QUANTITATIVE PALAEOECOLOGYLecture 4.Quantitative Environmental Reconstructions

BIO-351

CONTENTSIntroduction Indicator-species approach Assemblage approach Mutual Climate Range method Probability density functionsProxy data General theory Assumptions of transfer functions Linear-based methods Inverse linear regression Inverse multiple linear regression Principal components analysis regression Segmented inverse regression Partial least squaresRequirements biological and statistical

Non-linear (unimodal) based methods Maximum likelihood regression and calibration Weighted averaging regression and calibration Error estimationTraining set assessmentReconstruction evaluationReconstruction validationExamples of weighted-averaging reconstructionsWeighted-averaging assessmentCorrespondence analysis regressionWeighted-averaging partial least-squares (WA-PLS)Pollen-climate response surfacesAnalogue-based approachesConsensus reconstructions and smoothersUse of artificial simulated data setsNo analogue problemMultiple analogue problemMulti-proxy approachesSynthesisCONTENTS (2)

INTRODUCTION

TRANSFER FUNCTION or BIOTIC INDEXCALIBRATION or BIOINDICATION INDICATOR-SPECIES APPROACH = SINGLE SPECIES BIOASSAYASSEMBLAGE APPROACH = MULTI-SPECIES BIOASSAYBirks H. J. B. (1995) Quantitative palaeoenvironmental reconstructions. In Statistical modelling of Quaternary science data (ed D Maddy & J S Brew), Quaternary Research Association pp161254.

ter Braak C. J. F. (1995) Chemometrics and Intelligent Laboratory Systems 28, 165180.

GRADIENT ANALYSIS AND BIOINDICATION Gradient analysis:

Bioindication:Environment gradient

CommunityCommunity

EnvironmentRelation of species to environmental variables or gradientsIn bioindication, use species optima or indicator values to obtain an estimate of environmental conditions or gradient values. Calibration, bioindication, reconstruction.

The thermal-limit curves for Ilex aquifolium, Hedera helix, and Viscum album in relation to the mean temperatures of the warmest and coldest months. Samples 1,2,and 3 represent samples with pollen of Ilex, Hedera, and Viscum, Hedera and Viscum, and Ilex and Hedera, respectively. From Iversen 1944.INDICATOR-SPECIES APPROACH

ASSEMBLAGE APPROACHCompare fossil assemblage with modern assemblages from known environments.Identify the modern assemblages that are most similar to the fossil assemblage and infer the past environment to be similar to the modern environment of the relevant most similar modern assemblages.If done qualitatively, standard approach in Quaternary pollen analysis, etc., since 1950s.If done quantitatively, modern analogue technique or analogue matching.

MUTUAL CLIMATIC RANGE METHOD Grichuk et alUSSR1950s1960sAtkinson et alUK1986, 1987 Coleoptera TMAX - mean temperature of warmest month TMIN - mean temperature of coldest month TRANGE - TMAXTMIN Quote median values of mutual overlap and limits given by the extremes of overlap'.

Thermal envelopes for hypothetical species A, B, and CSchematic representation of the Mutual Climate Range method of quantitative temperature reconstructions (courtesy of Adrian Walking).

ASSUMPTIONS

Species distribution is in equilibrium with climate.Distribution data and climatic data are same age.Species distributions are well known, no problems with species introductions, taxonomy or nomenclature.All the suitable climate space is available for species to occur. ? Arctic ocean, ? Truncation of climate space.Climate values used in MCR are the actual values where the beetle species lives in all its known localities. Climate stations tend to be at low altitudes; cold-tolerant beetles tend to be at high altitudes. ? Bias towards warm temperatures. Problems of altitude, lapse rates.495 climate stations across Palaearctic region from Greenland to Japan.

Climate reconstructions from (a) British Isles, (b) western Norway, (c) southern Sweden and (d) central Poland. TMAX refers to the mean temperature of the warmest month (July). The chronology is expressed in radiocarbon years BPx1000 (ka). Each vertical bar represents the mutual climatic range (MCR) of a single dated fauna. The bold lines show the most probable value or best estimate of the palaeotemperature derived from the median values of the MCR estimates and adjusted with the consideration of the ecological preferences of the recorded insect assemblages.Coope & Lemdahl 1995

Khl et al. (2002) Quaternary Research 58; 381-392Khl (2003) Dissertations Botanicae 375; 149 pp.Khl & Litt (2003) Vegetation History & Archeobotany 12; 205-214Basic idea is the quantify the present-day distribution of plants that occur as Quaternary fossils (pollen and/or macrofossils) in terms of July and January temperature and probability density functions (pdf).Assuming statistical independence, a joint pdf can be calculated for a fossil assemblage as the product of the pdfs of the individual taxa. Each taxon is weighted by the extent of its climatic response range, so 'narrow' indicators receive 'high' weight.The maximum pdf is the most likely past climate and its confidence interval is the range of uncertainty.Can be used with pollen (+/-) and/or macrofossils (+/-).PROBABILITY DENSITY FUNCTIONS

Estimated probability density function of Ilex aquilifolium as an example for which the parametric normal distribution (solid line) fits well the non-parametric distribution (e.g., Kernel function (dashed line) histogram).Distribution of Ilex aquilifolium in combination with January temperature.

Estimated one- and two-dimensional pdfs of four selected species. The histograms (non-parametric pdf) and normal distributions (parametric pdf) on the left represent the one-dimensional pdfs. Crosses in the right-hand plots display the temperature values provided by the 0.5 x 0.5 gridded climatology (New et el., 1999). Black crosses indicate presence, grey crosses absence of the specific taxon. A small red circle marks the mean of the corresponding normal distribution and the ellipses represent 90% of the integral of the normal distribution centred on . Most sample points lie within this range. The interval, however, may not necessarily include 90% of the data points. Carex secalina as an example of an azonally distributed species is an exception. A normal distribution does not appear to be an appropriate estimating function for this species, and therefore no normal distribution is indicated.

Climate dependences of Carpinus (betulus) (C), Ilex (aquilifolium) (I), Hedera (helix) (H), and Tilia (T) and their combination. The pdf resulting from the product of the four individual pdfs (dotted) is similar to the ellipse calculated on the basis of the 216 points with common occurrences for the four taxa (dashed). No artificial narrowing of the uncertainty range is evident.Climate dependencies of Acer (A), Corylus (avellana) (C), Fraxinus (excelsior) (F), and Ulmus (U), and their combination. The pdf (dotted) resulting from the product of the four individual pdfs has a mean very similar to the mean of the pdf (dashed) calculated based on the 1667 points with common occurrences, but its variances are much smaller.

Reconstruction for the fossil assemblage of Grbern. The thin ellipses indicate the pdfs of the individual taxa included in the reconstruction, and the thick ellipse the 90% uncertainty range of the reconstruction result.

Simplified pollen diagram from Grbern (Litt 1994), reconstructed January and July temperature, and 18O (after Boettger et al. 2000).

Reconstructed most probable mean January (blue) and July (red) temperature and 90% uncertainty range (dotted lines)Khl & Litt (2003)

Comparison of the reconstructed mean January temperature using the pdf-method (green) and the analog technique (blue).Bispingen uncertainty range 90%; La Grande Pile 70%.

