Past: Paleontological Statistics Software Package for Education and Data Analysis

Palaeontologia Electronica http://palaeo-electronica.org

PAST: PALEONTOLOGICAL STATISTICS SOFTWARE PACKAGE FOR EDUCATION AND DATA ANALYSIS

Øyvind Hammer, David A.T. Harper, and Paul D. Ryan Øyvind Hammer. Paleontological Museum, University of Oslo, Sars gate1, 0562 Oslo, Norway David A. T. Harper. Geological Museum, Øster Voldgade 5-7, University of Copenhagen, DK-1350 Copen- hagen K, Denmark Paul D. Ryan. Department of Geology, National University of Ireland, Galway, Ireland

ABSTRACT

A comprehensive, but simple-to-use software package for executing a range of standard numerical analysis and operations used in quantitative paleontology has been developed. The program, called PAST (PAleontological STatistics), runs on standard Windows computers and is available free of charge. PAST integrates spreadsheet-type data entry with univariate and multivariate statistics, curve fitting, time- series analysis, data plotting, and simple phylogenetic analysis. Many of the functions are specific to paleontology and ecology, and these functions are not found in standard, more extensive, statistical packages. PAST also includes fourteen case studies (data files and exercises) illustrating use of the program for paleontological problems, making it a complete educational package for courses in quantitative methods. KEY WORDS: Software, data analysis, education Copyright: Palaeontological Association, 22 June 2001 Submission: 28 February 2001 Acceptance: 13 May 2001

INTRODUCTION vide students with a coherent, easy-to-use package that supported a wide range of Even a cursory glance at the recent algorithms while allowing hands-on experi- paleontological literature should convince ence with quantitative methods. The first anyone that quantitative methods in pale- PALSTAT version was programmed for the ontology have arrived at last. Neverthe- BBC microcomputer (Harper and Ryan less, many paleontologists still hesitate in 1987), while later revisions were made for applying such methods to their own data. the PC (Ryan et al. 1995). Incorporating One of the reasons for this has been the univariate and multivariate statistics and difficulty in acquiring and using appropri- other plotting and analytical functions spe- ate data-analysis software. The ‘PALSTAT’ cific to paleontology and ecology, PAL- program was developed in the 1980s in order to minimize such obstacles and pro-

Hammer, Øyvind, Harper, David A.T., and Paul D. Ryan, 2001. Past: Paleontological Statistics Software Package for Education and Data Analysis. Palaeontologia Electronica, vol. 4, issue 1, art. 4: 9pp., 178kb. http://palaeo-electronica.org/2001_1/past/issue1_01.htm. Øyvind Hammer, David A. T. Harper, and Paul D. Ryan: PALEONTOLOGICAL STATISTICS SOFTWARE

