C.J.F. ter Braak

Weighted averaging partial least squares regression (WA-PLS): an improved method for reconstructing environmental variables from species assemblages

Weighted averaging regression and calibration form a simple, yet powerful method for reconstructing environmental variables from species assemblages. Based on the concepts of niche-space partitioning and ecological optima of species (indicator values), it performs well with noisy, species-rich data that cover a long ecological gradient (>3 SD units). Partial least squares regression is a linear method for multivariate calibration that is popular in chemometrics as a robust alternative to principal component regression. It successively selects linear components so as to maximize predictive power. In this paper the ideas of the two methods are combined. It is shown that the weighted averaging method is a form of partial least squares regression applied to transformed data that uses the first PLS-component only. The new combined method, ast squares, consists of using further components, namely as many as are useful in terms of predictive power. The further components utilize the residual structure in the species data to improve the species parameters (‘optima’) in the final weighted averaging predictor. Simulations show that the new method can give 70% reduction in prediction error in data sets with low noise, but only a small reduction in noisy data sets. In three real data sets of diatom assemblages collected for the reconstruction of acidity and salinity, the reduction in prediction error was zero, 19% and 32%.

Canonical community ordination. Part I: Basic theory and linear methods

AbstractCanonical community ordination comprises a collection of methods that relate species assemblages to their environment, in both observational studies and designed experiments. Canonical ordination differs from ordination sensu stricto in that species and environment data are analyzed simultaneously. Part I reviews the theory in a non-mathematical way with emphasis on new insights for the interpretation of ordination diagrams. The interpretation depends on the ordination method used to create the diagram. After the basic theory, Part I is focused on the ordination diagrams in linear methods of canonical community ordination, in particular principal component analysis, redundancy analysis and canonical correlation analysis. Special attention is devoted to the display of qualitative environmental variables.

Inferring PH from diatoms: a comparison of old and new calibration methods

Two new methods for inferring pH from diatoms are presented. Both are based on the observation that the relationships between diatom taxa and pH are often unimodal. The first method is maximum likelihood calibration based on Gaussian logit response curves of taxa against pH. The second is weighted averaging. In a lake with a particular pH, taxa with an optimum close to the lake pH will be most abundant, so an intuitively reasonable estimate of the lake pH is to take a weighted average of the pH optima of the species present.Optima and tolerances of diatom taxa were estimated from contemporary pH and proportional diatom counts in littoral zone samples from 97 pristine soft water lakes and pools in Western Europe. The optima showed a strong relation with Hustedts pH preference groups. The two new methods were then compared with existing calibration methods on the basis of differences between inferred and observed pH in a test set of 62 additional samples taken between 1918 and 1983. The methods were ranked in order of performance as follows (between brackets the standard error of inferred pH in pH units); maximum likelihood (0.63) > weighted averaging (0.71) = multiple regression using pH groups (0.71) = the Gasse & Tekaia method (0.71) > Renberg & Hellbergs Index B (0.83) » multiple regression using taxa (2.2). The standard errors are larger than those usually obtained from surface sediment samples. The relatively large standard may be due to seasonal variation and to the effects of other factors such as humus content. The maximum likelihood method is statistically rigorous and can in principle be extended to allow for additional environmental factors. It is computer intensive however. The weighted averaging approach is a good approximation to the maximum likelihood method and is recommended as a practical and robust alternative.

The effects of car traffic on breeding bird populations in woodland. Ill. Reduction of density in relation to the proximity of main roads

1. This study investigated the effect of car traffic on the breeding density of birds in deciduous and coniferous woodland, and the importance of noise and visibility of cars as possible factors affecting density. 2. Of the 43 species analysed in both woodland types, 26 species (60%) showed evidence of reduced density adjacent to roads (based on analysis with Wilcoxon signed-ranks test and regression). 3. Regression models with noise load as the only independent variable gave the best overall results. Calculated «effect distances» (the distance from the road up to where a reduced density was present) based on these regressions varied between species from 40-1500 m for a road with 10 000 cars per day to 70-2800 m for a road with 60 000 cars per day (120 km h −1 and 70% amount of woodland along the road). For a zone of 250 m from the road the reduction of the density varied from 20 to 98%. 4. When visibility of cars was controlled for, the number of species showing density reductions was much higher on plots with a high noise load than on ones with a low noise load. When noise conditions were held constant, however, there was no difference in bird densities between plots with high and low visibility of cars. 5. It is argued that noise load is probably the most important cause of the reduced densities. Visibility of cars, direct mortality and pollution are considered unimportant. 6. The results of this study stress the importance of considering the effect of car traffic on the breeding density of birds in planning and constructing main roads

Matching species traits to environmental variables: a new three-table ordination method

This paper addresses the question of studying the joint structure of three data tablesR,L andQ. In our motivating ecological example, the central tableL is a sites-by-species table that contains the number of organisms of a set of species that occurs at a set of sites. At the margins ofL are the sites-by-environment data tableR and the species-by-trait data table Q. For relating the biological traits of organisms to the characteristics of the environment in which they live, we propose a statistical technique calledRLQ analysis (R-mode linked toQ-mode), which consists in the general singular value decomposition of the triplet (RtDILDJQ,Dq,Dp) whereDI,DJ,Dq,Dp are diagonal weight matrices, which are chosen in relation to the type of data that is being analyzed (quantitative, qualitative, etc.). In the special case where the central table is analysed by correspondence analysis,RLQ maximizes the covariance between linear combinations of columns ofR andQ. An example in bird ecology illustrates the potential of this method for community ecologists.

