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QUANTITATIVE PALAEOECOLOGY Lecture 3. Analysis of Stratigraphical Data BIO-351

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Introduction Temporal stratigraphical data Single sequence Partitioning or zonation Sequence splitting Rate-of-change analysis Gradient analysis and 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) CONTENTS

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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

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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”.

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CONSTRAINED CLASSIFICATIONS 1)Constrained agglomerative procedures CONSLINK 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

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3)Constrained optimal divisive analysisOPTIMAL 2 group______________________________ 3 group 4 group 4)Variable barriers approachBARRIER All methods in one program: ZONE n1n1 n1n1 n1n1 n2n2 n2n2 n3n3

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Pollen diagram and numerical zonation analyses for the complete Abernethy Forest 1974 data set. Birks & Gordon 1985

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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 259.315 328.41532 418.9153341 514.715334145 610.61532344145 78.1152632344145 85.88152632344145 94.7815242932344145 103.981524293233344145

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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

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Pollen percentage diagram plotted against depth. Lithostratigraphic column is represented; symbols are based on Troels-Smith (1995). Tzedakis 1994 Ioannina Basin

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Tzedakis 1994

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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

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Bennett 1996

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SEQUENCE SPLITTING Walker & Wilson1978J Biogeog 5, 1–21 Walker & Pittelkow1981J Biogeog 8, 37–51 SPLIT, SPLIT2 BOUND2 Need statistically ‘independent’ curves Pollen influx (grains cm–2 year–1) 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).

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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 & Grimm1986Ecology 67, 958-966 Grimm & Jacobson1992Climate Dynamics 6, 179-184 RATEPOL POLSTACK (TILIA) RATE OF CHANGE ANALYSIS

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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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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. - fertile - poor ONE PROXY, MANY SITES

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Pollen percentages from Loch Lang, Western Isles, plotted against age (radiocarbon years BP). Data from Bennett (1990).

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Pollen percentages from Hockham Mere, eastern England, plotted against age (radiocarbon years BP). Data from Bennett (1983).

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Comparison of Holocene rates of change at Loch Lang and Hockham Mere, with 2 - 2 dissimilarity coefficient on unsmoothed data, with a radiocarbon timescale. SE - continental NW - oceanic Rate x5 that at Loch Lang

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DATA SUMMARISATION BY ORDINATION OR GRADIENT ANALYSIS OF SINGLE SEQUENCE Ordination methodsCA/DCA orPCA joint plotbiplot Sample summary Species arrangement CA = correspondence analysis DCA = detrended correspondence analysis PCA = principal components analysis

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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. Biplot

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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

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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- stratigraphical 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

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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. Species arrangement

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Pollen types re-arranged on the basis of the weighted average for depth TRAN

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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 al1985Quat Res 23, 87–108 ANALOG MATCH MAT

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Compare fossil sample i with modern sample j. Calculate similarity between i and j Sij Find modern sample with highest similarity 'ANALOGUE‘ Repeat for all fossil samples Repeat for all modern samples ? Evaluation

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Dissimilarity coefficients, radiocarbon dates, pollen zones, and vegetation types represented by the top ten analogs from the Lake West Okoboji site.

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Maps of squared chord distance values with modern samples at selected time intervals

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Plots of the minimum squared chord-distance for each fossil spectrum at each of the eight sites.

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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 = <0.65), as defined by the mean squared Chi-squared estimate of dissimilarity (SCD, see text). Flower et al. 1997

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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)

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34% 16% 12%

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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)

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Pollen, oxygen-isotope stratigraphy, and sediment composition of Aegelsee core AE-1 (after Wegmüller and Lotter 1990)

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Pollen and oxygen-isotope stratigraphy of Gerzensee core G-III (after Eicher and Siegenthaler 1976)

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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’ SiteNo. of samples No. of taxa DCA Axis 1234 Aegelsee AE-11002657.212.02.31.4 Aegelsee AE-3543244.33.31.51.4 Gerzensee G-III652837.64.01.20.9 Faulenseemoos622544.118.85.03.8 Rotsee RL-250442338.213.33.12.3