STAT gained a wide user base among validation and correction of diversity both paleontologists and biologists. curves). After some years of service, however, One of the main ideas behind PAST is it was becoming clear that PALSTAT had to include many functions in a single pro- to undergo major revision. The DOS- gram package while providing for a con- based user interface and an architecture sistent user interface. This minimizes time designed for computers with miniscule spent on searching for, buying, and learn- memories (by modern standards) was ing a new program each time a new becoming an obstacle for most users. method is approached. Similar projects Also, the field of quantitative paleontology are being undertaken in other fields (e,g., has changed and expanded considerably systematics and morphometry). One in the last 15 years, requiring the imple- example is Wayne Maddison’s ‘Mesquite’ mentation of many new algorithms. There- package (http://mesquite.biosci.ari- fore, in 1999 we decided to redesign the zona.edu/mesquite/mesquite.html). program totally, keeping the general con- An important aspect of PALSTAT was cept but without concern for the original the inclusion of case studies, including source code. The new program, called data sets designed to illustrate possible PAST (PAleontological STatistics) takes uses of the algorithms. Working through full advantage of the Windows operating these examples allowed the student to system, with a modern, spreadsheet- obtain a practical overview of the different based, user interface and extensive methodologies in a very efficient way. graphics. Most PAST algorithms produce Some of these case studies have been graphical output automatically, and the adjusted and included in PAST, and new high-quality figures can be printed or case studies have been added in order to pasted into other programs. The function- demonstrate the new features. The case ality has been extended substantially with studies are primarily designed as student inclusion of important algorithms in the exercises for courses in paleontological standard PAST toolbox. Functions found data analysis. The PAST program, docu- in PAST that were not available in PAL- mentation, and case studies are available STAT include (but are not limited to) parsi- free of charge at http://www.nhm.uio.no/ mony analysis with cladogram plotting, ~ohammer/past. detrended correspondence analysis, principal coordinates analysis, time-series PLOTTING AND BASIC STATISTICS analysis (spectral and autocorrelation), geometrical analysis (point distribution Graphical plotting functions (see http:// and Fourier shape analysis), rarefaction, www.nhm.uio.no/~ohammer/past/ modelling by nonlinear functions (e.g., plot.html) in PAST include different types logistic curve, sum-of-sines) and quantita- of graph, histogram, and scatter plots. The tive biostratigraphy using the unitary asso- program can also produce ternary (trian- ciations method. We believe that the gle) plots and survivorship curves. functions we have implemented reflect the Descriptive statistics (see http:// present practice of paleontological data www.nhm.uio.no/~ohammer/past/ analysis, with the exception of some func- univar.html) include minimum, maximum, tionality that we hope to include in future and mean values, population variance, versions (e.g., morphometric analysis with sample variance, population and sample landmark data and more methods for the standard deviations, median, skewness, and kurtosis.

2 Øyvind Hammer, David A. T. Harper, and Paul D. Ryan: PALEONTOLOGICAL STATISTICS SOFTWARE

For associations or paleocommunity ables (components) that account for as data, several diversity statistics can be much of the variance in a multidimensional computed: number of taxa, number of indi- data set as possible (Davis 1986, Harper viduals, dominance, Simpson index, 1999). These new variables are linear Shannon index (entropy), Menhinick’s and combinations of the original variables. Margalef’s richness indices, equitability, PCA is a standard method for reducing the and Fisher’s a (Harper 1999). dimensionality of morphometric and eco- Rarefaction (Krebs 1989) is a method logical data. The PCA routine finds the for estimating the number of taxa in a eigenvalues and eigenvectors of the vari- small sample, when abundance data for a ance-covariance matrix or the correlation larger sample are given. With this method, matrix. The eigenvalues, giving a measure the number of taxa in samples of different of the variance accounted for by the corre- sizes can be compared. An example appli- sponding eigenvectors (components), are cation of rarefaction in paleontology is displayed together with the percentages of given by Adrain et al. (2000). variance accounted for by each of these The program also includes standard components. A scatter plot of these data statistical tests (see http:// projected onto the principal components is www.nhm.uio.no/~ohammer/past/ provided, along with the option of including twosets.html) for univariate data, includ- the Minimal Spanning Tree, which is the ing: tests for normality (chi-squared and shortest possible set of connected lines Shapiro-Wilk), the F and t tests, one-way joining all points. This may be used as a ANOVA, χ2 for comparing binned samples, visual aid in grouping close points (Harper Mann-Whitney’s U test and Kolmogorov- 1999). The component loadings can also Smirnov association test (non-parametric), be plotted. Bruton and Owen (1988) and both Spearman’s r and Kendall’s t describe a typical morphometrical applica- non-parametric rank-order tests. Dice and tion of PCA. Jaccard similarity indices are used for Principal coordinates analysis (PCO) comparing associations limited to is another ordination method, somewhat absence/presence data. The Raup-Crick similar to PCA. The PCO routine finds the randomization method for comparing eigenvalues and eigenvectors of a matrix associations (Raup and Crick 1979) is containing the distances between all data also implemented. Finally, the program points, measured with the Gower distance can also compute correlation matrices and or the Euclidean distance. The PCO algo- perform contingency-table analysis. rithm used in PAST was taken from Davis (1986), which also includes a more MULTIVARIATE ANALYSIS detailed description of the method and example analysis. Paleontological data sets, whether Correspondence analysis (CA) is a based on fossil occurrences or morphol- further ordination method, somewhat simi- ogy, often have high dimensionality. PAST lar to PCA, but for counted or discrete includes several methods for multivariate data. Correspondence analysis can com- data analysis (see http://www.nhm.uio.no/ pare associations containing counts of ~ohammer/past/multivar.html), including taxa or counted taxa across associations. methods that are specific to paleontology Also, CA is more suitable if it is expected and biology. that species have unimodal responses to Principal components analysis (PCA) the underlying parameters, that is they is a procedure for finding hypothetical vari- favor a certain range of the parameter and

