Lecture Notes on Compositional Data Analysis - Sedimentology ...

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4 Chapter 1. Introduction
Chapter 2 Compositional data and their sample space 2.1 Basic concepts Definition 2.1. A row vector, x = [x 1 , x 2 , . . .,x D ], is defined as a D-part composition when all its components are strictly positive real numbers and they carry only relative information. Indeed, that compositional information is relative is implicitly stated in the units, as they are always parts of a whole, like weight or volume percent, ppm, ppb, or molar proportions. The most common examples have a constant sum κ and are known in the geological literature as closed data (Chayes, 1971). Frequently, κ = 1, which means that measurements have been made in, or transformed to, parts per unit, or κ = 100, for measurements in percent. Other units are possible, like ppm or ppb, which are typical examples for compositional data where only a part of the composition has been recorded; or, as recent studies have shown, even concentration units (mg/L, meq/L, molarities and molalities), where no constant sum can be feasibly defined (Buccianti and Pawlowsky-Glahn, 2005; Otero et al., 2005). Definition 2.2. The sample space of compositional data is the simplex, defined as S D = {x = [x 1 , x 2 , . . ., x D ] |x i > 0, i = 1, 2, . . ., D; D∑ x i = κ}. (2.1) However, this definition does not include compositions in e.g. meq/L. Therefore, a more general definition, together with its interpretation, is given in Section 2.2. Definition 2.3. For any vector of D real positive components z = [z 1 , z 2 , . . .,z D ] ∈ R D + 5 i=1
Page 1 and 2: Lecture No
Page 3 and 4: Preface These notes have been prepa
Page 5 and 6: Contents Preface i 1 Introduction 1
Page 7 and 8: Chapter 1 Introduction The awarenes
Page 9: 3 centred logratio transformation (
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
Page 53 and 54: 5.7. Exercises 47 Table 5.4: Covari
Page 55 and 56: Chapter 6 Distributions on the simp
Page 57 and 58: 6.3. Tests of normality 51 which is
Page 59 and 60: 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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