Download Quantiative environmental reconstructions

QUANTITATIVE
PALAEOECOLOGY
Lecture 4.
Quantitative Environmental Reconstructions
BIO-351
CONTENTS
Introduction
Indicator-species approach
Assemblage approach
Mutual Climate Range method
Probability density functions
Proxy 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 squares
Requirements – biological and statistical
CONTENTS (2)
Non-linear (unimodal) based methods
Maximum likelihood regression and calibration
Weighted averaging regression and calibration
Error estimation
Training set assessment
Reconstruction evaluation
Reconstruction validation
Examples of weighted-averaging reconstructions
Weighted-averaging – assessment
Correspondence analysis regression
Weighted-averaging partial least-squares (WA-PLS)
Pollen-climate response surfaces
Analogue-based approaches
Consensus reconstructions and smoothers
Use of artificial simulated data sets
No analogue problem
Multiple analogue problem
Multi-proxy approaches
Synthesis
INTRODUCTION
‘TRANSFER FUNCTION’ or ‘BIOTIC INDEX’
CALIBRATION or BIOINDICATION
INDICATOR-SPECIES APPROACH = SINGLE SPECIES BIOASSAY
ASSEMBLAGE APPROACH = MULTI-SPECIES BIOASSAY
Birks H. J. B. (1995)
Quantitative palaeoenvironmental reconstructions.
In Statistical modelling of Quaternary science data
(ed D Maddy & J S Brew), Quaternary Research
Association pp161–254.
ter Braak C. J. F. (1995)
Chemometrics and Intelligent Laboratory
Systems 28, 165–180.
GRADIENT ANALYSIS AND BIOINDICATION
Relation of species to environmental variables or gradients
Gradient analysis:
Environment gradient
Community
Bioindication:
Community
Environment
In bioindication, use species optima or indicator values to
obtain an estimate of environmental conditions or gradient
values. Calibration, bioindication, reconstruction.
INDICATOR-SPECIES APPROACH
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.
ASSEMBLAGE APPROACH
Compare 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 al
Atkinson et al
Coleoptera
USSR
UK
TMAX
TMIN
TRANGE
1950s–1960s
1986, 1987
- mean temperature of warmest month
- mean temperature of coldest month
- TMAX–TMIN
Quote median values of mutual overlap and ‘limits given by the
extremes of overlap'.
Thermal envelopes for hypothetical species A, B, and
C
Schematic representation of the Mutual Climate Range
method of quantitative temperature reconstructions
(courtesy of Adrian Walking).
ASSUMPTIONS
1. Species distribution is in equilibrium with climate.
2. Distribution data and climatic data are same age.
3. Species distributions are well known, no problems with species
introductions, taxonomy or nomenclature.
4. All the suitable climate space is available for species to occur. ?
Arctic ocean, ? Truncation of climate space.
5. 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
PROBABILITY DENSITY FUNCTIONS
Kühl et al. (2002) Quaternary Research 58; 381-392
Kühl (2003) Dissertations Botanicae 375; 149 pp.
Kühl & Litt (2003) Vegetation History & Archeobotany 12; 205-214
Basic 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 (+/-).
Distribution of Ilex aquilifolium in
combination with January
temperature.
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).
Estimated one- and two-dimensional pdfs of
four selected species. The histograms (nonparametric pdf) and normal distributions
(parametric pdf) on the left represent the
one-dimensional pdfs. Crosses in the righthand plots display the temperature values
provided by the 0.5º x 0.5º 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 Gröbern. 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 Gröbern (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) Kühl & 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 DATA
• Biological data from palaeoecological studies
• Pollen, molluscs, foraminifera, macrofossil plant remains,
diatoms, chrysophytes, coleoptera, chironomids,
rhizopods, moss remains, ostracods
• Quantitative counts (usually %)
• Ordinal estimates (e.g. 1-5 scale)
• Presence-absence data (1/0) at different stratigraphical
intervals and hence times
GENERAL THEORY
Y - 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 = XB
If B = 0 or is constant, we can model Y in terms of X and Re, a set of ecological
response functions
Y = X (Re)
In palaeoecology we need to know Re. We cannot derive Re deductively from
ecological studies. 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
In practice, this is a two-step process
Regression in which we estimate Û m , modern calibration functions or
regression coefficients
Ym  Uˆm( X m )
or
Training set
Ym modern surface-sample data
Xm associated environmental data
X m  Uˆm1(Ym )
(inverse regression)
Calibration, in which we reconstruct X̂ f , past environment, from fossil core
data
Xˆ f  Uˆm1(Yf )
TRANSFER FUNCTION
Yf fossil core data fossil set
BIOLOGICAL DATA
ENVIRONMENTAL DATA
(e.g. Diatoms, pollen,
chironomids)
(e.g. Mean July temperature)
Modern data
”training set”
1,
,m
taxa
1 variable
Ym
Xm
n
samples
Fossil data
n
samples
1,
,m
taxa
1 variable
Yf
t
samples
Xo
t
samples
Unknown
To be reconstructed
Outline of the transfer function
approach to quantitative
palaeoenvironmental
reconstruction
GENERAL THEORY OF RECONSTRUCTION
Step 1 Regression to estimate modern optima for each
species
Y m  Uˆm( X m)
where
Ym = modern diatom abundance
Xm = modern chemical data (e.g. pH)
species
Ûm = estimated modern pH optimum for diatom
Step 2 Calibration to reconstruct past chemistry
*
X f  Uˆm(Y f )
where
Yf = fossil diatom abundance
Xf = reconstructed past chemistry (e.g. pH)
*
1
Uˆm  Uˆm = inverse of modern species optima from Step 1
REGRESSION ('CLASSICAL REGRESSION')
Y = f (X) + ERROR
Estimate f ( ) from training set by regression. The estimated f ( ) is then
‘inverted’ to find unknown x0 from fossil y0.
 