3 Øyvind Hammer, David A. T. Harper, and Paul D. Ryan: PALEONTOLOGICAL STATISTICS SOFTWARE become rare under for lower and higher Bray-Curtis, chord and Morisita indices for values (this is in contrast to PCA, that abundance data, and Dice, Jaccard, and assumes a linear response). The CA algo- Raup-Crick indices for presence-absence rithm employed in PAST is taken from data. Davis (1986), which also includes a more Seriation of an absence-presence detailed description of the method and matrix can be performed using the algo- example analysis. Ordination of both sam- rithm described by Brower and Kyle ples and taxa can be plotted in the same (1988). For constrained seriation, columns CA coordinate system, whose axes will should be ordered according to some normally be interpreted in terms of envi- external criterion (normally stratigraphic ronmental parameters (e.g., water depth, level) or positioned along a presumed fau- type of substrate temperature). nal gradient. Seriation routines attempt to The Detrended Correspondence reorganize the data matrix such that the (DCA) module uses the same ‘reciprocal presences are concentrated along the averaging’ algorithm as the program Dec- diagonal. Also, in the constrained mode, orana (Hill and Gauch 1980). It is special- the program runs a ‘Monte Carlo’ simula- ized for use on “ecological” data sets with tion to determine whether the original abundance data (taxa in rows, localities in matrix is more informative than a random columns), and it has become a standard matrix. In the unconstrained mode both method for studying gradients in such rows and columns are free to move: the data. Detrending is a type of normalization method then amounts to a simple form of procedure in two steps. The first step ordination. involves an attempt to “straighten out” The degree of separation between to points lying along an arch-like pattern (= hypothesized groups (e.g., species or Kendall’s Horseshoe). The second step morphs) can be investigated using dis- involves “spreading out” the points to criminant analysis (Davis 1986). Given two avoid artificial clustering at the edges of sets of multivariate data, an axis is con- the plot. structed that maximizes the differences Hierarchical clustering routines pro- between the sets. The two sets are then duce a dendrogram showing how and plotted along this axis using a histogram. where data points can be clustered (Davis The null hypothesis of group means equal- 1986, Harper 1999). Clustering is one of ity is tested using Hotelling’s T2 test. the most commonly used methods of multivariate data analysis in paleontology. CURVE FITTING AND TIME-SERIES ANALYSIS Both R-mode clustering (groupings of taxa), and Q-mode clustering (grouping Curve fitting (see http:// variables or associations) can be carried www.nhm.uio.no/~ohammer/past/fit- out within PAST by transposing the data ting.html) in PAST includes a range of lin- matrix. Three different clustering algo- ear and non-linear functions. rithms are available: the unweighted pair- Linear regression can be performed group average (UPGMA) algorithm, the with two different algorithms: standard single linkage (nearest neighbor) algo- (least-squares) regression and the rithm, and Ward’s method. The similarity- ”Reduced Major Axis” method. Least- association matrix upon which the clusters squares regression keeps the x values are based can be computed using nine dif- fixed, and it finds the line that minimizes ferent indices: Euclidean distance, correla- the squared errors in the y values. tion (using Pearson’s r or Spearman’s ρ, Reduced Major Axis minimizes both the x