ENVIRONMENTAL PROXY DATABiological data from palaeoecological studiesPollen, molluscs, foraminifera, macrofossil plant remains, diatoms, chrysophytes, coleoptera, chironomids, rhizopods, moss remains, ostracodsQuantitative counts (usually %)

Ordinal estimates (e.g. 1-5 scale)

Presence-absence data (1/0) at different stratigraphical intervals and hence times

GENERAL THEORYY - biological responses ("proxy data")X - set of environmental variables that are assumed to be causally related to Y (e.g. sea-surface temperatures)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)Y = XBIf B = 0 or is constant, we can model Y in terms of X and Re, a set of ecological response functionsY = X (Re)In palaeoecology we need to know Re. We cannot derive Re deductively from ecological studies. We 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 functions.Y = XU

BIOLOGICAL DATA(e.g. Diatoms, pollen, chironomids)ENVIRONMENTAL DATA(e.g. Mean July temperature)Modern datatraining setFossil dataUnknown To be reconstructed

Outline of the transfer function approach to quantitative palaeoenvironmental reconstruction

GENERAL THEORY OF RECONSTRUCTIONStep 1Regression to estimate modern optima for each specieswhereYm = modern diatom abundanceXm = modern chemical data (e.g. pH) = estimated modern pH optimum for diatom species

REGRESSION ('CLASSICAL REGRESSION')Y = f (X) + ERROREstimate f ( ) from training set by regression. The estimated f() is then inverted to find unknown x0 from fossil y0.INVERSE REGRESSION = CALIBRATION

Plug in estimate given Y0 and g

PROXY-DATA PROPERTIESContain many taxaContain many zero valuesCommonly expressed as percentages - "closed" compositional dataQuantitative data are highly variable, invariably show a skewed distributionNon-quantitative data are either presence / absence or ordinal ranksTaxa generally have non-linear relationship with their environment, and the relationship is often a unimodal function of the environmental variables

SPECIES RESPONSESSpecies nearly always have non-linear unimodal responses along gradientsJ. Oksanen 2002trees(m)

ASSUMPTIONS IN QUANTITATIVE PALAEOENVIRONMENTAL RECONSTRUCTIONSTaxa 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 (m) 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).Other environmental 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. The 'secret assumption' until Telford & Birks (2005).

LINEAR-BASED METHODSINVERSE REGRESSIONJuly temperature = b0 + b1y1 + b2y2 + ... bzyzPinusBetulaspecies parameter[ Y = UX ]'Classical' Regressionpredictor (e.g. environmental variables)response (e.g. biology)Inverse regression is most efficient if relation between each taxon and the environment is LINEAR and with a normal error distribution. Basically a linear model.

Light micrograph of the Quaternary fossil S. herbacea leaf showing epidermal cells and stomata (x40). The cuticle was macerated in sodium hypochlorine (8% w/v) for 2 min and mounted in glycerol jelly with safranin.

CLASSICAL REGRESSION e.g. GLM YXresponsepredictorvariablevariableStomatal density = a + b (CO2) + INVERSE REGRESSIONCO2 = a1+b1 (stomatal density) + YXresponsepredictorvariablevariableCO2 past = a1 + b1 (stomatal density of fossil leaves)

TermRegression StandardtpcoefficienterrorConstant (bo)294.9933.2488.87

Krkenes Lake

Late-glacial CO2 reconstructions at Krkenes, western Norway (38 m a.s.l.)

INVERSE MULTIPLE REGRESSION APPROACH Multiple regression of temperature (Xm) on abundance of taxa in core tops (Ym) (inverse regression). i.e.Approach most efficient if:relation between each taxon and environment is linear with normal error distribution environmental variable has normal distribution Usually not usable because:taxon abundances show multicollinearityvery many taxamany zero values, hence regression coefficients unstablebasically linear model Consider non-linear model and introduce extra terms:Can end up with more terms than samples. Cannot be solved.Hence "ad hoc" approach of Imbrie & Kipp (1971), and related approaches of Webb et al.

Location of 61 core top samples (Imbrie & Kipp 1971)61 core-top samples x 27 taxaPrincipal components analysis 61 samples x 4 assemblages (79%)PRINCIPAL COMPONENTS REGRESSION (PCR)

Abundance of the tropical assemblage versus winter surface temperature for 61 core top samples. Data from Tables 4 and 13. Curve fitted by eyeAbundance of the subtropical assemblage versus winter surface temperature for 61 core top samples. Data from Tables 4 and 13. Curve fitted by eyeAbundance of the subpolar assemblage versus winter surface temperature for 61 core top samples. Data from Tables 4 and 13. Curve fitted by eyeAbundance of the polar assemblage versus winter surface temperature for 61 core top samples. Data from Tables 4 and 13. Curve fitted by eye

General abundance trends for four of the varimax assemblages related to winter surface temperatures. Winter surface temperatures "measured" by Defant (1961) versus those estimated from the fauna in 61 core top samples by means of the transfer function.Imbrie & Kipp 1971

Average surface salinities measured by Defant (1961) versus those estimated from the fauna in 61 core top samples.Summer surface temperatures measured by Defant (1961) versus those estimated from the fauna in 61 core top samples.Imbrie & Kipp 1971

Palaeoclimatic estimates for 110 samples of Caribbean core V12-133, based on palaeoecological equations (Table 12) derived from 61 core tops. Tw = winter surface temperature; Ts = summer surface temperature; = average surface salinity. Salinity

APPROACH AD HOC BECAUSE Why 4 assemblages? Why not 3, 5, 6? No cross-validation

Assemblages inevitably unstable, because of many transformation, standardization, and scaling options in PCA

Assumes linear relationships between taxa and their environment

No sound theoretical basis

Scatter diagrams of: (A) the percent birch (Betula); and (B) the percent oak (Quercus) pollen versus latitude.The thirteen regions for which regression equations were obtained. Bartlein & Webb 1985SEGMENTED LINEAR INVERSE REGRESSION

Regression equations for mean July temperature from the thirteen calibration regions in eastern North AmericaRegion A: 54-71 N; 90-110 WPollen sum: Alnus + Betula + Cyperaceae + Forb sum + Gramineae + Picea + PinusJuly T (oC) = 12.39 + 0.50*Pinus.5 + 0.26*Forb sum + 0.15*Picea.5 (1.61) (.14) (.05) (.10) - 0.89*Cyperaceae.5 0.37*Gramineae 0.03*Alnus (.13) (.08) (.01) R2 = 0.80; adj. R2 = 0.78; Se = 0.96oC n = 114; F = 69.86; Pr = 0.0000Region B: 53-71 N; 50-80 WPollen sum: Abies + Alnus + Betula + Herb sum + Picea + PinusJuly T (oC) = 8.17 + 0.54*Picea.5 + 0.17*Betula.5 - 0.04*Herb sum 0.01*Alnus (2.27) (.19) (.14) (.01) (.01) R2 = 0.70; adj. R2 = 0.70; Se = 1.52oC n = 165; F = 95.48; Pr = 0.0000

Regression equations used to reconstruct mean July temperature at 6000 yr BP.Bartlein & Webb 1985"We selected the appropriate equation for each sample by identifying the calibration region that; (1) contains modern pollen data that are analogous to the fossil sample; and (2) has an equation that does not produce an unwarranted extrapolation when applied to the fossil sample."

Isotherms for estimated mean July temperatures (C) at 6000 yr BP.

