Lecture Notes on Compositional Data Analysis - Sedimentology ...

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28 Chapter 4. Coordinate representation In fact, they are coordinates in an oblique basis, something that affects distances if the usual Euclidean distance is computed from the alr coordinates. This approach is frequent in many applied sciences and should be avoided (see for example (Albarède, 1995), p. 42). 4.7 Simplicial matrix notation Many operations in real spaces are expressed in matrix notation. Since the simplex is an Euclidean space, matrix notations may be also useful. However, in this framework a vector of real constants cannot be considered in the simplex although in the real space they are readily identified. This produces two kind of matrix products which are introduced in this section. The first is simply the expression of a perturbationlinear combination of compositions which appears as a power-multiplication of a real vector by a compositional matrix whose rows are in the simplex. The second one is the expression of a linear transformation in the simplex: a composition is transformed by a matrix, involving perturbation and powering, to obtain a new composition. The real matrix implied in this case is not a general one but when expressed in coordinates it is completely general. Perturbation-linear combination of compositions For a row vector of l scalars a = [a 1 , a 2 , . . ., a l ] and an array of row vectors V = (v 1 ,v 2 , . . .,v l ) ′ , i.e. an (l, D)-matrix, ⎛ ⎞ v 1 v 2 a ⊙ V = [a 1 , a 2 , . . .,a l ] ⊙ ⎜ ⎟ ⎝ . ⎠ v l ⎛ ⎞ v 11 v 12 · · · v 1D v 21 v 22 · · · v 2D l⊕ = [a 1 , a 2 , . . .,a l ] ⊙ ⎜ ⎝ . . . .. ⎟ . ⎠ = a i ⊙ v i . i=1 v l1 v l2 · · · v lD The components of this matrix product are a ⊙ V = C [ l∏ j=1 v a j j1 , l∏ j=1 v a j j2 , . . ., l∏ j=1 v a j jD ] .
4.7. Matrix notation 29 In coordinates this simplicial matrix product takes the form of a linear combination of the coordinate vectors. In fact, if h is the function assigning the coordinates, ( l⊕ ) l∑ h(a ⊙ V) = h a i ⊙ v i = a i h(v i ) . i=1 Example 4.2. A composition in S D can be expressed as a perturbation-linear combination of the elements of the basis e i , i = 1, 2, . . .,D−1 as in Equation (4.6). Consider the (D −1, D)-matrix E = (e 1 ,e 2 , . . .,e D−1 ) ′ and the vector of coordinates x ∗ = ilr(x). Equation (4.6) can be re-written as x = x ∗ ⊙ E . Perturbation-linear transformation of S D : endomorphisms Consider a row vector of coordinates x ∗ ∈ R D−1 and a general (D −1, D −1)-matrix A ∗ . In the real space setting, y ∗ = x ∗ A ∗ expresses an endomorphism, obviously linear in the real sense. Given the isometric isomorphism of the real space of coordinates with the simplex, the A ∗ endomorphism has an expression in the simplex. Taking ilr −1 = h −1 in the expression of the real endomorphism and using Equation (4.8) i=1 y = C(exp[x ∗ A ∗ Ψ]) = C(exp[clr(x)Ψ ′ A ∗ Ψ]) (4.12) where Ψ is the clr matrix of the selected basis and the right-most member has been obtained applying Equation (4.7) to x ∗ . The (D, D)-matrix A = Ψ ′ A ∗ Ψ has entries a ij = D−1 ∑ D−1 ∑ Ψ ki Ψ mj a ∗ km , i, j = 1, 2, . . ., D . k=1 m=1 Substituting clr(x) by its expression as a function of the logarithms of parts, the composition y is [ D ] ∏ D∏ D∏ y = C x a j1 j , x a j2 j , . . ., x a jD j , j=1 j=1 which, taking into account that products and powers match the definitions of ⊕ and ⊙, deserves the definition y = x ◦ A = x ◦ (Ψ ′ A ∗ Ψ) , (4.13) where ◦ is the perturbation-matrix product representing an endomorphism in the simplex. This matrix product in the simplex should not be confused with that defined between a vector of scalars and a matrix of compositions and denoted by ⊙. An important conclusion is that endomorphisms in the simplex are represented by matrices with a peculiar structure given by A = Ψ ′ A ∗ Ψ, which have some remarkable properties: j=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 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: 4.6. Additive log-ratio coordinates
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
Page 61 and 62: 6.4. Exercises 55 Table 6.3: Critic
Page 63 and 64: Chapter 7 Statistical inference 7.1
Page 65 and 66: 7.1. Hypothesis about two groups 59
Page 67 and 68: 7.2. Probability and confidence reg
Page 69 and 70: Chapter 8 Compositional processes C
Page 71 and 72: 8.1. Linear processes 65 x1 2 t=0 1
Page 73 and 74: 8.2. Complementary processes 67 Exa
Page 75 and 76: 8.2. Complementary processes 69 0.5
Page 77 and 78: 8.3. Mixture processes 71 Example 8
Page 79 and 80: 8.4. Linear regression with composi
Page 81 and 82: 8.5. PRINCIPAL COMPONENT ANALYSIS 7
Page 83 and 84: 8.5. Principal component analysis 7
Page 85 and 86:
8.5. Principal component analysis 7
Page 87 and 88:
Bibliography Aitchison, J. (1981).
Page 89 and 90:
BIBLIOGRAPHY 83 Buccianti, A., V. P
Page 91 and 92:
BIBLIOGRAPHY 85 study of surface wa
Page 93 and 94:
A. Plotting a ternary diagram Denot
Page 95 and 96:
B. Parametrisation of an elliptic r
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