4 Øyvind Hammer, David A. T. Harper, and Paul D. Ryan: PALEONTOLOGICAL STATISTICS SOFTWARE and the y errors simultaneously. Both x of time series can be performed using the and y values can also be log-transformed, Lomb periodogram algorithm, which is in effect fitting the data to the “allometric” more appropriate than the standard Fast function y=10bxa. An allometric slope Fourier Transform for paleontological data value around 1.0 indicates that an “isomet- (which are often unevenly sampled; Press ric” fit may be more applicable to the data et al. 1992). Evenly-spaced data are of than an allometric fit. Values for the course also accepted. In addition to the regression slope and intercepts, their plotting of the periodogram, the highest errors, a χ2 correlation value, Pearson’s r peak in the spectrum is presented with its coefficient, and the probability that the col- frequency and power value, together with umns are not correlated are given. a probability that the peak could occur In addition, the sum of up to six sinu- from random data. The data set can be soids (not necessarily harmonically optionally detrended (linear component related) with frequencies specified by the removed) prior to analysis. Applications user, but with unknown amplitudes and include detection of Milankovitch cycles in phases, can be fitted to bivariate data. isotopic data (Muller and MacDonald This method can be useful for modeling 2000) and searching for periodicities in periodicities in time series, such as annual diversity curves (Raup and Sepkoski growth cycles or climatic cycles, usually in 1984). Autocorrelation (Davis 1986) can combination with spectral analysis (see be carried out on evenly sampled tempo- below). The algorithm is based on a least- ral-stratigraphical data. A predominantly squares criterion and singular value zero autocorrelation signifies random decomposition (Press et al. 1992). Fre- data—periodicities turn up as peaks. quencies can also be estimated by trial and error, by adjusting the frequency so GEOMETRICAL ANALYSIS that amplitude is maximized. Further, PAST allows fitting of data to PAST includes some functionality for geometrical analysis (see http:// the logistic equation y=a/(1+be-cx), using www.nhm.uio.no/~ohammer/past/mor- Levenberg-Marquardt nonlinear optimiza- pho.html), even if an extensive morpho- tion (Press et al. 1992). The logistic equa- metrics module has not yet been tion can model growth with saturation, and implemented. We hope to implement more it was used by Sepkoski (1984) to extensive functionality, such as landmark- describe the proposed stabilization of based methods, in future versions of the marine diversity in the late Palaeozoic. program. Another option is fitting to the von Berta- The program can plot rose diagrams lanffy growth equation y=a(1-be-cx). This (polar histograms) of directions. These equation is used for modeling growth of can be used for plotting current-oriented multi-celled animals (Brown and Rothery specimens, orientations of trackways, ori- 1993). entations of morphological features (e.g., Searching for periodicities in time trilobite terrace lines), etc. The mean series (data sampled as a function of time) angle together with Rayleigh’s spread are has been an important and controversial given. Rayleigh’s spread is further tested subject in paleontology in the last few against a random distribution using Ray- decades, and we have therefore imple- leigh’s test for directional data (Davis mented two methods for such analysis in 1986). A χ2 test is also available, giving the program: spectral analysis and autocorrelation. Spectral (harmonic) analysis

5 Øyvind Hammer, David A. T. Harper, and Paul D. Ryan: PALEONTOLOGICAL STATISTICS SOFTWARE the probability that the directions are ran- Character states are coded using inte- domly and evenly distributed. gers in the range 0 to 255. The first taxon Point distribution statistics using near- is treated as the outgroup and will be est neighbor analysis (modified from Davis placed at the root of the tree. Missing val- 1986) are also provided. The area is esti- ues are coded with a question mark. There mated using the convex hull, which is the are four algorithms available for finding smallest convex polygon enclosing the short trees: branch-and-bound (finds all points. The probability that the distribution shortest trees), exhaustive (finds all short- is random (Poisson process, giving an est trees, and allows the plotting of tree- exponential nearest neighbor distribution) length distribution), heuristic nearest is presented, together with the ‘R’ value. neighbor interchange (NNI) and heuristic Clustered points give R<1, Poisson pat- subtree pruning and regrafting (SPR). terns give R~1, while over-dispersed Three different optimality criteria are avail- points give R>1. Applications of this mod- able: Wagner (reversible and ordered ule include spatial ecology (are in-situ bra- characters), Fitch (reversible and unor- chiopods clustered) and morphology (are dered characters), and Dollo (irreversible trilobite tubercles over-dispersed; see and ordered). Bootstrapping can be per- Hammer 2000). formed with a given number of replicates. The Fourier shape analysis module All shortest (most parsimonious) trees (Davis 1986) accepts x-y coordinates digi- can be viewed. If bootstrapping has been tized around an outline. More than one performed, a bootstrap value is given at shape can be analyzed simultaneously. the root of the subtree specifying each Points do not need to be evenly spaced. group. The sine and cosine components are The consensus tree of all shortest given for the first ten harmonics, and the (most parsimonious) trees can also be coefficients can then be copied to the main viewed. Two consensus rules are imple- spreadsheet for further analysis (e.g., by mented: strict (groups must be supported PCA). Elliptic Fourier shape analysis is by all trees) and majority (groups must be also provided (Kuhl and Giardina 1982). supported by more than 50% of the trees). For an application of elliptic Fourier shape PAST can read and export files in the analysis in paleontology, see Renaud et al. NEXUS format, making it compatible with (1996). packages such as PAUP and MacClade.

