Preface

The delver into nature's aims

Seeks freedom and perfection;

Let calculation sift his claims

With faith and circumspection.

GOETHE

As a premise to this textbook on Numerical ecology, the authors wish to state their opinion concerning the role of data analysis in ecology. In the above quotation, Goethe cautioned readers against the use of mathematics in the natural sciences. In his opinion, mathematics may obscure, under an often esoteric language, the natural phenomena that scientists are trying to elucidate. Unfortunately, there are many examples in the ecological literature where the use of mathematics unintentionally lent support to Goethe’s thesis. This has become more frequent with the advent of computers, which facilitated access to the most complex numerical treatments. Fortunately, many other examples, including those discussed in the present book, show that ecologists who master the theoretical bases of numerical methods and know how to use them can derive a deeper understanding of natural phenomena from their calculations.

Numerical approaches can never dispense researchers from ecological reflection on observations. Data analysis must be seen as an objective and non-exclusive approach to carry out in-depth analysis of the data. Consequently, throughout this book, we put emphasis on ecological applications, which illustrate how to go from numerical results to ecological conclusions.

This book is written for the practising ecologists — graduate students and professional researchers. For this reason, it is organized both as a practical handbook and a reference textbook. Our goal is to describe and discuss the numerical methods which are successfully being used for analysing ecological data, using a clear and comprehensive approach. These methods are derived from the fields of mathematical physics, parametric and nonparametric statistics, information theory, numerical taxonomy, archaeology, psychometry, sociometry, econometry, and others. Some of these methods are presently only used by those ecologists who are especially interested

in numerical data analysis; field ecologists often do not master the bases of these techniques. For this reason, analyses reported in the literature are often carried out using techniques that are not fully adapted to the data under study, leading to conclusions that are sub-optimal with respect to the field observations. When we were writing the first English edition of Numerical ecology (Legendre & Legendre, 1983a), this warning mainly concerned multivariate versus elementary statistics. Nowadays, most ecologists are capable of using multivariate methods; the above remark now especially applies to the analysis of autocorrelated data (see Section 1.1; Chapters 12 and 13) and the joint analysis of several data tables (Sections 10.5 and 10.6; Chapter 11).

Computer packages provide easy access to the most sophisticated numerical methods. Ecologists with inadequate background often find, however, that using highlevel packages leads to dead ends. In order to efficiently use the available numerical tools, it is essential to clearly understand the principles that underlay numerical methods, and their limits. It is also important for ecologists to have guidelines for interpreting the heaps of computer-generated results. We therefore organized the present text as a comprehensive outline of methods for analysing ecological data, and also as a practical handbook indicating the most usual packages.

Our experience with graduate teaching and consulting has made us aware of the problems that ecologists may encounter when first using advanced numerical methods. Any earnest approach to such problems requires in-depth understanding of the general principles and theoretical bases of the methods to be used. The approach followed in this book uses standardized mathematical symbols, abundant illustration, and appeal to intuition in some cases. Because the text has been used for graduate teaching, we know that, with reasonable effort, readers can get to the core of numerical ecology. In order to efficiently use numerical methods, their aims and limits must be clearly understood, as well as the conditions under which they should be used. In addition, since most methods are well described in the scientific literature and are available in computer packages, we generally insist on the ecological interpretation of results; computation algorithms are described only when they may help understand methods. Methods described in the book are systematically illustrated by numerical examples and/or applications drawn from the ecological literature, mostly in English; references written in languages other than English or French are generally of historical nature.

The expression numerical ecology refers to the following approach. Mathematical ecology covers the domain of mathematical applications to ecology. It may be divided into theoretical ecology and quantitative ecology. The latter, in turn, includes a number of disciplines, among which modelling, ecological statistics, and numerical ecology.