Difference map for mean July temperatures (C) between 6000 yr BP and today. Positive values indicate temperatures that were higher at 6000 yr BP than today.

Reconstructions produced by the regression approachElk Lake, Minnesota

Regression equation applications to the Elk Lake pollen data.

Calibration RegionAge RangeR2(varve range)Mean January45-55N, 85-105W320-100840.876Temperature 45-55N, 95-105W10134-116380.842

Mean July40-50N,85-95W320-65620.799Temperature40-50N,85-105W6746-100840.78645-50N,95-105W10134-116380.701

Annual 40-55N,85-105W320-36920.578Precipitation40-50N,85-105W3794-76620.940 45-55N,85-105W7862-116380.578

APPROACHES TO MULTIVARIATE CALIBRATIONChemometrics predicting chemical concentrations from near Infra-red spectraResponsesPredictors

PARTIAL LEAST SQUARES REGRESSION PLSForm of PC regression developed in chemometrics

PCR -components are selected to capture maximum variance within the predictor variables irrespective of their predictive value for the environmental response variable PLS -components are selected to maximise the covariance with the response variables PLS usually requires fewer components and gives a lower prediction error than PCR.Both are biased inverse regression methods that guard against multi-collinearity among predictors by selecting a limited number of uncorrelated orthogonal components.(Biased because some data are discarded).

CONTINUUM REGRESSION

= 0 = normal least square regression = 0.5 = PLS = 1.0 = PCR

PLS is thus a compromise and performs so well by combining desirable properties of inverse regression (high correlation) and PCR (stable predictors of high variance) into one technique.

PLS will always give a better fit (r2) than PCR with same number of components.

BASIC REQUIREMENTS IN QUANTITATIVE PALAEOENVIRONMENTAL RECONSTRUCTIONS 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.Need good independent chronological control for fossil set.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.Need statistical estimation of standard errors of prediction for each constructed value.Need statistical and ecological evaluation and validation of the reconstructions.

A straight line displays the linear relation between the abundance value (y) of a species and an environmental variable (x), fitted to artificial data (). (a=intercept; b=slope or regression coefficient).A unimodal relation between the abundance value (y) of a species and an environmental variable (x). (u=optimum or mode; t=tolerance; c=maximum).

Outline of ordination techniques presented in this paper. DCA (detrended correspondence analysis) was applied for the determination of the length of gradient (LG). LG is important for choosing between ordination based on a linear or on an unimodal response model. Correspondence analysis (CA) is not considered any further because in the microcosm experiment discussed here LG was =

LINEAR OR UNIMODAL METHODSEstimate the gradient length for the environmental variable(s) of interest.Detrended canonical correspondence analysis with x as the only external or environmental predictor. Detrend by segments, non linear rescaling, ? rare taxa downweighted.Estimate of gradient length in relation to x in standard deviation (SD) units of compositional turnover. Length may be different for different environmental variables and the same biological data.pH 2.62 SDalkalinity2.76 SDcolour1.52 SDIf gradient length < 2 SD, taxa are generally behaving monotonically along gradient and linear-based methods are appropriate.If gradient length > 2 SD, several taxa have their optima located within the gradient and unimodal-based methods are appropriate.

Species Sample 'core tops'Imbrie & Kipp (1971) Core-top data

CANONICAL CORRESPONDENCE ANALYSIS1. Forward selection of environmental variables

Winter SST0.73p=0.0182.0%Salinity0.13p=0.0117.0%Summer SST0.02p=0.171.0%

2. Three environmental variables together explain 46.1% of the observed variation in the 61 core tops.3. First axis (1 = 0.75) is significantly different (p = 0.01) from random expectation, indicating that the taxa are significantly related to the environmental variables.

NON-LINEAR (UNIMODAL) METHODS

MAXIMUM LIKELIHOOD PREDICTION OF GRADIENT VALUESBioindication, Calibration, Transfer function, ReconstructionGaussian response model - regression+ We know observed abundances y+ We know gradient values x= Estimate or model the species response curves for all speciesBioindication - calibration+ We know observed abundances y+ We know the modelled species response curves for all species= Estimate the gradient value of xThe most likely value of the gradient is the one that maximises the likelihood function given observed and expected abundances of species

Can be generalised for any response function

Species - pH response curve

GAUSSIAN RESPONSE MODELCan be reparametrised as a generalised linear model:Gradient as a 2nd degree polynomialLogarithmic link functionJ. Oksanen 2002

Globigerina pachyderma (left coiling)Summer sea-surface temperature COptimum = 2GLIM

Globigerina pachyderma (right coiling)

Orbulina universa

Globigerina rubescens

GAUSSIAN LOGIT REGRESSION Imbrie and Kipp 197161 core tops27 taxa

Summer SSTWinter SSTSalinitySignificant Gaussian logit model 19 21 21Significant increasing linear logit model 6 3 4Significant decreasing linear logit model 1 1 0No relationship 1 2 2

MAXIMUM LIKELIHOOD PREDICTION OF GRADIENT VALUESBioindication, Calibration, Transfer function, ReconstructionGaussian response model - regression+ We know observed abundances y+ We know gradient values x= Estimate or model the species response curves for all speciesBioindication - calibration+ We know observed abundances y+ We know the modelled species response curves for all species= Estimate the gradient value of xThe most likely value of the gradient is the one that maximises the likelihood function given observed and expected abundances of species

Can be generalised for any response function

MAXIMUM LIKELIHOOD APPROACHLikelihood is the probability of a given observed value with a certain expected valueMaximum likelihood estimation: expected or reconstructed values that give the best likelihood for the observed fossil assemblages- ML estimates are close to observed values, and the proximity is measured with the likelihood function- commonly we use the negative logarithm for the likelihood, since combined probabilities may be very smallJ. Oksanen 2002

INFERRING PAST TEMPERATURE FROM MULTIVARIATE SPECIES COMPOSITIONModified from J. Oksanen 2002

ROOT MEAN SQUARED ERROR FOR WINTER SST, SUMMER SST, & SALINITY USING DIFFERENT PROCEDURES

(values in brackets are RMSE when taxa with significant fits only are used).

Winter SSTSummer SSTSalinityImbrie & Kipp 1971 linear2.572.550.573Imbrie & Kipp 1971 non-linear1.542.150.571WACALIB 2.1 3.3Maximum likelihood regression and ML calibration3.212.090.711Weighted averaging regression and WA calibration1.972.020.570WA regression and WA calibration with tolerance downweighting1.922.030.560ML regression, WA calibration1.56 (1.56)1.94 (1.94)0.557 (0.656)ML regression, WA calibration with tolerance downweighting1.25 (1.25)1.80 (1.80)0.534 (0.615)WACALIB 3.5 (debugged version!) Maximum likelihood regression and ML calibration1.201.630.54

Only three parameters:- u: location of the optimum on gradient x- h: modal height at the optimum- t: tolerance or width of responseParameters can be estimated with non-linear regression or generalised linear modelsWEIGHTED AVERAGES CAN APPROXIMATE U

WEIGHTED AVERAGINGThe basic idea is very simple.In a lake with a certain pH range, diatoms with their pH optima close to the lakes pH will tend to be the most abundant species present.A simple estimate of the species pH optimum is thus an average of all the pH values for lakes in which that species occurs, weighted by the species relative abundance. (WA regression)Conversely, an estimate of a lake's pH is the weighted average of the pH optima of all the species present. (WA calibration)

Weighted averaging regressionOptimumTolerancewhere Uk is the WA optimum of taxon k tk is WA standard deviation or tolerance of k Yik is percentage of taxon k in sample i Xi is environmental variable of interest in sample iAnd there are i=1,....,n samples and k=1, ....,m taxa

Weighted averaging calibration or reconstructionWAWAtol

Weighted averaging - the simple site average In the simple average all sites where the species is present have equal weight when calculating the optimum.However, the species is likely to be most abundant at sites near the optimum.Therefore samples with high abundance of the species should be given more weight. In weighted averaging this is achieved by weighting the environment variable by a measure of species abundance.