f Xˆ0  Y0
INVERSE REGRESSION = CALIBRATION
X  g Y   ERROR
Xˆ0  g(Y0 )
‘Plug in’ estimate given Y0 and g
PROXY-DATA PROPERTIES
Contain many taxa
Contain many zero values
Commonly expressed as percentages - "closed" compositional
data
Quantitative data are highly variable, invariably show a skewed
distribution
Non-quantitative data are either presence / absence or ordinal
ranks
Taxa generally have non-linear relationship with their
environment, and the relationship is often a unimodal function
of the environmental variables
SPECIES RESPONSES
Species nearly always have non-linear unimodal responses along
gradients
trees
(m)
J. Oksanen 2002
ASSUMPTIONS IN QUANTITATIVE
PALAEOENVIRONMENTAL RECONSTRUCTIONS
1.
Taxa in training set (Ym) are systematically related to the physical
environment (Xm) in which they live.
2.
Environmental variable (Xf , e.g. summer temperature) to be
reconstructed is, or is linearily related to, an ecologically important
variable in the system.
3.
Taxa in the training set (Ym) are the same as in the fossil data (Yf) and
their ecological responses (Ûm) have not changed significantly over the
timespan represented by the fossil assemblage.
4.
Mathematical methods used in regression and calibration adequately
model the biological responses (Um) to the environmental variable (Xm).
5.
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.
6.
In model evaluation by cross-validation, the test data are independent of
the training data. The 'secret assumption' until Telford & Birks (2005).
LINEAR-BASED METHODS
INVERSE REGRESSION
X m  UˆmYm
July temperature = b0 + b1y1 + b2y2 + ... bzyz
Pinus
Betula
species
parameter
[ Y = UX ]
‘response’
(e.g. biology)
Regression
'Classical'
‘predictor’ (e.g.
environmental
variables)
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
Stomatal density = a + b (CO2) + ε
Y
response
variable
X
predictor
variable
INVERSE REGRESSION
CO2 = a1+b1 (stomatal density) + ε
Y
response
variable

X
predictor
variable
CO2 past = a1 + b1 (stomatal density of fossil leaves)
X m  Y m Uˆm
ˆm
Xf YfU
Regression plot of the
total training set
(n=29) for stomatal
density of Salix
herbacea leaves and
the atmospheric CO2
concentration in
which they grew. The
regression details are
as follows:
Term
Regression
coefficienterror
Standard
t
p
Constant (bo)
294.99
33.248
8.87
<0.001
CO2 (b1)
-0.647
0.155
-5.61
<0.001
R2=0.538
R2adj=0.521
Both terms have regression coefficients significantly different from zero and the variance
ratio (F[1.27] =31.44) exceeds the critical value of F at the 0.01 significance level (7.68),
indicating that stomatal density has a strong statistical relationship with CO2
concentration.
Beerling et al. 1995
Kråkenes Lake
Late-glacial CO2 reconstructions at Kråkenes, 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).
ˆm Ym  b0  b1y1  b2 y 2  b3 y 3 ...  bm y m
Xm  U
Xˆ f  Uˆm Yf  b0  b1y 1  b2 y 2  b3 y 3 ...  bm y m
i.e. Xˆf   0 
m

