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Inferring Past Environments from Biological Data: Progress – Problems - Potentialities

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Inferring Past Environments from Biological Data: Progress – Problems - Potentialities

John Birks

University of Bergen, University College London, and University of Oxford

INQUA July 2011

Early attempts at quantitative environmental reconstructions used presence of one or more ‘indicator species’ (e.g. Andersson, Samuelsson, Iversen, Grichuk, Coope). Major development in Quaternary science occurred in 1971 with publication of the classic paper by Imbrie & Kipp.

Paper laid the foundation of calibration functions (transfer functions) as a tool for the quantitative reconstruction of past environments using the whole fossil assemblage, not just a few indicator species. Paradigm shift.

Y - biological responses (‘proxy data’)

X - set of environmental variables that are assumed to be causally related to Y(e.g.SST)

B - set of other environmental variables that together with X completely determine Y (e.g. trace nutrients)

If Y is totally explicable as responses to variables represented by X and B, we have a deterministic model (no allowance for random factors, historical influences, etc.)

Y = XB

Imbrie & Kipp (1971)

Y = XB

If B = 0 or is constant, we can model Y in terms of X and Re, a set of ecological response functions

Y = X (Re)

In palaeoecology we need to know Re. We cannot derive Re deductively from ecological studies and thus cannot build an explanatory model from our currently poor ecological knowledge.

Instead we have to use direct empirical models based on observed patterns of Y in modern surface-samples in relation to X, to derive U, our empirical calibration function.

Y = XU

Imbrie & Kipp (1971)

1,

, m taxa

Y

f

t

samples

Fossil data (e.g. pollen) ‘Proxy data’

Environmental variable (e.g. temperature)

1 variable

X

f

Unknown

To be estimated or reconstructed

t

samples

To solve for Xf, need modern data about species and climate from n samples

1,

, m taxa

Ym

n

samples

Modern biology (e.g. pollen)

Modern environment (e.g. temperature)

1 variable

Xm

n

samples

Model Ym in relation to Xm to derive modern calibration function Ûm

Apply Ûm to Yf to estimate past environment Xf

- Potential problems – discuss later

Calibration-in-space

Xm

Ym

transfer function

Ûm

Yf

Xf

Based on an unpublished diagram by Steve Juggins

Marine planktonic foraminifera - Imbrie & Kipp 1971

Foraminifera are a function of sea-surface temperature

Foraminifera can be used to reconstruct past sea-surface temperature

Pollen is a function of regional vegetation

Regional vegetation is a function of climate

Pollen is an indirect function of climate and can be used to reconstruct past regional climate

Chironomids (aquatic non-biting midges) are a function of lake-water temperature

Lake-water temperature is a function of climate

Chironomids are an indirect function of climate and can be used to reconstruct past climate

Freshwater diatoms are a function of lake-water chemistry

Diatoms can be used to reconstruct past lake-water chemistry

- Contain many taxa (200-300)
- Contain many zero values (absences)
- Commonly expressed as proportions or percentages - "closed" compositional data
- Quantitative data are highly variable, invariably show a skewed distribution. Few common taxa, many rare taxa
- Multicollinearity between variables
- Can show spatial autocorrelation e.g. forams, dinocysts, pollen
- Taxa generally have non-linear relationship with their environment, and the relationship is often a unimodal function of the environmental variables

A straight line displays the linear relation between the abundance value (y) of a species and an environmental variable (x). Modelled by linear regression.

LINEAR

A unimodal relation between the abundance value (y) of a species and an environmental variable (x). (u=optimum or mode; t=tolerance; c=maximum). Modelled by Gaussian logit regression (GLR)

UNIMODAL

- Generally few variables, often show a skewed distribution
- Strong multicollinearity (e.g. July mean temperature, growing season duration, annual mean temperature)
- Often difficult to obtain (few modern climate stations, corrections for altitude of sampling sites, etc.)
- Strong spatial autocorrelation (tendency of values at sites close to each other to resemble one another more than randomly selected sites). Values at one site can be partially predicted from its values at neighbouring sites. Problem of nearly all data in real world. Recognised by Francis Galton in 1889. First methods to eliminate spurious correlation due to spatial position developed by ‘Student’ in 1914.

