A Concise Guide To Compositional Data Analysis
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A Concise Guide to Compositional Data Analysis John Aitchison Honorary Senior Research Fellow Department of Statistics University of Glasgow Address for correspondence: Rosemount, Carrick Castle, Lochgoilhead Cairndow, Argyll, PA24 8AF, United Kingdom Email: john.aitchison@btinternet.com A Concise Guide to Compositional Data Analysis Contents Preface Why a course on compositional data analysis? 1. The nature of compositional problems 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 2. The simplex sample space and principles of compositional data analysis 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3. Some typical compositional problems A little bit of history: the perceived difficulties of compositional data An intuitive approach to compositional data analysis The principle of scale invariance Subcompositions: the marginals of compositional data analysis Compositional classes and the search for a suitable sample space Subcompositional coherence Perturbation as the operation of compositional change Power as a subsidiary operation of compositional change Limitations in the interpretability of compositional data Logratio analysis: a statistical methodology for compositional data analysis The unit simplex sample space and the staying- in the-simplex approach The algebraic-geometric structure of the simplex Useful parametric classes of distributions on the simplex Logratio analysis and the role of logcontrasts Simple estimation Simple hypothesis testing: the lattice approach Compositional regression, residual analysis and regression diagnostics Some other useful tools. From theory to practice: some simple applications 3.1 3.2 3.3 3.4 3.5 3.6 Simple hypothesis testing: comparison of hongite and kongite Compositional regression analysis: the dependence of Arctic lake sediments on depth Compositional invariance: economic aspects of household budget patterns Testing perturbation hypotheses: an application to change in cows’ milk Testing for distributional form Related types of data 4. Developing appropriate methodology for more complex compositional problems 4.1 Dimension reducing techniques: logcontrast principal components: application to hongite 4.2 Simplicial singular value decomposition 4.3 Compositional biplots and their interpretation 4.4 The Hardy-Weinberg law: an application of biplot and logcontrast principal component analysis 4.5 A geological example: interpretation of the biplot of goilite 4.6 Abstract art: the biplot search for understanding 4.7 Tektite mineral and oxide compositions 4.8 Subcompositional analysis 4.9 Compositions in an explanatory role 4.10 Experiments with mixtures 4.11 Forms of independence 5. A Compositional processes: a statistical search for understanding 5.1 5.2 5.3 5.4 5.5 5.6 Introduction Differential perturbation processes A simple example: Arctic lake sediment Exploration for possible differential processes Convex linear mixing processes Distinguishing between alternative hypothesis Postlude Pockets of resistance and confusion Appendix Tables Preface Why a course in compositional data analysis? Compositional data consist of vectors whose components are the proportion or percentages of some whole. Their peculiarity is that their sum is constrained to the be some constant, equal to 1 for proportions, 100 for percentages or possibly some other constant c for other situations such as parts per million (ppm) in trace element compositions. Unfortunately a cursory look at such vectors gives the appearance of vectors of real numbers with the consequence that over the last century all sorts of sophisticated statistical methods designed for unconstrained data have been applied to compositional data with inappropriate inferences. All this despite the fact that many workers have been, or should have been, aware that the sample space for compositional vectors is radically different from the real Euclidean space associated with unconstrained data. Several substantial warnings had been given, even as early as 1897 by Karl Pearson in his seminal paper on spurious correlations and then repeatedly in the 1960’s by geologist Felix Chayes. Unfortunately little heed was paid to such warnings and within the small circle who did pay attention the approach was essentially pathological, attempting to answer the question: what goes wrong when we apply multivariate statistical methodology designed for unconstrained data to our constrained data and how can the unconstrained methodology be adjusted to give meaningful inferences. Throughout all my teaching career I have emphasised to my students the importance of the first step in an statistical problem, the recognition and definition of a sensible sample space. The early modern statisticians concentrated their efforts on statistical methodology associated with the all- too-familiar real Euclidean space. The algebraicgeometric structure was familiar, at the time of development almost intuitive, and a huge array of meaningful, appropriate methods developed. After some hesitation the special problems of directional data, with the unit sphere as the natural sample space, were resolved mainly by Fisher and Watson, who recognised again the algebraicgeometric structure of the sphere and its implications for the design and implementation of an appropriate methodology. A remaining awkward problem of spherical regression was eventually solved by Chang, again recognising the special algebraic-geometric structure of the sphere. Strangely statisticians have been slow to take a similar approach to the problems of compositional data and the associated sample space, the unit simplex. This course is designed to draw attention to its special form, to principles which are based on logic al necessities for meaningful interpretation of compositional data and to the simple forms of statistical methodology for analysing real compositional data. Chapter 1 The nature of compositional problems Chapter 1 The nature of compositional problems 1.1 Some typical compositional problems In this section we present the reader with a series of challenging problems in compositional data analysis, with typical data sets and questions posed. These come from a number of different disciplines and will be used to motivate the concepts and principles of compositional data analysis, and will eventually be fully analysed to provide answers to the questions posed. The full data sets associated with these problems are set out in Appendix A. Problem 1 Geochemical compositions of rocks The statistical analysis of geochemical compositions of rocks is fundamental to petrology. Commonly such compositions are expressed as percentages by weight of ten or more major oxides or as percentages by weight of some basic minerals. As an illustration of the nature of such problems we present in Table 1.1.1a the 5-part mineral (A, B, C, D, E) compositions of 25 specimens of rock type hongite. Even a cursory examination of this table shows that there is substantial variation from specimen to specimen, and first questions are: In what way should we describe such variability? Is there some central composition around which this variability can be simply expressed? A further rock specimen has composition [A, B, C, D, E] = [44.0, 20.4, 13.9, 9.1, 12.6] and is claimed to be hongite. Can we say whether this is fairly typical of hongite? If not, can we place some measure on its atypicality? 7 Chapter 1 The nature of compositional problems Table 1.1.1b presents a set of 5-part (A, B, C, D, E) compositions for 25 specimens of rock type kongite. Some obvious questions are as follows. Do the mineral compositions of hongite and kongite differ and if so in what way? For a new specimen can a convenient form of classification be devised on the basis of the composition? If so, can we investigate whether a rule of classification based on only a selection of the compositional parts would be as effective as use of the full composition? Problem 2 Arctic lake sediments at different depths In sedimentology, specimens of sediments are traditionally separated into three mutually exclusive and exhaustive constituents -sand, silt and clay- and the proportions of these parts by weight are quoted as (sand, silt, clay) compositions. Table 1.1.2 records the (sand, silt, clay) compositions of 39 sediment samples at different water depths in an Arctic lake. Again we recognise substantial variability between compositions. Questions of obvious interest here are the following. Is sediment composition dependent on water depth? If so, how can we quantify the extent of the dependence? If we regard sedimentation as a process, do these data provide any information on the nature of the process? Even at this stage of investigation we can see that this may be a question of compositional regression. Problem 3 Household budget patterns An important aspect of the study of consumer demand is the analysis of household budget surveys, in which attention often focuses on the expenditures of a sample of households on a number of mutually exclusive and exhaustive commodity groups and their relation to total expenditure, income, type of housing, household composition and so on. In the investigation of such data the pattern or composition of expenditures, the proportions of total expenditure allocated to the commodity groups, can be shown to play a central role in a form of budget share approach to the analysis. Assurances of confidentiality and limitations of space preclude the publication of individual budgets from an actual survey, but we can present a reduced version of the problem, which retains its key characteristics. 8 Chapter 1 The nature of compositional problems In a sample survey of single persons living alone in rented accommodation, twenty men and twenty women were randomly selected and asked to record over a period of one month their expenditures on the following four mutually exclusive and exhaustive commodity groups: 1. Housing, including fuel and light. 2. Foodstuffs, including alcohol and tobacco. 3. Other goods, including clothing, footwear and durable goods. 