k 1
k
y ik  
Approach most efficient if:
1. relation between each taxon and environment is linear with normal error distribution
2. environmental variable has normal distribution
Usually not usable because:
1. taxon abundances show multicollinearity
2. very many taxa
3. many zero values, hence regression coefficients unstable
4. basically linear model
Consider non-linear model and introduce extra terms:
X m  b0  b1c1  b2c12  b3c2  b4c22  b5c3  b6c32  ...
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 taxa
Principal 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 eye
Abundance of the subtropical assemblage
versus winter surface temperature for 61
core top samples. Data from Tables 4 and
13. Curve fitted by eye
Abundance of the subpolar assemblage versus
winter surface temperature for 61 core top
samples. Data from Tables 4 and 13. Curve
fitted by eye
Abundance of the polar assemblage versus
winter surface temperature for 61 core top
samples. Data from Tables 4 and 13. Curve
fitted by eye
Now did inverse regression using 4 varimax assemblages rather than the 27
original taxa.
X m  b0  b1 A  b2 B  b3 C  b4 D
Linear
where A, B, C and D are varimax assemblages.
X m  b0  b1 A  b2 B  b3 C  b4 D  b5 AB  b6 AC 
b7 AD  b8 BC  b9 BD  b10 CD  b11 A 2  b12 B 2 
b13 C  b14 D
2
2
Non-linear
CALIBRATION STAGE using the fossil assemblages described as the 4 varimax
assemblages
X f  b0  b1 A f  b2 B f  b3 C f  b4 D f
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
Salinity
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.
APPROACH AD HOC BECAUSE
1. Why 4 assemblages? Why not 3, 5, 6? No crossvalidation
2. Assemblages inevitably unstable, because of many
transformation, standardization, and scaling options in
PCA
3. Assumes linear relationships between taxa and their
environment
4. No sound theoretical basis
SEGMENTED LINEAR INVERSE REGRESSION
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 1985
Regression equations for mean July temperature from the thirteen calibration
regions in eastern North America
Region A: 54-71 N; 90-110 W
Pollen sum: Alnus + Betula + Cyperaceae + Forb sum + Gramineae + Picea + Pinus
July 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 = 0.0000
Region B: 53-71 N; 50-80 W
Pollen sum: Abies + Alnus + Betula + Herb sum + Picea + Pinus
July 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
"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."
Regression equations used to reconstruct mean July
temperature at 6000 yr BP.
Bartlein & Webb 1985
Isotherms for estimated mean July temperatures (ºC) at 6000 yr BP.
Difference map for mean July temperatures (ºC) between 6000 yr BP and today.
Positive values indicate temperatures that were higher at 6000 yr BP than today.
Elk Lake, Minnesota
Reconstructions produced by the regression approach
Regression equation applications to the Elk Lake pollen data.
Mean January
Temperature
Calibration Region
Age Range
(varve range)
R2
45-55ºN, 85-105ºW
320-10084
0.876
10134-11638
0.842
45-55ºN, 95-105ºW
Mean July
40-50ºN,85-95ºW
320-6562
0.799
Temperature
40-50ºN,85-105ºW
6746-10084
0.786
45-50ºN,95-105ºW
10134-11638
0.701
Annual
40-55ºN,85-105ºW
320-3692
0.578
Precipitation
40-50ºN,85-105ºW
3794-7662
0.940
45-55ºN,85-105ºW
7862-11638
0.578
APPROACHES TO MULTIVARIATE CALIBRATION
Chemometrics – predicting chemical concentrations from near Infra-red spectra
Responses
Predictors
PARTIAL LEAST SQUARES REGRESSION – PLS
Form of PC regression developed in chemometrics
PCR
-
PLS
-
components are selected to capture maximum variance within
the predictor variables irrespective of their predictive value for
the environmental response variable
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
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 good independent chronological control for fossil set.
5.
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.
6.
Need statistical estimation of standard errors of prediction for each constructed
value.
7.
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
Gradient length
(detrended correspondence
estimation
analysis) was applied for the
determination of the length of
gradient (LG). LG is important
for choosing between ordination
INDIRECT
based on a linear or on an
GRADIENT
unimodal response model.
ANALYSIS
Correspondence analysis (CA) is
not considered any further
because in the microcosm
DIRECT
experiment discussed here LG
GRADIENT
was =<1.5 SD units. LG <3 SD
ANALYSIS
units are considered to be
typical in experimental
ecotoxicology. In cases where
LG<3, ordination based on
linear response models is
considered to be the most
appropriate. PCA (principal
component analysis) visualizes
variation in species data in
relation to best fitting
theoretical variables.
Environmental variables explaining this visualized variation are deduced afterwards, hence indirectly. RDA
(redundancy analysis) visualizes variation in species data directly in relation to quantified environmental
variables. Before analysis, covariables may be introduced in RDA to compensate for systematic differences
in experimental units. After RDA, a permutation test can be used to examine the significance of effects.
LINEAR OR UNIMODAL METHODS
Estimate 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
alkalinity
colour
2.62 SD
2.76 SD
1.52 SD
If 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.
Imbrie & Kipp (1971) Core-top data
• Species
•
Sample 'core tops'
CANONICAL CORRESPONDENCE ANALYSIS
1. Forward selection of environmental variables
Winter SST
Salinity
Summer SST
0.73
0.13
0.02
p=0.01 82.0%
p=0.01 17.0%
p=0.17 1.0%
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 VALUES
• Bioindication, Calibration, Transfer function, Reconstruction
• Gaussian response model - regression
+ We know observed abundances y
+ We know gradient values x
= Estimate or model the species response curves for all species
• Bioindication - calibration
+ We know observed abundances y
+ We know the modelled species response curves for all species
= Estimate the gradient value of x
• The 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 MODEL
Can be reparametrised as a
generalised linear model:
• Gradient as a 2nd degree
polynomial
• Logarithmic link function
(x  u)2
  h exp
 2t 2
log()  b0  b1x  b2 x 2
b1
u
2b2
t 
1
2b2

b12 

h  exp b0 
4 b2 

J. Oksanen 2002
Optimum = 2º
Summer sea-surface temperature ºC
GLIM
Globigerina pachyderma (left coiling)
Globigerina pachyderma (right coiling)
Orbulina universa
Globigerina rubescens
GAUSSIAN LOGIT REGRESSION
Imbrie and Kipp 1971
61 core tops
27 taxa
Summer SST
Winter SST
Salinity
Significant Gaussian logit model
19
21
21
Significant increasing
linear logit model
6
3
4
Significant decreasing
linear logit model
1
1
0
No relationship
1
2
2
MAXIMUM LIKELIHOOD PREDICTION OF
GRADIENT VALUES
• Bioindication, Calibration, Transfer function, Reconstruction
• Gaussian response model - regression
+ We know observed abundances y
+ We know gradient values x
= Estimate or model the species response curves for all species
• Bioindication - calibration
+ We know observed abundances y
+ We know the modelled species response curves for all species
= Estimate the gradient value of x
• The 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 APPROACH
•
Likelihood is the probability of a given observed value with a
certain expected value
•
Maximum 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 small
J. Oksanen 2002
INFERRING PAST TEMPERATURE FROM
MULTIVARIATE SPECIES COMPOSITION
Observed
Modern
responses
Inferred
9
10
11
12
13
14
Temperature (ºC)
Modified from J. Oksanen 2002
ROOT MEAN SQUARED ERROR FOR WINTER SST, SUMMER
SST, & SALINITY USING DIFFERENT PROCEDURES
Winter SST
Summer SST
Salinity
Imbrie & Kipp 1971 linear
2.57
2.55
0.573
Imbrie & Kipp 1971 non-linear
1.54
2.15
0.571
Maximum likelihood regression and ML calibration 3.21
2.09
0.711
Weighted averaging regression and WA
calibration
1.97
2.02
0.570
WA regression and WA calibration with tolerance
downweighting
1.92
2.03
0.560
ML regression, WA calibration
1.56
(1.56)
1.94
(1.94)
0.557
(0.656)
ML regression, WA calibration with tolerance
downweighting
1.25
(1.25)
1.80
(1.80)
0.534
(0.615)
WACALIB 3.5 (debugged version!) Maximum
likelihood regression and ML calibration
1.20
1.63
0.54
WACALIB 2.1 – 3.3
(values in brackets are RMSE when taxa with significant fits only are used).
•
Only three parameters:
- u: location of the optimum on gradient x
- h: modal height at the optimum
- t: tolerance or width of response
•
Parameters can be estimated with non-linear
regression or generalised linear models
•
WEIGHTED AVERAGES CAN APPROXIMATE U
WEIGHTED AVERAGING
The basic idea is very simple.
In a lake with a certain pH range, diatoms with their pH optima close to the
lake’s 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 regression
n
Uˆk 
 Y ik X i
i 1
n
Optimum
 Y ik
i 1
  X  ˆ 2 
 i 1Y ik i U k 
tˆ  
n

