Lecture Notes on Compositional Data Analysis - Sedimentology ...

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Prof. Dr. Vera Pawlowsky-Glahn Catedrática de Universidad (Full professor) University of Girona Dept. of Computer Science and Applied Mathematics Campus Montilivi — P-1, E-17071 Girona, Spain vera.pawlowsky@udg.edu Prof. Dr. Juan José Egozcue Catedrático de Universidad (Full professor) Technical University of Catalonia Dept. of Applied Mathematics III Campus Nord — C-2, E-08034 Barcelona, Spain juan.jose.egozcue@upc.edu Dr. Raimon Tolosana-Delgado Wissenschaftlicher Mitarbeiter (fellow scientist) Georg-August-Universität Göttingen Dept. of Sedimentology and Environmental Geology Goldschmidtstr. 3, D-37077 Göttingen, Germany raimon.tolosana@geo.uni-goettingen.de
Preface These notes have been prepared as support to a short course on compositional data analysis. Their aim is to transmit the basic concepts and skills for simple applications, thus setting the premises for more advanced projects. One should be aware that frequent updates will be required in the near future, as the theory presented here is a field of active research. The notes are based both on the monograph by John Aitchison, Statistical analysis of compositional data (1986), and on recent developments that complement the theory developed there, mainly those by Aitchison, 1997; Barceló-Vidal et al., 2001; Billheimer et al., 2001; Pawlowsky-Glahn and Egozcue, 2001, 2002; Aitchison et al., 2002; Egozcue et al., 2003; Pawlowsky-Glahn, 2003; Egozcue and Pawlowsky-Glahn, 2005. To avoid constant references to mentioned documents, only complementary references will be given within the text. Readers should be aware that for a thorough understanding of compositional data analysis, a good knowledge in standard univariate statistics, basic linear algebra and calculus, complemented with an introduction to applied multivariate statistical analysis, is a must. The specific subjects of interest in multivariate statistics in real space can be learned in parallel from standard textbooks, like for instance Krzanowski (1988) and Krzanowski and Marriott (1994) (in English), Fahrmeir and Hamerle (1984) (in German), or Peña (2002) (in Spanish). Thus, the intended audience goes from advanced students in applied sciences to practitioners. Concerning notation, it is important to note that, to conform to the standard praxis of registering samples as a matrix where each row is a sample and each column is a variate, vectors will be considered as row vectors to make the transfer from theoretical concepts to practical computations easier. Most chapters end with a list of exercises. They are formulated in such a way that they have to be solved using an appropriate software. A user friendly, MS-Excel based freeware to facilitate this task can be downloaded from the web at the following address: http://ima.udg.edu/Recerca/EIO/inici eng.html Details about this package can be found in Thió-Henestrosa and Martín-Fernández (2005) or Thió-Henestrosa et al. (2005). There is also available a whole library of i
Page 1: Lecture No
Page 5 and 6: Contents Preface i 1 Introduction 1
Page 7 and 8: Chapter 1 Introduction The awarenes
Page 9 and 10: 3 centred logratio transformation (
Page 11 and 12: Chapter 2 Compositional data and th
Page 13 and 14: 2.2 Principles of compositional ana
Page 15 and 16: 2.3 Exercises 9 Definition 2.6. a f
Page 17 and 18: Chapter 3 The Aitchison geometry 3.
Page 19 and 20: 3.3 Inner product, norm and distanc
Page 21 and 22: 3.4. GEOMETRIC FIGURES 15 3.4 Geome
Page 23 and 24: Chapter 4 Coordinate representation
Page 25 and 26: 4.3. Generating systems 19 where (
Page 27 and 28: 4.4 Orthonormal coordinates 21 been
Page 29 and 30: 4.4 Orthonormal coordinates 23 Exam
Page 31 and 32: 4.5. WORKING IN COORDINATES 25 4.5
Page 33 and 34: 4.6. Additive log-ratio coordinates
Page 35 and 36: 4.7. Matrix notation 29 In coordina
Page 37 and 38: 4.8. EXERCISES 31 4.8 Exercises Exe
Page 39 and 40: Chapter 5 Exploratory data analysis
Page 41 and 42: 5.3 Centring and scaling 35 Definit
Page 43 and 44: 5.4. The biplot 37 same. As a resul
Page 45 and 46: 5.5. EXPLORATORY ANALYSIS OF COORDI
Page 47 and 48: 5.6. Illustration 41 0.00 0.02 0.04
Page 49 and 50: 5.6. Illustration 43 Table 5.2: Nor
Page 51 and 52: 5.6. Illustration 45 Figure 5.4: Pl
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5.7. Exercises 47 Table 5.4: Covari
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Chapter 6 Distributions on the simp
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6.3. Tests of normality 51 which is
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6.3. Tests of normality 53 Tolosana
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6.4. Exercises 55 Table 6.3: Critic
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Chapter 7 Statistical inference 7.1
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7.1. Hypothesis about two groups 59
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7.2. Probability and confidence reg
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Chapter 8 Compositional processes C
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8.1. Linear processes 65 x1 2 t=0 1
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8.2. Complementary processes 67 Exa
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8.2. Complementary processes 69 0.5
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8.3. Mixture processes 71 Example 8
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8.4. Linear regression with composi
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8.5. PRINCIPAL COMPONENT ANALYSIS 7
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8.5. Principal component analysis 7
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8.5. Principal component analysis 7
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Bibliography Aitchison, J. (1981).
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BIBLIOGRAPHY 83 Buccianti, A., V. P
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BIBLIOGRAPHY 85 study of surface wa
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A. Plotting a ternary diagram Denot
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B. Parametrisation of an elliptic r
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