Lecture Notes on Compositional Data Analysis - Sedimentology ...

More documents

Recommendations

Info

44 Chapter 5. Exploratory data analysis −1.0 −0.5 0.0 0.5 1.0 Variances 0.00 0.05 0.10 0.15 0.20 71 % 90 % 98 % 100 % Comp.2 −0.4 −0.2 0.0 0.2 0.4 16 2 10 5 Na2O TiO2 P2O5Al2O3 CaO K2O 17 14 15 13 1 11 8 12 7 SiO2 MnO FeO Fe2O3 9 6 3 MgO 4 −1.0 −0.5 0.0 0.5 1.0 −0.4 −0.2 0.0 0.2 0.4 Comp.1 Figure 5.3: Biplot of data of Table 5.1 (right), and scree plot of the variances of all principal components (left), with indication of cumulative explained variance. with CaO (as they would probably come from different rock types, e.g. siliciclastic against carbonate). Using the sequential binary partition given in Table 5.3, Figure 5.5 represents the balance-dendrogram of the sample, within the range (−3, +3). This range translates for two part compositions to proportions of (1.4, 98.6)%; i.e. if we look at the balance MgO- MnO the variance bar is placed at the lower extreme of the balance axis, which implies that in this subcomposition MgO represents in average more than 98%, and MnO less than 2%. Looking at the lengths of the several variance bars, one easily finds that the balances P 2 O 5 -TiO 2 and SiO 2 -Al 2 O 3 are almost constant, as their bars are very short and their box-plots extremely narrow. Again, the balance between the subcompositions (P 2 O 5 , TiO 2 ) vs. (SiO 2 , Al 2 O 3 ) does not display any box-plot, meaning that it is above +3 (thus, the second group of parts represents more than 98% with respect to the first group). The distribution between K 2 O, Na 2 O and CaO tells us that Na 2 O and CaO keep a quite constant ratio (thus, we should interpret that there are no strong variations in the plagioclase composition), and the ratio of these two against K 2 O is also fairly constant, with the exception of some values below the first quartile (probably, a single value with an particularly high K 2 O content). The other balances are well equilibrated (in particular, see how centred is the proxy balance between feldspar and
5.6. Illustration 45 Figure 5.4: Plot of subcomposition (Fe 2 O 3 ,K 2 O,Na 2 O). Left: before centring. Right: after centring. mafic minerals), all with moderate dispersions. K2O Na2O CaO MgO MnO FeO Fe2O3 P2O5 TiO2 Al2O3 SiO2 0.00 0.05 0.10 0.15 0.20 Figure 5.5: Balance-dendrogram of data from Table 5.1 using the balances of Table 5.3. Once the marginal empirical distribution of the balances have been analysed, we can use the biplot to explore their relations (Figure 5.6), and the conventional covariance or correlation matrices (Table 5.4). From these, we can see, for instance: • The constant behaviour of v3 (balance TiO 2 -P 2 O 5 ), with a variance below 10 −4 , and in a lesser degree, of v5 (anortite-albite relation, or balance CaO-Na 2 O). • The orthogonality of the pairs of rays v1-v2, v1-v4, v1-v7, and v6-v8, suggests the lack of correlation of their respective balances, confirmed by Table 5.4, where correlations of less than ±0.3 are reported. In particular, the pair v6-v8 has a correlation of −0.029. These facts would imply that silica saturation (v2), the
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 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: 5.6. Illustration 43 Table 5.2: Nor
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

Lecture Notes on Compositional Data Analysis - Sedimentology ...

Create successful ePaper yourself

Delete template?

Save as template?