4. Services, including transport and vehicles. The results are recorded in Table 1.1.3. Interesting questions are readily formulated. To what extent does the pattern of budget share of expenditures for men depend on the total amount spent? Are there differences between men and women in their expenditure patterns? Are there some commodity groups which are given priority in the allocation of expenditure? Problem 4 Milk composition study In an attempt to improve the quality of cow milk, milk from each of thirty cows was assessed by dietary composition before and after a strictly controlled dietary and hormonal regime over a period of eight weeks. Although seasonal variations in milk quality might have been regarded as negligible over this period it was decided to have a control group of thirty cows kept under the same conditions but on a regular established regime. The sixty cows were of course allocated to control and treatment groups at random. Table 1.1.4 provides the complete set of before and after milk compositions for the sixty cows, showing the protein, milk fat, carbohydrate, calcium, sodium, potassium proportions by weight of total dietary content. The purpose of the experiment was to determine whether the new regime has produced any significant change in the milk composition so it is essential to have a clear idea of how change in compositional data is characterised by some meaningful operation. A main question here is therefore how to formulate hypotheses of change of compositions, and indeed how we may investigate the full lattice of such hypotheses. Meanwhile we note that because of the before and after nature of the data within each experimental unit we have for compositional data the analogue of a paired comparison situation for real 9 Chapter 1 The nature of compositional problems measurements where traditionally the differences in pairs of measurements are considered. We have thus to find the counterpart of difference for paired compositions. Problem 5 Analysis of an abstract artist The data of Table 1.1.5 show six-part colour compositions in 22 paintings created by an abstract artist. Each painting was in the form of a square, divided into a number of rectangles, in the style of a Mondrian abstract painting and the rectangles were each coloured in one of six colours: black and white, the primary colours blue, red and yellow, and one further colour, labelled ‘other’, which varied from painting to painting. An interesting question posed here is to attempt to see whether there is any pattern discernible in the construction of the paintings. There is considerable variability from painting to painting and the challenge is to describe the pattern of variability appropriately in as simple terms as possible. Problem 6 A statistician’s time budget Time budgets, how a day or a period of work is divided up into different activities, have become a popular source of data in psychology and sociology. To illustrate such problems we consider six daily activities of an academic statistician: T, teaching; C, consultation; A, administration; R, research; O, other wakeful activities; S, sleep. Table 1.1.6 records the proportions of the 24 hours devoted to each activity, recorded on each of 20 days, selected randomly from working days in alternate weeks so as to avoid possible carry-over effects such as a short-sleep day being compensated by make-up sleep on the succeeding day. The six activities may be divided into two categories: ‘work’ comprising activities T, C, A, R, and ‘leisure’ comprising activities O, S. Our analysis may then be directed towards the work pattern consisting of the relative times spent in the four work activities, the leisure pattern, and the division of the day into work time and leisure time. Two obvious questions are as follows. To what extent, if any, do the patterns of work and of leisure depend on the times allocated to these major divisions of the day? Is the ratio of sleep to other wakeful activities dependent on the times spent in the various work activities? 10 Chapter 1 The nature of compositional problems Problem 7 Sources of pollution in a Scottish loch A Scottish loch is supplied by three rivers, here labelled 1, 2, 3. At the mouth of each 10 water samples have been taken at random times and analysed into 4-part compositions of pollutants a, b, c, d. Also available are 20 samples, again taken at random times, at each of three fishing locations A, B, C. Space does not allow the publication of the full data set of 90 4-part compositions but Table 1.1.7, which records the first and last compositions in each of the rivers and fishing locations, gives a picture of the variability and the statistical nature of the problem. The problem here is to determine whether the compositions at a fishing location may be regarded as mixtures of compositions from the three sources, and what can be inferred about the nature of such a mixture. Other typical problems in different disciplines The above seven problems are sufficient to demonstrate that compositional problems arise in many different forms in many different disciplines, and as we develop statistical methodology for this particular form of variability we shall meet a number of other compositional problems to illustrate a variety of forms of statistical analysis. We list below a number of disciplines and some examples of compositional data sets within these disciplines. The list is in no way complete. Agriculture and farming Fruit (skin, stone, flesh) compositions Land use compositions Effects of GM Archaeology Ceramic compositions Developmental biology Shape analysis: (head, trunk, leg) composition relative to height Economics Household budget compositions and income elasticities of demand Portfolio compositions 11 Chapter 1 The nature of compositional problems Environometrics Pollutant compositions Geography US state ethnic compositions, urban-rural compositions Land use compositions Geology Mineral compositions of rocks Major oxide compositions of rocks Trace element compositions of rocks Major oxide and trace element compositions of rocks Sediment compositions such as (sand, silt, clay) compositions Literary studies Sentence compositions Manufacturing Global car production compositions Medicine Blood compositions Renal calculi compositions Urine compositions Ornithology Sea bird time budgets Plumage colour compositions of greater bower birds Palaeontology Foraminifera compositions Zonal pollen compositions Psephology US Presidential election voting proportions 12 Chapter 1 The nature of compositional problems Psychology Time budgets of various groups Waste disposal Waste composition 1.2 A little bit of history: the perceived difficulties of compositional analysis We must look back to 1897 for our starting point. Over a century ago Karl Pearson published one of the clearest warnings (Pearson, 1897) ever issued to statisticians and other scientists beset with uncertainty and variability: Beware of attempts to interpret correlations between ratios whose numerators and denominators contain common parts. And of such is the world of compositional data, where for example some rock specimen, of total weight w, is broken down into mutually exclusive and exhaustive parts with component weights w1 , . . . , wD and then transformed into a composition (x 1 , . . . , x D ) = (w1 , . . . , wD )/(w1 + . . . + wD ). Our reason for forming such a composition is that in many problems composition is the relevant entity. For example the comparison of rock specimens of different weights can only be achieved by some form of standardization and composition (per unit weight) is a simple and obvious concept for achieving this. Equivalently we could say that any meaningful statement about the rock specimens should not depend on the largely accidental weights of the specimens. It appears that Pearson’s warning went unheeded, with raw components of compositional data being subjected to product moment correlation analysis with unsound interpretation based on methods of ‘standard’ multivariate analysis designed for unconstrained multivariate data. In the 1960’s there emerged a number of scientists who warned against such methodology and interpretation, in geology mainly Chayes, Krumbein, Sarmanov and Vistelius, and in biology mainly Mosimann: see, for example, Chayes (1956, 1960, 1962, 1971), Krumbein (1962), 13 Chapter 1 The nature of compositional problems Sarmanov and Vistelius (1959), Mosimann (1962,1963). The main problem was perceived as the impossibility of interpreting the product moment correlation coefficients between the raw components and was commonly referred to as the negative bias problem. For a D-part composition [ x1 , . . . , x D ] with the component sum x1 + . . . + x D = 1 , since cov( x1 , x1 + . . . + x D ) = 0 we have cov( x1 , x2 ) +...+ cov( x1 , x D ) = − var( x1 ) . The right hand side here is negative except for the trivial case where the first component is constant. Thus at least one of the covariances on the left must be negative or, equivalently, there must be at least one negative element in the first row of the raw covariance matrix. The same negative bias must similarly occur in each of the other rows so that at least D of the elements of the raw covariance matrix. Hence correlations are not free to range over the usual interval (-1, 1) subject only to the non-negative definiteness of the covariance or correlation matrix, and there are bound to problems of interpretation. The problem was described under different headings: the constant-sum problem, the closure problem, the negative bias problem, the null correlation difficulty. Strangely no attempt was made to try and establish principles of compositional data analysis. The approach was essentially pathological with attempts to see what went wrong when standard multivariate analysis was applied to compositional data in the hope that some corrective treatments could be applied; see, for example, Butler (1979), Chayes (1971, 1972), Chayes and Kruskal (1966), Chayes and Trochimczyk (1978), Darroch and James (1975), Darroch and Ratcliff (1970, 1978). An appropriate methodology, taking account of some logically necessary principles of compositional data analysis and the special nature of compositional sample spaces, began to emerge in the 1980’s with, for example, contributions from Aitchison and Shen (1980), Aitchison (1982, 1983, 1985), culminating in the methodological 14 Chapter 1 The nature of compositional problems monograph Aitchison (1986) on The Statistical Analysis of Compositional Data. This course is largely based on that monograph and the many subsequent developments of the subject. 