Since 1971, calibration functions widely used in palaeoceanography, terrestrial palaeoecology, and palaeolimnology

Used with wide range of biological proxies

- foraminifera, radiolaria, marine diatoms, coccolithophores
- pollen, testate amoebae, mollusca, bryophytes, plant macrofossils
- diatoms, chrysophytes, chironomids, ostracods, cladocerans

Now many different numerical methods – at least 26 methods published

Reconstruction methods can be divided into three main types(Birks et al. 2010)

- Indicator-species approach – one or many taxa considered as presence/absence
- Similarity-based assemblage methods involving a quantitative comparison between past assemblages Yf and modern assemblages Ym (e.g. MAT, response surfaces)
- Multivariate calibration methods involving a quantitative calibration functionÛm estimated from Xm and Ym, modern calibration or training data-set (e.g. weighted averaging regression and calibration)

Concentrate on calibration-function approach

1. Basic Numerical Models

- Classical Approach

(1)Y = f(X) + error

Biology

Environment

(2)Estimate f by some mathematical procedure and 'invert' estimated (f) to find unknown past environment Xffrom fossil data Yf

Xf f-1(Yf)

- Inverse Approach

In practice, for various mathematical reasons, do an inverse regression or calibration

(3)X = g(Y) + error

(4)Xf = g(Yf)

Obtain 'plug-in' estimate of past environment Xf from fossil data Yf

f or g are calibration functions

Easier to compute and nearly always performs as well as classical approach

- 2. Assumed Species Response Model
- Linear or unimodal
- No response model assumed (linear or non-linear)
- 3. Dimensionality of Model
- Full (all species considered)
- Reduced (selected components of species used)
- 4. Estimation Procedure for Model
- Global (estimate parametric functions, extrapolation possible)
- Local (estimate non-parametric functions, extrapolation not possible)

I = inverse; C = classical

L = linear; U = unimodal; NA = not assumed;

R = reduced dimensionality; F = full dimensionality;

G = global parametric estimation; Ln = local non-parametric estimation

CF = calibration-function based; S = similarity-based

Good reasons for preferring methods with assumed biological response model, full dimensionality, and global parametric estimation(ter Braak (1995), ter Braak et al. (1993), etc.)

- Can test statistically if taxon A has a statistically significant relation to particular environmental variables
- Can develop ‘artificial’ simulated data with realistic assumptions for numerical ‘experiments’
- Such methods have clear and testable assumptions – less of a ‘black box’ than e.g. artificial neural networks
- Can develop model evaluation or diagnostic procedures analogous to regression diagnostics in statistical modelling
- Having a statistical basis, can adopt well-established principles of statistical model selection and testing. Minimises ‘ad hoc’ aspects of MAT

“To make sense of an observation, everyone needs a model … whether he or she knows it or not” Marc Kéry (2010)

- Need biological system with abundant fossils that is responsive and sensitive to environmental variables of interest.
- Need a large, high-quality training set of modern samples. Should be representative of the likely range of variables, be of consistent taxonomy and nomenclature, be of highest possible taxonomic detail, be of comparable quality (methodology, count size, etc.), and be from the same sedimentary environment.
- Need fossil set of comparable taxonomy, nomenclature, quality, and sedimentary environment.

4.Need robust statistical methods for regression and calibration that can adequately model taxa and their environment with the lowest possible error of prediction and the lowest bias possible and sound methods for model selection.

5.Need means of establishing if reconstruction is statistically significant.

6.Need statistical estimation of standard errors of prediction for each reconstructed value.

7.Need statistical and ecological evaluation and validation of the reconstruction and of each reconstructed value.

PC1

PC2

Ym

Xm

PC3

Major Methods Used

Principal components regression (PCR) = Imbrie & Kipp (1971) approach

Multiple linear regression or quadratic regression of X on PC1, PC2, PC3, etc, to derive Ûm. Express Yf as principal components and apply Ûm to estimate Xf

Principal components maximise variance withinYonly

Selection of PCA components done visually until recently. Now cross-validation is used to select model with fewest components, lowest root mean square error of prediction (RMSEP), & lowest maximum bias. ‘Minimal adequate model’ in statistical modelling

Inverse, linear, reduced dimensionality, global estimation. Linear response model is assumed, although non-linear responses are possible.

