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Title: Quantitative%20Methods%20in%20Palaeoecology%20and%20Palaeoclimatology%20PAGES%20Valdivia%20October%202010

1
Quantitative Methods in Palaeoecology and
Palaeoclimatology PAGES Valdivia October 2010
• Analysis of Stratigraphical Data

John Birks
2
CONTENTS
Introduction Temporal stratigraphical
data Single sequence Partitioning or
zonation Sequence splitting Rate-of-change
summarisation Analogue matching Relationships
between two or more sets of variables in same
sequence Two or more sequences Sequence
comparison and correlation Combined
scaling Variance partitioning space and
time Difference diagrams Mapping Locally
weighted regression (LOWESS) Summary
3
INTRODUCTION
Analysis of quadrats, lakes, streams, etc.
Assume no autocorrelation, namely cannot predict
the values of a variable at some point in space
from known values at other sampling
points.   PALAEOCOLOGY fixed sample order in
time. strong autocorrelation temporal
autocorrelation   STRATIGRAPHICAL
DATA biostratigraphic, lithostratigraphic,
geochemical, geophysical, morphometric,
isotopic multivariate continuous or discontinuous
time series ordering very important display,
partitioning, trends, interpretation
4
ZONATION OR PARTITIONING OF STRATIGRAPHICAL DATA
Useful for 1) description 2) discussion and
interpretation 3) comparisons in time and
space   sediment body with a broadly similar
composition that differs from underlying and
overlying sediment bodies in the kind and/or
amount of its composition.
5
CONSTRAINED CLASSIFICATIONS 1) Constrained
CONISS  2) Constrained binary divisive
procedures Partition into g groups by
placing g 1 boundaries. Number of
possibilities Compared with non-constrained
situation. Criteria within-group
sum-of-squares or variance SPLITLSQ
within-group information
SPLITINF
6
n3
7
Pollen diagram and numerical zonation analyses
for the complete Abernethy Forest 1974 data set.
Birks Gordon 1985
8
CONISS constrained incremental sum-of-squares
( constrained Word's minimum variance)
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OPTIMAL SUM OF SQUARES PARTITIONS OF THE
ABERNETHY FOREST 1974 DATA
Number of groups g (zones) Percentage of total sum-of-squares Markers Markers Markers Markers Markers Markers Markers Markers Markers
2 59.3 15
3 28.4 15 32
4 18.9 15 33 41
5 14.7 15 33 41 45
6 10.6 15 32 34 41 45
7 8.1 15 26 32 34 41 45
8 5.8 8 15 26 32 34 41 45
9 4.7 8 15 24 29 32 34 41 45
10 3.9 8 15 24 29 32 33 34 41 45
11
HOW MANY ZONES?
K D Bennett (1996) Determination of the number of
zones in a biostratigraphical sequence. New
Phytologist 132, 155-170
Broken stick model
12
Ioannina Basin
Tzedakis 1994
Pollen percentage diagram plotted against depth.
Lithostratigraphic column is represented symbols
are based on Troels-Smith (1995).
13
Ioannina Basin
Tzedakis 1994
14
Variance accounted for by the nth zone as a
proportion of the total variance (fluctuating
curve) compared with values from a broken-stick
model (smooth curve) (a) randomized data set,
(b) original data set. Zonation method binary
divisive using the information content statistic.
Data set Ioannina.
Original data
Broken stick model
15
Bennett 1996
16
SEQUENCE SPLITTING
Walker Wilson 1978 J Biogeog 5, 121 Walker
Pittelkow 1981 J Biogeog 8, 3751 SPLIT,
SPLIT2 BOUND2 Need statistically independent
curves   Pollen influx (grains cm2 year1) PCA
or CA or DCA axes CANOCO Aitchison
log-ratio transformation LOGRATIO
where
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Correlograms of sequence splits with charcoal,
inorganic matter and total pollen influxes for
three sections of the pollen record. The vertical
scales give correlations the horizontal scales
give time lag in years (assuming a sampling
interval of 50 years).