Weighted averaging of species indicator values: Its efficiency in environmental calibration

Abstract A common bioassay problem in applied ecology is to estimate values of an environmental variable from species incidence or abundance data. An example is the problem of reconstructing past changes in acidity (pH) in lakes from diatom assemblages found in successive strata of the bottom sediment. The method of weighted averaging is based on indicator values, the indicator value of a species being, intuitively, the value of the environmental variable most preferred by that species. Indicator values of all species present in a site are averaged to give an estimate of the value of the environmental variable at the site. The average is weighted by species abundances, if known, with absent species having zero weight. Using field data, several authors have compiled lists of indicator values of species for various environmental variables for use in weighted averaging, e.g. pH indicator values of diatom species. In this paper the properties of the method of weighted averaging are studied, starting from the idea that indicator values are parameters of response curves that describe the expected abundance of each species in relation to the environmental variable. In practice the response curves must be estimated by regression methods, but here they are assumed to be known in advance. Conditions are derived under which the weighted average is a consistent and efficient estimator for the value of an environmental variable at a site. Because weighted averaging is central to the ordination technique known as reciprocal averaging or correspondence analysis, the conditions also define models that are implicitly invoked when reciprocal averaging is used in ecological ordination studies.

Non-linear methods for multivariate statistical calibration and their use in palaeoecology: a comparison of inverse (k-nearest neighbours, partial least squares and weighted averaging partial least squares) and classical approaches

Current environmental problems, such as acid rain and global warming, have greatly increased interest in fossil species assemblages as indicators of the palaeoenvironment and thus in quantitative methods for reconstructing environmental variables from species assemblage data. The ensuing multivariate calibration problem appears to be even harder than that of spectroscopic calibration, primarily because the basic model is unimodal (Shelfords law of tolerance) instead of being linear (Beers law). The strong non-linearity has led to the use of non-parametric calibration methods, in particular the smooth response surface method (SRS) and the method of best modern analogues, alias k-nearest neighbours (k-NN), and to a form of non-linear partial least squares (PLS), called weighted averaging partial least squares (WA-PLS), specially designed to analyze unimodal data. SRS and k-NN are recognized as non-parametric smoothing versions of the classical and inverse approach to linear calibration, respectively, whereas PLS and WA-PLS are inverse methods that bring in the aspect of dimension reduction. In a comparison on ‘realistically looking’ simulated compositional data with 100 training samples and 500 independent evaluation samples, WA-PLS and k-NN outperformed PLS when the species response functions were unimodal. For such data, k-NN resisted the curse of dimensionality. However, when the response functions were near-linear, WA-PLS and PLS performed about equally and clearly outperformed k-NN. On other simulated data, simultaneous calibration of two climate variables via a parametric non-linear classical method was compared with individual calibrations via inverse methods. The simultaneous calibration method was better at the border of the sampled space than the best inverse method (WA-PLS) and much better than k-NN. The simulations demonstrated the limitations of the leave-one-out estimate of prediction error: it showed severe method-dependent bias.

Model-Free Estimation from Spatial Samples: a Reappraisal of Classical Sampling Theory

A commonly held view among geostatisticians is that classical sampling theory is inapplicable to spatial sampling because spatial data are dependent, whereas classical sampling theory requires them to be independent. By comparing the assumptions and use of classical sampling theory with those of geostatistical theory, we conclude that this view is both false and unfortunate. In particular, estimates of spatial means based on classical sampling designs require fewer assumptions for their validity, and are therefore more robust, than those based on a geostatistical model.

An experimental manipulation of oligochaete communities in mesocosms treated with chlorpyrifos or nutrient additions: multivariate analyses with Monte Carlo permutation tests

Oligochaete communities were monitored under semi-natural conditions in experimental ditch mesocosms. Twelve ditches were used in a Before-Control-After-Impact (BACI) experiment to study the effect of the insecticide chlorpyrifos. Another eight ditches were used in a randomized experiment to study the effect of eutrophication. Oligochaete communities were sampled by deploying trays of substratum for colonization over a 20-week period. The experiments were analyzed by multivariate analysis using redundancy analysis and Monte Carlo permutation to assess statistical significance. These novel methods have the advantage over classical multivariate analysis of variance (MANOVA) of being distribution-free and of having no restrictive upper limit on the number of species that can be analyzed simultaneously. In the BACI-experiment no significant effect of chlorpyrifos on oligochaete communities was detected. Eutrophication effects were observed at the higher eutrophication levels in clay ditches. Oligochaete abundances decreased in those ditches. Considerable variation was attributed to stochastic factors given that the ditches were in an early developmental stage when the experiments were initiated.Large-scale experiments such as the ones that we describe require time to develop and stabilize before parameters of community structure like population abundance, can be employed to detect changes associated with water quality manipulations.

Permutation Versus Bootstrap Significance Tests in Multiple Regression and Anova

Kempthorne’s (1952) formulation of the randomization test is extended to yield a permutational analog of the bootstrap significance test. In the new test, residuals of a multiple regression are permuted instead of being bootstrapped. The test is an attractive alternative for Oja’s test that permutes predictors (Austr. J. Statist. 29, 91–100, 1987).