Gaussian logit regression (GLR) and maximum likelihood (ML) calibration

ter Braak & van Dam (1989)

b0, b1, b2

ML calibration

Ym + Xm

Yf

Xf

b0, b1, b2

environmental reconstruction

modern data

fossil data

b0, b1, b2

taxon GLR regression coefficients for all taxa Ûm

Classical, unimodal, full dimensionality, global estimation. Robust to spatial autocorrelation. Can be computationally difficult. ML finds the most likely value of Xf that maximises the likelihood function given Yf and Ûm

Two-way weighted averaging regression and calibration (WA)

ter Braak & van Dam (1989); Birks et al. (1990)

U1

WA regression

WA calibration

Ym + Xm

Yf

Xf

U2

modern data

environmental reconstruction

fossil data

Ut

taxa WA optima ‘calibration function’ Ûm

Inverse, unimodal, full dimensionality, global parametric estimation. Robust to spatial autocorrelation. First used in Quaternary science by Lynts and Judd (1971) Science 171: 1143-1144

- Ecologically plausible – based on unimodal species response model.
- Mathematically simple but has a rigorous mathematical theory. Properties fairly well known now.
- Empirically powerful:
- does not assume linear responses
- not hindered by too many taxa, in fact helped by many taxa! Full dimensionality
- relatively insensitive to outliers

- Tests with simulated and real data – at its best with noisy, taxon-rich compositional percentage data with many zero values over long environmental gradients.
- Because of its computational simplicity, can derive error estimates for predicted inferred values by bootstrapping.
- Does well in ‘non-analogue’ situations as it is not based on the assemblage as a whole but on INDIVIDUAL taxa optima and/or tolerances. Robust to spatial autocorrelation. Globalparametric estimation.
- Ignores absences of taxa.

WA

WA GLR

GLR

pH

- Weaknesses
- Sensitive to distribution of environmental variable in training set, leading to ‘edge effects’ where responses are truncated.

J. Oksanen (2002)

2.Disregards residual correlations in biological data.

Can extend WA to WA-partial least squares to include residual correlations in biological data in an attempt to improve estimates of taxon optima

Weighted averaging partial least squares regression and calibration (WA-PLS)ter Braak & Juggins (1993) and ter Braak et al. (1993)

WA-PLS regression

PLS1

WA-PLS calibration

Ym

Xm

βm

Yf

Xf

PLS2

coefficients (Ûm)

PLS3

Components selected to maximise covariance between taxon weighted averages and environmental variable X

Selection of number of PLS components to include based on cross-validation. Model selected should have fewest components possible and low RMSEP and maximum bias – minimal adequate model. Inverse, unimodal, reduced dimensionality, global parametric estimation. Can be sensitive to spatial autocorrelation.

Comparison of different methods

Imbrie & Kipp (1971) data

Model performance statistic is root mean squared error of prediction (RMSEP) based on leave-one-out cross-validation

Linear

Unimodal

Shows importance of using a unimodal-based method(ter Braak et al. (1993))

Besides the development of new methods for deriving calibration functions and of modern calibration data-sets, there have been major developments in model evaluation and selection and in reconstruction assessment, namely statistics of calibration functions and in understanding the strengths and weaknesses of different methods and in their underlying theory

1. Model evaluation and selection

Tendency to use several different methods and to select so-called ‘best’ method. Resulted in a shift from an obsession with the model with lowest RMSEP or, even worse, the highest r2.

More concern with model performance statistics including estimates of bias and number of components fitted (e.g. in WA-PLS).

Model performance usually based on some form of internal cross-validation (leave-one-out, n-fold cross-validation, or bootstrapping) or external cross-validation with independent test-set.

van der Voet (1994) randomisation test of models helps find ‘minimal adequate model’ (MAM).

Model with good performance statistics and fewest number of fitted parameters. May be more than one MAM.

More work needed on model selection using criteria like Akaike Information Criterion (AIC) where unnecessary parameters are penalised. Active research area in ecology and evolutionary biology today.

Of course, performance of modern model is being assessed with other modern data, not with fossil data! Major problem.

2. Effects of spatial autocorrelation

Estimating model performance in terms of RMSEP, r2, maximum bias, etc, assumes that the test-set is statistically independent of the training-set. Cross-validation in presence of spatial autocorrelation violates this assumption as test samples are not spatially and statistically independent.

Spatial autocorrelation property of almost all environmental data and much ecological and biological data.

