A generalized variogram-based framework for multiscale ordination

Multi-scale ordination (MSO) deals with potential scale dependence in species assemblages by studying how results from multivariate ordination may be different at different spatial scales. MSO methods were initially based on two-term local covariances between species and, therefore, required sampling designs composed of adjacent quadrats. A variogram-based MSO, recently introduced by H. H. Wagner, is applicable to very diverse sampling designs and for use with principal-components analysis, correspondence analysis, and derived, "two-table" (also called "direct") ordination methods, i.e., redundancy analysis and canonical correspondence analysis. In this paper we put forward an enlarged framework for variogram-based MSO that relies on a generalized definition of inter-species covariance and on matrix expression of spatial contiguity between sampling units. This enables us to provide distance-explicit decompositions of variances and covariances (in their generalized meaning) that are consistent with many ordination methods in both their single- and two-table versions. A spatially explicit apportioning of diversity indices is proposed for some particular definitions of variance. Referring to two-table ordination methods allowed the multi-scale study of residual spatial patterns after factoring out available environmental variables. Some aspects of the approach are briefly illustrated with vegetation data from a Neotropical rain forest in French Guian

Orthonormal transform to decompose the variance of a life-history trait across a phylogenetic tree

In recent years, there has been an increased interest in studying the variability of a quantitative life history trait across a set of species sharing a common phylogeny. However, such studies have su.ered from an insu.cient development of statistical methods aimed at decomposing the trait variance with respect to the topological structure of the tree. Here we propose, a new and generic approach that expresses the topological properties of the phylogenetic tree via an orthonormal basis, which is further used to decompose the trait variance. Such a decomposition provides a structure function, referred to as "orthogram," which is relevant to characterize in both graphical and statistical aspects the dependence of trait values on thetopology of the tree ("phylogenetic dependence"). We also propose four complementary test statistics to be computed from orthogram values that help to diagnose both the intensity and the nature of phylogenetic dependence. The relevance of the method is illustrated by the analysis of three phylogenetic data sets, drawn from the literature and typifying contrasted levels and aspects of phylogenetic dependence. Freely available routines which have been programmed in the R framework are also proposed

Textural Ordination Based on Fourier Spectral Decomposition: A Method to Analyze and Compare Landscape Patterns

We propose an approach to texture characterization and comparison that directly uses the information of digital images of the earth surface without requesting a prior distinction of structural ‘patches'. Digital images are partitioned into square ‘windows' that define the scale of the analysis and which are submitted to the two-dimensional Fourier transform for extraction of a simplified textural characterization (in terms of coarseness) via the computation of a ‘radial' power spectrum. Spectra computed from many images of the same size are systematically compared by means of a principal component analysis (PCA), which provides an ordination along a limited number of coarseness vs. fineness gradients. As an illustration, we applied this approach to digitized panchromatic air photos depicting various types of land cover in a semiarid landscape of northern Cameroon. We performed ‘textural ordinations' at several scales by using square windows with sides ranging from 120 m to 1 km. At all scales, we found two coarseness gradients (PCA axes) based on the relative importance in the spectrum of large (> 50 km−1), intermediate (30–50 km−1), small (10–25 km−1) and very small (<10 km−1) spatial frequencies. Textural ordination based on Fourier spectra provides a powerful and consistent framework to identifying prominent scales of landscape patterns and to compare scaling properties across landscapes

Testing Pairwise Association between Spatially Autocorrelated Variables: A New Approach Using Surrogate Lattice Data