22
RATE OF CHANGE ANALYSIS
Amount of palynological compositional change
per unit time. Calculate dissimilarity between
pollen assemblages of two adjacent samples and
standardise to constant time unit, e.g. 250 14C
years. Jacobson Grimm 1986 Ecology 67,
958-966 Grimm Jacobson 1992 Climate Dynamics
6, 179-184 RATEPOL POLSTACK (TILIA)
23
Graph of distance (number of standard deviations)
moved every 100 yr in the first three dimensions
of the ordination vs age. Greater distance
indicates greater change in pollen spectra in
100yr.
Jacobson Grimm 1986
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MANY PROXIES, ONE SITE
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ONE PROXY, MANY SITES
- fertile
- poor
Chord distance between samples at Solsø, Skånsø,
and Kragsø, calculated on smoothed data with 35
taxa and interpolated at 400 year and 1,000 year
intervals.
- poor
32
Pollen percentages from Loch Lang, Western Isles,
plotted against age (radiocarbon years BP). Data
from Bennett (1990).
33
Pollen percentages from Hockham Mere, eastern
England, plotted against age (radiocarbon years
BP). Data from Bennett (1983).
34
Rate x5 that at Loch Lang
Comparison of Holocene rates of change at Loch
Lang and Hockham Mere, with ?2 - 2 dissimilarity
coefficient on unsmoothed data, with a
SE - continental
NW - oceanic
35
DATA SUMMARISATION BY ORDINATION OR GRADIENT
ANALYSIS OF SINGLE SEQUENCE
Ordination methods CA/DCA or PCA joint
plot biplot     Sample summary   Species
arrangement CA correspondence analysis DCA
detrended correspondence analysis PCA principal
components analysis
36
Biplot
PCA Biplot of the Kirchner Marsh data C2
0.746. The lengths of the Picea and Quercus
vectors have been scaled down relative to the
other vectors, in the manner described in the
text. Stratigraphically neighbouring levels are
joined by a line.
37
Joint plot
Correspondence analysis representation of the
Kirchner Marsh data C2 0.620.
Stratigraphically neighbouring levels are joined
by a line. Joint plot. Gordon 1982
38
Stratigraphical plot of sample scores on the
first correspondence analysis axis (left) and of
rarefaction estimate of richness (E(Sn)) (right)
for Diss Mere, England. Major pollen-stratigraphi
cal and cultural levels are also shown. The
vertical axis is depth (cm). The scale for sample
scores runs from 1.0 (left) to 1.2
(right).
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The 1st and 2nd axis of the Detrended
Correspondence Analysis for Laguna Oprasa and
Laguna Facil plotted against calibrated calendar
age (cal yr BP). The 1st axis contrasts taxa from
warmer forested sites with cooler herbaceous
sites. The 2nd axis contrasts taxa preferring
wetter sites with those preferring drier sites
Haberle Bennett 2005
42
Species arrangement
Percentage pollen and spore diagram from
Abernethy Forest, Inverness-shire. The
percentages are plotted against time, the age of
each sample having been estimated from the
deposition time. Nomenclatural conventions follow
Birks (1973a) unless stated in Appendix 1. The
sediment lithology is indicated on the left side,
using the symbols of Troels-Smith (1995). The
pollen sum, ?P, includes all non-aquatic taxa.
Aquatic taxa, pteridophytes, and algae are
calculated on the basis of ?P ? group as
indicated.
43
Pollen types re-arranged on the basis of the
weighted average for depth TRAN
44
ANALOGUE ANALYSIS
Modern training set similar taxonomy
similar sedimentary environment   Compare fossil
sample 1 with all modern samples, use appropriate
DC, find sample in modern set most like (i.e.
lowest DC) fossil sample 1, call it closest
analogue, repeat for fossil sample 2,
etc. Overpeck et al 1985 Quat Res 23,
87108   ANALOG MATCH MAT
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pollen zones, and vegetation types represented by
the top ten analogs from the Lake West Okoboji
site.
47
Maps of squared chord distance values with modern
samples at selected time intervals
48
Plots of the minimum squared chord-distance for
each fossil spectrum at each of the eight sites.
