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 R functions e
, a set of ecological response
Y = X (R e
)
In palaeoecology we need to know R derive R e currently poor ecological knowledge.
e
. We cannot deductively from ecological studies and thus cannot build an explanatory model from our
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)

Fossil data (e.g. pollen)
‘Proxy data’
1, , m taxa t samples
Y f
Environmental variable
(e.g. temperature)
1 variable t samples
X f Unknown
To be estimated or reconstructed
To solve for X f
, need modern data about species and climate from n samples

Modern biology
(e.g. pollen)
1, n samples
Y m
, m taxa
Modern environment
(e.g. temperature)
1 variable n samples
X m
Model Y m in relation to X m calibration function Û m to derive modern
Apply Û m to Y f to estimate past environment X f
- Potential problems – discuss later

X m
Calibration-inspace
Y m
Y f
Û m transfer function
X f
Based on an unpublished diagram by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 lakewater 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

LINEAR
A straight line displays the linear relation between the abundance value (y) of a species and an environmental variable (x). Modelled by
linear regression.
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)
1. Indicator-species approach – one or many taxa considered as presence/absence
2. Similarity-based assemblage methods involving a quantitative comparison between past assemblages Y f and modern assemblages
(e.g. MAT, response surfaces)
Y m
3. Multivariate calibration methods involving a quantitative calibration function Û from X m and Y m m estimated
, 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 X f from fossil data Y f
X f
 f -1 (Y f
)

• Inverse Approach
In practice, for various mathematical reasons, do an inverse regression or calibration
(3)
(4)
X = g(Y) + error
X f
= g(Y f
)
Obtain 'plug-in' estimate of past environment X f fossil data Y f f or g are calibration functions from
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)

Principal components regression
(PCR)
Segmented linear inverse regression
I
I
L (U) R
L
Partial least squares (PLS)
Guassian logit regression (GLR)
I L
C U
Two-way weighted averaging (WA) I U
WA-PLS I U
F
R
F
F
R
G
Ln
G
G
G
G
CF
CF
CF
CF
CF
CF
Artificial neural networks (ANN) I NA F Ln CF
Modern analogue technique (MAT) I NA F Ln S
Smooth response surfaces C NA F Ln S
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.)
1. Can test statistically if taxon A has a statistically significant relation to particular environmental variables
2. Can develop ‘artificial’ simulated data with realistic assumptions for numerical ‘experiments’
3. Such methods have clear and testable assumptions
– less of a ‘black box’ than e.g. artificial neural networks
4. Can develop model evaluation or diagnostic procedures analogous to regression diagnostics in statistical modelling
5. 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)

1. Need biological system with abundant fossils that is responsive and sensitive to environmental variables of interest.
2. 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.
3. 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.

Y m
Principal components regression (PCR)
= Imbrie & Kipp (1971) approach
PC1
PC2
PC3
X m
Multiple linear regression or quadratic regression of X on
PC1, PC2, PC3, etc, to derive
Û m
. Express Y components and apply Û estimate X f f as principal m to
Principal components maximise variance within Y only
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) b
0
, b
1
, b
2
Y m
+ X m modern data b
0
, b
1
, b
2
Y f fossil data
ML calibration
X f environmental reconstruction b
0
, b
1
, b
2 taxon GLR regression coefficients for all taxa Û m
Classical, unimodal, full dimensionality, global estimation. Robust to spatial autocorrelation. Can be computationally difficult. ML finds the most likely value of X f that maximises the likelihood function given Y f and Û m

Two-way weighted averaging regression and calibration (WA) ter Braak & van Dam
(1989); Birks et al. (1990)
WA regression
U
1
Y m
+ X m modern data
U
2
U t
Y f fossil data
WA calibration
X f environmental reconstruction taxa WA optima
‘calibration function’
Û m
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

1. Ecologically plausible – based on unimodal species response model.
2. Mathematically simple but has a rigorous mathematical theory. Properties fairly well known now.
3. Empirically powerful: a.does not assume linear responses b.not hindered by too many taxa, in fact helped by many taxa! Full dimensionality c. relatively insensitive to outliers
4. Tests with simulated and real data – at its best with noisy, taxon-rich compositional percentage data with many zero values over long environmental gradients.
5. Because of its computational simplicity, can derive error
estimates for predicted inferred values by bootstrapping.
6. 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. Global
parametric estimation.
7. Ignores absences of taxa.

