Lecture Notes on Compositional Data Analysis - Sedimentology ...

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24 Chapter 4. Coordinate representation wing ones (there are more than one party in each wing). If, from a journal, you get only the percentages within each group, you are unable to know which wing, and obviously which party, has won the elections. You probably are going to ask for the balance between the two wings as the information you need to complete the actual state of the elections. The balance, as defined here, permits you to complete the information. The balance is the remaining relative information about the elections once the information within the two wings has been removed. To be more precise, assume that the parties are six and the composition of the votes is x ∈ S 6 ; assume the left wing contested with 4 parties represented by the group of parts {x 1 , x 2 , x 5 , x 6 } and only two parties correspond to the right wing {x 3 , x 4 }. Consider the sequential binary partition in Table 4.1. The first partition just separates the two wings and thus the balance informs us about the equilibrium between the two wings. If one leaves out this balance, the remaining balances inform us only about the left wing (balances 3,4) and only about the right wing (balance 5). Therefore, to retain only balance 5 is equivalent to know the relative information within the subcomposition called right wing. Similarly, balances 2, 3 and 4 only inform about what happened within the left wing. The conclusion is that the balance 1, the forgotten information in the journal, does not inform us about relations within the two wings: it only conveys information about the balance between the two groups representing the wings. Many questions can be stated which can be handled easily using the balances. For instance, suppose we are interested in the relationships between the parties within the left wing and, consequently, we want to remove the information within the right wing. A traditional approach to this is to remove parts x 3 and x 4 and then close the remaining subcomposition. However, this is equivalent to project the composition of 6 parts orthogonally on the subspace associated with the left wing, what is easily done by setting b 5 = 0. If we do so, the obtained projected composition is x proj = C[x 1 , x 2 , g(x 3 , x 4 ), g(x 3 , x 4 ), x 5 , x 6 ] , g(x 3 , x 4 ) = (x 3 x 4 ) 1/2 , i.e. each part in the right wing has been substituted by the geometric mean within the right wing. This composition still has the information on the left-right balance, b 1 . If we are also interested in removing it (b 1 = 0) the remaining information will be only that within the left-wing subcomposition which is represented by the orthogonal projection x left = C[x 1 , x 2 , g(x 1 , x 2 , x 5 , x 6 ), g(x 1 , x 2 , x 5 , x 6 ), x 5 , x 6 ] , with g(x 1 , x 2 , x 5 , x 6 ) = (x 1 , x 2 , x 5 , x 6 ) 1/4 . The conclusion is that the balances can be very useful to project compositions onto special subspaces just by retaining some balances and making other ones null.
4.5. WORKING IN COORDINATES 25 4.5 Working in coordinates Coordinates with respect to an orthonormal basis in a linear vector space underly standard rules of operation. As a consequence, perturbation in S D is equivalent to translation in real space, and power transformation in S D is equivalent to multiplication. Thus, if we consider the vector of coordinates h(x) = x ∗ ∈ R D−1 of a compositional vector x ∈ S D with respect to an arbitrary orthonormal basis, it holds (Property 4.2) h(x ⊕ y) = h(x) + h(y) = x ∗ + y ∗ , h(α ⊙ x) = α · h(x) = α · x ∗ , (4.11) and we can think about perturbation as having the same properties in the simplex as translation has in real space, and of the power transformation as having the same properties as multiplication. Furthermore, d a (x,y) = d(h(x), h(y)) = d(x ∗ ,y ∗ ), where d stands for the usual Euclidean distance in real space. This means that, when performing analysis of compositional data, results that could be obtained using compositions and the Aitchison geometry are exactly the same as those obtained using the coordinates of the compositions and using the ordinary Euclidean geometry. This latter possibility reduces the computations to the ordinary operations in real spaces thus facilitating the applied procedures. The duality of the representation of compositions, in the simplex and by coordinates, introduces a rich framework where both representations can be interpreted to extract conclusions from the analysis (see Figures 4.1, 4.2, 4.3, and 4.4, for illustration). The price is that the basis selected for representation should be carefully selected for an enhanced interpretation. Working in coordinates can be also done in a blind way, just selecting a default basis and coordinates and, when the results in coordinates are obtained, translating the results back in the simplex for interpretation. This blind strategy, although acceptable, hides to the analyst features of the analysis that may be relevant. For instance, when detecting a linear dependence of compositional data on an external covariate, data can be expressed in coordinates and then the dependence estimated using standard linear regression. Back in the simplex, data can be plotted with the estimated regression line in a ternary diagram. The procedure is completely acceptable but the visual picture of the residuals and a possible non-linear trend in them can be hidden or distorted in the ternary diagram. A plot of the fitted line and the data in coordinates may reveal new interpretable features.
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: 4.4 Orthonormal coordinates 23 Exam
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
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
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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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