49
A schematic representation of how fossil diatom
zones/samples in a sediment core from an
acidified lake can be compared numerically with
modern surface sediment samples collected from
potential modern analogue lakes. In this
space-for-time model the vertical axis represents
sedimentary diatom zones defined by depth and
time the horizontal axis represents spatially
distributed modern analogue lakes and the dotted
lines indicate good floristic matches (dij
lt0.65), as defined by the mean squared
Chi-squared estimate of dissimilarity (SCD, see
text).
Flower et al. 1997
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Flower et al. 1997
51
COMPARISON AND CORRELATION BETWEEN TIME SERIES
Two or more stratigraphical sets of variables
from same sequence. Are the temporal patterns
similar? (1) Separate ordinations Oscillation
log - likelihood G-test or ?2 test (2) Constrained
ordinations Pollen data - 3 or 4 ordination
axes or major patterns of variation Y Chemical
data - 3 or 4 ordination axes X Depth as a
covariable Does 'chemistry' explain or predict
'pollen'? i.e. is variance in Y well explained by
X? Lotter et al., 1992 J. Quat. Sci. Pollen
16O/18O (depth)
52
34 16 12
53
79 12 4 1
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COMPARISON AND CORRELATION BETWEEN TIME SERIES
Two or more stratigraphical sets of variables
from same sequence. Are the temporal patterns
similar? (1) Separate ordinations Oscillation
log - likelihood G-test or ?2 test (2) Constrained
ordinations Pollen data - 3 or 4 ordination
axes or major patterns of variation Y Chemical
data - 3 or 4 ordination axes X Depth as a
covariable Does 'chemistry' explain or predict
'pollen'? i.e. is variance in Y well explained by
X? Lotter et al., 1992 J. Quat. Sci. Pollen
16O/18O (depth)
56
Pollen, oxygen-isotope stratigraphy, and sediment
composition of Aegelsee core AE-1 (after
Wegmüller and Lotter 1990)
57
Pollen and oxygen-isotope stratigraphy of
Gerzensee core G-III (after Eicher and
Siegenthaler 1976)
58
Is there a statistically significant relationship
between the pollen stratigraphy and the
stable-isotope record? Summary of the results
from detrended correspondence analysis (DCA) of
late-glacial pollen spectra from five sequences.
The percentage variance represented by each DCA
axis is listed. Reduce pollen data to DCA axes.
Use these then as responses
Site No. of samples No. of taxa DCA Axis DCA Axis DCA Axis DCA Axis
Site No. of samples No. of taxa 1 2 3 4
Aegelsee AE-1 100 26 57.2 12.0 2.3 1.4
Aegelsee AE-3 54 32 44.3 3.3 1.5 1.4
Gerzensee G-III 65 28 37.6 4.0 1.2 0.9
Faulenseemoos 62 25 44.1 18.8 5.0 3.8
Rotsee RL-250 44 23 38.2 13.3 3.1 2.3
59
Results of redundancy analysis and partial
redundancy analysis permutation tests for the
significance of axis 1 when oxygen isotopes and
depth are predictor variables, when oxygen is the
only predictor, and when oxygen isotopes are the
predictor variable and depth is a covariable.
Site Predictor variable ? 18O and depth Predictor variable ? 18O Covariable depth Predictor variable ? 18O Number of response variables (DCA axes) Pollen DCA axes
Aegelsee AE-1 0.01a 0.01a 0.02a 2
Aegelsee AE-3 0.01a 0.16 0.20 1
Gerzensee G-III 0.01a 0.46 0.57 1
Faulenseemoos 0.01a 0.01a 0.01a 3
Rotsee RL-250 0.01a 0.21 0.08 2
a Significant at plt 0.05 a Significant at plt 0.05
(Lotter et al. 1992)
60
ANALYSIS OF TWO OR MORE SEQUENCES
Regional zones, description of common features,
interpretation, detection of unique
features. Sequence comparison and
correlation. Sequence slotting
SLOTSEQ FITSEQ CONSSLOT   Combined scaling of
two or more sequences.
CANOCO Variance Partitioning
CANOCO Difference diagrams Mapping procedures
61
SLOTSEQ
Slotting of the sequences S1 (A1, A2, ..., A10)
and S2 (B1, B2, ..., B7), illustrating the
contributions to the measure of discordance ?