 Y ik


i 1


n
1
2
Tolerance
k
where
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 i
And there are i=1,....,n samples
and k=1, ....,m taxa
Weighted averaging calibration or reconstruction
m
 Y ik Uˆk
Xˆi 
k 1
m
WA
 Y ik
k 1
m
Xˆ
i

2
Y Uˆ tˆ
k 1
m
ik
k
k
2
Y tˆ
k 1
ik
k
WAtol
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
Weighted averaging calibration
A lake will tend to be dominated by taxa with chemical optima close to the
lake's chemistry
Estimate of this chemistry is given by averaging the optima of all taxa present in
the lake.
If a species' abundance data are available these can be used as weights:
m
WA Calibration
xˆi 
y
k 1
m
y
k 1
where
ik
uˆk
ik
x̂i = estimate of environmental variable for fossil sample i
yik= abundance of species k in fossil sample i
uk= optima of species k
ESTIMATION OF SAMPLE-SPECIFIC ERRORS
BASIC IDEA OF COMPUTER RE-SAMPLING PROCEDURES
TRAINING SET - 178 modern diatom samples and lake-water pH
Jack-knifing
Do 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 jackknifing estimate of pH and its variance and hence its standard error.
Bootstrap
Draw 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 testset 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 prediction 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.
The bootstrap algorithm for estimating the standard error of a statistic ˆ  s ( x ); each
bootstrap sample is an independent random sample of size n from F̂ . The number of
bootstrap replications B for estimating a standard error is usually between 25 and 200. As B

 , seB approaches the plug-in estimate of se f ˆ

ERROR ESTIMATION BY BOOTSTRAPPING
WACALIB 3.1+ AND C2
61 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

+
(xˆi, boot  xi, boot )2
n
boot
error due to variation in abundances
at a given temperature
(xi, boot  xi, boot )2

n
boot
(actual prediction error differences between
observed xi and mean bootstrap estimate
(s.e. of bootstrap estimates)
s1
( xi, boot is mean of
s2
xi, boot for all cycles when sample i is in test set).
For a fossil sample
MSEP 

boot

xˆ
i ,boot  x i,boot
n
 S
2
2
2
RMSE = (S1 + S2)½
ROOT MEAN SQUARE ERRORS OF PREDICTION
ESTIMATED BY BOOTSTRAPPING
WA
W Atol
Summer sea-surface temperature C Training set
RMSE total
2.31
2.37
RMSE S1
0.63
0.70
RMSE S2
2.22
2.27
Fossil samples
2.2252.251
2.2832.296
Winter sea-surface temperature C Training set
RMSE total
2.23
2.19
RMSE S1
0.62
0.7
RMSE S2
2.14
2.07
Fossil samples
Salinity ‰ Training set
RMSE total
RMSE S1
RMSE S2
Fossil samples
2.1562.201
2.1062.249
0.61
0.11
0.60
0.60
0.13
0.59
0.6030.607
0.5990.606
TRAINING SET ASSESSMENT
ROOT MEAN SQUARED ERROR (RMSE) of
xi  xˆi 
ˆi
CORRELATION BETWEEN xi and x
r
COEFFICIENT OF DETERMINATION
r2

xˆi  xi 2
n
r or r2 measures strength between observed and inferred values and
allows comparison between transfer functions for different variables.
RMSE2 = error2 + bias2
Error
= SE xi  xˆi 
Bias
= Mean xi  xˆi  SYSTEMATIC PREDICTION ERROR
(Mean of prediction errors)
RANDOM PREDICTION ERROR ABOUT BIAS
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 ERROR
Good: Prediction root mean squared error (RMSEP)
Correlation unreliable: depends on the range of observations
•
Root mean squared error
RMSE


N
2
(
x

x
)
/N
i
i
i 1
•
Bias b: systematic difference
•
Error : random error about
bias.
•
RMSE2 = b2 + 2
Must be cross-validated or
will be badly biased
J. Oksanen 2002
ACCURACY OF PREDICTION
• Root Mean Squared Error
RMSE 