PHYLOGENETIC ANALYSIS (PARSIMONY) BIOSTRATIGRAPHICAL CORRELATION WITH UNITARY ASSOCIATIONS The cladistics package (see http:// www.nhm.uio.no/~ohammer/past/cla- Quantitative or semi-quantitative dist.html) in PAST is fully operational, but methods for biostratigraphy are not yet in is lacking comprehensive functionality. For common use, except for the relatively sub- example, there is no character reconstruc- jective approach of graphical correlation. tion (plotting of steps on the cladogram). Such methods are, however, well devel- The use of PAST in parsimony analysis oped, and we hope that the inclusion of should probably be limited to entry-level one method in PAST will help introduce education and preliminary investigations. more paleontologists to this field. We have The parsimony algorithms used in PAST chosen to implement Unitary Associations are from Kitching et al. (1998). analysis (see http://www.nhm.uio.no/ ~ohammer/past/unitary.html) (Guex 1991)

6 Øyvind Hammer, David A. T. Harper, and Paul D. Ryan: PALEONTOLOGICAL STATISTICS SOFTWARE because of its solid theoretical basis and in the program. The cases are taken from minimum of statistical assumptions. such diverse fields as morphology, taxon- The data input consists of a presence- omy, paleoecology, paleoclimatology, sedi- absence matrix with samples in rows and mentology, extinction studies, and taxa in columns. Samples belong to a set biostratigraphy. The examples are taken of sections (localities), where the strati- from both vertebrate and invertebrate graphical relationships within each section paleontology, and they cover the whole of are known. The basic idea is to generate a the Phanerozoic. These case studies are set of assemblage zones (similar to ‘Oppel well suited for an introductory course in zones’) that are optimal in the sense that paleontological data analysis and have they give maximal stratigraphic resolution been tested in classroom situations. The with a minimum of superpositional contra- cases are organized into four main subject dictions. An example of such a contradic- areas: morphology and taxonomy, bioge- tion would be a section containing species ography and paleoecology, time-series A above species B, while assemblage 1 analysis, and biostratigraphy. (containing species A) is placed below Case studies 1-51 involve the descrip- assemblage 2 (containing species B). The tion and analysis of morphological varia- method of Unitary Associations is a logical tion of different sorts, while case study 6 but somewhat complicated procedure, targets some phylogenetic problems in a consisting of several steps. Its implemen- group of Cambrian trilobites and the mam- tation in PAST does not include all the fea- mals. tures found in the standard program, Case Study 1 investigates the external called BioGraph (Savary and Guex 1999), morphology of the Permian brachiopod and advanced users are referred to that Dielasma, developing ontogenic models package. for the genus and comparing the growth PAST produces a detailed report of rates and outlines of different samples the analysis, including maximal cliques, from in and around a Permian reef com- unitary associations, correlation table, plex. In a more focused exercise, Case reproducibility matrix, contradictions Study 2 uses spatial statistics to assess between cliques, biostratigraphic graph, the mode of distribution of tubercles on the graph of superpositional relationships cranidium of the trilobite Paradoxides from between maximal cliques, and strong the middle Cambrian. components (cycles) in the graphs (Guex Case Study 3 tackles the multivariate 1991). It is important to inspect these morphometrics of the Ordovician illaenid results thoroughly in order to assess the trilobite Stenopareia using Principal Com- quality of the correlation and to improve ponents Analysis (PCA), Principal Coordi- the quality of the data, if necessary. Angio- nate Analysis (PCO), cluster and lini and Bucher (1999) give an example of discriminant analyses to determine the such careful use of the method of Unitary validity of two species from Scandinavia. Associations.