(S1, S2) and the 'length' of the sequences, ?(S1,
S2).
The results of sequence-slotting of the Wolf
Creek and Horseshoe Lake pollen sequences (?
2.095). Radiocarbon dates for the pollen zone
boundaries are also given, expressed as
Birks Gordon 1985
62
Comparison of oxygen-isotope records from Swiss
lakes Aegelsee (AE-3), Faulenseemoos (FSM) and
Gerzensee (G-III) with the Greenland Dye 3 record
(Dansgaard et al, 1982). LST marks the position
of the Laacher See Tephra (11,000 yr BP). Letters
and numbers mark the position of synchronous
events (for details see text).
63
Lotter et al 1992
Psi values for pair-wise sequence slotting of the
stable-isotope stratigraphy at five Swiss
late-glacial sites and the Dye 3 site in
Greenland. Values above the diagonal are
constrained slotting, using the three major
shifts shown in previous figure values below the
diagonal are for sequence slotting in the absence
of any external constraints. The mean ? 18O and
standard deviation for each sequence is also
listed.
CONSLOXY
64
FUGLA NESS, Shetland
65
Pollen diagram from Sel Ayre showing the
frequencies of all determinable and
indeterminable pollen and spores expressed as
percentages of total pollen and spores (?P).
Abbreviations undiff. undifferentiated, indet
indeterminable.
66
COMBINED SCALING
67
Comparison of Bjärsjöholmssjön and Färskesjön
using principal component analysis. The mean
scores of the local pollen zones and the ranges
of the sample scores in each zone are plotted on
the first and second principal components, and
are joined up in stratigraphic order. The
Blekinge regional pollen assemblage zones are
also shown.
68
Comparison of Färskesjön and Lösensjön using
principal component analysis. The mean scores of
the local pollen zones and the ranges of the
sample scores in each zone are plotted on the
first and second principal components, and are
joined up in stratigraphic order. The regional
pollen assemblage zones are also shown.
Birks Berglund, 1979
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SWISS LATE-GLACIAL
7 sites, 357 samples
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Summary of results of detrended correspondence
analyses of the biozone II and III assemblages at
the seven individual sites. Gradient length is
given in standard deviation units. The contrast
statistic is explained in the text.
Site Number of biozone II and III samples Sum of eigenvalues (total variance) Gradient length Biozone II/III contrast
Lobsigensee 32 0.18 1.14 0.58
Murifeld 22 0.12 0.63 0.24
Aegelsee 60 0.12 0.62 0.24 R
Saanenmöser 21 0.15 0.73 0.14
Zeneggen (Hellelen) 16 0.29 0.88 0.47 R
Hopschensee 22 0.21 0.77 0.46 R
Lago di Ganna 21 0.46 1.36 0.06
R revertence
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VARIANCE PARTITIONING
Total variance between-site variance
within-site variance   unexplained (error)
component
73
VARIANCE PARTITIONING
Use partial constrained ordinations to partition
variance into a) Unexplained variance
13.8 b) Between-site spatial variance 13.2 p
0.01 c) Within-site temporal variance 73.0 p
0.01 Within a sequence variance
partitioning a) Unexplained variance not captured
by zonation 39.7 b) Variance captured by zone
II 33.2 c) Variance captured by zone
III 17.9 d) Variance captured by zone I 9.2
Can now do a partial ordination of 39.7
unexplained variance to see what sort of patterns
remain. Noise, chaos, trends or what?
74
Tzedakis Bennett 1995
Pollen percentage diagram of selected taxa
plotted against depth. Lithostratigraphic symbols
are based on Troels-Smith (1995). For
correlations and ages see Tzedakis (1993, 1994).
75
Pollen percentage diagrams of selected arboreal
taxa of the Metsovon, Zista, Pamvotis and Dodoni
I and II forest periods of Ioannina 249
5e
5e
7c
7c
9c
9c
11a b c
11a b c
76
Tzedakis Bennett 1995
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Tzedakis Bennett 1995
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Tzedakis Bennett 1995
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DIFFERENCE DIAGRAMS
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Pollen percentage difference diagram for the
Hockham Mere and Stow Bedon sequences for
selected taxa, plotted against radiocarbon age.