Telford & Birks 2005 Quat. Sci. Rev. 24: 2173-2179

Telford 2006 Quat. Sci. Rev. 25: 1375-1382

Telford & Birks 2009 Quat. Sci. Rev. 28: 1309-1316

Results show the apparent performance of some models is enhanced as a result of spatial autocorrelation in oceans and on land

Problems in finding spatially independent test-sets.

Telford & Birks (2009) have developed methods for cross-validating a calibration function in presence of spatial autocorrelation, h-block cross-validation

3. Partitioning Root Mean Square Error of Prediction

Model uncertainty commonly expressed as RMSEP

4. Testing the statistical significance of a quantitative palaeoenvironmental reconstruction

All calibration-function programs will produce output or ‘reconstruction’

Does the resulting reconstruction explain more of the variance in the fossil data than most (say 95%) reconstructions derived from calibration functions trained on random environmental data?

If it does, then it is statistically significant.

Telford & Birks 2011 Quat. Sci. Rev. 30: 1272-1278

5. Evaluation of individual reconstructed estimates

Assuming overall reconstruction is statistically significant, some individual estimates may be less reliable than others (poor preservation, unusual composition or peak, etc). Need to evaluate individual reconstructed values.

- Goodness-of-fit measures for each individual fossil sample, as in regression modelling (Birks et al. 1990)
- Analogue statistics (Birks et al. 1990; Simpson 2007)
- Proportions of taxa in fossil assemblage absent or rare in modern training data with no or poorly estimated taxon parameters (Birks 1998)
- Sample-specific errors for reconstructed values estimated by bootstrapping or Monte Carlo simulation (Birks et al. 1990)

What to do with sample-specific errors?

Has a statistically significant (p=0.009) reconstruction but there is also a continuous overlap in RMSEP. Problems of temporal autocorrelation in assessing RMSEP for samples.

Birks & Peglar (unpub.)

6. Highlighting ‘signal’ from ‘noise’ in reconstructions

Use of LOESS smoother a great help

Sample-specific errors

or

LOESS smoother

Seppä & Birks (2002)

Brooks & Birks (2001)

7. Ecological validation

Compare reconstructed values with historical data. Rarely possible as few historical data exist.

Renberg & Hultberg (1992)

But when done, sometimes the model that gives the closest correspondence is not the model with lowest RMSEP or maximum bias!

Conflict between model performance and selection based on cross-validation of modern data and validation results using independent historical test-sets

8. Palaeoecological validation by multi-proxy data

Birks & Ammann (2000)

Similar trends, different absolute values. Not surprising, given different biology of different groups of organisms

- Violation of assumptions
- Multiple –variable reconstructions
- Spatial autocorrelation and non-independent test-set
- Confounding effects of correlated environmental variables
- Assumption of uniformitarianism
- Different results from different proxies

- The biggest set of problems is that the calibration-function approach, like any other quantitative procedure, makes assumptions, as originally stated by Imbrie & Kipp (1971), Imbrie & Webb (1981), and Birks et al. (1990).
- These assumptions are being increasingly violated, especially in the last 5-10 years.
- What are these assumptions?

- Assumptions in quantitative palaeoenvironmental reconstructions

- Taxa in training set (Ym) are systematically related to the physical environment (Xm) in which they live
- Environmental variable (Xf , e.g. summer temperature) to be reconstructed is, or is linearily related to, an ecologically important variable in the system
- Taxa in the training set (Ym) are the same as in the fossil data (Yf) and their ecological responses (Ûm) have not changed significantly over the timespan represented by the fossil assemblage
- Mathematical methods used in regression and calibration adequately model the biological responses (Um) to the environmental variable (Xm)
- Otherenvironmental variables than, say, summer temperature have negligible influence, or their joint distribution with summer temperature in the fossil set is the same as in the training set
- In model evaluation by cross-validation, the test-data are independent of the training data
- Imbrie & Kipp (1971), Imbrie & Webb (1981), Birks et al. (1990), Telford & Birks (2005)

- Multiple-variable reconstructions
- Increasing tendency to reconstruct 2 or 3, even 7-8, environmental variables that on the basis of current ecological knowledge of, e.g., vegetation, chironomids, or diatoms, cannot all be ‘ecologically important’ (assumption 2)
- e.g. mean January, mean July, mean annual temperature, growing degree days above 0C and above 5C, annual precipitation, and evaporation : potential evaporation.
- Ecological data are not usually influenced by 8 independent ‘ecologically important’ variables. Usually only 1-3 significant ordination axes.
- All variables may be statistically significant in a RDA or CCA when considered individually(‘marginal’ effects) but almost certainly not significant when considered together (‘conditional’ effects, high multicollinearity, variance inflation factors). Many reconstructions of, for example, ‘distance to littoral vegetation’ suspect.