Numerical ecology is the field of quantitative ecology devoted to the numerical analysis of ecological data sets. Community ecologists, who generally use multivariate data, are the primary users of these methods. The purpose of numerical ecology is to describe and interpret the structure of data sets by combining a variety of numerical approaches. Numerical ecology differs from descriptive or inferential biological statistics in that it extensively uses non-statistical procedures, and systematically

combines relevant multidimensional statistical methods with non-statistical numerical techniques (e.g. cluster analysis); statistical inference (i.e. tests of significance) is seldom used. Numerical ecology also differs from ecological modelling, even though the extrapolation of ecological structures is often used to forecast values in space or/and time (through multiple regression or other similar approaches, which are collectively referred to as correlative models). When the purpose of a study is to predict the critical consequences of alternative solutions, ecologists must use predictive ecological models. The development of models that predict the effects on some variables, caused by changes in others (see, for instance, De Neufville & Stafford, 1971), requires a deliberate causal structuring, which is based on ecological theory; it must include a validation procedure. Such models are often difficult and costly to construct. Because the ecological hypotheses that underlay causal models (see for instance Gold, 1977, Jolivet, 1982, or Jørgensen, 1983) are often developed within the context of studies using numerical ecology, the two fields are often in close contact.

Loehle (1983) reviewed the different types of models used in ecology, and discussed some relevant evaluation techniques. In his scheme, there are three types of simulation models: logical, theoretical, and “predictive”. In a logical model, the representation of a system is based on logical operators. According to Loehle, such models are not frequent in ecology, and the few that exist may be questioned as to their biological meaningfulness. Theoretical models aim at explaining natural phenomena in a universal fashion. Evaluating a theory first requires that the model be accurately translated into mathematical form, which is often difficult to do. Numerical models (called by Loehle “predictive” models, sensu lato) are divided in two types: application models (called, in the present book, predictive models, sensu stricto) are based on well-established laws and theories, the laws being applied to resolve a particular problem; calculation tools (called forecasting or correlative models in the previous paragraph) do not have to be based on any law of nature and may thus be ecologically meaningless, but they may still be useful for forecasting. In forecasting models, most components are subject to adjustment whereas, in ideal predictive models, only the boundary conditions may be adjusted.

Ecologists have used quantitative approaches since the publication by Jaccard (1900) of the first association coefficient. Floristics developed from this seed, and the method was eventually applied to all fields of ecology, often achieving high levels of complexity. Following Spearman (1904) and Hotelling (1933), psychometricians and social scientists developed non-parametric statistical methods and factor analysis and, later, nonmetric multidimensional scaling (MDS). During the same period, anthropologists (e.g. Czekanowski, 1909) were interested in numerical classification. The advent of computers made it possible to analyse large data sets, using combinations of methods derived from various fields and supplemented with new mathematical developments. The first synthesis was published by Sokal & Sneath (1963), who established numerical taxonomy as a new discipline.

Numerical ecology combines a large number of approaches, derived from many disciplines, in a general methodology for analysing ecological data sets. Its chief characteristic is the combined use of treatments drawn from different areas of mathematics and statistics. Numerical ecology acknowledges the fact that many of the existing numerical methods are complementary to one another, each one allowing to explore a different aspect of the information underlying the data; it sets principles for interpreting the results in an integrated way.

The present book is organized in such a way as to encourage researchers who are interested in a method to also consider other techniques. The integrated approach to data analysis is favoured by numerous cross-references among chapters and the presence of sections presenting syntheses of subjects. The book synthesizes a large amount of information from the literature, within a structured and prospective framework, so as to help ecologists take maximum advantage of the existing methods.