International audienceBackground: Independence between observations is a standard prerequisite of traditional statistical tests of association. This condition is, however, violated when autocorrelation is present within the data. In the case of variables that are regularly sampled in space (i.e. lattice data or images), such as those provided by remote-sensing or geographical databases, this problem is particularly acute. Because analytic derivation of the null probability distribution of the test statistic (e.g. Pearson's r) is not always possible when autocorrelation is present, we propose instead the use of a Monte Carlo simulation with surrogate data. Methodology/Principal Findings: The null hypothesis that two observed mapped variables are the result of independent pattern generating processes is tested here by generating sets of random image data while preserving the autocorrelation function of the original images. Surrogates are generated by matching the dual-tree complex wavelet spectra (and hence the autocorrelation functions) of white noise images with the spectra of the original images. The generated images can then be used to build the probability distribution function of any statistic of association under the null hypothesis. We demonstrate the validity of a statistical test of association based on these surrogates with both actual and synthetic data and compare it with a corrected parametric test and three existing methods that generate surrogates (randomization, random rotations and shifts, and iterative amplitude adjusted Fourier transform). Type I error control was excellent, even with strong and long-range autocorrelation, which is not the case for alternative methods. Conclusions/Significance: The wavelet-based surrogates are particularly appropriate in cases where autocorrelation appears at all scales or is direction-dependent (anisotropy). We explore the potential of the method for association tests involving a lattice of binary data and discuss its potential for validation of species distribution models. An implementation of the method in Java for the generation of wavelet-based surrogates is available online as supporting material

Predicting tropical forest stand structure parameters from Fourier transform of very high-resolution remotely sensed canopy images

1. Predicting stand structure parameters for tropical forests from remotely sensed data has numerous important applications, such as estimating above-ground biomass and carbon stocks and providing spatial information for forest mapping and management planning, as well as detecting potential ecological determinants of plant species distributions. As an alternative to direct measurement of physical attributes of the vegetation and individual tree crown delineation, we present a powerful holistic approach using an index of canopy texture that can be extracted from either digitized air photographs or satellite images by means of two-dimensional spectral analysis by Fourier transform. 2. We defined an index of canopy texture from the ordination of the Fourier spectra computed for 3545 1-ha square images of an undisturbed tropical rain forest in French Guiana. This index expressed a gradient of coarseness vs. fineness resulting from the relative importance of small, medium and large spatial frequencies in the Fourier spectra. 3. Based on 12 1-ha control plots, the canopy texture index showed highly significant correlations with tree density (R2 = 0·80), diameter of the tree of mean basal area (R2 = 0·71), distribution of trees into d.b.h. classes (R2 = 0·64) and mean canopy height (R2 = 0·57), which allowed us to produce reasonable predictive maps of stand structure parameters from digital aerial photographs. 4. Synthesis and applications. Two-dimensional Fourier analysis is a powerful method for obtaining quantitative characterization of canopy texture, with good predictive ability on stand structure parameters. Forest departments should use routine forest inventory operations to set up and feed regional databases, featuring both tree diameter figures and digital canopy images, with the ultimate aims of calibrating robust regression relationships and deriving predictive maps of stand structure parameters over large areas of tropical forests. Such maps would be particularly useful for forest classification and to guide field assessment of tropical forest resources and biodiversity

A tribute to the use of minimalistic spatially-implicit models of savanna vegetation dynamics to address broad spatial scales in spite of scarce data

The savanna biome encompasses a variety of vegetation physiognomies that traduce complex dynamical responses of plants to the rainfall gradients leading from tropical forests to hot deserts. Such responses are shaped by interactions between woody and grassy plants that can be either direct, disturbance-mediated or both. There has been increasing evidence that several vegetation physiognomies, sometimes highly contrasted, may durably coexist under similar rainfall conditions suggesting multi-stability or at least not abrupt transitions. These fascinating questions have triggered burgeoning modelling efforts which have, however, not yet delivered an integrated picture liable to furnish sensible predictions of potential vegetation at broad scales. In this paper, we will recall the key ecological processes and resulting vegetation dynamics that models should take into account. We will also present the main modelling options present in the literature and advocate the use of minimalistic models, capturing only the essential processes while retaining sufficient mathematical tractability and restricting themselves to a minimal set of parameters assessable from the overall literature