1. SPATIAL ANALYSIS IN MACROECOLOGY

2. INTRODUCTION
Spatial Processes
Spatial data

3. General technique for exploratory spatial data analysis
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

4. 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

5. First Applications... A. D.Cliff J. K. Ord
First applications in Population Genetics and Ecology
Robert Sokal
Sokal, R. R. & Oden, N. L Spatial autocorrelation in biology:
1. methodology
2. Some biological implications and four applications of evolutionary and ecological interest
Biological Journal of Linnean Society 10:
A. D.Cliff
J. K. Ord

6. Marie Jose Fortin
Pierre Legendre
Legendre, P. & Fortin, M.J Spatial pattern and ecological analysis. Vegetatio 80:
Legendre, P Spatial autocorrelation: trouble or new paradigm? Ecology 74:

7. Spatial Structure SPATIAL DEPENDENCE, EXOGENOUS OR EXTRINSIC STRUCTURE
SPATIAL AUTOCORRELATION, ENDOGENOUS OR INTRINSIC STRUCTURE

8. Spatial autocorrelation
New Paradigm ?
Pierre Legendre
Spatial autocorrelation
Problem ?

9. SPATIAL DATA AND STRUCTURE
Point pattern
Spatial Data
Surface

10. Range of Moran’s I – maximum and minimum estimates
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 =

11. Spatial correlogram (Moran’s I) with SAM
Classe W
Dmax
I de Moran
se(I) P
Bonf.
Bonf. Seq.
I(max)
I / I(max) 1
1799
3.000
0.830
0.026
0.001
0.005
0.050
0.898
0.924 2
1079
4.123
0.634
0.037
0.025
0.728
0.871 3
1889
5.385
0.427
0.024
0.017
0.537
0.796 4
1924
6.403
0.182
0.013
0.363
0.502 5
1601
7.616
-0.049
0.029
0.129
0.010
0.331
-0.144 6
1564
8.602
-0.215
0.008
0.403
-0.533 7
1681
10.000
-0.382
0.028
0.007
0.515
-0.742 8
1722
11.402
-0.535
0.027
0.006
0.642
-0.834 9
1445
13.342
-0.643
0.798
-0.805 10
1586
21.932
-0.326
1.039
-0.314

12. Interpreting correlograms
Positive autocorrelation in short distances...
E(I) = - 1 / (n – 1)
Negative autocorrelation in long distances...

13. Positive autocorrelation in short distances...
E(I) = - 1 / (n – 1)
Null autocorrelation in long distances...

14. Interpreting the correlograms
CLINES – Positive Moran’s I in the first distances classes associated with long distances negative autocorrelation;
PATCHES – Positive Moran’s 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 .

16. Exploratory data analysis
(EDA)
Spatial data analysis
Modeling & Inference

17. Correlations Y2 Y2 Y2 Y1 Y1 Y1 r
positive r
null r
negative

18. Relationship between bird species richness and AET in South America

19. HOW MANY DEGREES OF FREEDOM?
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?

20. So, in the presence of spatial autocorrelation...
Degrees of freedom
(upward biased)
Confidence intervals (downward biased)

21. estimated correlation (r)
 = 1
 = 0 A)
DF upward biased
IC (1-) narrow
True DF
IC (1-) wide B)
Situation A) – significant correlation at a given 
Situation B) – non-significant correlation at a given 

22. OLS MODELING ON SPATIAL DATA
OLS model residuals are usually not independent...
Use Moran`s I or correlograms of model residuals as a diagnosis for randomness and independence of residuals

23. Correlograms of the two variables...
Richness
Class 1 (0-700 km):
Moran’s I =  0.016
AET
Class 1 (0-700 km):
Moran’s I =  0.013

24. 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 = (P <<0.001)

25. Spatial correlogram of model residuals...
Significant Moran’s I at the first distance class
(I =  ; P < 0.01)

26. Spatial distribution of Y, estimated Y and model residuals

27. 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

28. First, compute the following regressions:
(1) Y and XE (“environmental variables”); R2 = [a]+[b]
(2) Y and S (TSA); R2 = [b]+[c]
(3) Y = f (XE, S); R2 = [a]+[b]+[c]
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 ([a]+[b]+[c]) – ([b]+[c])
[b] = R2 (step 1) + R2 (step 2) – R2 (step 3)
[c] = R2 (step 3) - R2 (step 1)
[d] = 1-(a+b+c)

29. Partial regression of bird richness in South America (5 environmental variables - AET, PET, range in elevation, annual temperature and precipitation)
1st order

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

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

32. The ‘SAM’ Team