SPATIAL ANALYSIS IN MACROECOLOGY

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SPATIAL ANALYSIS IN MACROECOLOGY
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INTRODUCTION
Spatial data
Spatial Processes
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SPATIAL AUTOCORRELATION
Spatial autocorrelation may be loosely defined
as the property of random variables which take
values, at pairs of sites a given distance apart,
that are more similar (positive autocorrelation)
or dissimilar (negative autocorrelation) than
expected for randomly associated pairs of
observations (Legendre Legendre 1998)
General technique for exploratory spatial data
analysis
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Basically, autocorrelation measures the
correlation of the variable with itself, but
considering variable distances (spatial, temporal
or phylogenetic) among units... Time series
r 1.0
Lag 1
Lag 2
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First Applications...
First applications in Population Genetics and
Ecology
Robert Sokal
Sokal, R. R. Oden, N. L. 1978. Spatial
autocorrelation in biology 1. methodology 2.
Some biological implications and four
applications of evolutionary and ecological
interest Biological Journal of Linnean Society
10 199-249.
A. D.Cliff
J. K. Ord
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Marie Jose Fortin
Pierre Legendre
Legendre, P. Fortin, M.J. 1989. Spatial pattern
and ecological analysis. Vegetatio 80
107-138. Legendre, P. 1993. Spatial
autocorrelation trouble or new paradigm? Ecology
74 1659-1673.
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SPATIAL DEPENDENCE, EXOGENOUS OR EXTRINSIC
STRUCTURE
Spatial Structure
SPATIAL AUTOCORRELATION, ENDOGENOUS OR INTRINSIC
STRUCTURE
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New Paradigm ?
Pierre Legendre
Spatial autocorrelation
Problem ?
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SPATIAL DATA AND STRUCTURE
Point pattern
Spatial Data
Surface
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Range of Morans I maximum and minimum estimates
-1.0 lt Morans I lt 1.0
Positive autocorrelation
Negative autocorrelation
Maximum and minimum are a function of eigenvalues
extracted from W (see Lichstein et al. 2002)
IMAX 0.513 I / IMAX -0.286
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Spatial correlogram (Morans I) with SAM
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Interpreting correlograms
Positive autocorrelation in short distances...
E(I) - 1 / (n 1)
Negative autocorrelation in long distances...
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Positive autocorrelation in short distances...
E(I) - 1 / (n 1)
Null autocorrelation in long distances...
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Interpreting the correlograms
CLINES Positive Morans I in the first
distances classes associated with long distances
negative autocorrelation PATCHES Positive
Morans I in the short distances without long
distance negative autocorrelation (i.e.,
stabilization). The intercept of the correlogram
is the diameter of the patch Long-distances
differentiation Null autocorrelation at short
distances associated with significant long
distance negative autocorrelation Null no
significant autocorrelation .
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Exploratory data analysis (EDA)
Spatial data analysis
Modeling Inference
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Correlations
Y2
Y2
Y2
Y1
Y1
Y1
r 1 positive
r 0 null
r -1 negative
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Relationship between bird species richness and
AET in South America
richness
AET
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If there is spatial autocorrelation in the two
variables, then spatial units close in geographic
space tend to be redundant in the sense of
providing evidence of the relationship between
richness and AET...
The assumption of statistical independence is
violated...
HOW MANY DEGREES OF FREEDOM?
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So, in the presence of spatial autocorrelation...
Degrees of freedom (upward biased)
Confidence intervals (downward biased)
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estimated correlation (r)
? 1
? 0
DF upward biased IC (1-?) narrow
A)
True DF IC (1-?) wide
B)
Situation A) significant correlation at a given
? Situation B) non-significant correlation at a
given ?
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• OLS MODELING ON SPATIAL DATA
• OLS model residuals are usually not
  independent...

Use Morans I or correlograms of model residuals
as a diagnosis for randomness and independence of
residuals
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Correlograms of the two variables...
Richness Class 1 (0-700 km) Morans I 0.695 ?
0.016
AET Class 1 (0-700 km) Morans I 0.673 ? 0.013
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Results of multiple regression of bird species
richness against 6 environmental predictors (AET,
PET, rainfall, elevation, temperature and
interaction between topography and
temperature) - R2 0.856 F 363.3 (P
ltlt0.001)
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Spatial correlogram of model residuals...
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Spatial distribution of Y, estimated Y and model
residuals
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Spatial partial regression uses trend surface
analysis

• Problem with TSA
• Broad-scale trends only

1st. Order I 0.477
2nd. Order I 0.393
3rd. Order I 0.229
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First, compute the following regressions
(1) Y and XE (environmental variables) R2
ab (2) Y and S (TSA) R2 bc (3) Y
f (XE, S) R2 abc
The individual values of a, b, and c can be
obtained by subtraction from the previous results
a R2 (step 3) - R2 (step 2) or (abc)
(bc) b R2 (step 1) R2 (step 2) R2
(step 3) c R2 (step 3) - R2 (step 1) d
1-(abc)
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• Partial regression of bird richness in South
  America (5 environmental variables - AET, PET,
  range in elevation, annual temperature and
  precipitation)
• 1st order

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• Partial regression of bird richness in South
  America (5 environmental variables - AET, PET,
  range in elevation, annual temperature and
  precipitation)
• 2nd order

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• Partial regression of bird richness in South
  America (5 environmental variables - AET, PET,
  range in elevation, annual temperature and
  precipitation)
• 3rd order

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The SAM Team