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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. SitePredictor variable: 18O and depth Predictor variable: 18O Covariable: depth Predictor variable: 18O Number of response variables (DCA axes) Pollen DCA axes Aegelsee AE-10.01a 0.02a2 Aegelsee AE-30.01a0.160.201 Gerzensee G- III 0.01a0.460.571 Faulenseemoos0.01a 3 Rotsee RL-2500.01a0.210.082 a Significant at p< 0.05 (Lotter et al 1992)

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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 ANALYSIS OF TWO OR MORE SEQUENCES

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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 radiocarbon years before present (BP). SLOTSEQ Birks & Gordon 1985

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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).

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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 Lotter et al 1992

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FUGLA NESS, Shetland

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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.

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COMBINED SCALING

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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.

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Birks & Berglund, 1979 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.

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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. SiteNumber of biozone II and III samples Sum of eigenvalues (total variance) Gradient length Biozone II/III contrast Lobsigensee320.181.140.58 Murifeld220.120.630.24 Aegelsee600.120.62 0.24 R Saanenmöser210.150.730.14 Zeneggen (Hellelen) 160.290.88 0.47 R Hopschensee220.210.77 0.46 R Lago di Ganna210.461.360.06 R = revertence

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VARIANCE PARTITIONING Total variance= between-site variance + within-site variance + unexplained (‘error’) component

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VARIANCE PARTITIONING Use partial constrained ordinations to partition variance into: a)Unexplained variance 13.8% b)Between-site spatial variance13.2%p = 0.01 c)Within-site temporal variance73.0%p = 0.01 Within a sequence variance partitioning a)Unexplained variance not captured by zonation39.7% b)Variance captured by zone II33.2% c)Variance captured by zone III17.9% d)Variance captured by zone I9.2% Can now do a partial ordination of 39.7% unexplained variance to see what sort of patterns remain. Noise, chaos, trends or what?

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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). Tzedakis & Bennett 1995

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5e 7c 9c 11a + b + c Pollen percentage diagrams of selected arboreal taxa of the Metsovon, Zista, Pamvotis and Dodoni I and II forest periods of Ioannina 249 5e 7c 9c 11a + b + c

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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.

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Location of the two coring sites, Rezina Marsh and Gramousti Lake, in relation to altitude.

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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.

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Distribution in northern England of maximum values for pollen of Tilia during the period 5000 to 3000 B.C. MAPPING

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Maps of pollen frequencies 5,000 years B.P. PinusBetula

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Maps of pollen frequencies 5,000 years B.P. UlmusCorylus

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Maps of pollen frequencies 5,000 years B.P. QuercusTilia

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Map of pollen frequencies 5,000 years B.P. Alnus

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Map of scores of pollen spectra on the first principal component, 5,000 years B.P.

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Map of scores of pollen spectra on the second principal component, 5,000 years B.P.

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Map of scores of pollen spectra on the third principal component, 5,000 years B.P.

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Provisional map of wood- land types for the British Isles 5,000 years ago.

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Vegetation regions reconstructed from pollen data for 9,000, 6,000, 3,000, and 0 yr B.P.

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LOCALLY WEIGHTED REGRESSION W.S. ClevelandLOWESS locally weighted regression or LOESSscatterplot 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 y t-1, y, y t+1 or add weights to y t-1, y, y t+1 )

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(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. Linear

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(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. QuadraticLOWESS

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LOWESS - more general 1.Decide how “smooth” the fitted relationship should be. 2.Each observation given a weight depending on distance to observation x 1 for all adjacent points considered. 3.Fit simple linear regression for adjacent points using weighted least squares. 4.Repeat for all observations. 5.Calculate residuals (difference between observed and fitted y). 6.Estimate robustness weights based on residuals, so that well-fitted points have high weight. 7.Repeat LOWESS procedure but with new weights based on robustness weights and distance weights. Repeat for different degree of smoothness, to find “optimal” smoother.

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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. tri-cube function linear regression target value

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Round Loch of Glenhead LOWESS curve

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