RECONSTRUCTING AN ENVIRONMENTAL VARIABLE FROM A FOSSIL ASSEMBLAGE

ESTIMATION OF SAMPLE-SPECIFIC ERRORS BASIC IDEA OF COMPUTER RE-SAMPLING PROCEDURESTRAINING SET - 178 modern diatom samples and lake-water pHJack-knifingDo reconstruction 177 times. Leave out sample 1 and reconstruct pH; add sample 1 but leave out sample 2 and reconstruct pH. Repeat for all 177 reconstructions using a training set of size 177 leaving out one sample every time. Can derive jack-knifing estimate of pH and its variance and hence its standard error.BootstrapDraw at random a training set of 178 samples using sampling with replacement so that same sample can, in theory, be selected more than once. Any samples not selected form an independent test set. Reconstruct pH for both modern test-set samples and for fossil samples. Repeat for 1000 bootstrap cycles.Mean square error of prediction =1. error due to variability in estimating species parameters in training set (i.e. s.e. of bootstrap estimates)+2. error due to variation in species abundances at a given pH (i.e. actual predic-tion error differences between observed pH and the mean bootstrap estimate of pH for modern samples when in the independent test).Birks et al. 1990

Use of data and the bootstrap distribution to infer a sampling distribution. The bootstrap procedure estimates the sampling distribution of a statistic in two steps. The unknown distribution of population values is estimated from the sample data, then the estimated population is repeatedly sampled to estimate the sampling distribution of the statistic.

ERROR ESTIMATION BY BOOTSTRAPPING WACALIB 3.1+ AND C261 sample training set, draw 61 samples at random with replacement to give a bootstrap training set of size 61. Any samples not selected form a test set.Mean square error of prediction =

error due to variability in estimates of optima and/or tolerances in training set error due to variation in abundances at a given temperature +(s.e. of bootstrap estimates) (actual prediction error differences between observed xi and mean bootstrap estimate s1 s2 +( is mean of for all cycles when sample i is in test set).

For a fossil sample

RMSE = (S1 + S2)

ROOT MEAN SQUARE ERRORS OF PREDICTION ESTIMATED BY BOOTSTRAPPING W AW AtolSummer sea-surface temperature C Training setRMSE total2.312.37RMSE S10.630.70RMSE S22.222.27Fossil samples2.225-2.283-2.2512.296Winter sea-surface temperature C Training setRMSE total2.232.19RMSE S10.620.7RMSE S22.142.07Fossil samples2.156-2.106-2.2012.249Salinity Training setRMSE total0.610.60RMSE S10.110.13RMSE S20.600.59Fossil samples0.603-0.599-0.6070.606

TRAINING SET ASSESSMENTROOT MEAN SQUARED ERROR (RMSE) of CORRELATION BETWEEN rCOEFFICIENT OF DETERMINATIONr2r or r2 measures strength between observed and inferred values and allows comparison between transfer functions for different variables.Also Maximum bias divide sampling interval of xi into equal intervals (usually 10), calculate mean bias for each interval, and the largest absolute value of mean bias for an interval is used as a measure of maximum bias.Note in RMSE the divisor is n, not (n - 1) as in standard deviation. This is because we are using the known gradient values only.

BIAS AND ERRORGood: Prediction root mean squared error (RMSEP)Correlation unreliable: depends on the range of observationsRoot mean squared error RMSEBias b: systematic differenceError : random error about bias.RMSE2 = b2 + 2Must be cross-validated or will be badly biasedJ. Oksanen 2002

ACCURACY OF PREDICTIONRoot Mean Squared ErrorTwo components- Error- BiasRMSE2 = bias2 + error2 Correlation coefficient is dependent on the range of observations - Large range: Large part of variance explainedCross-validation must be used in assessing the prediction accuracy 1: Split sample- Divide your data into training and test data sets2: Jack-knife- For every site i repeat: Remove site i from the data set Estimate species response curves Do the calibration for site i

CROSS-VALIDATIONLeave-one-out ('jack-knife'), each in turn, or divide data into training and test data sets. Leave-one-out changes the data too little, and hence exaggerates the goodness of prediction. K-fold cross-validation leaves out a certain proportion (e.g. 1/10) and evaluates the model for each of the data sets left out.Badly biased unless one does cross-validationJ. Oksanen 2002

CROSS-VALIDATION STATISTICSPARTITIONING RMSEPRMSEPcf.RMSEr jackrr2 jackr2PREDICTED VALUESAPPARENT VALUES or ESTIMATED VALUESmean biasmean biasmaximum biasmaximum bias TRAINING SET ASSESSMENT AND SELECTIONLowest RMSEP, highest r or r2 jack, lowest mean bias, lowest maximum bias.Often a compromise between RMSEP and bias.

SWAP (= Surface Waters Acidification Project) Diatom pH Training Set 178 surface sediments267 taxa in 2 or more samples with 1% or more in sample pH arithmetic mean 4.33 7.25mean = 5.59median = 5.51

Screened to 167 samplespH 4.33 7.25mean = 5.56median = 5.27262 taxaRMSE = 0.297r = 0.933 RMSEP (bootstrapping) = 0.32RMSEP (split-sampling) = 0.31

EnglandNorwayWalesScotlandSweden 551326030

ROOT MEAN SQUARED ERRORS OF PREDICTION FOR THE TRAINING SETWAWAtolRMSEsi10.0720.305RMSES20.3120.371Total RMSE of prediction0.3200.480________________________________________________Cross-validation0.3080.376RMSE(0.269-0.338)(0.287-0.541)

The Round Loch of Glenhead, Galloway

WA pH reconstructions with bootstrap standard errors of prediction

STATISTICAL AND ECOLOGICAL EVALUATION OF RECONSTRUCTIONSINITIAL ASSUMPTIONSTaxa related to physical environment.Modern and fossil taxa have same ecological responses.Mathematical methods adequately model the biological responses.Reconstructions have low errors.Training set is representative of the range of variation in the fossil set.