This second English edition of Numerical ecology is a revised and largely expanded translation of the second edition of Écologie numérique (Legendre & Legendre, 1984a, 1984b). Compared to the first English edition (1983a), there are three new chapters, dealing with the analysis of semiquantitative data (Chapter 5), canonical analysis (Chapter 11), and spatial analysis (Chapter 13). In addition, new sections have been added to almost all other chapters. These include, for example, new sections (numbers given in parentheses) on: autocorrelation (1.1), statistical testing by randomization (1.2), coding (1.5), missing data (1.6), singular value decomposition (2.11), multiway contingency tables (6.3), cophenetic matrix and ultrametric property (8.3), reversals (8.6), partitioning by K-means (8.8), cluster validation (8.12), a review of regression methods (10.3), path analysis (10.4), a review of matrix comparison methods (10.5), the 4th-corner problem (10.6), several new methods for the analysis of data series (12.3-12.5), detection of discontinuities in multivariate series (12.6), and Box-Jenkins models (12.7). There are also sections listing available computer programs and packages at the end of several Chapters.

The present work reflects the input of many colleagues, to whom we express here our most sincere thanks. We first acknowledge the outstanding collaboration of Professors Serge Frontier (Université des Sciences et Techniques de Lille) and F. James Rohlf (State University of New York at Stony Brook) who critically reviewed our manuscripts for the first French and English editions, respectively. Many of their suggestions were incorporated into the texts which are at the origin of the present edition. We are also grateful to Prof. Ramón Margalef for his support, in the form of an influential Preface to the previous editions. Over the years, we had fruitful discussions on various aspects of numerical methods with many colleagues, whose names have sometimes been cited in the Forewords of previous editions.

During the preparation of this new edition, we benefited from intensive collaborations, as well as chance encounters and discussions, with a number of people who have thus contributed, knowingly or not, to this book. Let us mention a few. Numerous discussions with Robert R. Sokal and Neal L. Oden have sharpened our

understanding of permutation methods and methods of spatial data analysis. Years of discussion with Pierre Dutilleul and Claude Bellehumeur led to the Section on spatial autocorrelation. Pieter Kroonenberg provided useful information on the relationship between singular value decomposition (SVD) and correspondence analysis (CA). Peter Minchin shed light on detrended correspondence analysis (DCA) and nonmetric multidimensional scaling (MDS). A discussion with Richard M. Cormack about the behaviour of some model II regression techniques helped us write Subsection 10.3.2. This Subsection also benefited from years of investigation of model II methods with David J. Currie. In-depth discussions with John C. Gower led us to a better understanding of the metric and Euclidean properties of (dis)similarity coefficients and of the importance of Euclidean geometry in grasping the role of negative eigenvalues in principal coordinate analysis (PCoA). Further research collaboration with Marti J. Anderson about negative eigenvalues in PCoA, and permutation tests in multiple regression and canonical analysis, made it possible to write the corresponding sections of this book; Dr. Anderson also provided comments on Sections 9.2.4, 10.5 and 11.3. Cajo J. F. ter Braak revised Chapter 11 and parts of Chapter 9, and suggested a number of improvements. Claude Bellehumeur revised Sections 13.1 and 13.2; FrançoisJoseph Lapointe commented on successive drafts of 8.12. Marie-Josée Fortin and Daniel Borcard provided comments on Chapter 13. The ÉCOTHAU program on the Thau lagoon in southern France (led by Michel Amanieu), and the NIWA workshop on soft-bottom habitats in Manukau harbour in New Zealand (organized by Rick Pridmore and Simon Thrush of NIWA), provided great opportunities to test many of the ecological hypothesis and methods of spatial analysis presented in this book.

Graduate students at Université de Montréal and Université Laval have greatly contributed to the book by raising interesting questions and pointing out weaknesses in previous versions of the text. The assistance of Bernard Lebanc was of great value in transferring the ink-drawn figures of previous editions to computer format. Philippe Casgrain helped solve a number of problems with computers, file transfers, formats, and so on.

While writing this book, we benefited from competent and unselfish advice … which we did not always follow. We thus assume full responsibility for any gaps in the work and for all the opinions expressed therein. We shall therefore welcome with great interest all suggestions or criticisms from readers.

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