- Problems of spatial autocorrelation and lack of independence in cross-validation test-data (assumption 6)
- Spatial autocorrelation results in highly optimistic model performance statistics when data are spatially autocorrelated, especially when MAT (or ANN) are used (local non-parametric estimation procedures)
- h-block cross-validation – allows for spatial autocorrelation, unlike leave-one-out c-v
- Increase of RMSEP in h-block c-v for all data except diatoms & pH.Greatest increase in RMSEP always in MAT and in training-sets with the highest spatial autocorrelation in the X variable (forams & salinity; forams & SST; pollen & July sunshine)

4.Confounding effects of correlated environmental variables

Present in all studies, starting with Imbrie & Kipp (1971) with reconstructions of summer and winter sea-surface temperature and salinity.

Covarying environmental variables e.g. temperature and lake trophic status (e.g. total N or P) or temperature and lake depth and chironomids. Is the fossil chironomid signal temperature or trophic status?

Broderson & Anderson (2002)

In almost all ecological systems, assemblages are a complex function of multiple climatic, edaphic, land-use, biotic, and historical factors.

First part of assumption 5 (environmental variables other than the variable being reconstructed have negligible influence) is therefore almost never met. Need very careful design of modern training-set and rigorous statistical analysis to establish what can reliably and significantly be reconstructed.

Second part of assumption 5 (the joint distribution of additional variables with the variable of interest does not change with time) is also violated in many cases.

Climate model and glaciological results suggest that the joint distribution between summer temperature and winter accumulation has not been the same in the past 11,000 years.

Good evidence to suggest that lake-water pH has decreased naturally (soil deterioration) whilst summer temperature rose and then fell in the last 11,000 years. pH-climate relationship changed with time.

In Norway today, lake-water pH is negatively correlated with summer temperature because lakes of pH 6-7.5 are on basic rock and this happens in Norway to occur mainly at high altitudes and hence at low temperatures. In the past after deglaciation, almost all lakes had a higher pH than today, so the pH-temperature relationship in the past was different than today.

5.Assumption 3 “Taxa in the training-set are the same as in the fossil data and their ecological responses have not changed significantly over the timespan represented by the fossil assemblage”

Assumption not unique to calibration functions. Basic assumption of all Quaternary palaeoecology, namely uniformitarianism.

Considerable interest in niche-conservatism amongst biogeographers and conservation and evolutionary biologists. Increasing evidence for conservatism of ecological niche characteristics in the timespan of last 20,000 years.

Problems of ‘cryptic’ species and of taxa like Saxifraga oppositifolia-type in environmental reconstructions currently unresolved.

6.Use of different proxies can give different reconstructions

Mean July temp, Bjørnfjell

p = 0.001

p = 0.183 ns

Validate using another proxy – e.g. macrofossils of tree birch

Validate using second proxy – e.g. chironomids

Importance of independent validation and establishing what is statistically significant

Quantitative palaeoenvironmental reconstructions in the context of Quaternary palaeoecology are not really an end in themselves (in contrast to Quaternary palaeoclimatology) but they are a meansto an end.

Use the reconstructions based on one proxy (e.g. chironomids) to provide an environmental history against which observed biological changes in another, independent proxy (e.g. pollen) can be viewed and interpreted as biological responses to environmental change.

Minden Bog, Michigan Booth & Jackson (2003)

Black portions = wet periods, grey = dry periods

Major change 1000 years ago towards drier conditions, decline in Fagus and rise in Pinus in charcoal

Climate vegetation fire frequency

These approaches involving environmental reconstructions independent of the main fossil record can be used as a long-term ecological observatory or laboratory to study long-term ecological dynamics under a range of environmental conditions, not all of which exist on Earth today (e.g. lowered CO2 concentrations, low human impact).

Can begin to study the Ecology of the Past.