1. PE Note: The Case Study files are avail- CASE STUDIES able from the PE site, and also directly from the author. The links below point to the The fourteen case studies have been author's site, which will, as time and the designed to demonstrate both the use of author proceed, contain updates and newer different data analysis methods in paleon- versions. The author’s site is: http:// tology and the specific use of the functions www.nhm.uio.no/~ohammer/past/.

7 Øyvind Hammer, David A. T. Harper, and Paul D. Ryan: PALEONTOLOGICAL STATISTICS SOFTWARE

Case Study 4 demonstrates the use of variate techniques (similarity and distance Elliptic Fourier shape analysis and princi- coefficients, cluster analysis, detrended pal components for detecting changes in correspondence analysis, and seriation) trilobite cephalon shape through ontogeny. the reality and mutual relationships of In Case Study 5, aspects of the allom- these benthic associations can be tested etric growth of the Triassic rhynchosaur using a modified dataset. Scaphonyx are investigated using regres- Case Study 10 discusses some well- sion analysis. known Jurassic shelly faunas from Case Study 6 investigates the phylo- England and France. The integrity and genetic structure of the middle Cambrian onshore – offshore distribution of six Cor- Paradoxididae through cladistic analysis, allian bivalve-dominated communities is using parsimony analysis and bootstrap- investigated with diversity measures, clus- ping. Similar techniques can be applied to ter analysis and detrended correspon- a matrix of 20 taxa of mammal; cla- dence analysis. dograms generated by the program can Case Study 11 completes the analysis be compared with a cluster analysis of the of biotic assemblages with an investigation data matrix. of the direction and orientation of a bed- Case studies 7-11 cover aspects of ding-plane sample of brachiopod shells paleobiogeography and paleoecology. from the upper Ordovician rocks of Scot- Case Study 7 analyzes a global dataset of land. late Ordovician brachiopod distributions. A Two cases involve the study of time series of provincial faunas were developed series data. Case Study 12 investigates against a background of regression and the periodicity of mass extinctions during cooler surface waters during the first strike the Permian to Recent time interval using of the late Ordovician (Hirnantian) glacia- spectral analysis. A number of diversity tion. Through the calculation of similarity curves can be modeled for the Paleozoic and distance coefficients together with and post-Paleozoic datasets available in cluster analysis, these data can be orga- Fossil Record 2, and turnover rates can be nized into a set of latitudinally controlled viewed for Phanerozoic biotas. provinces. Seriation helps to develop any Case Study 13 addresses the period- faunal, possibly climatically generated, icity of oxygen isotope data from ice cores gradients within the data structure. representing the last million years of Earth In Case Study 8 faunal changes history. through a well-documented section in the The final case study demonstrates the upper Llanvirn rocks of central Wales are use of quantitative biostratigraphical corre- investigated graphically and by the calculation with the method of Unitary Associa- lation of diversity, dominance, and related tions. Eleven sections from the Eocene of parameters for each of ten horizons in the Slovenia are correlated using alveolinid sections. The changes in faunas finger- foraminiferans studied by Drobne. print environmental shifts through the section, shadowed by marked changes in CONCLUSION lithofacies. This dataset is ripe for consid- erable experimentation. Statistical and other quantitative meth- Case Study 9 involves a re-evaluation ods are now very much part of the paleon- of Ziegler’s classic Lower Paleozoic tologists’ tool kit. PAST is a free, user- depth-related communities from the friendly and comprehensive package of Anglo-Welsh area. Using a range of multi- statistical and graphical algorithms, tailor

