface to the trend using regression
and then working with regression residuals
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Regionalized Variable Zi
Function Z in domain D a set of space
dependent values
Histogram of samples zi
Cov(Z(x),Z(xh))
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Correlation
A Statement of the extent to which two data sets
agree.
Determined by the extent to which the two
regressions lines depart from the horizontal and
vertical.
Correlation Coefficient
Two distributions
One distribution
?2
)
a
a
?1
)
)
?
b
b
As ? decreases, a/b goes to 0
data
deviates
x
y
xy
x2
y2
X 1 3 2 1 µ
Y 3 2 5 3 µ
If you were to calculate correlation by hand .
You would produce these Terms.
Deviations
Sum of squares
Product of Deviations
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CorrelationCoefficient

Spatial auto-correlation
Briggs UT-Dallas GISC 6382 Spring 2007
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Spatial Structure
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Spatial Structure
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J I H B

G C F
D A E
Space
Topological vs Euclidean
First Order Neighbors Topology Binary
Connectivity Matrix
Distance Class Connectivity Matrix
A B C D E F G H I J
A B C D E F G H I J
A B C D E F G H I J
A B C D E F G H I J
1 connected, 0not connected
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Spatial Autocorrelation
Positive autocorrelation Negative
autocorrelation No autocorrelation
A variable is thought to be autocorrelated if it
is possible to predict its value at a given
location, by knowing its value at other nearby
locations.

• Autocorrelation is evaluated using structure
  functions that assess the spatial structure or
  dependency of the variable.
• Two of these functions are autocorrelation and
  semivariance which are graphed as a correlogram
  and semivariogram, respectively.
• Both functions plot the spatial dependence of
  the variable against the spatial separation or
  lag distance.

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Space
Euclidean Distance Matrix
Connectivity Matrix
A B C D . J
A B C D . J
A 0.0 B 2.00 0.00 C 1.41 3.16 0.00 J
A 0 B 0 0 C 1 0 0 J
Euclidean Distance Matrix
Weighted Matrix
A B C D . J
A B C D . J
A 0 B 2 0 C 1 3 0 J
A 0 B 0 0 C 0.7 0 0 D 0.7
0.7 0 J
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Moran I (1950)

• A cross-product statistic that is used to
  describe autocorrelation
• Compares value of a variable at one location
  with values at all other locations

Where n is the number of pairsZi is the
deviation from the mean for value at location i
(i.e., Zi xi x for variable x) Zj is
the deviation from the mean for value at location
j (i.e., Zj xj x for variable x) wij is an
indicator function or weight at distance d (e.g.
wij 1, if j is in distance class d from point
i, otherwise 0)

Wij is the sum of all weights (number
of pairs in distance class)
The numerator is a covariance (cross-product)
term the denominator is a variance term.
Values range from -1, 1
Value 1 Perfect positive correlation
Value -1 Perfect negative correlation
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Moran I (1950)
Again where for variable x n is the number of
pairswij(d) is the distance class connectivity
matrix (e.g. wij 1, if j is in distance class d
from point i, otherwise 0)

W(d) is the sum of all
weights (number of pairs in distance class)
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GEARY C (1954)

• A squared difference statistic for assessing
  spatial autocorrelation
• Considers differences in values between pairs
  of observations, rather than the covariation
  between the pairs (Moran I)

The numerator in this equation is a defference
term that gets squared. The Geary C statistic
is more sensitive to extreme values clustering
than the Moran I, and behaves like a distance
measure
Values range from 0,3 Value 0 Positive
autocorrelation Value 1 No autocorrelation
Value gt 1 Negative
autocorrelation
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Ripleys K (1976) The L (d) transformations
Determines if features are clustered at multiple
different distance. Sensitive to study area
boundary. Conceptualized as number of points
within a set of radius sets. If events follow
complete spatial randomness, the number of points
in a circle follows a Poisson distribution (mean
less then 1) and defines the expected.
Where A area N nuber of points D
distance K(i,j) the weight, which is 1 when
i-j lt d, 0 when i-j gt d
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General G
Effectively Distinguishes between hot and cold
spots. G is relatively large if high values
cluster, low if low values cluster. Numerator
are within a distance bound (d), expressed
relative to the entire study area.
Where d distance class Wij weight matrix,
which is 1 when i-j lt d, 0 when i-j gt d
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Semivariance
The geostatistical measure that describes the
rate of change of the regionalized variable is
known as the semivariance. Semivariance is
used for descriptive analysis where the spatial
structure of the data is investigated using the
semivariogram and for predictive applications
where the semivariogram is fitted to a
theoretical model, parameterized, and used to
predict the regionalized variable at other
non-measured points (kriging).
Where j is a point at distance d from i nd is
the number of points in that distance class
(i.e., the sum of the weights wij for that
distance class) wij is an indicator function set
to 1 if the pair of points is within the distance
class.
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A semivariogram is a plot of the structure
function that, like autocorrelation, describes
the relationship between measurements taken some
distance apart. Semivariograms define the range
or distance over which spatial dependence exists.

• The nugget is the semivariance at a distance 0.0,
  (the y intercept)

• The sill is the value at which the semivariogram
  levels off (its asymptotic value)

• The range is the distance at which the
  semivariogram levels off (the spatial extent of
  structure in the data)

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Stationarity

• Autocorrelation assumes stationarity, meaning
  that the spatial
• structure of the variable is consistent over the
  entire domain of the dataset.
• The stationarity of interest is second-order
  (weak) stationarity, requiring that
• the mean is constant over the region
• variance is constant and finite and
• covariance depends only on between-sample spacing
• In many cases this is not true because of larger
  trends in the data
• In these cases, the data are often detrended
  before analysis.
• One way to detrend data is to fit a regression to
  the trend, and use only
• the residuals for autocorrelation analysis

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Anistotropy
Autocorrelation also assumes isotropy, meaning
that the spatial structure of the variable is
consistent in all directions. Often this is
not the case, and the variable exhibits
anisotropy, meaning that there is a
direction-dependent trend in the data.
If a variable exhibits different ranges in
different directions, then there is a geometric
anisotropy. For example, in a dune deposit,
larger range in the wind direction compared to
the range perpendicular to the wind direction.
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For predictions, the empirical semivariogram is
converted to a theoretic one by fitting a
statistical model (curve) to describe its range,
sill, nugget.
There are four common models used to fit
semivariograms
Assumes no sill or range
Linear
Exponential
Spherical
Where c0 nugget b regression slope a
range c0 c sill
Gaussian
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Variogram Modeling Suggestions

• Check for enough number of pairs at each lag
  distance (from 30 to 50).
• Removal of outliers
• Truncate at half the maximum lag distance to
  ensure enough pairs
• Use a larger lag tolerance to get more pairs and
  a smoother variogram
• Start with an omnidirectional variogram before
  trying directional variograms
• Use other variogram measures to take into account
  lag means and variances
• (e.g., inverted covariance,
  correlogram, or relative variograms)
• Use transforms of the data for skewed
  distributions (e.g. logarithmic transforms).
• Use the mean absolute difference or median
  absolute difference to derive the range