1.3 An intuitive approach to compositional data analysis A typical composition is a (sand, silt, clay) sediment composition such as the percentages [77.5 19.5 3.0] of the first sediment in Table 1.1.2. Standard terminology is to refer to sand, silt and clay as the labels of the three parts of the composition and the elements 77.5, 19.5, 3.0 of the vector as the components of the composition. A typical or generic composition [ x1 x 2 . . . x D ] will therefore consist D parts with labels 1, . . , D and components x1 , x 2 , . . . , x D The components will have a constant sum, 1 when the components are proportions of some unit, 100 when these are expressed as percentages, and so on. We shall find that the particular value of constant sum is of no relevance in compositional data analysis and in much of our theoretical development we shall standardise to a constant sum of 1. Note that we have set out a typical composition as a row vector. This seems a sensible convention and is common in much modern practice as, for example, in MSExcel where the practice is to have rows as cases. In the early 1980’s it seemed to the writer that there was an obvious way of analysing compositional data. Since compositional data provide information only about the relative magnitudes of the parts, not their absolute values, then the information provided is essentially about ratios of the components. Therefore it seemed to make sense to think in terms of ratios. There is clearly a one-to-one correspondence between compositions and a full set of ratios. Moreover, since ratios are awkward to handle mathematically and statistically (for example there is no exact relationship between var( xi / x j ) and var( x j / x i ) ) it seems sensible to work in terms of logratios, for example reaping the benefit of simple relationships such as var{log( xi / x j )} = var{log( x j / xi )} . 15 Chapter 1 The nature of compositional problems Since there is also a one-to-one correspondence between compositions and a full set of logratios, for example, [ y 1 . . . y D−1 ] = [log( x1 / x D ) . . . log( x D −1 / x D )] with inverse [ x1 x 2 . . . x D ] = [exp( y1 ) . . . . exp( y D−1 ) 1] / {exp( y1 ) +. . . .+ exp( y D −1 ) + 1} any problem or hypothesis concerning compositions can be fully expressed in terms of logratios and vice versa. Therefore, since a logratio transformation of compositions takes the logratio vector onto the whole of real space we have available, with a little caution, the whole gamut of unconstrained multivariate analysis. The conclusions of the unconstrained multivariate analysis can then be translated back into conclusions about the compositions, and the analysis is complete. This proposed methodology, essentially a transformation technique, is in line with a long tradition of statistical methodology, starting with McAlister (1879) and his logarithmic transformation, the lognormal distribution and the importance of the geometric mean, and more recently the Box-Cox transformations and the transformations involved in the general linear model approach to statistical analysis. There has always been opposition, sometimes fierce, to transformation techniques. For example, Karl Pearson became involved in a heated controversy with Kapteyn on the relative merits of his system of curves and the lognormal curve; see Kapteyn (1903, 1905), Pearson (1905, 1906). With a general mistrust of the technique of transformations Pearson would pose such questions as: what is the meaning of the logarithm of weight? History has clearly come down on the side of Wicksell and the logarithmic transformation and the lognormal distribution are long established useful tools of statistical modelling. One might therefore have expected the logratio transformation technique to have been an immediate happy and successful end of story. While it has eventually become so, immediate opposition along Pearsonian lines undoubtedly came to the fore. The 16 Chapter 1 The nature of compositional problems reader interested in pursuing the kinds of anti- transformation and other arguments against logratio analysis may find some entertainment in the following sequence of references published in the Mathematical Geology: Watson and Philip (1989), Aitchison (1990a), Watson (1990), Aitchison (1990b), Watson (1991), Aitchison (1991, 1992), Woronow (1997a, 1997b), Aitchison (1999), Zier and Rehder (1998), Aitchison et al (2000), Rehder and Zier (2001), Aitchison et al (2001). While much of this argumentative activity has been unnecessary and time-consuming, there have been episodes of progress. While the transformation techniques of Aitchison (1986) are still valid and provide a comprehensive methodology for compositional data analysis, there is now a better understanding of the fundamental principles which any compositional data methodology must adhere to. Moreover, there is now an alternative approach to compositional data analysis which could be termed the staying- in-the-simplex approach, whereby the tools introduced by Aitchison (1986) are adapted to defining a simple algebraic-geometric structure on the simplex, so that all analysis may be conducted entirely within this framework. This makes the analysis independent of transformations and results in unconstrained multivariate analysis. It should be said, however, that inferences will be identical whether a transformation technique or a staying- in-the-simplex approach is adopted. Which approach a compositional data analyst adopts will largely depend on the analyst’s technical understanding of the algebraic-geometric structure of the simplex. In this guide we will adopt a bilateral approach ensuring that we provide examples of interpretations in both ways. 1.4 The principle of scale invariance One of the disputed principles of compositional data analysis in the early part of the sequence above is that of scale invariance. When we say that a problem is compositional we are recognizing that the sizes of our specimens are irrelevant. This trivial admission has far-reaching consequences. A simple example can illustrate the argument. Consider two specimen vectors 17 Chapter 1 The nature of compositional problems w = (1.6, 2.4, 4.0) and W = (3.0, 4.5, 7.5) in R+3 as in Figure 1.4, representing the weights of the three parts (a, b, c) of two specimens of total weight 8g and 15g, respectively. If we are interested in compositional problems we recognize that these are of the same composition, the difference in weight being taken account of by the scale relationship W =(15/8) w. More generally two specimen vectors w and W in R+D are compositionally equivalent, written W ∼ w, when there exists a positive proportionality constant p such that W= pw. The fundamental requirement of compositional data analysis can then be stated as follows: any meaningful function f of a specimen vector must be such that f(W)=f(w) when W∼w, or equivalently f(pw) = f(w), for every p>0. In other words, the function f must be invariant under the group of scale transformations. Since any group invariant function can be expressed as a function of any maximal invariant h and since h(w) = (w1 / wD , . . . , wD-1 / wD) is such a maximal invariant we have the following important consequence: Any meaningful (scale-invariant) function of a composition can be expressed in terms of ratios of the components of the composition. Note that there are many equivalent sets of ratios which may be used for the purpose of creating meaningful functions of compositions. For example, a more symmetric set of ratios such as w/g(w), where g(w) = (w1 . . . wD ) 1/D is the geometric mean of the components of w, would equally meet the scale- invariant requireme nt. 18 Chapter 1 The nature of compositional problems Fig. 1.4 Representation of equivalent specimen vectors as points on rays of the positive orthant 1.5 Subcompositions: the marginals of compositional data analysis The marginal or projection concept for simplicial data is slightly more comple x than that for unconstrained vectors in RD , where a marginal vector is simply a subvector of the full D-dimensional vector. For example, a geologist interested only in the parts (Na2 O, K2O, Al2 O3 ) of a ten-part major oxide composition of a rock commonly forms the subcomposition based on these parts. Formally the subcomposition based on parts (1, 2, . . . ,C) of a D-part composition [x 1 , ... , x D ] is the (1, 2, . . . ,C)-subcomposition [s1 , . . . , sC ] defined by [s1 , . . . , sC ] = [x 1 , . . . , xC ] / (x 1 + . . . + xC). Note that this operation is a projection from a subsimplex to another subsimplex. See, for example, Aitchison (1986, Section 2.5). 1.6 Compositional classes and the search for a suitable sample space In my own teaching over the last 45 years I have issued a warning to all my students, similar to that of Pearson. Ignore the clear definition of your sample space at your 19 Chapter 1 The nature of compositional problems peril. When faced with a new situation the first thing you must resolve before you do anything else is an appropriate sample space. On occasions when I have found some dispute between students over some statistical issue the question of which of them had appropriate sample spaces has almost always determined which students are correct in their conclusions. If, for example, it is a question of association between the directions of departure and return of migrating New York swallows then an appropriate sample space is a doughnut. We must surely recognize that a rectangular box, a tetrahedron, a sphere and a doughnut look rather different. It should come as no surprise to us therefore that problems with four different sample spaces might require completely different statistical methodologies. It has always seemed surprising to this writer that the direction data analysts had little difficulty in seeing that the sphere and the torus require their own special methodology, whereas for so long statisticians and scientists seemed to think that what was good enough for a box was good enough for a tetrahedron. In the first step of statistical modelling, namely specifying a sample space, the choice is with the modeller. It is how the sample space is used or exploited to answer relevant problems that is important. We might, as in our study of scale invariance above, take the set of rays through the origin and in the positive orthant as our sample space. The awkwardness here is that the notion of placing a probability measure on a set of rays is less familiar than on a set of points. Moreover we know that as far as the study of compositions is concerned any point on a ray can be used to represent the corresponding composition. The selection of each representative point x where the rays meet the unit hyperplane w1 + . . . + wD = 1 with x = w/(w1 + . . . + wD) is surely the simplest form of standardization possible. We shall thus adopt the unit simplex D S = { [x 1 , . . . , x D ]: x i>0 (i = 1, . . . , D) , x 1 + . . . + x D = 1}. To avoid any confusion on terminology for our generic composition x we reiterate that we refer to the labels 1, . . . , D of the parts and the proportions x 1 , ... , x D as the components of the composition x. With this representation we shall continue to ensure 20 Chapter 1 The nature of compositional problems scale invariance by formulating all our statements concerning compositions in terms of ratios of components. Note the one-to-one correspondence between the components of x and a set of independent and exhaustive ratios such as ri = x i /(x 1 + . . . + xD ) (i = 1, ... , D-1), rD = 1 /(x 1 + . . . + xD ), with the components of x determined by these ratios as x i = ri /(r1 + . . . + rD-1 +1) (i = 1, ... , D-1), x D = 1/(r1 + . . . + rD-1 +1). Our next logical requirement will reinforce the good sense of this formulation in terms of ratios. 