i
(xˆi  xi )2 n
• Two components
- Error
RMSE2 = bias2 + error2
- Bias
• Correlation coefficient is dependent on the range of observations
- Large range: Large part of variance explained
• Cross-validation must be used in assessing the prediction accuracy
1: Split sample
- Divide your data into training and test data sets
2: 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-VALIDATION
Leave-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-validation
J. Oksanen 2002
CROSS-VALIDATION STATISTICS
RMSEP
r jack
r2 jack
PREDICTED VALUES
mean bias
maximum bias
cf.RMSE
r
r2
APPARENT VALUES or ESTIMATED VALUES
mean bias
maximum bias
TRAINING SET ASSESSMENT AND SELECTION
Lowest RMSEP, highest r or r2 jack, lowest mean bias, lowest maximum bias.
Often a compromise between RMSEP and bias.
PARTITIONING RMSEP
RMSEP2 = ERROR2 + BIAS2
s12  s22
Error due to estimating optima and tolerances
Error due to variations in abundance of
taxa at given environmental value
SWAP (= Surface Waters Acidification Project) Diatom – pH
Training Set
England
5
Norway
51
Wales
32
178 surface
sediments
Scotland 60
Sweden
30
267 taxa –
pH –
in 2 or more samples with 1% or more in sample
arithmetic mean 4.33 – 7.25
mean = 5.59
median = 5.51
Screened to 167 samples pH 4.33 – 7.25
mean = 5.56
262 taxa
RMSE = 0.297
r = 0.933
RMSEP (bootstrapping) = 0.32
RMSEP (split-sampling) = 0.31
median = 5.27
ROOT MEAN SQUARED ERRORS OF PREDICTION
FOR THE TRAINING SET
WA
WAtol
RMSE
si1
0.072
0.305
RMSE
S2
0.312
0.371
Total RMSE of prediction
0.320
0.480
________________________________________________
Cross-validation
0.308
RMSE
(0.269-0.338)
0.376
(0.287-0.541)
The Round Loch of Glenhead, Galloway
WA pH reconstructions
with bootstrap standard
errors of prediction
STATISTICAL AND ECOLOGICAL EVALUATION OF
RECONSTRUCTIONS
INITIAL ASSUMPTIONS
1. Taxa related to physical environment.
2. Modern and fossil taxa have same ecological responses.
3. Mathematical methods adequately model the biological
responses.
4. Reconstructions have low errors.
5. Training set is representative of the range of variation in the
fossil set.
RECONSTRUCTION EVALUATION
1. RMSEP for individual fossil samples
Monte Carlo simulation using leave-one-out initially to estimate standard errors of
taxon coefficient and then to derive specific sample standard errors, or
bootstrapping.
2. Goodness-of-fit statistics
CCA 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.
3. Analogue statistics
Good and close analogues. Extreme 5% and 2.5% of modern DCs.
4. 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 present
reliable
< 10% not present
okay
< 25% not present
possibly okay
> 25% not present
not reliable
ASSESSMENT OF ANALOGUES
ANALOG, MAT
Chord distance or chi-squared distances.
Select first
fifth
VERY GOOD or CLOSE ANALOGUE
GOOD
tenth percentiles of all pairs of DC values
n samples  1 2 n n  1
FAIR
DC values
RANDOMISATION TESTS
ANALOG
Poor fit
Chironomids
and climate
Ordination of (a) chironomid taxa and environmental variables in
Labrador, Canada, and (b) lakes.
Relationship between actual
and chironomid-inferred
summer surface-water
temperatures for Labrador
lakes.
Walker et al. 1991
Walker et al. 1996
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.
Summer surface-water
paleotemperature
reconstruction for Splan
Pond. For comparison
names of climatic
events for correlative
European time intervals
are included. The
apparent root-meansquare error of the
temperature estimates
is 1.32ºC (10).
Walker et al 1997
VALIDATION
Diatoms and pH
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. Diatominferred values (thick solid line) were obtained by weighted averaging.
Diatoms and total P validation
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.
Measured annual mean
TP concentrations (line
with open circles)
compared with the
diatom-inferred TP
values calculated as 3year running means
(single line), for the
period 1975-93. All
model values are back
transformed to g l-1.
Measured
Diatominferred
total P
Baldeggersee
frozen core
Baldeggersee
Lotter 1998
Diatom succession in Baldeggersee freeze-core BA93-C between 1885 & 1993.
Only major taxa shown
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 uppermost 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 climate
Diatom-inferred mean July air
temperature (black dots) from
sediments of the three study
lakes Alanen Laanijärvi, Lake
850, and Lake Njulla including
sample specific error estimates
(vertical error bars) and 210Pbdating errors (horizontal error
bars) compared with
measured July T (grey dots) in
Kiruna (for Alanen Laanjärvi)
and in Abisko (for Lakes 850
and Njullla) during the past
century. Measured July T are
corrected for elevation
(0.57ºC 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 climate
Comparison 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
0.57ºC 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
corresponding to the date obtained at each level.The black line is the chironomid-inferred temperatures with
the estimated errors as vertical bars (mean±SSE). 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 (mean±SSE). 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 2003
WEIGHTED AVERAGING – AN ASSESSMENT
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 species, in fact helped by many species!
c.
relatively insensitive to outliers
4.
Tests 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).
5.
Because of its computational simplicity, can derive error estimates for
predicted inferred values.
6.
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.
7.
Ignores absences.
8.
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
1.
Sensitive to distribution of environmental variable in training set.
2.
Considers each environmental variable separately.
3.
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 AVERAGES
•
1.
2.
WA estimate of species optimum (u) is good if:
Sites are uniformly distributed over species range
Sites are close to each other
•
1.
2.
3.
4.
WA estimates of gradient values (x) are good if:
Species optima are dispersed uniformly around x
All species have equal tolerances
All species have equal modal abundances
Optima are close together
u~i 
y x
y
ij
i
j
y
u
x
=
=
=
x~ j 
j
ij
abundance
optimum
gradient value
y u
y
ij
i
j
i
j
~
=
=
=
i
ij
species
site
WA estimate
These conditions are only true for infinite species packing conditions!
WEIGHTED AVERAGING CONDITIONS JOINTLY
1.
Both species and sites must have uniform and dense distribution over the
gradient
2.
To estimate values at gradient ends, some species optima must be
outside the gradient endpoints. Result is bias and truncation
3.
To estimate extreme species optima, some sites must be outside the most
extreme species optima. Result is bias and truncation
4.
Conditions 2 and 3 can be satisfied simultaneously only with infinite
gradients
5.
WA equations define the two-way reciprocal averaging algorithm of CA -
6.
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.
x  u~, u~  x~, x~  u~...
BIAS AND TRUNCATION IN WEIGHTED
AVERAGES
Weighted averages are good estimates of Gaussian optima, unless
the response in truncated.
Bias towards the gradient centre: shrinking.
WA
WA GLR
GLR
pH
J. Oksanen 2002
APPROACHES TO MULTIVARIATE CALIBRATION
Chemometrics – predicting chemical concentrations from near infra-red
spectra
Responses
Predictors
CORRESPONDENCE ANALYSIS REGRESSION
Roux 1979
Reduced Imbrie & Kipp (1971) modern foraminifera data to 3 CA axes.
Then used these in inverse regression.
RMSE apparent
Summer temp
Winter temp
PC regression
2.55°C
2.57°C
CA regression
1.72°C
1.37°C
WA-PLS
1.53°C
1.17°C
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 y
PLS 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 PLS
WA = WA-PLS if only first WA-PLS component is used
WA-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 AVERAGING
WA
1. 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  x i  –
WA calibration.
4. Regress the environmental variable (xi) on the preliminary new site
scores and take the fitted values as the estimate of x i – deshrinking regression.
x i on xi
[Regression
or
CLASSICAL

xi on x i
INVERSE
good for ‘ends’
lower RMSE]
y ikuk
xˆi  b0  b x  b0  b1 
y ik
k
m

