SPATIAL ANALYSIS IN MACROECOLOGY

INTRODUCTION Spatial Processes Spatial data

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

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

First Applications... A. D.Cliff J. K. Ord 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

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.

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

Spatial autocorrelation New Paradigm ? Pierre Legendre Spatial autocorrelation Problem ?

SPATIAL DATA AND STRUCTURE Point pattern Spatial Data Surface

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 = -0.286

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

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

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

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 .

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

Correlations Y2 Y2 Y2 Y1 Y1 Y1 r 1 positive r 0 null r -1 negative

Relationship between bird species richness and AET in South America

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?

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

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 

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

Correlograms of the two variables... Richness Class 1 (0-700 km): Moran’s I = 0.695  0.016 AET Class 1 (0-700 km): Moran’s I = 0.673  0.013

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 <<0.001)

Spatial correlogram of model residuals... Significant Moran’s I at the first distance class (I = 0.213  0.014 ; P < 0.01)

Spatial distribution of Y, estimated Y and model residuals

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

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)

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

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

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

The ‘SAM’ Team