1.7 Subcomposional coherence Less familiar than scale invariance is another logical necessity of compositional analysis, namely subcompositional coherence. Consider two scientists A and B interested in soil samples, which have been divided into aliquots For each aliquot A records a 4-part composition (animal, vegetable, mineral, water); B first dries each aliquot without recording the water content and arrives at a 3-part composition (animal, vegetable, mineral). Let us further assume for simplicity the ideal situation where the aliquots in each pair are identical and where the two scientists are accurate in their determinations. Then clearly B's 3-part composition [s1 , s2 , s3 ] for an aliquot will be a subcomposition of A's 4-part composition [x 1 , x2 , x3 , x 4 ] for the corresponding aliquot related as in the definition of subcomposition in Section 1.5 above with C = 3, D = 4. It is then obvious that any compositional statements that A and B make about the common parts, animal, vegetable and mineral, must agree. This is the nature of subcompositional coherence. 21 Chapter 1 The nature of compositional problems The ignoring of this principle of subcompositional coherence has been a source of great confusion in compositional data analysis, The literature, even currently, is full of attempts to explain the dependence of components of compositions in terms of product moment correlation of raw components. Consider the simple data set: Full compositions [ x1 x 2 x 3 x 4 ] Subcompositions [ s1 s 2 s3 ] [0.1, 0.2, 0.1, 0.6] [0.25, 0.50, 0.25] [0.2, 0.1, 0.1, 0.6] [0.50, 0.25, 0.25] [0.3, 0.3, 0.2, 0.2] [0.375, 0.375, 0.25] Scientist A would report the correlation between animal and vegetable as corr(x 1 , x2 ) = 0.5 whereas B would report corr(s1 , s2 ) = -1. There is thus incoherence of the product- moment correlation between raw components as a measure of dependence. Note, however, that the ratio of two components remains unchanged when we move from full composition to subcomposition: si / s j = xi / x j , so that as long as we work with scale invariant functions, or equivalently express all our statements about compositions in terms of ratios, we shall be subcompositionally coherent. 1.8 Perturbation as the operation of compositional change 1.8.1 The role of group operations in statistics For every sample space there are basic group operations which, when recognized, dominate clear thinking about data analysis. In RD the two operations, translation (or displacement) and scalar multiplication, are so familiar that their fundamental role is often overlooked. Yet the change from y to Y = y + t by the translation t or to Y = ay by the scalar multiple a are at the heart of statistical methodology for RD sample spaces. For example, since the translation relationship between y1 and Y1 is the same as that between y2 and Y2 if and only if Y1 and Y2 are equal translations t of y1 and y2 , any definition of a difference or a distance measure must be such that the measure is the same for (y1 , Y1 ) as for (y1 + t, Y1 + t) for every translation t. Technically this is a 22 Chapter 1 The nature of compositional problems requirement of invariance under the group of translations. This is the reason, though seldom expressed because of its obviousness in this simple space, for the use of the mean vector ?µ = E ( y ) and the covariance matrix Σ = cov( y ) = E {( y − µ )( y − µ ) T } as meaningful measures of ‘central tendency’ and ‘dispersion’. Recall also, for further reference, two basic properties: for a fixed translation t, E( y + t) = E(y) + t ; V(y + t) = V(y). The second operation, that of scalar multiplication, also plays a substantial role in, for example, linear forms of statistical analysis such as principal component analysis, where linear combinations a1 y1 + . . . + a D y D with certain properties are sought. Recall, again for further reference, that for a fixed scalar multiple a, E(ay) = aE(y) ; V(ay) = a2 V(y). Similar considerations of groups of fundamental operations are essential for other sample spaces. For example, in the analysis of directional data, as in the study of the movement of tectonic plates, it was recognition that the group of rotations on the sphere plays a central role and the use of a satisfactory representation of that group that led Chang (1988) to the production of the essential statistical tool for spherical regression. 1.8.2 Perturbation: a fundamental group operation in the simplex By analogy with the group operation arguments for RD the obvious questions for a simplex sample space are whether there is an operation on a composition x, analogous to translation in R D , which transforms it into X, and whether this can be used to characterize ‘difference’ between compositions or change from one composition to another. The answer is to be found in the perturbation operator as defined in Aitchison (1986, Section 2.8). The perturbation operator can be motivated by the following observation within the positive orthant representation of compositions. For any two equivalent compositions w and W on the same ray there is a scale relationship W = pw for some p > 0, where 23 Chapter 1 The nature of compositional problems each component of w is scaled by the same factor p to obtain the corresponding component of W. For any two non-equivalent compositions w and W on different rays a similar, but differential, scaling relationship W 1 = p1 w1 , . . . , WD = pDwD reflects the change from w to W. Such a unique differential scaling can always be found by taking pi = Wi / wi (i = 1, . . . , D). We can translate this into terms of the compositional representations x and X within the unit simplex sample space S D . If we define a perturbation p as a differential scaling operator p = [ p1 . . . pD ] ∈ S D and denote by ⊕ the perturbation operation, then we can define the perturbation operation in the following way. The perturbation p applied to the composition x = [ x1 . . . x D ] produces the composition X given by X = p ⊕ x = [ p1 x1... p D x D ] /( p1 x1 +...+ p D x D ) = C[ p1 x1... p D x D ], where C is the so-called closure operation which divides each component of a vector by the sum of the components, thus scaling the vector to the constant sum 1. Note that because of the nature of the scaling in this relationship it is not strictly necessary for the perturbation p to be a vector in S D . In mathematical terms the set of perturbations in S D form a group with the identity perturbation e = [1 / D . . . 1 / D ] and the inverse of a perturbation p being the closure p −1 = C[ p1−1 ... p D−1 ] . We use the notation x Θ p to denote the operation of this inverse on x giving x Θ p = C[ x1 / p1 . . . x D / p D ] . The relation between any two compositions x and X can always be expressed as a perturbation operation X = ( X Θ x ) ⊕ x , where X Θ x is a perturbation in the group of perturbations in the the simplex S D . Similarly the change from X to x is expressed by the perturbation x Θ X . Thus any measure of difference between compositions x and X must be expressible in terms of one or other of these perturbations. A consequence of this is that if we wish to define any scalar measure of distance between two compositions x and X , say ∆( x , X ) then we must ensure that it is a function of the ratios x 1 /X1 . . . . , x D/XD . As we shall see later this, together with attention to the need for scale 24 Chapter 1 The nature of compositional problems invariance, subcompositional coherence and some other simple requirements, has led Aitchison (1992) to advocate the follolowing definition: x X ∆ ( x , X ) = ∑ log i − log i xj Xj i< j 2 2 as a simplicial metric, reinforcing an intuitive equivalent choice in Aitchison (1986, Section 8.3). 1.8.3 Some familiar perturbations In relation to probability statements the perturbation operation is a standard process. Bayesians perturb the prior probability assessment x on a finite number D of hypotheses by the likelihood p to obtain the posterior assessment X through the use of Bayes’s formula. Again, in genetic selection, the population composition x of genotypes of one generation is perturbed by differential survival probabilities represented by a perturbation p to obtain the composition X at the next generation, again by the perturbation probabilistic mechanism. In certain geological processes, such as metamorphic change, sedimentation, crushing in relation to particle size distributions, change may be best modelled by such perturbation mechanisms, where an initial specimen of composition x 0 is subjected to a sequence of perturbations p1 , . . . , pn in reaching its current state x n : x1 = p1 ⊕ x 0 , x2 = p2 ⊕ x1 , . . . , xn = pn ⊕ x n−1 so that x n = ( p1 ⊕ p2 ⊕ . . . ⊕ pn ) ⊕ x 0 . It is clear that in this mechanism we have the makings of some form of central limit theorem but we delay consideration of this until we have completed the more mathematical aspects of the simplex sample space. A further role which perturbation plays in simplicial inference is in characterizing 25 Chapter 1 The nature of compositional problems imprecision or error. A simple example will suffice for the moment. In the process of replicate analyses of aliquots of some specimen in an attempt to determine its composition ξ ?we may obtain different compositions x 1 , . . . , x N because of the imprecision of the analytic process. In such a situation we can model by setting x n = ξ ⊕ pn ( n = 1, . . . , N ) , where the pn are independent error perturbations characterizing the imprecision. 