1 i
m

k
y ikuˆk
y ik
where
uˆk  b0  b1uk
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 motivates
Part 1 (WA regression): Estimate species optima (u*k) by weighted averaging of
the x-value of the sites, i.e.
*
u   y ik x y
k
i
i
k
Species present and abundant in a particular site will tend to have optima close
to its x-value. This motivates
Part 2 (WA calibration): Estimate the x-value of the sites by weighted averaging
of the species optima, i.e.
*
*
x  y u y
i
k
ik
k
i
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
xˆ0  a0  a1 x 0  a0  a1k y 0k xk
*
 k y 0k u
ˆk
y
*
y
0
0
where a0 and a1 are the coefficients of the deshrinking regression
and ûk = a0 + a1û*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
reconstructions
Solution – deshrinking inverse regression
Derive inverse regression coefficients
initial xi = a + bxi
Apply regression to reconstructed values to ”deshrink”
final xi = (initiali – a)/b
Where 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
AVERAGING
1.
Estimate species optima (ûk) by weighted averaging of the environmental
variables (x) at the sites
n
uˆk   y ik xi y k
i 1
where y+k has + to replace the summation over the subscript,
in this case i = 1, ...., n sites.
2.
Estimate the x values of the sites by weighted averaging of the species
optima
m
initial xˆi   y ikuˆk y i 
k 1
3.
Because averages are taken twice, the range of the estimated initial xvalues (x) is shrunken. Need to deshrink using either
(a) Inverse linear regression
xi  b0  b1(initial xˆi )  εi

final xˆi  b0  b1(initial xˆi )
This minimises RMSE in the training set
or
(b) Classical linear regression
initial xˆi  b0  b1xi  εi

final xˆi  (initial xˆi  b0 ) b1
This deshrinks more than inverse regression and takes inferred values
further away from the mean.
For inverse regression and two-way WA
m
xˆ0  b0  b1xˆ0  b0  b1  y 0 kuˆk y 0 
k 1
m
  y 0 k uk* y 0 
k 1
where uk*  b0  b1uˆk
For classical regression and two-way WA
 m


xˆ0     y 0 kuˆk y 0    b0  b1

  k 1

where
uk*  uˆk  b0  b1
m
  y 0 k uk* y 0 
k 1
Can also estimate for each species its WA tolerance or standard deviation
(niche breadth)
as


2
ˆ
tk    y ik x i  uˆk  y k 
 i 1

n
1
2
and use these in a tolerance-weighted estimate of x
m
2
ˆ
  y ikuˆk tk 

xˆi   k 1
y i  tˆk2


WEIGHTED AVERAGING PARTIAL LEAST SQUARES –
WA-PLS
1.
Centre the environmental variable by subtracting weighted mean.
2.
Take the centred environmental variable (xi) as initial site scores – (cf.
WA/CA)
3.
Calculate new species scores by WA of site scores.
4.
Calculate new site scores by WA of species scores.
5.
For axis 1, go to 6. For axes 2 and more, make site scores uncorrelated
with previous axes.
6.
Standardise new site scores and (cf. WA/CA) use as new component.
7.
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-PLS
Step 0
Centre the environmental variable by subtracting the weighted mean, i.e.
xi :  xi  i yi xi y
This simplifies the formulae.
Step 1
Take the centred environmental variable (xi) as initial site scores (ri)
Do steps 2 to 7 for each component:
Step 2
Calculate new species scores (u*k) by weighted averaging of the site scores, i.e.
Step 3
uk*  i yik ri y k
Calculate new site scores (ri) by weighted averaging of the species scores, i.e. new
ri  k y ik u k* y i 
Step 4
For 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 5
Standardise the new site scores (ri).
Step 6
Take the standardised scores as the new component.
Step 7
Regress 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 ( x̂i). Go to step 2 with the residuals of the regression
as the new site scores (ri).
Method for calculating inferred temperatures
Using WA inverse deshrinking models, inferred summer surface water
temperatures (°C) for shallow lakes may be calculated as:
xˆ
i
or
xˆ
i
a b
 m



y
  ik u
ˆ
k
k

1



a b
 m
  y ik u
ˆk
 k 1

m
y
k 1
tˆ 
2
k

ik 

(without tolerance down-shrinking)
 y



m
k 1


tˆ 
2
ik
k
(with tolerance down-weighting)
With WA classical deshrinking models, the inferred summer surface water
temperatures (°C) are calculated as:
xˆ
  m

    y u

ˆ
k
ik
k

1



xˆ
  m
   
  k 1
i
or
i
y
ik
uˆ
k
m
y
k 1




a
ik 




ˆ
t   y
2
k
m
k 1
b
(without tolerance down-shrinking)




tˆ  a 
2
ik
k

b
(with tolerance down-weighting)
Using the WA-PLS models, inferred summer surface water temperatures (°C)
may be calculated as:
xˆ
i
m
   y ik 
ˆk 
 k 1