RECONSTRUCTION EVALUATIONRMSEP for individual fossil samplesMonte Carlo simulation using leave-one-out initially to estimate standard errors of taxon coefficient and then to derive specific sample standard errors, or bootstrapping.Goodness-of-fit statisticsCCA of calibration set, fit fossil sample passively on axis (environmental variable of interest), examine squared residual distance to axis, see if any fossil samples poorly fitted.Analogue statisticsGood and close analogues. Extreme 5% and 2.5% of modern DCs.Percentages of total fossil assemblage that consist of taxa not represented in all calibration data set and percentages of total assemblage that consist of taxa poorly represented in training set (e.g. < 10% occurrences) and have coefficients poorly estimated in training set (high variance) of beta values in cross-validation).< 5% not presentreliable< 10% not presentokay< 25% not presentpossibly okay > 25% not presentnot reliable

ASSESSMENT OF ANALOGUESANALOG, MATChord distance or chi-squared distances.SelectfirstVERY GOOD or CLOSE ANALOGUEfifthGOODtenthpercentiles of all pairs of DC valuesFAIRn samples DC valuesRANDOMISATION TESTSANALOG

Poor fit

Chironomids and climate

Walker et al. 1996

Summer surface-water paleotemperature reconstruction for Splan Pond. For comparison names of climatic events for correlative European time intervals are included. The apparent root-mean-square error of the temperature estimates is 1.32C (10).Percentage abundance of common midge taxa in sediments of Splan Pond, New Brunswick, Canada. For comparison, names of climatic events for correlative European time intervals are included.

Walker et al 1997

Reconstructions of the pH history of Lysevatten based on historical data and inference from the subfossil diatoms in the sediment. Historical data are pH measurements (thin solid line) and indirect data from fish reports and data from other similar lakes (thin broken line). The insert, showing pH variations from April 1961 to March 1962, is based on real measurements. Diatom-inferred values (thick solid line) were obtained by weighted averaging.VALIDATIONDiatoms and pH

Plot of observed vs. inferred annual mean TP concentrations (log g l-1) based on simple WA classical regression of 44 lakes.Comparison of the measured seasonal range in TP concentrations for Mondsee (mean is shown by a line with open circles; minimum and maximum are shown as single lines) with the bootstrap RMSE of prediction for each individual reconstructed TP value (Est_se_p) using the diatom model (Mean boot is shown by a line with filled circles and the lower and upper errors are shown as single lines). All model values are back-transformed to g l-1. Diatoms and total P validation

Measured annual mean TP concentrations (line with open circles) compared with the diatom-inferred TP values calculated as 3-year running means (single line), for the period 1975-93. All model values are back transformed to g l-1.

Baldeggersee frozen core

BaldeggerseeDiatom succession in Baldeggersee freeze-core BA93-C between 1885 & 1993. Only major taxa shownLotter 1998

Measured total phosphorus (TP) during spring circulation compared to diatom-inferred TP values and median grain-size distribution in the Baldeggersee annual layers (see Lotter et al. 1997c). The large filled circles show the measured spring circulation TP values for the upper-most 15m, whereas the horizontal lines represent the annual TP range in the uppermost 15m of the water column. The dots on the right side of the graph represent samples with close (filled dots; 2nd percentile) and good modern analogues (open dots; 5th percentile).

Diatoms and climateDiatom-inferred mean July air temperature (black dots) from sediments of the three study lakes Alanen Laanijrvi, Lake 850, and Lake Njulla including sample specific error estimates (vertical error bars) and 210Pb-dating errors (horizontal error bars) compared with measured July T (grey dots) in Kiruna (for Alanen Laanjrvi) and in Abisko (for Lakes 850 and Njullla) during the past century. Measured July T are corrected for elevation (0.57C per 100m; Laaksonen, 1976) and smoothed (grey line) with a running mean (n = 13). The stippled lines separate periods with apparent 'good' and 'poor' correspondence between diatom-inferred and measured July T in Lakes 850 and Njulla.Bigler and Hall 2003

Chironomids and climatecorresponding to the date obtained at each level.The black line is the chironomid-inferred temperatures with the estimated errors as vertical bars (meanSSE). The horizontal error bars represent an estimated error in dating. The open stars indicate sediment intervals where the instrumental values fall outside the range of chironomid-inferred temperature (meanSSE). The Pearson correlation coefficient r, and associated p-values are presented and indicate statistically significant correlations between measured and chironomid-inferred mean July air temperature at all study sites. The arrows indicate the climate normals (mean 1960-1999).Larocque & Hall 2003Comparison between meteorological data and chironomid-inferred temperatures at each of the 4 study sites. The blue line represents the 5- or 2-year running means of the meteorological data at Abisko and Kiruna respectively, corrected using a lapse rate of 0.57C per 100m. The red line represents the 5-year (for lakes Njulla, 850, and Vuoskkujarvi) or 2-year (for Alanen Laanijavri) running means of the meteorological data

WEIGHTED AVERAGING AN ASSESSMENT

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 responsesnot hindered by too many species, in fact helped by many species!relatively insensitive to outliersTests with simulated and real data at its best with noisy, species-rich compositional percentage data with many zero values over long environmental gradients (> 3 standard deviations).Because of its computational simplicity, can derive error estimates for predicted inferred values.Does well in non-analogue situations as it is not based on the assemblage as a whole but on INDIVIDUAL species optima and/or tolerances.Ignores absences.Weaknesses.

Species packing model: Gaussian logit curves of the probability (p) that a species occurs at a site, against environmental variable x. The curves shown have equispaced optima (spacing = 1), equal tolerances (t = 1) and equal maximum probabilities occurrence (pmax = 0.5). xo is the value of x at a particular site.

Diatoms and pH

Sensitive to distribution of environmental variable in training set.

Considers each environmental variable separately.

Disregards residual correlations in species data.Can extend WA to WA-partial least squares to include residual correlations in species data in an attempt to improve our estimates of species optima.

WEIGHTED AVERAGESWA estimate of species optimum (u) is good if:Sites are uniformly distributed over species rangeSites are close to each otherWA estimates of gradient values (x) are good if:Species optima are dispersed uniformly around xAll species have equal tolerancesAll species have equal modal abundancesOptima are close together y=abundancei=speciesu=optimumj=sitex=gradient value~=WA estimateThese conditions are only true for infinite species packing conditions!

WEIGHTED AVERAGING CONDITIONS JOINTLYBoth species and sites must have uniform and dense distribution over the gradientTo estimate values at gradient ends, some species optima must be outside the gradient endpoints. Result is bias and truncationTo estimate extreme species optima, some sites must be outside the most extreme species optima. Result is bias and truncationConditions 2 and 3 can be satisfied simultaneously only with infinite gradientsWA equations define the two-way reciprocal averaging algorithm of CA -

Ranges and variances of weighted averages are smaller than the range of values that they are based on. Need to 'deshrink' to restore the original range and variance.

BIAS AND TRUNCATION IN WEIGHTED AVERAGESWeighted averages are good estimates of Gaussian optima, unless the response in truncated.Bias towards the gradient centre: shrinking.J. Oksanen 2002

APPROACHES TO MULTIVARIATE CALIBRATIONChemometrics predicting chemical concentrations from near infra-red spectra

CORRESPONDENCE ANALYSIS REGRESSIONRoux 1979Reduced Imbrie & Kipp (1971) modern foraminifera data to 3 CA axes. Then used these in inverse regression.RMSE apparentSummer tempWinter tempPC regression2.55C2.57CCA regression1.72C1.37CWA-PLS1.53C1.17C

WEIGHTED AVERAGING PARTIAL LEAST SQUARES (WA-PLS)Extend simple WA to WA-PLS to include residual correlations in species data in an attempt to improve our estimates of species optima.Partial least squares (PLS)Form of PCA regression of x on yPLS components selected to show maximum covariance with x, whereas in PCA regression components of y are calculated irrespective of their predictive value for x.Weighted averaging PLSWA = WA-PLS if only first WA-PLS component is usedWA-PLS uses further components, namely as many as are useful in terms of predictive power. Uses residual structure in species data to improve our estimates of species parameters (optima) in final WA predictor. Optima of species that are abundant in sites with large residuals are likely to be updated most in WA-PLS.

