Weighted Averaging Partial Least Squares Research Articles

Cladoceran and chironomid assemblages as qualitative indicators of water depth in subarctic Fennoscandian lakes

The relationship between surface-sediment cladoceran and chironomid communities to lake depth was analysed in 53 lakes distributed across timberline in northern Fennoscandia using multivariate statistical approaches. The study sites are small and bathymerically simple, with water depth ranging from 0.85-27.0 m (mean 6.36 m). Maximum lake depth was the most important factor in explaining the cladoceran distributions and the second most important factor in explaining the chironomid distributions in these subarctic lakes, as assessed on the basis of a series of constrained RDAs, Monte Carlo permutation tests, and variance partitioning. Quantitative inference models for maximum lake depth were created for both groups of animals. Well-performing calibration functions for predicting lake depth were obtained in each case using linear partial least squares (PLS) regression and calibration, weighted averaging (WA) with an 'inverse' deshrinking regression, and weighted averaging partial least squares (WA-PLS). Quantitative reconstructions of lake level fluctuations should be possible from cladoceran and chironomid core data with a root mean squared error of prediction (RMSEP), as estimated by jack-knifing, of about 1.6-3.0 m.

An intercontinental comparison of chironomid palaeotemperature inference models: Europe vs North America

Chironomid-temperature inference models based on North American, European and combined surface sediment training sets were compared to assess the overall reliability of their predictions. Between 67 and 76% of the major chironomid taxa in each data set showed a unimodal response to July temperature, whereas between 5 and 22% of the common taxa showed a sigmoidal response. July temperature optima were highly correlated among the training sets, but the correlations for other taxon parameters such as tolerances and weighted averaging partial least squares (WA-PLS) and partial least squares (PLS) regression coefficients were much weaker. PLS, weighted averaging, WA-PLS, and the Modern Analogue Technique, all provided useful and reliable temperature inferences. Although jack-knifed error statistics suggested that two-component WA-PLS models had the highest predictive power, intercontinental tests suggested that other inference models performed better. The various models were able to provide good July temperature inferences, even where neither good nor close modern analogues for the fossil chironomid assemblages existed. When the models were applied to fossil Lateglacial assemblages from North America and Europe, the inferred rates and magnitude of July temperature changes varied among models. All models, however, revealed similar patterns of Lateglacial temperature change. Depending on the model used, the inferred Younger Dryas July temperature decrease ranged between 2.5 and 6°C.

Quantitative estimates of water tables and soil moisture in Holocene peatlands from testate amoebae

Changes in surface wetness on Holocene ombrotrophic mires have principally been estimated from plant macrofossils and humification. Testate amoebae (Protozoa: Rhizopoda) provide an additional technique and have the potential to provide improved quantitative estimates of water-table depths and soil moisture. The relationship between hydrology and testate amoebae assemblages from 163 samples on nine British mires is explored using canonical correspondence analysis (CCA). Mean annual water-table depth and percentage soil moisture are two of the most important environmental variables related to the distribution of testate amoebae within peat. Transfer functions for these variables are developed using four underlying models; weighted aver aging (WA), tolerance downweighted weighted averaging (WA-Tol), weighted averaging partial least squares (WA-PLS) and partial least squares (PLS). In ‘jack-knifed’ validation, WA produced the lowest prediction errors for water table, but was outperformed by WA-Tol for percentage moisture. WA and WA-Tol based transfer functions are then applied to a fossil data set from Bolton Fell Moss, Cumbria. This methodology offers a new technique for reconstructing surface wetness changes on British ombrotrophic and oligotrophic mires and provides data in terms of a meaningful environmental parameter. The cosmopolitan distribution of testate amoebae species suggests that the technique has a much wider geographical potential.

An expanded surface-water palaeotemperature inference model for use with fossil midges from eastern Canada

Using an expanded surface sample data set, representing lakes distributed across a transect from southernmost Canada to the Canadian High Arctic, a revised midge-palaeotemperature inference model was developed for eastern Canada. Modelling trials with weighted averaging (with classical and inverse deshrinking; with and without tolerance downweighting) and weighted averaging partial least squares (WA-PLS) regression, with and without square-root transformation of the species data, were used to identify the best model. Comparison of measured and predicted temperatures revealed that a 2 component WA-PLS model for square-root transformed percentage species data provided the model with the highest explained variance (r $$_{jack}^2 $$ = 0.88) and the lowest error estimate (RMSEP jack = 2.26 °C). Comparison of temperature inferences based on the new and old models indicates that the original model may have seriously under-estimated the magnitude of late-glacial temperature oscillations in Atlantic Canada. The new inferences suggest that summer surface water temperatures in Splan Pond, New Brunswick were approximately 10 to 12 °C immediately following deglaciation and during the Younger Dryas. During the Allerod and early Holocene, surface water temperatures of 20 to 24 °C were attained. The new model thus provides the basis for more accurate palaeotemperature reconstructions throughout easternmost Canada.

Predicting Epilimnetic Phosphorus Concentrations Using an Improved Diatom-Based Transfer Function and Its Application to Lake Eutrophication Management

A diatom transfer function based on data from 152 lakes in northwest Europe is derived using the technique of weighted-averaging partial least squares (WAPLS) and applied to Lake Søbygård, a shallow, hypertrophic Danish lake, to infer past changes in epilimnetic total phosphorus (TP) concentrations since the 1930s. The results show that the two-component WAPLS model has low prediction errors (RMSEP = 0.21 log10 TP units) and is applicable to lakes distributed throughout northwest Europe, covering a TP range 10−1000 μg TP L-1. The example shows how the method can provide an estimate of baseline conditions and rates of enrichment, allowing managers to set realistic targets for lake restoration.

Diatom-based water chemistry reconstructions from northern Sweden: a comparison of reconstruction techniques

The diatom composition in surface sediments from 119 northern Swedish lakes was studied to examine the relationship with lake-water pH, alkalinity, and colour. Diatom-based predictive models, using weighted-averaging (WA) regression and calibration, partial least squares (PLS) regression and calibration, and weighted-averaging partial least squares (WA-PLS) regression and calibration, were developed for inferences of water chemistry conditions. The non-linear response between the diatom assemblages and pH and alkalinity was best modelled by weighted-averaging methods. The lowest prediction error for pH was obtained using weighted averaging, with or without tolerance downweighting. For alkalinity there was still some information in the residual structure after extracting the first weighted-averaging component, which resulted in a slight improvement of predictions when using a two component WA-PLS model. The best colour predictions were obtained using a two component PLS model. Principal component analysis (PCA) of the prediction errors, with some characteristics of the training set included as passive variables, was performed to compare the results for the different alkalinity predictive models. The results indicate that calibration techniques utilizing more than one component (PLS and WA-PLS) can improve the predictions for lakes with diatom taxa that have broad tolerances. Furthermore, we show that WA-PLS performs best compared with the other techniques for those lakes that have a high relative abundance of the most dominant taxa and a corresponding low sample heterogeneity.

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 (Shelford's law of tolerance) instead of being linear (Beer's 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.

Rescaling species optima estimated by weighted averaging

The common practice of linear deshrinking in weighted averaging is known to be equivalent to a linear rescaling of the estimated species optima. In published lists of species optima, the use of rescaling is recommended, as it allows values derived from different data sets to be compared and used for new inferences, assuming that taxonomic consistency is assured. Rescaling optima is also shown to influence WA estimates of species tolerances. Non-linear rescaling is also discussed, in the form of cubical rescaling and weighted averaging-partial least squares (WA-PLS). The use of a different deshrinking equation in a small data set did lead to similar prediction errors, probably because of the small size of the data set used.