m
y
k 1
ik
where xˆi 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, ûk is the temperature
optimum (°C) of species, ˆ , is the Beta of species k, and tˆk is the tolerance
(°C) of species k (Fritz et al. 1991; ter Braak 1987; Birks, pers.comm.).
k
LEAVE-ONE-OUT AND TEST SET
CROSS-VALIDATION
Performance 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.
s
1
2
3
4
5
6
Apparent
RMSE
6.14
3.37
2.87
2.22
2.01
1.82
Training set
Leave-one-out
RMSEP
6.22
4.24
4.16*
4.65
4.65
4.50
Test set
RMSEP
6.61
4.40*
4.57
4.94
5.11
5.62
ter 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
s
1
2
3
4
5
6
SWAP
RMSE RMSEP
Bergen
RMSE RMSEP
Thames
RMSE RMSEP
0.276
0.232
0.194
0.173
0.153
0.134
0.353
0.256
0.213
0.192
0.174
0.164
0.341
0.238
0.196
0.166
0.153
0.140
0.310*
0.302
0.315
0.327
0.344
0.369
Reduction in
prediction error (%)
0
0.394
0.318*
0.330
0.335
0.359
0.374
19
0.354
0.279
0.239*
0.224
0.219
0.219
32
Bergen data set: predicted pH and bias as a function of observed pH for components 1 and
2 in WA-PLS. Solid lines represent Cleveland’s 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 Cleveland’s LOESS scatterplot
smooth (1979). Salinity is g-1 and transformed as log10 (salinity – 0.08). 32% gain.
NW Europe
Total P
152 lakes
The relationship between
(a) diatom-inferred TP and
(b) residuals (inferred TP
– observed TP) and
observed TP for the oneand two-component WAPLS models. Solid lines
show LOWESS scatter plot
smoothers.
Summary diatom diagram and
reconstructed annual mean TP
concentrations using one- and twocomponent WA-PLS models for
Lake SøbyGård, showing standard
errors of prediction for the twocomponent model. 210Pb dates (AD)
are shown on the right hand side.
Bennion et al. 1996
Imbrie &
Kipp
(1971)
data
WA
WA-PLS
ALPE - DIATOM - pH TRAINING SET
Italian and Austrian Alps (Aldo Marchetto & Roland Schmidt)
Spanish Pyrenees (Jordi Catalan & Joan Garcia)
ALPE sites (Nigel Cameron & Viv Jones)
Norway (Frode Berge & John Birks)
SWAP Norway & UK (Frode Berge, Roger Flower, Viv Jones)
Total
One 'rogue' sample detected
118 samples
527 diatom taxa
pH 4.48 - 8.04
median 6.10
mean 6.15
Gradient length 5.19 standard deviations
31
28
30
10
20
119
ALPE TRAINING SET - 118 SAMPLES
Square root transformation
Components
WA-PLS
-1
-2
-3
-4
-5
RMSE
0.299
0.178
0.131
0.100
0.075
r2
0.85
0.96
0.97
0.98
0.99
RMSEP
0.359
0.337
0.331
0.331
0.339
r2 (jack)
0.78
0.81
0.81
0.81
0.80
Select WA-PLS model with 3 components as simplest model (least
parameters) that gives lowest RMSEP.
NORWEGIAN CHIRONOMID – CLIMATE TRAINING SET
Leave-one-out cross validation
Predicted air temperature.
1:1
RMSEP = 0.89ºC
109 samples
Bias = 0.61ºC
Predicted – observed air temperature
Inferred mean July
air temperature
Oxygen isotope
ratios
NORWEGIAN POLLEN
AND CLIMATE
Precipitation
300 - 3537mm
Mean July
7.7 - 16.4ºC
Mean January
-17.8 - 1.1ºC
Root mean squared errors of prediction (RMSEP) based on leave-oneout jack-knifing cross-validation for annual precipitation, mean July
temperature, and mean January temperature using five different statistical
models.
Pptn
(mm)
July
(C)
January
(C)
Weighted averaging (WA) (classical)
486.5
1.33
2.86
Weighted averaging (WA) (inverse)
427.2
1.07
2.61
Partial least squares (PLS)
420.1
0.94
2.82
WA-PLS
417.5
1.03
2.57
Modern analogue technique (MAT)
385.3
0.91
2.42
Vuoskojaurasj, Abisko, Sweden
Vuoskojaurasj consensus reconstructions
Tibetanus, Abisko Valley
Inferred from
pollen
Inferred from
pollen
Hammarlund et al. 2002
Björnfjelltjörn, N. Norway
Björnfjelltjörn consensus reconstructions
LINEAR AND UNIMODAL-BASED NUMERICAL METHODS
Response model
Problem
Linear
Unimodal
Regression
Multiple linear regression
Weighted averaging (WA) of
sample scores
Calibration
Linear calibration 'inverse
regression'
WA of taxa scores and
simple two-way WA
Principal components
regression
Correspondence analysis
regression
Partial least squares (PLS-1)
WA-PLS (WAPLS-1)
Multivariate calibration
(PLS-2)
WAPLS-2
Ordination
Principal components analysis
(PCA)
Correspondence analysis
(CA)
Constrained ordination
(= reduced rank regression)
Redundancy analysis (RDA)
Canonical correspondence
analysis (CCA)
Partial ordination
Partial PCA
Partial CA
Partial constrained
ordination
Partial RDA
Partial CCA
RESPONSE SURFACES
Pollen percentages in modern samples plotted in ‘climate space’ (cf Iversen’s
thermal limit species +/– plotted in climate space).
Contoured
Trend-surface analysis R2
Bartlein et al. 1986
Contoured only
Webb et al. 1987
Reconstruction purposes – grid, analogue matching
Simulation purposes
PROBLEMS
1. Need large high-quality modern data for large geographical areas.
2. No error estimation for reconstruction purposes.
3. 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 1020%, ’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 localarea 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.
Elk Lake,
Minnesota
Reconstruction produced by the response surface approach
Simulation purposes
Fossil and simulated isopoll map sequences for Betula. Isopolls are drawn
at 5, 10, 25, 50 and 75% using an automatic contouring program.
Fossil and simulated isopoll map sequences from Quercus (deciduous). Maps are drawn at 3000year intervals between 12000 yr BP and the present. The upper map sequence presents the observed 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 hoc’
1. Choice of how much or how little smoothing.
2. Choice of scale of grid for reconstructions.
3. No statistical measure of ‘goodness-of-fit’.