Note different percentage scale for each taxon.
81
Location of the two coring sites, Rezina Marsh
and Gramousti Lake, in relation to altitude.
82
Pollen percentage difference diagram to compare
results between the pollen percentage values of
selected taxa at Rezina Marsh and Gramousti Lake.
The values are plotted against an estimated time
scale and have been calculated at a time interval
of 250 yr. Values to the right of the axis (blue)
indicate a higher recorded percentage of a taxon
at Rezina Marsh, values to the left (red)
indicate a higher recorded percentage of the
taxon at Gramousti Lake.
83
MAPPING
Distribution in northern England of maximum
values for pollen of Tilia during the period 5000
to 3000 B.C.
84
Pinus
Betula
Maps of pollen frequencies 5,000 years B.P.
85
Ulmus
Corylus
Maps of pollen frequencies 5,000 years B.P.
86
Quercus
Tilia
Maps of pollen frequencies 5,000 years B.P.
87
Alnus
Map of pollen frequencies 5,000 years B.P.
88
Map of scores of pollen spectra on the first
principal component, 5,000 years B.P.
89
Map of scores of pollen spectra on the second
principal component, 5,000 years B.P.
90
Map of scores of pollen spectra on the third
principal component, 5,000 years B.P.
91
Provisional map of wood-land types for the
British Isles 5,000 years ago.
92
Vegetation regions reconstructed from pollen data
for 9,000, 6,000, 3,000, and 0 yr B.P.
93
LOCALLY WEIGHTED REGRESSION
W.S. Cleveland LOWESS locally weighted
regression or LOESS scatterplot smoothing May
be unreasonable to expect a single functional
relationship between Y and X throughout range of
X. (Running averages for time-series smooth by
average of yt-1, y, yt1 or add weights to yt-1,
y, yt1)
94
Linear
(A) Survival rate (angularly transformed) of
tadpoles in a single enclosure plotted as a
function of the average body mass of the
survivors in the enclosure. Data from Travis
(1983). Line indicates the normal least-squares
regression. (B) Residuals from the linear
regression depicted in part A plotted as a
function of the independent variable, average
body mass.
95
LOWESS
LOWESS
(A) DATA from previous graph A with a line
depicting a least-square quadratic model. (B)
Data from previous graph A with a line depicting
LOWESS regression model with f 0.67. (C) Data
from previous graph A with a line depicting a
LOWESS regression model with f 0.33.
96
LOWESS - more general
• Decide how smooth the fitted relationship
should be.
• Each observation given a weight depending on
distance to observation x1 for all adjacent
points considered.
• Fit simple linear regression for adjacent points
using weighted least squares.
• Repeat for all observations.
• Calculate residuals (difference between observed
and fitted y).
• Estimate robustness weights based on residuals,
so that well-fitted points have high weight.
• Repeat LOWESS procedure but with new weights
based on robustness weights and distance weights.
• Repeat for different degree of smoothness, to
find optimal smoother.

97
linear regression
tri-cube function
target value
How the LOESS smoother works. The shaded region
indicates the window of values around the target
value (arrow). A weighted linear regression
(broken line) is computed, using weights given by
the 'tri-cube' function (dotted curve). Repeating
this process for all target values gives the
solid curve.
98
LOWESS curve
99
SUMMARY
1. Stratigraphical data have special numerical
properties fixed order of samples, often closed
percentage data, many variables, many samples
2. Numerical procedures that take account of these
properties are available for partitioning
(zonation), sequence splitting, rate-of-change
analysis, summarisation of stratigraphical
patterns, analogue matching, and establishing
relationships between two or more sets of
variables in the same sequence
3. Numerical procedures for analysing two or more
sequences from different sites are less well
developed but there are robust techniques for
sequence comparison and correlation, examining
differences, and displaying spatial pattern at
particular times
4. Locally weighted regression (LOWESS) is a very
valuable technique for highlighting signal in
stratigraphical data
5. Palaeoecologists now have a valuable set of
robust numerical tools available for summarising
patterns in stratigraphical data