1.9 Power as a subsidiary operation in the simplex The simplicial operation analogous to scalar multiplication in real space is the power operation. First we define the power operatio n and then consider its relevance in compositional data analysis. For any real number a ∈ R1 and any composition x ∈ S D we define X = a ⊗ x = C [ x1a ...x aD ] as the a-power transform of x. Such an operation arises in compositional data analysis in two distinct ways. First it may be of relevance directly because of the nature of the sampling process. For example, in grain size studies of sediments, sediment samples may be successively sieved through meshes of different diameters and the weights of these successive separations converted into compositions based on proportions by weight. Thus though separation is based on the linear measurement diameter the composition is based essentially on a weight, or equivalently a volume measurement, with a power transformation being the natural connecting concept. More indirectly the power transformation can be useful in describing regression relations for compositions. For example, the finding of Aitchison (1986, Section 7.7) of the relationship of a (sand, silt, clay) sediment x to depth d can be expressed in the form x = ξ ⊕{log d ⊗ β } ⊕ p , 26 Chapter 1 The nature of compositional problems where β is a composition playing the counterpart of regression coefficients and p is a perturbation playing the role of error in more familiar regression situations. It must be clear that together the operations perturbation ⊕ and power ⊗ play roles in the geometry of S D analogous to translation and scalar multiplication in R D and indeed can be used to define a vector space in S D . We shall take up the full algebraicgeometric structure of the simplex sample space later in this guide. 1.10 Limitations in the interpretability of compositional data There is a tendency in some compositional data analysts to expect too much in their inferences from compositional data. For these the following situation may show the nature of the limitations of compositional data. Outside my home I have a planter consisting of water, soil and seed. One evening before bedtime I analyse a sample and determine its (water, soil, seed) composition as x = [3/6 2/6 1/6]. I sleep soundly and in the morning again analyse a sample finding X = [6/9 2/9 1/9]. I measure the change as the perturbation X Θ x = C[( 6 / 9) / ( 3 / 6) ( 2 / 9 ) / (2 / 6) (1 / 9) / (1 / 6)] = [1 / 2 1 / 4 1 / 4] . Now I can picture two simple scenarios which could describe this change. Suppose that the planter last evening actually contained [18 12 6] kilos of (water, soil, seed), corresponding to the evening composition [3/6 2/6 1/6], and it rained during the night increasing the water content only so that the morning content was [36 12 6] kilos, corresponding to the morning composition [6/9 2/9 1/9]. Although this rain only explanation may be true, is it the only explanation? Obviously not, because the change could equally be explained by a wind only scenario, in which the overnight wind has swept away soil and seed resulting in content of [18 6 3] kilos and the same morning composition [6/9 2/9 1/9]. Even more complicated scenarios will produce a similar change. For example a combination of rain and wind might have resulted in a 27 Chapter 1 The nature of compositional problems combination of increased water and decreased soil and seed, say to a content of [27 9 4.5] kilos, again with morning composition [6/9 2/9 1/9]. The point here is that compositions provide information only about the relative magnitudes of the compositional components and so interpretations involving absolute values as in the above example cannot be justified. Only if there is evidence external to the compositional information would such inferences be justified. For example, if I had been wakened by my bedroom windows rattling during the night and I found my rain gauge empty in the morning would I be justified in painting the wind only scenario. But I slept soundly during the night. A consequence of this example is that we must learn to phrase our inferences from compositional data in terms which are meaningful and we have seen that the meaningful operations are perturbation and power. In subsequent chapters we shall how we may use these operations successfully. 28 Chapter 2 The simplex sample space Chapter 2 The simplex sample space and principles of compositional data analysis 2.1 Logratio analysis: a statistical methodology for compositional data analysis What has come to be known as logratio analysis for compositional data problems arose in the 1980’s out of the realisation of the importance of the principle of scale invariance and that its practical implementation required working with ratios of components, This, together with an awareness that logarithms of ratios are mathematically more tractable than ratios led to the advocacy of a transformation technique involving logratios of the components. There were two obvious contenders for this. Let x = [ x1 . . . x D ] ∈ S D be a typical D-part composition. Then the so-called additive logratio transformation alr: S D → R D−1 is defined by y = alr ( x ) = [log( x1 / x D ) log( x 2 / x D )...log( x D −1 / x D )] , where the ratios involve the division of each of the first D – 1 components by the final component. The inverse transformation alr −1: R D−1 → S D is x = alr −1 ( y ) = C [exp( y1 ) exp( y 2 )...exp( y D −1 )1] , where C denotes the closure operation. Note that this transformation takes the composition into the whole of R D −1 and so we have the prospect of using standard unconstrained multivariate analysis on the transformed data, and because of the oneto-one nature of the transformation transferring any inferences back to the simplex and to the components of the composition. One apparent drawback with this technique is the choice of the final component as divisor, with a much asked question. Would we obtain the same inference if we chose 29 Chapter 2 The simplex sample space another component as divisor, or more generally if we permuted the parts? The answer to this question is yes. We shall not go into any details of the proofs of this assertion, but the interested reader may find these in Aitchison (1986, Chapter 5). The alr transformation is asymmetric in the parts and it is sometimes convenient to treat the parts symmetrically. This can be achieved by the so-called centred logratio transformation clr: S D → U D : z = clr ( x ) = [log{ x1 / g ( x )} ... log{ x D / g ( x)}] , where U D = {[ u1 ...u D ] :u1 +...u D = 0} . a hyperplane of R D .The inverse transformation clr −1 :U D → S D takes the form x = C [exp( z1 )...exp( z D )] . This transformation to a real space again opens up the possibility of using standard unconstrained multivariate methods. We note here that the mean vector µ = E {alr ( x)} and covariance matrix Σ = cov{alr ( x )} of the logratio vector alr ( x ) will play an important role in our compositional data analysis, as will do the centred logratio analogues λ = E {clr ( x )} and cov{clr ( x)} . So the philosophy of logratio analysis can be stated simply. 1. Formulate the compositional problem in terms of the components of the composition. 2. Translate this formulation into terms of the logratio vector of the composition. 30 Chapter 2 The simplex sample space 3. Transform the compositional data into logratio vectors. 4. Analyse the logratio data by an appropriate standard multivariate statistical method. 5. Translate back into terms of the compositions the inference obtained at step 4. We shall see later many examples of this compositional methodology. 2.2 The unit simplex sample space and the staying-in the-simplex approach Logratio analysis emerged in the 1980’s in a series of papers Aitchison (1981a, 1981b, 1981c, 1982, 1983, 1984a, 1984b, 1985), Aitchison and Bacon-Shone (1984), Aitchison and Lauder (1985), Aitchison and Shen (1980, 1984) and in the monograph Aitchison (1986); and has been successfully applied in a wide variety of disciplines. Since, however, there seem an appreciable number of statisticians and scientists who seem, for whatever reason, uncomfortable with transformation techniques it seems worth considering what are the alternatives. In the discussion of Aitchion (1982), Fisher made the following comment: Clearly the speaker has been very successful in fitting simple models to normal transformed data, the counterpart to the simplicity of these models is the complexity of corresponding relationships among the untransformed components. This is hardly an original observation. Yet there are certain aromas rising from the murky potage of compositional data problems which are redolent of some aspects of problems with directional data, and herein lies the point. When attacking these latter problems, one is ultimately better off working within the confines of the original geometry (of the circle, sphere cylinder, . . .) and with techniques particular thereto (vector methods, etc), in terms of perceiving simple underlying ideas and modelling them in a natural way. Mapping from, say, the sphere into the plane, and then back, rarely produces these elements, and usually introduces unfortunate distortion. I still hold out some hope that simple models of dependence can be found, peculiar to the simplex. . . . Meanwhile, I shall analyse data with the normal transform method. 31 Chapter 2 The simplex sample space The lack of success in transforming the sphere into the plane is that the spaces are topologically different whereas the simplex and real space are topologically equivalent. Nevertheless there is a challenge to confine the statistical argument to the geometry of the simplex, and this approach has been emerging over the last decade, based on the operations of perturbation and power and on the already indicated simplicial metric. It is now certainly possible to analyse compositional data entirely within simplicial geometry. Clearly the success of such an approach must depend largely on the mathematical sophistication of the user. In the remainder of this guide we shall adopt a bilateral approach, attempting to interpret inferences from our compositional data problems both from the logratio analysis approach and the staying- in-the-simplex approach. First in the next section we give a concise account of the algebraic-geometric structure of the simplex. 