4. No reliable error estimation for predicted values.
ANALOGUE-BASED APPROACH
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.
PROBLEMS
1.
Assessment of ‘most similar’?
2.
1, 2, 9, 10 most similar?
3.
No-analogues for past assemblages.
4.
Choice of similarity measure.
5.
Require huge set of modern samples of comparable site type, pollen
morphological quality, etc, as fossil samples. Must cover vast geographical
area.
6.
Human impact.
Elk Lake, Minnesota
Reconstructions produced using the analogue approach
MODIFIED MODERN ANALOGUE APPROACHES
Joel Guiot
1.
Taxon weighting
Palaeobioclimatic operators (PBO) computed from either a timeseries 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.2
Guiot et al. 1989
Cor 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.
MODERN ANALOGUE TECHNIQUES FOR
ENVIRONMENTAL RECONSTRUCTION
= K – NEAREST NEIGHBOURS (K – NN)
MAT, ANALOG, C2
1.
Modern 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.
2.
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.
CONSENSUS RECONSTRUCTIONS
Elk Lake climate reconstruction 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 (19781984) for Itasca Park are
also shown.
PLOTTING OF RECONSTRUCTED VALUES
1.
Plot against depth or age the reconstructed values, indicate the observed
modern value if known.
2.
Plot deviations from the observed modern value or the inferred modern
value against depth or age.
3.
Plot centred values (subtract the mean of the reconstructed values)
against depth or age to give relative deviations.
4.
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 FUNCTIONS
Telford & Birks (2005) Quaternary Science Reviews 24:
2173-2179
Estimating the predictive power and performance of a
training set as RMSEP, maximum bias, r2, etc., by crossvalidation 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 autocorrelation seriously violates this assumption.
See Richard Telford's lecture after this lecture.
USE OF ARTIFICIAL, SIMULATED DATA-SETS
SIMULATED DATA-SETS
Generate 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
1. Probably widespread.
2. Does it matter?
3. Analog-based techniques for reconstruction - YES!
Modern analog technique
Response surfaces
4. WA and related inverse regression methods
What we need are ‘good’ (i.e. reliable) estimates of ûk. Apply them to
same taxa but in no-analog conditions in the past.
Assume that the realised niche parameter ûk 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.
5. Simulated data
ter Braak, 1995. Chemometrics & Intelligent Laboratory Systems 28,
165–180
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 C
Inverse 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).
100
No analogue test
set
Set C
Z
Training set
Set A
0
Set
X
t
B
100
A
C
Inverse approach
WA-PLS
2.97-9
3.0
2.8
5.9
k-NN
4.43-5
2.5
2.9
13.5
a
Classical approach
MLM w.r.t. x
4.3
4.4
3.5
10.6
MLM w.r.t. x
and z
2.8
3.0
2.9
4.6
ter Braak 1995
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).
Significant difference (P<0.01)
between leave-out validation and
validation by the independent
evaluation set B.
a
6. General conclusions from simulated data experiments
WA, 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 10–20) for each fossil sample, devise dynamic
training set, use linear PLS methods, avoids edge effects,
truncated responses, etc.
MULTIPLE ANALOG PROBLEM
Fossil 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 – Guiot
e.g.
constrain pollen choices on basis of inferred biome, fossil beetles,
inferred lake-level changes
Constrained response surfaces (  analog matching) – Huntley
e.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)
a) using beetles
alone,
b) using pollen alone,
(b)
c) using pollen
constrained by
beetles
(c)
Guiot et al. 1993
MULTI-PROXY APPROACHES
Swiss surface
pollen samples –
lake sediments
Selected trees and
shrubs
Swiss surface lake sediments.
Selected herbs and pteridophytes
Root mean squared errors of prediction (RMSEP) based on leave-one-out jackknifing cross-validation for mean summer temperature (June, July, August),
mean winter temperature (December, January, February) and mean annual
precipitation using WA-PLS model.
RMSEP
R2
No. of comps.
Mean summer temperature
1.252C
0.90
3
Mean winter temperature
1.025C
0.88
3
Mean annual precipitation
194.1mm
0.57
2
Modern Swiss pollen - climate
Gerzensee, Bernese Oberland, Switzerland
Gerzensee
PB-O
YD-PB Tr
YD
AL-YD Tr
G-O
Lotter et al. 2000
Gerzensee
O16/O18
Pollen
Chydorids
Lotter et al. 2000
Birks & Ammann 2000
GRADIENT LENGTH
(Compositional turnover along environmental gradient – SD units)
SHORT (<2sd)
LINEAR
NOISE
OF
TRAINING SET
DATA †
MEDIUM (2-4sd)
LONG (>4sd)
UNIMODAL-BASED METHODS
VERY
LOW
Least squares linear
regression and
calibration (inverse
regression) GLM
Gaussian logit or
multinomial regression
and calibration GLM
? Generalised
additive models
GAM
LOW
Partial least squares PLS
regression and
calibration PLS
Weighted averaging PLS
regression and
calibration WA-PLS
? WA-PLS
MEDIUM
Partial least squares PLS
or robust linear
regression and
calibration
Weighted averaging
regression and
calibration WA
? WA or WA-PLS
HIGH
PCA regression
CA-regression
? DCA-regression
? IDEAL
PROBLEMS AT
GRADIENT ENDS
TRY TO AVOID,
CAUSES MULTIPLE
ANALOG PROBLEM?
†
HOW TO ESTIMATE NOISE IN REAL DATA?
1. Zero values
2. High sample heterogeneity (root mean squared deviation for
samples)
3. High taxon tolerances (root mean squared deviation)
4. Rare taxa
5. % variance in Y explained by X, constrained 1 relative to
unconstrained 2 .