WEIGHTED AVERAGINGWA1. Take the environmental variable (xi) as the site scores.2. Calculate species scores (optima) (uk) by weighted averaging of site scores WA regression.3. Calculate new site scores by weighted averaging of species scores WA calibration.4. Regress the environmental variable (xi) on the preliminary new site scores and take the fitted values as the estimate of deshrinking regression. where

The weighted averaging (WA) method thus consists of three parts: WA regression, WA calibration and a deshrinking regression. The parts are motivated as follows. A species with a particular optimum will be most abundant in sites with x-values close to its optimum. This motivatesPart 1 (WA regression): Estimate species optima (u*k) by weighted averaging of the x-value of the sites, i.e.

Species present and abundant in a particular site will tend to have optima close to its x-value. This motivatesPart 2 (WA calibration): Estimate the x-value of the sites by weighted averaging of the species optima, i.e.

Because averages are taken twice, the range of the estimated x-values (x*i) is shrunken. The amount of shrinking can be estimated from the training set by regression either (x*i) on (xi) or (xi) on (x*i) proposed by ter Braak (1988) and ter Braak & Van Dam (1989), respectively. Birks et al. (1990a) discuss the virtue of these two deshrinking methods. For establishing the link with PLS we need the latter, inverse deshrinking regression. This method also has the attractive property of giving minimum root mean squared error in the training set. This motivates

Part 3 (deshrinking regression): Regress the environmental variable (xi) on the preliminary estimates (x*i) and take the fitted values as the estimates of (xi).The final prediction formula for inferring the value of the environmental value from a fossil species assemblage is thus

where a0 and a1 are the coefficients of the deshrinking regression and k = a0 + a1*k. The final prediction formula is thus again a weighted average, but one with updated species optima.

The problem of weighted averaging shrinking of range of environmental reconstructionsSolution deshrinking inverse regressionDerive inverse regression coefficientsinitial xi = a + bxiApply regression to reconstructed values to deshrinkfinal xi = (initiali a)/bWhere xi = the measured env var; initial xi = the initial WA estimate of the env var; final xi = the final, deshrunk env var; and a and b are regression coefficients.

FULL DEFINITION OF TWO-WAY WEIGHTED AVERAGINGEstimate species optima (k) by weighted averaging of the environmental variables (x) at the sites

where y+k has + to replace the summation over the subscript, in this case i = 1, ...., n sites.

Estimate the x values of the sites by weighted averaging of the species optima

Because averages are taken twice, the range of the estimated initial x-values (x) is shrunken. Need to deshrink using either(a) Inverse linear regressionThis minimises RMSE in the training setor(b) Classical linear regressionThis deshrinks more than inverse regression and takes inferred values further away from the mean.

For inverse regression and two-way WAFor classical regression and two-way WA

Can also estimate for each species its WA tolerance or standard deviation (niche breadth)

asand use these in a tolerance-weighted estimate of x

WEIGHTED AVERAGING PARTIAL LEAST SQUARES WA-PLS

Centre the environmental variable by subtracting weighted mean.Take the centred environmental variable (xi) as initial site scores (cf. WA/CA)Calculate new species scores by WA of site scores.Calculate new site scores by WA of species scores.For axis 1, go to 6. For axes 2 and more, make site scores uncorrelated with previous axes.Standardise new site scores and (cf. WA/CA) use as new component.Regress environmental variable on the components obtained so far using a weighted regression (inverse) and take fitted values as current estimate of estimated environmental variable. Go to step 2 and use the residuals of the regression as new site scores (hence name partial) (cf WA/CA).Optima of species that are abundant in sites with large residuals likely to be most updated.

DEFINITION OF WA-PLSStep 0Centre the environmental variable by subtracting the weighted mean, i.e.

This simplifies the formulae.Step 1Take the centred environmental variable (xi) as initial site scores (ri)Do steps 2 to 7 for each component:Step 2Calculate new species scores (u*k) by weighted averaging of the site scores, i.e.

Step 3Calculate new site scores (ri) by weighted averaging of the species scores, i.e. new

Step 4For the first axis go to step 5. For second and higher components, make the new site scores (ri) uncorrelated with the previous components by orthogonalization.Step 5Standardise the new site scores (ri).Step 6Take the standardised scores as the new component.Step 7Regress the environmental variable (xi) on the components obtained so far using weights (yi+/y++) in the regression and take the fitted values as current estimates ( ). Go to step 2 with the residuals of the regression as the new site scores (ri).

Method for calculating inferred temperaturesUsing WA inverse deshrinking models, inferred summer surface water temperatures (C) for shallow lakes may be calculated as:

(without tolerance down-shrinking)or(with tolerance down-weighting)

With WA classical deshrinking models, the inferred summer surface water temperatures (C) are calculated as:

(without tolerance down-shrinking)or(with tolerance down-weighting)

Using the WA-PLS models, inferred summer surface water temperatures (C) may be calculated as:

where is the inferred temperature for sample i, a and b are the intercept and slope for the deshrinking equations, yik is the abundance (depending on the model, either expressed as a percent of the total identifiable Chironomidae, or as the square-root of this value) of taxon k in sample i, k is the temperature optimum (C) of species, , is the Beta of species k, and is the tolerance (C) of species k (Fritz et al. 1991; ter Braak 1987; Birks, pers.comm.).

LEAVE-ONE-OUT AND TEST SET CROSS-VALIDATIONPerformance of WA-PLS in relation to the number of components (s): apparent error (RMSE) and prediction error (RMSEP) in simulated data (R = 1 from simulation series III). The estimated optimum number of components is 3 because three components give the lowest RMSEP in the training set. The last column is not available for real data. sTraining setTest setApparentLeave-one-outRMSERMSEPRMSEP16.146.226.6123.374.244.40*32.874.16*4.5742.224.654.9452.014.655.1161.824.505.62ter Braak & Juggins, 1993

The performance of WA-PLS applied to the three diatom data sets in number of components (s) in terms of apparent RMSE and leave-one-out (RMSEP) (selected model). Dataset SWAP Bergen ThamesRMSERMSEPRMSERMSEPRMSERMSEPs10.2760.310*0.3530.3940.3410.35420.2320.3020.2560.318*0.2380.27930.1940.3150.2130.3300.1960.239*40.1730.3270.1920.3350.1660.22450.1530.3440.1740.3590.1530.21960.1340.3690.1640.3740.1400.219Reduction in prediction error (%)01932

Bergen data set: predicted pH and bias as a function of observed pH for components 1 and 2 in WA-PLS. Solid lines represent Clevelands LOESS scatterplot smooth (1979). 19% gain.

Thames data set: predicted salinity and bias as a function of observed salinity for components 1 and 3 in WA-PLS. Solid lines represents Clevelands LOESS scatterplot smooth (1979). Salinity is g-1 and transformed as log10 (salinity 0.08). 32% gain.

