A new multivariate analysis technique, developed to relate community composition to known variation in the environment, is described. The technique is an extension of correspondence analysis (reciprocal averaging), a popular ordination technique that extracts continuous axes of variation from species occurrence or abundance data. Such ordination axes are typically interpreted with the help of external knowledge and data on environmental variables; this two-step approach (ordination followed by environmental gradient identification) is termed indirect gradient analysis. In the new technique, called canonical correspondence analysis, ordination axes are chosen in the light of known environmental variables by imposing the extra restriction that the axes be linear combinations of environmental variables. In this way community variation can be directly related to environmental variation. The environmental variables may be quantitative or nominal. As many axes can be extracted as there are environmental variables. The method of detrending can be incorporated in the technique to remove arch effects.(Detrended) canonical correspondence analysis is an efficient ordination technique when species have bell-shaped response curves or surfaces with respect to environmental gradients, and is therefore more appropriate for analyzing data on community composition and environmental variables than canonical correlation analysis. The new technique leads to an ordination diagram in which points represent species and sites, and vectors represent environmental variables. Such a diagram shows the patterns of variation in community composition that can be explained best by the environmental variables and also visualizes approximately the "centers" of the species distributions along each of the environmental variables. Such diagrams effectively summarized relationships between community and environment for data sets on hunting spiders, dyke vegetation, and algae along a pollution gradient.

Ecological data has several special properties: the presence or absence of species on a semi-quantitative abundance scale; non-linear relationships between species and environmental factors; and high inter-correlations among species and among environmental variables. The analysis of such data is important to the interpretation of relationships within plant and animal communities and with their environments. In this corrected version of Data Analysis in Community and Landscape Ecology, without using complex mathematics, the contributors demonstrate the methods that have proven most useful, with examples, exercises and case-studies. Chapters explain in an elementary way powerful data analysis techniques such as logic regression, canonical correspondence analysis, and kriging.

Differential Evolution (DE) is a simple genetic algorithm for numerical optimization in real parameter spaces. In a statistical context one would not just want the optimum but also its uncertainty. The uncertainty distribution can be obtained by a Bayesian analysis (after specifying prior and likelihood) using Markov Chain Monte Carlo (MCMC) simulation. This paper integrates the essential ideas of DE and MCMC, resulting in Differential Evolution Markov Chain (DE-MC). DE-MC is a population MCMC algorithm, in which multiple chains are run in parallel. DE-MC solves an important problem in MCMC, namely that of choosing an appropriate scale and orientation for the jumping distribution. In DE-MC the jumps are simply a fixed multiple of the differences of two random parameter vectors that are currently in the population. The selection process of DE-MC works via the usual Metropolis ratio which defines the probability with which a proposal is accepted. In tests with known uncertainty distributions, the efficiency of DE-MC with respect to random walk Metropolis with optimal multivariate Normal jumps ranged from 68% for small population sizes to 100% for large population sizes and even to 500% for the 97.5% point of a variable from a 50-dimensional Student distribution. Two Bayesian examples illustrate the potential of DE-MC in practice. DE-MC is shown to facilitate multidimensional updates in a multi-chain "Metropolis-withinGibbs" sampling approach. The advantage of DE-MC over conventional MCMC are simplicity, speed of calculation and convergence, even for nearly collinear parameters and multimodal densities.

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Palaeolimnological diatom data comprise counts of many species expressed as percentages for each sample. Reconstruction of past lake-water pH from such data involves two steps; (i) regression, where responses of modern diatom abundances to pH are modelled and (ii) calibration where the modelled responses are used to infer pH from diatom assemblages preserved in lake sediments. In view of the highly multivariate nature of diatom data, the strongly nonlinear response of diatoms to pH, and the abundance of zero values in the data, a compromise between ecological realism and computational feasability is essential. The two numerical approaches used are (i) the computationally demanding but formal statistical approach of maximum likelihood (m l) Gaussian logit regression and calibration and (ii) the computationally straightforward but heuristic approach of weighted averaging (w a ) regression and calibration. When the Surface Water Acidification Project (SWAP) modern training set of 178 lakes is reduced by data-screening to 167 lakes, w a gives superior results in terms of lowest root mean squared errors of prediction in cross-validation. Bootstrapping is also used to derive prediction errors, not only for the training set as a whole but also for individual pH reconstructions by WA for stratigraphic samples from Round Loch of Glenhead, southwest Scotland covering the last 10000 years. These reconstructions are evaluated in terms of lack-of-fit to pH and analogue measures and are interpreted in terms of rate of change by using bootstrapping of the reconstructed pH time-series.