Exciting prospect, many potentialities in future research, as outlined by Flessa and Jackson (2005)and discussed by Birks et al. (2010 Open Ecol J 3: 68-110)

- Effective use of calibration functions needs
- good understanding of underlying ecology, mathematics, and principles of statistical modelling and cross-validation
- good quality modern and fossil data

Bayesian framework is an important future research direction but it presents very difficult and time-consuming computational problems.

Importance of research collaboration between palaeoecologists and applied statisticians

Simple two-way weighted-averaging hard to beat

Takes account of % data, ignores zero values, assumes unimodal responses, can handle several hundred species, and gives calibration functions of high precision (0.8ºC), low bias, and high robustness.

Xm = g(Y1, Y2, Y3, ... ... ..., Yp)Modern data WA regression

Xf = g (Yf1, Yf2, Yf3, ... ... ..., Yfp)Fossil data WA calibration

g is our calibration function for Xm and Ym

^

Simple, ecologically realistic, and robust

WA is robust to spatial autocorrelation, as are Gaussian logit regression and ML calibration

Lynts and Judd 1971 Science 171: 1143-1144

Late Pleistocene Paleotemperatures at Tongue of the Ocean, Bahamas

It too is 40 years old! Has 20 citations (cf. 652)

John Imbrie

Cajo ter Braak

Tom Webb

Svante Wold

Steve Juggins

Richard Telford

One cannot do calibration-function research without high quality data and these need skilled palaeoecologists. Many colleagues have contributed to the development of calibration functions by creating superb modern-environmental data sets

Nilva Kipp

Heikki Seppä

Andy Lotter

Oliver Heiri

Sylvia Peglar

Steve Brooks

Viv Jones

Ulrike Herzschuh

X

X

0

f

Y

Y

0

f

To solve for Xf, model Y0 in relation toX0, derive and apply calibration function F0 to Yf to estimate Xf

^

Fossil data (e.g. pollen)

Environmental variable (e.g. temperature)

1

1,

, m taxa

Known from historical data

p observations

p samples

Unknown, to be reconstructed

t samples

t samples

All done at one site

- Potential problems
- Temporal autocorrelation in Y0 and X0. How many independent samples are there? What is n?
- Chronological – samplecorrelation between Y0 and X0
- Applicability – can the model be applied to other sites other than the site where the calibration is made? Similar problem of applicability with intra-lake approach (Ym and Xm to derive Ûm from one lake applied to other non-training set lakes).

Modern analogue technique (MAT) = k-nearest neighbours (k-NN)

(e.g. Hutson (1980), Guiot (1985), ter Braak (1995), Simpson (2007)

Compare fossil sample (Yf) t with modern sample i

Repeat for all modern samples (Ym)

Calculate DC between t and i

DC=dissimilarity coefficient = proximity measure

Repeat for all fossil samples

Value of k estimated by visual inspection, arbitrary rules (e.g. 10, 20, etc.), or cross-validation

Select k-closest analogues for fossil sample t

Estimate past environment (Xf) for sample t as (weighted) mean of the modern environment (Xm) of the kanalogues

Inverse, no assumed model, full dimensionality, local non-parametric estimation, similarity-based, many tuneable parameters. Very sensitive to spatial autocorrelation. Smooth response surfaces and ANN related to MAT

Bayesian approaches with an implicit consideration of uncertainties potentially valuable but currently only possible at very considerable computer costs.

Relatively small (20-25%) s1 past of RMSEP can be ‘improved’ by new methodological developments, nature monopolises 75-80% of the RMSEP as s2.

Leave-one-out (L-O-O) – no allowance for spatial autocorrelation and h-block cross-validation – allowance for spatial ac

Telford & Birks (2009)

Increase of RMSEP in h-block c-v for all data except diatoms & pH

Greatest increase in RMSEP always in MAT and in training-sets with the highest spatial autocorrelation in the X variable

Diatoms, pH, and climate

Is the reconstruction a reconstruction of pH or climate?

Anderson (2000)

6.Different methods can give very different reconstructions, even though they have similar modern model performances

Birks (2003)

Methods in use

In comparisons using simulated and real data, WA and WA-PLS usually outperform PCR and MAT but not always.

Classical methods of Gaussian logit regression and calibration currently rarely used (freshwater, terrestrial). Some applications of ANNs and few studies within a Bayesian framework.

Bayesian framework is an important future research direction but it presents very difficult and time-consuming computational problems.