2.3 The algebraic-geometric structure of the simplex 2.3.1 Introduction The purpose of this section is to provide a reasonably agreed account in terms of terminology and notation of the algebraic-geometric structure of the unit simplex as a standard sample space for those compositional data analysts wishing to adopt a staying- in-the-simplex approach as an alternative to logratio transformation techniques. Emphasis is placed on the metric vector space structure of the simplex, with perturbation and power operations, the associated metric, the importance of bases, power-perturbation combinations, and simplicial subspaces in range and null space terms. Concepts of rates of compositional change, including compositional differentiation and integration are also considered. For compositional data sets, some basic ideas are discussed including concepts of distributional centre and dispersion, and the fundamental simplicial singular value decomposition. The sources of the ideas are dispersed through the References and will not be cited throughout the text. 32 Chapter 2 The simplex sample space 2.3.2 Compositional vectors Compositions, positive vectors with unit, 100 per cent or some other constant sum, are a familiar, important data source for geologists. Since in compositional problems the magnitude of the constant sum is irrelevant we assume that the data vectors have been standardised to be of unit sum; we then regard a generic D-part composition, such as ten major oxides or sedimental sand, silt, clay, to take the form of a row vector x = [ x1 , . . . , x D ], where the x i (i = 1, . . . , D ) are the components, proportions of the available unit, and the integers 1, . . . D act as labels for the parts. We have chosen the convention of recording compositions as row vectors since this conforms with the common practice of setting out compositional data with cases set out in rows and parts such as major oxides in columns. Such a convention also conforms with practice in such software as MSExcel. Thus a data set consisting of N D-part compositions x1 , . . . , xN may be set out as an N × D matrix X = [x1 ; . . . ; xN], where the semi-colon is used to indicate that the next vector occurs in the next row. As in standard multivariate analysis marginal concepts are important. For compositions and the simplex the marginal concept is a subcomposition, such as the CNK (CaO, N2 O, K2O)-subcomposition of a major oxide composition. For example the (1, . . . C)-subcomposition of a D-part composition [ x1 , . . . , x D ] is defined as [ s1 ,..., sC ] = C [ x1 ,..., xC ] = [ x1 ,..., xC ] /( x1 +... + xC ). Note that the ‘closure’ operator C standardises the contained vector by dividing by the sum of its components so that a subcomposition consists of components summing to unity. In geometric terms formation of a subcomposition is geometrically a projection. 2.3.3 The algebraic-geometric structure of the unit simplex The sample space associated with D-part compositions is the unit simplex: S D = {[ x1 ,..., x C ] :x i > 0 ( i = 1,..., D ), x 1 +...+ x D = 1}. The fundamental operations of change in the simplex are those of perturbation and power transformation. In their simplest forms these can be defined as follows. Given 33 Chapter 2 The simplex sample space any two D-part compositions x , y ∈ S D their perturbation is x ⊕ y = C [ x1 y1 ,..., x D y D ], where C is the well known closure or normalizing operation in which the elements of a positive vector are divided by their sum; and given a D-part composition x ∈ S D and a real number, a the power transformed composition is a ⊗ x = C [ x1a ,..., x aD ] Note that we have used the operator symbols ⊕ and ⊗ to emphasize the analogy with the operations of displacement or translation and scalar multiplication of vectors in R D . It is trivial to establish that the internal ⊕ operation and the external ⊗ ?operation ?define a vector or linear space structure on S D . In particular the ⊕ operation defines an abelian group with identity e = [1, . . . ,1] / D . We record a few of the simple properties of ⊕ and ⊗ : x ⊕ y = y ⊕ x , ( x ⊕ y ) ⊕ z = x ⊕ ( y ⊕ z), a ⊗ ( x ⊕ y ) = ( a ⊗ x ) ⊕ (a ⊗ y ). The operator Θ , the inverse of ⊕ , is simply defined by xΘ y = C [ x1 / y1 ,..., x D / y D ] and plays an important role in compositional data analysis, for example in the construction of compositional residuals. The structure can be extended by the introduction of the simplicial metric ∆: S D × S D → R≥0 defined as follows: 2 D xi yi ∆ ( x, y ) = ∑ log − log g( y) i=1 g ( x) 1/ 2 2 D xi yi = ∑ log − log y j i< j x j 34 1/2 ( x , y ∈ S D ), Chapter 2 The simplex sample space where g( ) is the geometric mean of the components of the composition. The metric ∆ satisfies the usual metric axioms: M1 Positivity : ∆ ( x, y ) > 0( x ≠ y), ∆( x , y ) = 0 ( x = y) M2 Symmetry: ∆ ( x, y ) = ∆ ( y , x) M3 Power relationship: ∆ (a ⊗ x, a ⊗ y ) =| a | ∆ ( x, y ) M4 Triangular inequality: ∆ ( x, z ) + ∆ ( z , y ) ≥ ∆( x, y) The fact that this metric has also desirable properties relevant and logically necessary, such as scale, permutation and perturbation invariance and subcompositional dominance, for meaningful statistical analysis of compositional data is now well established and the relevant properties are recorded briefly here: M5 Permutation invariance: ∆ ( xP, yP) = ∆ ( x, y) , for any permutation matrix P. M6 Perturbation ∆ ( x ⊕ p , y ⊕ p ) = ∆ ( x, y ) , invariance: where p is any perturbation. M7 Subcompositional dominance: if sx and sy are similar, say (1, . . . , C)- subcompositions of x and y, the n ∆ S C ( s x , s y ) ≤ ∆ S D ( x, y) . It is possible to go to even more mathematical sophistication for the unit simplex if either theoretical or practical requirements demand it. For example, consistent with the metric ∆ is the norm ||x||, defined by x || x || = ∆ ( x, e) = ∑ log i g (x ) i =1 D 2 2 2 and the inner product, defined by D 〈 x , y 〉 = ∑ log i =1 xi yi log , g( x) g( x) where e is the identity perturbation [1, . . . , 1]/D. An interesting aspect of these extensions is that an inner product 〈 b, x 〉 can be expressed as 35 Chapter 2 The simplex sample space D D bi xi ∑ log g (b) log g (x ) = ∑ ai log x i i =1 i =1 where a = log{ b / g (b)} and so a1 + . . . + a D = 0 . Thus inner products play the role of logcontrasts, well established as the compositional ‘linear combinations’ required in many forms of compositional data analysis such as principal component analysis and investigation of subcompositions as concomitant or explanatory vectors. 2.3.4 Generators, orthonormal basis and subspaces As for any vector space generating vectors, bases, linear dependence, orthonormal bases and subspaces play a fundamental role and this is equally true for the simplex vector space. In such concepts the counterpart of ‘linear combination’ is a powerperturbation combination such as x = ( u1 ⊗ β 1) ⊕...⊕ (u C ⊗ β C ) and such combinations play a central role. In such a specification the β ’s are compositions regarded as generators, and the combination generates some subspace of the unit simplex as the real number u-coefficient vary. When this subspace is the whole of the unit simplex then the β ’s form a basis. Generally a basis should be chosen such that the generators are ‘linearly independent’ in the sense that β 1,..., β C are linearly independent if and only if (u1 ⊗ β 1) ⊕...⊕ ( uC ⊗ β C )= e ⇒ u1 =...= uC = 0 . For S D , which is essentially a (D –1)-dimensional space, a linearly independent basis has D –1 generators, and important among such basis are those which form an orthonormal basis, say with generators β 1,..., β D−1 which have unit norm || β i||= 1 (i = 1,..., D − 1) , and are orthogonal in the sense that 〈 β i , β j 〉 = 0 (i ≠ j ) . As on any vector space a set of C orthonormal generators can be easily extended to form an orthonormal basis of S D . Later in Section 2.3.7 we shall see that orthonormal 36 Chapter 2 The simplex sample space bases a central role in a data-analytic sense in terms of the simplicial singular value decomposition of a compositional data set. As in standard multivariate analysis range and null spaces play an important and complementary role in such areas of data investigation as compositional regression , principal logcontrast component analysis and in the study of compositional processes. The set Β = [ β 1;...; β C ] of linearly independent generators identifies a range space range (Β) = {x : x = (u1 ⊗ β 1) ⊕...⊕ (u C ⊗ β C ), u i ∈ R1 (i = 1,..., C )} namely the subspace of dimension C generated by the compositions in Β . Similarly associated with Β can be defined a null space null ( Β) = {x : 〈 β 1, x 〉 = 0,..., 〈 β C , x〉 = 0} a subspace of dimension D – C – 1. Range and null spaces are essentially equivalent ways of expressing certain constraints which may apply to compositions. The relationship of these equivalences is simple. For example, the null space corresponding to range(Β ) above is null( Β⊥ ) where Β ⊥ is the comple tion of a basis orthogonal to Β ; similarly null( Β) = range (Β ⊥ ) . As defined, these range and null spaces contain the identity e of S D . It is often convenient to allow a displacement so that they contain a specified composition ξ : all that this requires is the specification of the range space above to start with ξ -i.e., x = ξ ⊕ range(B ) -, and the zero values of the inner products in the specification of the null space to be replaced by 〈 β 1, x − ξ 〉 ,..., 〈 β C , x − ξ 〉 . 