The relationship between (a) diatom-inferred TP and (b) residuals (inferred TP observed TP) and observed TP for the one- and two-component WA-PLS models. Solid lines show LOWESS scatter plot smoothers.Summary diatom diagram and reconstructed annual mean TP concentrations using one- and two-component WA-PLS models for Lake SbyGrd, showing standard errors of prediction for the two-component model. 210Pb dates (AD) are shown on the right hand side.Bennion et al. 1996NW EuropeTotal P152 lakes

Imbrie & Kipp (1971) dataWAWA-PLS

ALPE - DIATOM - pH TRAINING SETItalian and Austrian Alps (Aldo Marchetto & Roland Schmidt)31Spanish Pyrenees (Jordi Catalan & Joan Garcia)28ALPE sites (Nigel Cameron & Viv Jones)30Norway (Frode Berge & John Birks)10SWAP Norway & UK (Frode Berge, Roger Flower, Viv Jones)20Total119One 'rogue' sample detected118 samples527 diatom taxapH 4.48 - 8.04median 6.10mean 6.15Gradient length 5.19 standard deviations

ALPE TRAINING SET - 118 SAMPLESSquare root transformationComponentsRMSEr2RMSEPr2 (jack)WA-PLS-10.2990.850.3590.78-20.1780.960.3370.81-30.1310.970.3310.81-40.1000.980.3310.81-50.0750.990.3390.80Select WA-PLS model with 3 components as simplest model (least parameters) that gives lowest RMSEP.

NORWEGIAN CHIRONOMID CLIMATE TRAINING SETLeave-one-out cross validationPredicted air temperature.RMSEP = 0.89CBias = 0.61CPredicted observed air temperature109 samples

Inferred mean July air temperatureOxygen isotope ratios

Precipitation300 - 3537mmMean July7.7 - 16.4CMean January-17.8 - 1.1CNORWEGIAN POLLEN AND CLIMATE

Root mean squared errors of prediction (RMSEP) based on leave-one-out jack-knifing cross-validation for annual precipitation, mean July temperature, and mean January temperature using five different statistical models.PptnJulyJanuary(mm)(C)(C)Weighted averaging (WA) (classical)486.51.332.86Weighted averaging (WA) (inverse)427.21.072.61Partial least squares (PLS)420.10.942.82WA-PLS417.51.032.57Modern analogue technique (MAT)385.30.912.42

Vuoskojaurasj, Abisko, Sweden

Vuoskojaurasj consensus reconstructions

Tibetanus, Abisko ValleyInferred from pollenInferred from pollenHammarlund et al. 2002

Bjrnfjelltjrn, N. Norway

Bjrnfjelltjrn consensus reconstructions

LINEAR AND UNIMODAL-BASED NUMERICAL METHODS

Response modelProblemLinearUnimodalRegressionMultiple linear regressionWeighted averaging (WA) of sample scoresCalibrationLinear calibration 'inverse regression'WA of taxa scores and simple two-way WAPrincipal components regressionCorrespondence analysis regressionPartial least squares (PLS-1)WA-PLS (WAPLS-1)Multivariate calibration(PLS-2)WAPLS-2OrdinationPrincipal components analysis (PCA)Correspondence analysis (CA)Constrained ordination (= reduced rank regression)Redundancy analysis (RDA)Canonical correspondence analysis (CCA)Partial ordinationPartial PCAPartial CAPartial constrained ordinationPartial RDAPartial CCA

RESPONSE SURFACESPollen percentages in modern samples plotted in climate space (cf Iversens thermal limit species +/ plotted in climate space).ContouredTrend-surface analysis R2Bartlein et al. 1986Contoured onlyWebb et al. 1987Reconstruction purposes grid, analogue matchingSimulation purposesPROBLEMS Need large high-quality modern data for large geographical areas. No error estimation for reconstruction purposes. Reconstruction procedure ad hoc grid size, etc.

Response surfaces for individual pollen types. Each point is labelled by the abundance of the type. Many points are hidden only the observation with the highest abundance was plotted at each position. For (a) to (e) + denotes 0%, 0 denotes 0-10%, 1 denotes 10-20%, 2 denotes 20-30% etc. For (f) to (h), + denotes 0%, 0 denotes 0-1%, 1 denotes 1-2%, 2 denotes 2-3%, etc. H denotes greater than 10%.

Percentage of spruce (Picea) pollen at individual sites plotted in climate space along axes for mean July temperature and annual precipitation. (B) Grid laid over the climate data to which the pollen percentage are fitted by local-area regression. The box with the plus sign is the window used for local-area regression. (C) Spruce pollen percentages fitted onto the grid. (D) Contours representing the response surface and pollen percentages shown in part C.

Scatter diagram showing the smoothed distribution of percentages of spruce (Picea) and beech (Fagus) pollen from sediment with modern pollen data in eastern North America when the pollen percentages are plotted at coordinates for modern January and July mean temperature (P.J. Bartlein, unpublished). The arrow indicates the direction and approximate magnitude of temperature change at Montreal since 6000 yr BP.

Reconstruction produced by the response surface approachElk Lake, Minnesota

Fossil and simulated isopoll map sequences for Betula. Isopolls are drawn at 5, 10, 25, 50 and 75% using an automatic contouring program.Simulation purposes

Fossil and simulated isopoll map sequences from Quercus (deciduous). Maps are drawn at 3000-year intervals between 12000 yr BP and the present. The upper map sequence presents the obser-ved fossil and contemporary pollen values. The lower map sequence presents the pollen values simulated, by means of the pollen-climate response surface from the climate conditions obtained by applying to the measured contem-porary climate the palaeoclimate anomalies that Kutzbach & Guetter (1986) simulated using the NCAR CCM, for 12000 to 3000 yr BP. The map for the present is simulated from the measured contemporary climate. Isopolls are drawn at 2, 5, 10, 25, and 50% using an automatic contouring program. Huntley 1992

RESPONSE SURFACES - Ad hocChoice of how much or how little smoothing.

Choice of scale of grid for reconstructions.

No statistical measure of goodness-of-fit.

No reliable error estimation for predicted values.

ANALOGUE-BASED APPROACHPROBLEMSAssessment of most similar?1, 2, 9, 10 most similar?No-analogues for past assemblages.Choice of similarity measure.Require huge set of modern samples of comparable site type, pollen morphological quality, etc, as fossil samples. Must cover vast geographical area.Human impact. Do an analogue-matching between fossil sample i and available modern samples with associated environmental data. Find modern sample(s) most similar to i, infer the past environment for sample i to be the modern environment for those modern samples. Repeat for all fossil samples.

Reconstructions produced using the analogue approachElk Lake, Minnesota

MODIFIED MODERN ANALOGUE APPROACHESJoel GuiotTaxon weightingPalaeobioclimatic operators (PBO) computed from either a time-series of fossil sequence or from a PCA of fossil pollen data from large spatial array of sites.Weights are selected to 'emphasis the climate signal within the fossil data and to 'highlight those taxa that show the most coherent behaviour in the vegetational dynamics', 'to minimise the human action which has significantly disturbed the pollen spectra', 'to reduce noise'.2.Environmental estimates are weighted means of estimates based on 20, 40 or 50 or so most similar assemblages.3.Standard deviations of these estimates give an approximate standard error.