8 Øyvind Hammer, David A. T. Harper, and Paul D. Ryan: PALEONTOLOGICAL STATISTICS SOFTWARE made for the scientific investigation of dence of lateral inhibition. Acta Palaeontologica paleontological material. PAST provides a Polonica, 45:251-270. Harper, D.A.T. (ed.). 1999. Numerical Palaeobiology. window on current and future develop- John Wiley & Sons, New York. ments in this rapidly evolving research Harper, D.A.T. and Ryan, P.D. 1987. PALSTAT. A statisti- area. Together with a simple manual and cal package for palaeontologists. Lochee Publica- tions and the Palaeontological Association. linked case histories and datasets, the Hill, M.O. and Gauch Jr, H.G. 1980. Detrended Corre- package is an ideal educational aid and spondence analysis: an improved ordination tech- first-approximation research tool. Planned nique. Vegetation, 42:47-58. future developments include extended Kitching, I.J., Forey, P.L., Humphries, C.J. and Williams, D.M. 1998. Cladistics. Oxford University Press, functionality for morphometrics and the Oxford. extension of available algorithms within Krebs, C.J. 1989. Ecological Methodology. Harper & the cladistics and unitary associations Row, New York. modules. Kuhl, F.P. and Giardina, C.R. 1982. Elliptic Fourier analysis of a closed contour. Computer Graphics and Image Processing, 18:259-278. REFERENCES Muller, R.A. and MacDonald, G.J. 2000. Ice ages and astronomical causes: Data, Spectral Analysis, and Adrain, J.M., Westrop, S.R. and Chatterton, D.E. 2000. Mechanisms. Springer Praxis, Berlin. Silurian trilobite alpha Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flan- diversity and the end-Ordovician mass extinction. Paleo- nery, B.P. 1992. Numerical Recipes in C. Cambridge biology, 26:625-646. University Press, Cambridge. Angiolini, L. and Bucher, H. 1999. Taxonomy and quanti- Raup, D. and Crick, R.E. 1979. Measurement of faunal tative biochronology of similarity in paleontology. Journal of Paleontology, Guadalupian brachiopods from the Khuff Formation, 53:1213-1227. Southeastern Oman. Raup, D. and Sepkoski, J.J. 1984. Periodicities of extinc- Geobios, 32:665-699. tions in the geologic past. Proceedings of the Brower, J.C. and Kyle, K.M. 1988. Seriation of an original National Academy of Science, 81:801-805. data matrix as applied to Renaud, S., Michaux, J., Jaeger, J.-J. and Auffray, J.-C. palaeoecology. Lethaia, 21:79-93. 1996. Fourier analysis applied to Stephanomys Brown, D. and Rothery, P. 1993. Models in biology: (Rodentia, Muridae) molars: nonprogressive evolu- mathematics, statistics and computing. John Wiley & tionary pattern in a gradual lineage. Paleobiology, Sons, New York. 22:255-265. Bruton, D.L. and Owen, A.W. 1988. The Norwegian Ryan, P.D., Harper, D.A.T. and Whalley, J.S. 1995. PAL- Upper Ordovician illaenid trilobites. Norsk Geolo- STAT, Statistics for palaeontologists. Chapman & Hall gisk Tidsskrift, 68:241-258. (now Kluwer Academic Publishers). Davis, J.C. 1986. Statistics and Data Analysis in Geol- Sepkoski, J.J. 1984. A kinetic model of Phanerozoic tax- ogy. John Wiley & Sons, New York. onomic diversity. Paleobiology, 10:246-267. Guex, J. 1991. Biochronological Correlations. Springer Savary, J. and Guex, J. 1999. Discrete Biochronological Verlag, Berlin. Scales and Unitary Associations: Description of the Hammer, Ø. 2000. Spatial organisation of tubercles and BioGraph Computer Program. Mémoires de Geolo- terrace lines in Paradoxides forchhammeri - evi- gie (Lausanne), 34.