2.3.5 Differentiation, integration, rates of change Clearly in compositional processes rates of change of compositions are important and here we define the basic ideas. Suppose that a composition x (t ) depends on some continuous variable t such as time or depth. Then the rate of change of the composition with respect to t can be defined as the limit 37 Chapter 2 The simplex sample space Dx (t ) = lim dt − > 0 1 d ⊗ ( x(t + dt )Θx(t)) = C (exp( log x( t ))) dt dt where d/dt denotes ‘ordinary’ derivation with respect to t. Thus, for example, if x ( t) = ξ ⊕ h( t) ⊗ β , then Dx( t ) = h'( t ) ⊗ β . There are obvious extensions through partial differentiation to compositional functions of more than one variable. We note also that the inverse operation of integration of a compositional function x (t ) over an interval (t 0 , t) can be expressed as ∫ x(t )dt = C (exp( ∫ log x(t ) dt )). t t0 2.3.6 Distributional concepts in the simplex For statistical modelling we have to consider distributions on the simplex and their characteristics. The well-established ‘measure of central tendency’ ξ ∈S D which minimizes E(∆(x, ξ)) is ξ = cen ( x ) = C (exp( E (log x ))) satisfying certain necessary requirements, such as cen (a ⊗ x ) = a ⊗ cen ( x ) and cen ( x ⊕ y) = cen ( x ) ⊕ cen( y ). There are a number of criteria which dictate the choice of any measure V(x) of dispersion and dependence which forms the basis of characteristics of compositional variability in terms of second order moments: (a) Is the measure interpretable in relation to the specific hypotheses and problems of interest in fields of application? (b) Is the measure conformable with the definition of center associated with the sample space and basic algebraic operation? (c) Is the measure invariant under the group of basic operations, in our case the group of perturbations? Is dis ( p ⊕ x) = dis ( x) for every constant perturbation 38 Chapter 2 The simplex sample space p? (Recall the result in Section 1.8.1 that for y ∈ R D the covariance matrix V is invariant under translation: V(t + y) = V(y)). (d) Is the measure tractable mathematically? To ensure a positive answer to (a) we must clearly work in terms of ratios of the components of compositions to ensure scale invariance. At first thought this might suggest the use of variances and covariances of the form var(x i/x j) and cov(x i /x j , xk / x l ). Unfortunately these are mathematically intractable because, for example, there is no exact or even simple approximate relationship between var(x i/x j) and var(x j/x i). Fortunately we already have a clue as to how to overcome this difficulty in the appearance of logarithms of ratios of components both in the central limit theorem at Section 1.8.3 and in the definition of the center of compositional variability. It seems worth the risk therefore of apparently complicating the definition of dispersion and dependence by considering such dispersion characteristics as var{log(x i /x j )}, cov{log(x i /x j )} , log(x k /x l ) . Obvious advantages of this are simple relationships such as var{log(x i/x j)} = var{log(x j/x i)} cov{log(x i /x j ) , log(x l /x k )} = cov{log(x i /x j ) , log(x l /x k ). There are a number of useful and equivalent ways (Aitchison, 1986, Chapter 4) in which to summarize such a sufficient set of second-order moment characteristics. For example, the logratio covariance matrix? Σ ( x ) = cov( alr ( x )) = [cov{log( xi / x D ), log( x j / x D )}] using only the final component x D as the common ratio divisor, or the centered 39 Chapter 2 The simplex sample space logratio covariance matrix G(x) = cov{clr(x)} = [cov{log(x i /g(x)), log(x j /g(x))}]. My preferred summarizing characteristic is what I have termed the variation matrix T(x) = [t ij ] = [var{log(x i /x j )}]. Note that T is symmetric, has zero diagonal elements, and cannot be expressed as the standard covariance matrix of some vector. It is a fact, however, that S, G and T are equivalent: each can be derived from any other by simple matrix operations (Aitchison, 1986, Chapter 4). A first reaction to this variation matrix characterization is surprise because it is defined in terms of variances only. The simplest statistical analogue is in the use of a completely randomized block design in, say, an industrial experiment . From such a situation information about var(yi - yj) for all i, j is a sufficient description of the variability for purposes of inference. So far we have emphasized criteria (a), (b) and (d). Fortunately criterion (c) is automatically satisfied since, for each of the dispersion measures dis ( p ⊕ x) = dis ( x) for any constant perturbation p. We should also note here that the dimensionality of the covariance parameter is ½ D(D -1) and so is as parsimonious as corresponding definitions in other essentially (D-1)-dimensional spaces. To sum up, importantly these dispersion characteristics are consistent with the simplicial metric defined above and satisfy the following properties: dis( a ⊗ x ) =| a|2 dis( x) , for any scalar a in R; dis( x ⊕ p) = dis( x ) , for any constant perturbation p; dis( x ⊕ y) = dis( x) + dis( y ) , for independent x, y. 2.3.7 Relevance to compositional data sets 40 Chapter 2 The simplex sample space There are substantial implications in the above development for the analysis of a N × D compositional data set X = [ x1 ; . . . ; x N ] . A main feature is that the estimate of the centre ξ is given by ξ$ = C ( g1 , . . . , g D ) , gi is the geometric mean of the ith component of the N compositions. Measures of dispersion are simply estimated from the estimated variances of the appropriate logratios. There is for the simplex a central result, analogous to the singular value decomposition for data sets associated with the sample space RD, which plays a central role. Any N x D compositional data matrix X with nth row composition x n can be decomposed in a perturbation-power form as follows x n = ξ ⊕ (u n1s1 ⊗ β 1 ) ⊕ . . . ⊕ ( u nm s m ⊗ β m ) where ξ is the centre of the data set, the s’s are positive ‘singular values’ in descending order of magnitude, the β ’s are compositions, m ( ≤ D − 1 ) is a readily defined rank of the compositional data set, and the u’s are power components specific to each composition. In a way similar to that for data sets in RD we may consider an approximation of order rf ( x * | data )} , R and this is easily evaluated in terms of standard incomplete beta functions; for details see Aitchison (1986, Section 7.10). Atypicality indices lie between 0 and 1, with nearzero corresponding to a composition near the center of the distribution and near 1 corresponding to an extremely atypical composition lying in a region of very low density. Atypicality indices are therefore useful in detecting possible outliers or anomalous compositions. For inspection of a given data set it is advisable to use the now standard jackknife or leaving-one-out technique to avoid resubstitution bias in assessing the atypicality index of any composition in the data set. Again atypicality indices for such a procedures are readily computable. 61 62 Chapter 3 From theory to practice: some simple apllications Chapter 3 From theory to practice: some simple applications 3.1 Simple hypothesis testing: comparison of hongite and kongite A general question that we asked in Example 1 of Section 1.1 was whether any differences could be detected between the hongite and kongite compositional experience. After an alr logratio transformation of the compositional vectors we are then faced with two multivariate normal samples with questions about equality of mean vectors and covaria nce matrices. We have already obtained the estimates for hongite in Section 2.6. The corresponding estimations for kongite are as follows. The kongite centre is [0.486 0.201 0.114 0.105 0.094], again quite different from the arithmetic mean [0.438 0.214 0.165 0.097 0.086]. The estimates of Σ , Γ , Τ for the kongite compositional data matrix are: ∧ Σ= 0.1131 0.2352 -0.2008 0.0961 0.2352 0.6554 -0.7231 0.1061 -0.2008 -0.7231 1.0504 0.0911 0.0961 0.1061 0.0911 0.1951 0.0646 0.1807 0.2441 -0.0014 0.0002 0.1807 0.5949 -0.7724 0.0026 -0.0058 -0.2441 -0.7724 1.0123 -0.0012 0.0054 -0.0014 0.0026 -0.0012 0.0487 -0.0487 ∧ Γ= 0.0002 -0.0058 0.0054 -0.0487 0.0489 ∧ Τ= 0 0.2981 1.5652 0.1161 0.1131 0.2981 0 3.1520 0.6384 0.6554 1.5652 3.1520 0 1.0634 1.0504 0.1161 0.6384 1.0634 0 0.1951 0.1131 0.6554 1.0504 0.1951 0 Following the lattice strategy we can set out the model of two completely different distributions and the hypotheses within that model in a self explanatory lattice diagram (Figure 3.1). We are now within the structure of standard multivariate 63 Chapter 3 From theory to practice: some simple apllications analysis apart from the constraints of the simplicity postulate in the order and nature of the hypothesis testing within the model. To simplify matters here we use the asymptotic forms of the generalised likelihood ratio test statistics, the Q of Section 2.7, to be compared against appropriate chi-squared percentiles.. The computational procedures are uninteresting and can be found in Aitchison (1986, Section 7.5). The only unusual feature is the computation for the hypothesis µ1 = µ 2 with different covariance matrices, commonly referred to as the Fisher-Behrens problem. Model L (π1 , Σ1 ) 5 L5 ( µ2 , Σ 2 ) No of parameters 28 µ1 = µ 2 No of parameters 24 Test statistic 160.8 Level 2 Σ1 = Σ 2 No of parameters 18 Test statistic 10.7 Level 1 µ1 = µ 2 Σ1 = Σ 2 Test statistic 46.7 Fig. 3.1 Lattice of hypotheses for comparison of hongite and kongite compositions 64 Chapter 3 From theory to practice: some simple apllications The sequence of tests are then as follows. The hypothesis at level 1, that the hongite and kongite distributions are identical compares the value of the Q-statistic 46.7 against the 95 percentile of χ 2 (14) , namely 23.7, and so we reject this hypothesis and move up to testing the hypotheses at level 2. The hypotheses that the mean vectors are equal, allowing different covariance matrices, has a Q-statistic value of 160.8, to be compared with the 95 percentile of χ 2 (34 ) , namely 36.4, and so again this hypothesis has to be rejected. Finally the hypothesis that the covariance matrices are equal but that the mean vectors are different has a Q-statistic value of 10.7 to be compared with the 95 percentile of χ 2 (6) , namely 12.6. Thus we cannot reject this hypothesis and so would conclude that a reasonable working model would assume equal covariance structure for hongite and kongite but with different mean vectors. Along the lines of Section 2.9 we could apply the leaving-one-out technique to compute the atypicality indices of the hongite and kongite sets. For example, for the hongite set theses are: Speciment 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 A 48,8 48,2 37,0 50,9 44,2 52,3 44,6 34,6 41,2 42,6 49,9 45,2 32,7 41,4 46,2 32,3 43,2 49,5 42,3 44,6 45,8 49,9 48,6 45,5 45,9 B 31,7 23,8 9,1 23,8 38,3 26,2 33,0 5,2 11,7 46,6 19,5 37,3 8,5 12,9 17,5 7,3 44,3 32,3 15,8 11,5 16,6 25,0 34,0 16,6 24,9 C 3,8 9,0 34,2 7,2 2,9 4,2 4,6 42,9 26,7 0,7 11,4 2,7 38,9 23,4 15,8 40,9 1,0 3,1 20,4 23,8 16,8 6,8 2,5 17,6 9,7 65 D 6,4 9,2 9,5 10,1 7,7 12,5 12,2 9,6 9,6 5,6 9,5 5,5 8,0 15,8 8,3 12,9 7,8 8,7 8,3 11,6 12,0 10,9 9,4 9,6 9,8 E 9,3 9,8 10,2 8,0 6,9 4,8 5,6 7,7 10,8 4,5 9,7 9,3 11,9 6,5 12,2 6,6 3,7 6,3 13,2 8,5 8,8 7,4 5,5 10,7 9,7 Atypicality index 0,7122 0,0171 0,1990 0,0318 0,7086 0,8284 0,8822 0,9689 0,1000 0,9873 0,1678 0,7012 0,8148 0,7405 0,2679 0,7508 0,7298 0,1454 0,5135 0,3501 0,0990 0,0739 0,8282 0,2800 0,2864 Chapter 3 From theory to practice: some simple apllications The hongite specimens with atypicality index in excess of 0.95 are specimens no 8 (0.97) and no 10 (0.99). From inspection of the components we can see that for no 8 the atypicality probably arises from the relatively low value of the B component, and for no 10 the relatively low value of the C component. Clearly we should place these in a possible outlier category and refer the question of possible reasons to the geologist. What remains is to answer the question as to whether the new composition [44.0, 20.4, 13.9, 9.1, 12.6] can be regarded as typical of the hongite experience. Here a straightforward application of the new case atypicality computation of Section 2.9 produces an atypicality index of 0.997, raising substantial doubt as to the hongiteness of this new specimen. 