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Canonical Analysis: A Review with Applications in Ecology PDF

360 Pages·1985·10.006 MB·English
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Biomathematics Volume 12 Managing Editor S.A. Levin Editorial Board M. Arbib H.J. Bremermann J. Cowan W.M. Hirsch S. Karlin J. Keller K. Krickeberg R.C. Lewontin R.M. May J.D. Murray L.A. Segel R Gittins Canonical Analysis A Review with Applications in Ecology With 16 Figures Springer-Verlag Berlin Heidelberg NewY ork Tokyo Robert Gittins Department of Plant Sciences University of Western Ontario London, Ontario, Canada N6A 5B7 AMS-MOS Classification (1980): 62-02, 62H20, 62H 17, 62J05, 92-02, 92A17 ISBN -13 :978-3-642-69880-4 e-ISBN -13 :978-3-642-69878-1 DOl: 10.1007/978-3-642-69878-1 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustra tions, broadcasting, reproduction by photocopying machine or similar means, and stor age in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © Springer-Verlag Berlin Heidelberg 1985 Softcover reprint of the hardcover 1st edition 1985 2141/3020-543210 For my wife Ong and daughter Genevieve Preface Relationships between sets of variables of different kinds are of interest in many branches of science. The question of the analysis of relationships of this sort has nevertheless rather surprisingly received less attention from statisticians and others than it would seem to deserve. Of the available methods, that address ing the question most directly is canonical correlation analysis, here referred to for convenience as canonical analysis. Yet canonical analysis is often coolly received despite a lack of suitable alternatives. The purpose of this book is to clarify just what may and what may not be accomplished by means of canoni cal analysis in one field of scientific endeavor. Canonical analysis is concerned with reducing the correlation structure be tween two (or more) sets of variables to its simplest possible form. After a review of the nature and properties of canonical analysis, an assessment of the method as an exploratory tool of use in ecological investigations is made. Applications of canonical analysis to several sets of ecological data are described and discussed with this objective in mind. The examples are drawn largely from plant ecology. The position is adopted that canonical analysis exists primarily to be used; the examples are accordingly worked through in some detail with the aim of showing how canonical analysis can contribute towards the attainment of ecological goals, as well as to indicate the kind and extent of the insight afforded. The applicability of canonical analysis is of course not confined to plant ecology. Indeed, a surprisingly wide class of problems that arise in such allied disciplines as animal ecology, biogeography, climatology, epidemiology, palaeoclimatology, geology, psychology, taxonomy, agriculture, forestry and fisheries can be formulated and analyzed in terms of relationships between two or more sets of variables. Potential applications in these and other fields no doubt will occur to readers with interests in particular areas. The book is intended primarily for ecologists and others interested in rela tionships between variables of different kinds. More generally, the aim is to raise issues encountered in analyzing systems of this sort so as to encourage interaction between statisticians and ecologists. The work may appeal to statisti cians by exposing areas where further theoretical work is needed and by provid ing a source of worked numerical examples. The book may also serve a useful purpose by drawing the attention of ecologists to opportunities afforded by canonical analysis not only as a means of addressing a varied assortment of ecological problems, but also as a unifying structure in terms of which many statistical methods can be compared and related. VIII Preface The book has been aimed at the graduate or research level. Suitable prerequi sites would be a course in linear algebra and one in multivariate analysis. Nahik ian's A Modern Algebra for Biologists or G. Strang's Linear Algebra and its Applications would provide much of the necessary mathematical background. With respect to statistical requirements, P.E. Green's Analyzing Multivariate Data covers the relevant material at a level appropriate to understanding the book. The same author's Mathematical Tools for Applied Multivariate Analysis provides a useful exposition of the same ground from a geometric and more intuitively appealing viewpoint. The text is in three parts. Part I reviews the theoretical foundations of the subject, Part II consists of accounts of applications of canonical analysis to real data, and Part III is devoted to assessing the applications and to a consideration of the prospects for canonical analysis in ecology and more generally. Chapter 1 provides a general introduction to the study of relationships between two sets of variables. Chapter 2 deals with the formulation and derivation of canonical analysis before proceeding to review properties of the solution and the computation of canonical correlations and variates. In Chap. 3 several numerical indices which are helpful in interpreting the results of analysis are introduced and a number of extensions and generaliza tions of canonical analysis described. The chapter closes with an account of hypothesis testing in canonical analysis. Two special cases of canonical analysis important in applications form the subject matter of Chaps. 