## Structure functions

Ecologists are interested in describing spatial structures in quantitative ways and testing for the presence of spatial autocorrelation in data. The primary objective is to:

Table 13.1 Surface pattern analysis: research objectives and related numerical methods. Modified from Legendre & Fortin (1989).

Research objective

### Numerical methods

1) Description of spatial structures and testing for the presence of spatial autocorrelation (Descriptions using structure functions should always be complemented by maps.)

2) Mapping; estimation of values at given locations

3) Modelling species-environment relationships while taking spatial structures into account

4) Performing valid statistical tests on autocorrelated data

Univariate structure functions: correlogram, variogram, etc. (Section 13.1)

Multivariate structure functions: Mantel correlogram (Section 13.1)

Testing for a gradient in multivariate data: (1) constrained (canonical) ordination between the multivariate data and the geographic coordinates of the sites (Section 13.4). (2) Mantel test between ecological distances (computed from the multivariate data) and geographic distances (Subsection 10.5.1)

Univariate data: local interpolation map; trend-surface map (global statistical model) (Sect. 13.2)

Multivariate data: clustering with spatial contiguity constraint, search for boundaries (Section 13.3); interpolated map of the 1st (2nd, etc.) ordination axis (Section 13.4); multivariate trend-surface map obtained by constrained ordination (canonical analysis) (Section 13.4)

Raw data tables: partial canonical analysis (Section 13.5)

Distance matrices: partial Mantel analysis (Section 13.6)

### Subsection 1.1.1

• either support the null hypothesis that no significant spatial autocorrelation is present in a data set, or that none remains after detrending (Subsection 13.2.1), thus insuring valid use of the standard univariate or multivariate statistical tests of hypotheses.

• or reject the null hypothesis and show that significant spatial autocorrelation is present in the data, in order to use it in conceptual or statistical models.

Tests of spatial autocorrelation coefficients may only support or reject the null hypothesis of the absence of significant spatial structure. When significant spatial structure is found, it may correspond, or not, to spatial autocorrelation (Section 1.1, model b) — depending on the hypothesis of the investigator.

Spatial structures may be described through structure functions, which allow one to quantify the spatial dependency and partition it amongst distance classes.

Map Interpretation of this description is usually supported by maps of the univariate or multivariate data (Sections 13.2 to 13.4). The most commonly used structure functions are correlograms, variograms, and periodograms.

A correlogram is a graph in which autocorrelation values are plotted, on the ordinate, against distance classes among sites on the abscissa. Correlograms (Cliff & Ord 1981) can be computed for single variables (Moran's I or Geary's c autocorrelation coefficients, Subsection 1) or for multivariate data (Mantel correlogram, Subsection 5); both types are described below. In all cases, a test of significance is available for each individual autocorrelation coefficient plotted in a correlogram.

Variogram Similarly, a variogram is a graph in which semi-variance is plotted, on the ordinate, against distance classes among sites on the abscissa (Subsection 3). In the geostatistical tradition, semi-variance statistics are not tested for significance, although they could be through the test developed for Geary's c, when the condition of second-order stationarity is satisfied (Subsection 13.1.1). Statistical models may be fitted to variograms (linear, exponential, spherical, Gaussian, etc.); they allow the investigator to relate the observed structure to hypothesized generating processes or to produce interpolated maps by kriging (Subsection 13.2.2).

Because they measure the relationship between pairs of observation points located a certain distance apart, correlograms and variograms may be computed either for preferred geographic directions or, when the phenomenon is assumed to be isotropic in space, in an all-directional way.

2-D A two-dimensional Schuster (1898) periodogram may be computed when the periodogram structure under study is assumed to consist of a combination of sine waves propagated through space. The basic idea is to fit sines and cosines of various periods, one period at a time, and to determine the proportion of the series' variance (r2) explained by each period. In periodograms, the abscissa is either a period or its inverse, a frequency; the ordinate is the proportion of variance explained. Two-dimensional periodograms may be plotted for all combinations of directions and spatial frequencies. The technique is described Priestley (1964), Ripley (1981), Renshaw and Ford (1984) and Legendre & Fortin (1989). It is not discussed further in the present book.