Weaknesses
1. Sensitive to distribution of environmental variable in training set, leading to ‘edge effects’ where responses are truncated.
WA GLR WA GLR pH
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)
Y m
PLS1
PLS2
PLS3
WA-PLS regression
X m
β m coefficients ( Û m
)
Y f
WA-PLS calibration
X f
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 crossvalidation
RMSEP
PC regression
PC regression with quadratic terms
CA regression
GLR (ML)
WA
WA-PLS
Summer SST Winter SST
2.55
 C 2.57
 C
2.15
 C 1.54
 C
1.72
 C
1.63
 C
2.02
 C
1.53
 C
1.37
 C
1.20
 C
1.07
 C
1.17
 C
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 r 2 .
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, r 2 , 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
Effect of spatial autocorrelation
MAT, ANN High Local, non-parametric estimation
WA-PLS Some Global, parametric + potentially some local estimation
GLR, WA Low Global parametric estimation
Problems in finding spatially independent testsets.
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 s1 Error due to variability in estimates of taxon parameters in training-set (model error or lack of fit) s2 Error due to variation in taxon abundances at a given environmental value
1. Within-lake variability (Heiri et al. 2002)
2. Variability in modern environmental data
(Nilsson et al. 1996)
Models cannot, at present, take account of variation in environmental data
3. Variability in assemblages at a given environmental value due to unknown historical, ecological, stochastic, taphonomic, etc, processes. Unexplained variation
20-25%
75-80% c. 15-20%
25-40% (up to 60%)
10-35%

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
Seppä & Birks (2002)
Brooks & Birks (2001)
Sample-specific errors or
LOESS smoother

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

1. Violation of assumptions
2. Multiple –variable reconstructions
3. Spatial autocorrelation and nonindependent test-set
4. Confounding effects of correlated environmental variables
5. Assumption of uniformitarianism
6. Different results from different proxies

1. The biggest set of problems is that the calibrationfunction 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?

1. Assumptions in quantitative palaeoenvironmental reconstructions
1.
2.
3.
4.
5.
6.
Taxa in training set ( Y m environment ( X m
) are systematically related to the physical
) in which they live
Environmental variable ( X reconstructed is, or is linearily related to, an ecologically important variable in the system f , e.g. summer temperature) to be
Taxa in the training set ( Y m
) are the same as in the fossil data ( Y f their ecological responses ( Û m timespan represented by the fossil assemblage
) and
) have not changed significantly over the
Mathematical methods used in regression and calibration adequately
model the biological responses ( U m
) to the environmental variable ( X m
)
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
Imbrie & Kipp (1971), Imbrie & Webb (1981), Birks et al. (1990),
Telford & Birks (2005)

2. 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 0  C and above 5  C, 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.

3. 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
means to 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 CO
2 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.
X m
= g(Y
1
, Y
2
, Y
3
, ... ... ..., Y p
) Modern data WA regression
X f
= g (Y f1
^
, Y f2
, Y f3
, ... ... ..., Y fp
) Fossil data WA calibration g is our calibration function for X m and Y m
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

Fossil data
(e.g. pollen)
1, , m taxa
Y
0 p samples
Y f t samples
Environmental variable
(e.g. temperature)
1
X
0
Known from historical data p observations
X f
Unknown, to be reconstructed t samples
To solve for X f
, model Y
0 apply calibration function in relation to X
0
F
0 to Y f
, derive and to estimate X f
All done at one site

Potential problems
1. Temporal autocorrelation in n ?
Y
0 and X
0
. How many independent samples are there? What is
2. Chronological – sample correlation between
Y
0 and X
0
3. 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 ( Y
X m to derive Û m other non-training set lakes).
m from one lake applied to and

Modern analogue technique (MAT)
= k-nearest neighbours (k-NN)
(e.g. Hutson (1980), Guiot (1985), ter Braak (1995), Simpson (2007)

Repeat for all fossil samples
Compare fossil sample (Y f with modern sample i
) t
Repeat for all modern samples (Y m
)
Calculate DC between t and i DC=dissimilarity coefficient
= proximity measure
Select k -closest analogues for fossil sample t
Value of k estimated by visual inspection, arbitrary rules (e.g. 10, 20, etc.), or cross-validation
Estimate past environment ( X f
) for sample t as (weighted) mean of the modern environment ( X m
) of the k analogues
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
Diatoms - pH L-O-O
h-block
MAT
0.37
0.37
% change 0%
Benthic forams - salinity L-O-O
h-block
0.14
0.36
% change 171%
Planktonic forams - SST L-O-O 1.2ºC
Pollen – July sunshine
h-block 3.0ºC
% change 150%
L-O-O 2.3%
h-block 7.6%
% change 230%
WA-PLS
0.36
0.36
0%
0.17
0.34
100%
1.7ºC
2.7ºC
58%
3.6%
9.0%
150%
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
Marine studies
(foraminifera, diatoms, radiolaria, coccoliths, dinoflagellate cysts)
Freshwater studies
(diatoms, chironomids, ostracods, cladocera)
Terrestrial studies
(pollen)
PCR, much MAT plus variants, some ANN and response surfaces, very few WA or WA-PLS
WA or WA-PLS, very few MAT or
ANN, no PCR
MAT plus variants, response surfaces, a few ANN, increasingly more WA-PLS, no PCR
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.