Reconstruction of variations in annual total precipitation and mean temperature expressed as deviations from the modern values (1080 mm and 9.5oC for La Grande Pile. 800 mm and 11oC for Les Echets). The error bars are computed by simulation. The vertical axis is obtained by linear interpretation from the dates indicated in Fig.2Cor is the correlation between estimated and actual data. +ME is the mean upper standard deviation associated to the estimates, -ME is the lower standard deviation. These statistics are calculated on the fossil data and on the modern data. In this case, R must be replaced by C.Guiot et al. 1989

MODERN ANALOGUE TECHNIQUES FOR ENVIRONMENTAL RECONSTRUCTION= K NEAREST NEIGHBOURS (K NN)MAT, ANALOG, C2Modern data and environmental variable(s) of interest.Do analog matches and environmental prediction for all samples but with cross-validation jack-knifing.Find number of analogues to give lowest RMSEP for environmental variable based on mean or weighted mean of estimates of environmental variable. Can calculate bias statistics as well.Reconstruct using fossil data using the optimal number of analogues (lowest RMSEP, lowest bias).Advise chord distance or chi-squared distance as dissimilarity measure. Optimises signal to noise ratio.

Elk Lake climate reconstruc-tion summary. The three series plotted with red, green and blue lines show the reconstructions produced by the individual approaches, the series plotted with the thin black line show the envelope of the prediction intervals, and the series plotted with a thick purple line represents the stacked and smoothed reconstruction of each variable (constructed by simple averaging of the individual reconstructions for each level, followed by smoothing [Velleman, 1980]). The modern observed values (1978-1984) for Itasca Park are also shown.CONSENSUS RECONSTRUCTIONS

PLOTTING OF RECONSTRUCTED VALUESPlot against depth or age the reconstructed values, indicate the observed modern value if known.Plot deviations from the observed modern value or the inferred modern value against depth or age.Plot centred values (subtract the mean of the reconstructed values) against depth or age to give relative deviations.Plot standardised values (subtract the mean of the reconstructed values and divide by the standard deviation of the reconstructed values) against depth or age to give standardised deviations.Add LOESS smoother to help highlight major trends.

LOESS smoother

THE SECRET ASSUMPTION OF TRANSFER FUNCTIONSTelford & Birks (2005) Quaternary Science Reviews 24: 2173-2179Estimating the predictive power and performance of a training set as RMSEP, maximum bias, r2, etc., by cross-validation ASSUMES that the test set (one or many samples) is INDEPENDENT of the training set (The Secret or Totally Ignored Assumption).Cross-validation in the presence of spatial auto-correlation seriously violates this assumption.See Richard Telford's lecture after this lecture.

USE OF ARTIFICIAL, SIMULATED DATA-SETSSIMULATED DATA-SETSGenerate many training sets (different numbers of samples and taxa, different gradient lengths, vary extent of noise, absences, etc) and evaluation test sets, all under different species response models.

NO-ANALOG PROBLEM Probably widespread. Does it matter? Analog-based techniques for reconstruction - YES!Modern analog techniqueResponse surfaces WA and related inverse regression methodsWhat we need are good (i.e. reliable) estimates of k. Apply them to same taxa but in no-analog conditions in the past.Assume that the realised niche parameter k is close to the potential or theoretical niche parameter uk*.WA and WA-PLS are, in reality, additive indicator species approaches rather than strict multivariate analog-based methods. Simulated datater Braak, 1995. Chemometrics & Intelligent Laboratory Systems 28, 165180

L-shaped climate configuration of samples (circles) in the training set (Table 3), with x the climate variable to be calibrated and z another climate variable. Also indicated are the regions of the samples in evaluation set A and set CInverse versus classical methods; method-dependent bias in the leave-one-out error estimate. Comparison of the prediction error of inverse (WA-PLS and k-NN) and classical (MLM) approaches in the training set (t) and the three evaluation sets (B, A and C). Set B is a five time replication of t, set A is a subset of t and set C is an extrapolation set. The data are from simulation series 3 of Ref [53] in which species composition is governed by two climate variable (x and z) with an intermediate amount of unimodality (Rx = Rz = 1).Numbers are geometric means of root mean squared errors of prediction of x in four replications. The coefficients of variation of each mean is ca. 10%. Coefficient of variation of the ratio of 2 means within a column is ca. 15%. The range of x is [0, 100]. The number in superscript is the range of optimal number of components in WA-PLS and the optimal number of nearest neighbours in k-NN in the four replicates. k-NN uses Eq.(3) & (5).a Significant difference (P

General conclusions from simulated data experimentsWA, WA-PLS, Maximum likelihood and MAT all perform poorly and no one method performs consistently better than other methods.For strong extrapolation, WA performed best. Appears WA-PLS deteriorates quicker than WA with increasing extrapolation.Hutson (1977) no-analog conditions WA outperformed inverse regression and PCR.Important therefore to assess analog status of fossil samples as well as best training set in terms of RMSEP, bias, etc.Dynamic training set concept.Analogues (say 1020) for each fossil sample, devise dynamic training set, use linear PLS methods, avoids edge effects, truncated responses, etc.

MULTIPLE ANALOG PROBLEMFossil assemblage is similar to a number of modern samples that differ widely in their modern environment. Happens in pollen studies with training sets covering Europe, N America and parts of Asia. Major taxa only included, e.g. Pinus pollen may dominate northern, Mediterranean and southern assemblages.Constrained analog matching Guiote.g.constrain pollen choices on basis of inferred biome, fossil beetles, inferred lake-level changesConstrained response surfaces ( analog matching) Huntleye.g.constrain area of search on the basis of inferred biome or plant macrofossils

Reconstructed range of July temperatures (oC) at La Grande Pile (Vosges, France) from three methods:a) using beetles alone, b) using pollen alone,c) using pollen constrained by beetles Guiot et al. 1993(a)(b)(c)

Swiss surface pollen samples lake sediments Selected trees and shrubsMULTI-PROXY APPROACHES

Swiss surface lake sediments. Selected herbs and pteridophytes

Root mean squared errors of prediction (RMSEP) based on leave-one-out jack-knifing cross-validation for mean summer temperature (June, July, August), mean winter temperature (December, January, February) and mean annual precipitation using WA-PLS model.RMSEPR2No. of comps.Mean summer temperature1.252C0.903Mean winter temperature1.025C0.883Mean annual precipitation194.1mm0.572

Modern Swiss pollen - climate

Gerzensee, Bernese Oberland, Switzerland

Lotter et al. 2000PB-OYD-PB TrYDAL-YD TrG-OGerzensee

Lotter et al. 2000Gerzensee

Birks & Ammann 2000

GRADIENT LENGTH(Compositional turnover along environmental gradient SD units)

SHORT (4sd)LINEARUNIMODAL-BASED METHODSNOISE OF TRAIN-ING SET DATA VERY LOWLeast squares linear regression and calibration (inverse regression) GLMGaussian logit or multinomial regression and calibration GLM? Generalised additive models GAMLOWPartial least squares PLS regression and calibration PLSWeighted averaging PLS regression and calibration WA-PLS? WA-PLSMED-IUMPartial least squares PLS or robust linear regression and calibrationWeighted averaging regression and calibration WA? WA or WA-PLSHIGHPCA regressionCA-regression? DCA-regression? IDEALPROBLEMS AT GRADIENT ENDSTRY TO AVOID, CAUSES MULTIPLE ANALOG PROBLEM?

Zero valuesHigh sample heterogeneity (root mean squared deviation for samples)High taxon tolerances (root mean squared deviation)Rare taxa% variance in Y explained by X, constrained 1 relative to unconstrained 2 .

HOW TO ESTIMATE NOISE IN REAL DATA?