Workshop 2: Spatial scale and dependence in biogeographical patterns

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.

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

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

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

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

Correlation coefficient -1 1 0

Correlation coefficient Correct confidence interval -1 1 0

Correlation coefficient Correct confidence interval Confidence interval affected byspatial correlation -1 1 0

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

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}

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

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

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

Workshop 2: Spatial scale and dependence in biogeographical patterns