4 and 5, namely canonical variate analysis and dual scaling, respectively. The second of these chapters concludes the first part of the book. The applications of Part II cover field surveys and designed experiments though the balance between the two is heavily in favor of the former. Chapter 6 describes a canonical analysis of spatial variation in the representation of three plant species. Soil/vegetation relationships are the focus of attention in Chaps. 7 and 8. In Chap. 9 canonical analysis is applied in a study of the temporal stability of vegetation. Chapter 10 deals with the analysis of vegetation structure. Chapter 11 reports a comparative study of the multivariate responses of several grass species to an imposed treatment regime in a controlled experi ment. In Chap. 12, the final chapter of Part II, canonical analysis is used to explore relationships between the occurrence of several species of large herbi vores and environmental conditions in an East African rangeland. Part III con sists of Chaps. 13 and 14. In Chap. 13 the results of the seven applications are reviewed and the worth of the analyses in ecological terms assessed. Chapter 14 is to some extent speculative. In recent years statisticians have devoted a great deal of attention to two topics which have a strong bearing on canonical analysis - statistical data analysis and linear regression methodology. The spirit of this work is well-conveyed by Gnanadesikan's text Methods for Statistical Data Analysis of Multivariate Observations. In Chap. 14 developments in statistical data analysis and modern regression methodology are reviewed and an attempt is made to embed canonical analysis in the context provided by them. In effect, the chapter seeks to anticipate and delineate changes in the way in which canoni cal analysis will come to be used during the course of the next decade. The text closes with Appendices devoted to multivariate regression and to tabulations of the data and other information pertaining to the worked examples. Preface IX Part of the book was written while I was visiting the Department of Plant Sciences at the University of Western Ontario. My debt to Professor L. Orl6ci for making my visit to Canada possible and for his support and encouragement during this period is considerable. I am also much indebted to Dr. N.A. Camp bell of the Division of Mathematics and Statistics, CSIRO, Australia who read the theoretical chapters in manuscript and made valuable suggestions for their improvement. Portions of the manuscript were also read by Professor S.W. Nash of the Department of Mathematics, University of British Columbia and by Mr. J.C. Gower of the Statistics Department, Rothamsted Experimental Station. Both suggested improvements for which I am most grateful. Naturally, responsibility for all errors and obscurities which remain is mine. Dr. J. Ogden kindly placed the rain forest data used in the analyses of Chaps. 8 and 9 at my disposal and helped in interpreting the results obtained. Finally, it is a pleasure to thank Mrs. Eleanor Lowther for her skill and patience in so expertly preparing the typescript for publication. October 1984 Robert Gittins Contents 1. Introduction . . . . . . . . 1 1.1 The study of relationships 1 1.2 Objectives ..... . 3 1.3 Canonical analysis: overview 4 PART I. THEORY 11 2. Canonical correlations and canonical variates 13 2.1 Introduction 13 2.2 Formulation 13 2.3 Derivation of canonical correlation coefficients and canonical variates ........... . 15 2.3.1 Eigenanalysis . . . . . . . 15 2.3.2 Singular value decompositon 17 2.3.3 Other derivations .... . 18 2.3.4 Concluding remarks ... . 20 2.4 Properties of canonical correlation coefficients, weights and variates ................... 22 2.4.1 Properties of canonical correlation coefficients 22 2..4.2 Properties of canonical weights 25 2.4.3 Properties of canonical variates 27 2.5 Computation . . . . . 31 2.5.1 Numerical methods 31 2.5.2 Further remarks . 35 3. Extensions and generalizations 37 3.1 Introduction ..... 37 3.2 Further interpretive devices 37 3.2.1 Correlations between canonical variates and the original variables . . . . . . . . . . . . . . 38 3.2.2 Variance extracted by a canonical variate 40 3.2.3 Redundancy 40 3.2.4 Total redundancy .......... 41 XII Contents 3.2.5 Variable communalities 42 3.2.6 Concluding remarks 43 3.3 Extensions and generalizations 45 3.3.1 Redundancy analysis: an alternative to canonical analysis . 45 3.3.2 Improving the interpretability of canonical weights 45 3.3.3 Rotation of canonical variates 50 3.3.4 Validation 50 3.3.5 Predicting a criterion of maximum utility 52 3.3.6 Generalizations of canonical analysis 53 3.3.7 Concluding remarks 55 3.4 Hypothesis testing 56 3.4.1 Independence 56 3.4.2 Dimensionality 59 3.4.3 The contribution of particular variables 62 3.4.4 Hypothesis tests for nonnormal data 63 3.4.5 Residuals from a fitted model 63 4. Canonical variate analysis 67 4.1 Introduction 67 4.2 Binary-valued dummy variables 68 4.3 Formulation and derivation 69 4.3.1 Point conceptualizations of and 70 NXp NZq 4.3.2 Derivation 73 4.4 Further aspects of canonical variate analysis 75 4.5 Hypothesis testing 80 4.5.1 Equality of g vector-means 81 4.5.2 Dimensionality 81 4.6 Affinities with other methods 82 4.6.1 Canonical variate analysis, multivariate analysis of variance and multiple discriminant analysis 82 4.6.2 Canonical variate analysis and principal component analysis . 87 4.7 Imposition of structure 88 4.7.1 Designed comparisons 89 4.7.2 Separating the sources of variation 90 4.7.3 Further comments 91 4.8 Concluding remarks 92 5. Dual scaling 96 5.1 Introduction 96 5.2 Formulation and derivation 98 5.2.1 Maximizing the correlation between rows and columns 99 5.2.2 Maximizing the separation between rows and columns 103

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