1. Workshop 2: Spatial scale and dependence in biogeographical patterns Objective: introduce the following fundamental concepts on spatial data analysis: - What is spatial data analysis and why it is important - Pattern vs process - Spatial correlation, induced spatial dependence and autocorrelation - Stationary process - Scale and its components Main sources: Legendre and Legendre (2012) Numerical Ecology. Elsevier. Fortin and Dale (2005) Spatial analysis: a guide for ecologists. Cambridege University Press Bailey and Gatrell (1995) Interactive spatial data analysis. Prentice Hall.

2. Bird endemism in Northern Melanesian islands (Mayr and Dimond 2001)

4. Interim General Model 2 (“IGM2”, Field et al. 2005)

6. energy and water availability topographic heterogeneity

7. Interim General Model 2 (“IGM2”, Field et al. 2005) energy and water availability topographic heterogeneity ++−+

8. 0 1 Correlation coefficient

9. 0 1 Correlation coefficient Correct confidence interval

10. 0 1 Correlation coefficient Confidence interval affected by spatial correlation Correct confidence interval

11. Workshop 2: Spatial scale and dependence in biogeographical patterns Objective: introduce the following fundamental concepts on spatial data analysis: - What is spatial data analysis and why it is important - Pattern vs process - Spatial correlation, induced spatial dependence and autocorrelation - Stationary process - Scale and its components

12. Pattern: a single realization of or a “snapshot” of a process or combination of processes at one given time (Fortin and Dale 2005) Process: a phenomenon (response variable), or a set of pehnomena, which are organized along some independent axis (Legendre and Legendre 2012) Spatial process: a set of possibly non-independent random variables (Bailey and Gatrell 1995): {Y(s) = s belongs to region R}

13. Workshop 2: Spatial scale and dependence in biogeographical patterns Objective: introduce the following fundamental concepts on spatial data analysis: - What is spatial data analysis and why it is important - Pattern vs process - Spatial correlation, induced spatial dependence and autocorrelation - Stationary process - Scale and its components

14. -Spatial correlation: relationships between values observed at neighboring points in space, hence lack of independence of values of the observed variables. - Induced spatial dependence: functional dependence of the response variables (Y) on explanatory variables (X) that are themselves spatially correlated. - Autocorrelation: spatial correlation in the error component of a response variable

15. Workshop 2: Spatial scale and dependence in biogeographical patterns Objective: introduce the following fundamental concepts on spatial data analysis: - What is spatial data analysis and why it is important - Pattern vs process - Spatial correlation, induced spatial dependence and autocorrelation - Stationary process - Scale and its components

16. - Stationary process: informally, a spatial process is stationary (or homogeneous) if its statistical properties are independent of the absolute location in region R. Mean, variance, covariance

17. Workshop 2: Spatial scale and dependence in biogeographical patterns Objective: introduce the following fundamental concepts on spatial data analysis: - What is spatial data analysis and why it is important - Pattern vs process - Spatial correlation, induced spatial dependence and autocorrelation - Stationary process - Scale and its components

18. - Scale and its components in sampling theory: Grain: size of elementary sampling units Sampling interval: average distance between neighboring sampling units Extent: total length, area or volume included in the study - Scale of pattern or process: Size of “unit object” or “unit process” Distance between unit objects of unit process Space over which unit objects or unit process exist