3.2 Compositional regression analysis: the dependence of Arctic lake sediments on depth We asked the question in Example 2 of Section 1.1 whether the Arctic lake sediments of Table 1.1.2 were dependent of depth and, if so, what is the nature of the dependence. Again the alr transformation of the 3-part sediment compositions produces two-dimensional vectors in real space and we can then consider a straightforward multivariate (bivariate) regression on some function of depth. In this respect we follow the two-dimensional counterpart of the lattice used as an introductory example in Section 2.7. This is shown in Fig 3.2.a in which the various hypotheses on the functional form of the regressand and their rela tionships to each other are detailed. The tests of these hypotheses within the model (based on ratios of determinants of residual matrices) are a familiar part of unconstrained multivariate analysis and need not be dwelt on here; details can be found in Aitchison (1986, Sections 7.6-7.9). 66 Chapter 3 From theory to practice: some simple apllications Fig. 3.2.a Lattice of hypotheses for investigation of the dependence of Arctic lake sediments on water depth, showing residual determinants |Rh | and significance probabilities Ph . The test at level 1 dismisses the hypothesis of no dependence on depth. Stepping up to level 2 the linear hypothesis is dismissed, but the simple logarithmic hypothesis cannot be dismissed. If we had in fact reached level 3 we would have found the quadratic and linear- logarithmic dependence hypotheses also acceptable. Lattice testing prefers the logarithmic dependence because it provides a simpler working model with only four regression parameters, compared with the six parameters of the hypotheses at level 3. The working model in alr logratio terms can thus be expressed in the following terms: log( sand / clay) = α 1 + δ1 log( depth ) + error1 , log( silt / clay ) = α 2 + δ2 log( depth) + error2 67 Chapter 3 From theory to practice: some simple apllications with fitted model: log( sand / clay) = 9.70 − 2.74 log(depth) log( silt / clay) = 4.80 − 110 . log( depth ) . It should be emphasised here that this compositional regression is permutation invariant. In particular, a different choice of divisor in the alr transformation would have led to compatible results. Indeed the log(sand/silt) regression expression can be obtained by a simple subtraction of the two forms above, giving log( sand / silt ) = 4.90 − 1.64 log(depth) . The working model here clearly conforms to theories that as depth increases sand gives way to silt and more so to clay with these differential effects decreasing with depth. It would be interesting to compare the rates of change associated with such processes for different locations. The stay-in-the-simplex versions of these are the compositional regression composition = [0.9928 0.0071 0.0001] ⊕ log( depth) ⊗ [ 0.046 0.238 0.716] . This provides the same interpretation of the regression as the transformation regression: as depth increases sand gives way to silt and more so to clay with differential effects decreasing with depth. Which characterisation of the regression is chosen may well depend on personal choice. Here we can show the regression line within the [sand silt clay] ternary diagram as in Figure 3.2.b. The fit is obviously reasonably convincing. 68 Chapter 3 From theory to practice: some simple apllications Sand Direction of increasing depth Silt Fig. 3.2.b Clay Arctic sediments and regression line of sediment on logarithm of depth. Residual analysis can obviously be carried out either in terms of the transformed regression or in a stay- in-the simplex format. Since the latter is less familiar we demonstrate it briefly. This simply involves computation and investigation of the residual compositions, namely composition Θ fitted composition . The residual compositions here are: sediment 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 sand 77,5 71,9 50,7 52,2 70,0 66,5 43,1 53,4 15,5 31,7 65,7 70,4 17,4 10,6 38,2 10,8 18,4 4,6 15,6 31,9 9,5 17,1 10,5 4,8 2,6 silt 19,5 24,9 36,1 40,9 26,5 32,2 55,3 36,8 54,4 41,5 27,8 29,0 53,6 69,8 43,1 52,7 50,7 47,4 50,4 45,1 53,5 48,0 55,4 54,7 45,2 clay 3,0 3,2 13,2 6,6 3,5 1,3 1,6 9,8 30,1 26,8 6,5 0,6 29,0 19,6 18,7 36,5 30,9 48,0 34,0 23,0 37,0 34,9 34,1 41,0 52,2 69 res_sand res_silt 0,3661 0,2596 0,3766 0,3028 0,1583 0,2257 0,2304 0,3523 0,5081 0,2751 0,5550 0,3614 0,3727 0,5461 0,3692 0,2728 0,0837 0,2665 0,2044 0,2180 0,6040 0,2036 0,7652 0,2181 0,1518 0,2952 0,1670 0,4747 0,5052 0,2329 0,1816 0,3118 0,3110 0,2885 0,0760 0,2743 0,3182 0,2887 0,6110 0,2031 0,2530 0,3338 0,4127 0,2595 0,2996 0,3424 0,1518 0,3736 0,1206 0,3371 res_clay 0,3744 0,3206 0,6160 0,4173 0,2168 0,0836 0,0812 0,3580 0,6499 0,5776 0,1924 0,0166 0,5530 0,3583 0,2619 0,5066 0,4005 0,6497 0,3931 0,1858 0,4132 0,3278 0,3580 0,4746 0,5423 Chapter 3 From theory to practice: some simple apllications sediment 26 27 28 29 30 31 32 33 34 35 36 37 38 39 sand 11,4 6,7 6,9 4,0 7,4 4,8 4,5 6,6 6,7 7,4 6,0 6,3 2,5 2,0 silt 52,7 46,9 49,7 44,9 51,6 49,5 48,5 52,1 47,3 45,6 48,9 53,8 48,0 47,8 clay 35,9 46,4 43,4 51,1 40,9 45,7 47,0 41,3 45,9 46,9 45,1 39,9 49,5 50,2 res_sand res_silt 0,4164 0,3019 0,2910 0,3060 0,3047 0,3236 0,2353 0,3277 0,4057 0,3178 0,3051 0,3473 0,3157 0,3438 0,4375 0,3190 0,4725 0,2808 0,5020 0,2585 0,4587 0,2994 0,4711 0,3210 0,2877 0,3893 0,2677 0,4088 res_clay 0,2818 0,4030 0,3716 0,4371 0,2764 0,3476 0,3406 0,2435 0,2467 0,2395 0,2418 0,2080 0,3229 0,3235 These should be spread around the centre of a ternary diagram, as in Figure 3.2.c. The question of outliers among these residuals obviously arises. We report that there are two sediment compositions -S12 and S7- with residual atypicality indices of 0.9998 and 0.9990, respectively. sand Arithmetic Mean Geometric Mean silt Fig. 3.2.c clay Residuals of the Arctic sediments fitted by the regression line of sediment on logarithm of depth 70 Chapter 3 From theory to practice: some simple apllications 3.3 Compositional invariance: Economic aspects of household budget patterns In the literature of consumer demand analysis there have been only a few attempts to incorporate compositional analysis directly into the analysis of household budgets. This technique has many advantages and provides opportunities for new forms of investigation. Suppose the w is a record of household expenditure on D mutually exclusive and exhaustive commodity groups so that t = w1 + . . . + w D is total expenditure and x = C( w) is the proportional pattern of allocation to the groups. Logcontrast linear modelling with p( x| t ) of LD (α + β log t , Σ ) form has interesting consequences. First, the sometimes troublesome budget constraint or Engel aggregation (Brown and Deaton, 1972, 1163), that for each household total expenditure should equal the sum of all commodity expenditures, is automatically satisfied. Secondly, the hypothesis of compositional invariance β = 0 , that composition is independent of size, has a direct interpretation in terms of the income elasticities ε i = ∂ log wi / ∂ log t (i = 1, . . , D) of demand, if for the moment and for simplicity we identify household total expenditure with household income. In expectation terms β i = εi − ε D (i = 1, . . ., D − 1) so that compositional invariance, not surprisingly, corresponds to equality of all D income elasticities. Thirdly, whether or not there is compositional invariance, the modelling can clearly be extended to a full consumer demand analysis by the incorporation of commodity prices and other covariates such as household type and household composition into the mean parameter of the logistic normal distribution. Indeed, such an extension can be shown to be identical with the Houthakker (1960) indirect addilog model of consumer demand (Brown and Deaton, 1972, Equation 115), In the above discussion we have identified household total expenditure t with household income s. This is not an essential feature of the modelling since we could approach it through the conditioning p( s, t , x ) = p ( s) p( t | s) p( x| s, t ) with perhaps a 71 Chapter 3 From theory to practice: some simple apllications reasonable assumption that, for given total expenditure t, the pattern x is independent of income s, leading to the above concentration on p( x| t ) . As a simple start to our analysis of the household budgets of Table 1.1.3 let us first apply tests of compositional invariance separately to the 20 single male households and to the 20 single female households. Estimation and testing follows standard unconstrained multivariate analysis, with the result that we reject the hypothesis of compositional invariance for both single male and single female households. Thus for each set there is strong evidence against the hypothesis of compositional invariance: in other words, the patterns of expenditures do appear to depend on total expenditure. From the relations hip above we see that although the D ‘income’ elasticities are not determined by the D – 1 regression coefficients they can at least be placed in order of magnitude. The commodity groups arranged in increasing order of elasticity, that is, in conventional economic terminology from necessity to increasing luxury groups are (for each gender): 1. Foodstuffs, including alcohol and tobacco 2. Housing, including fuel and light 3. Services, including transport and vehicles 4. Other goods, including clothing, footwear and durable goods The fact that the ordering is the same for males and females raises the question of whether the dependence of pattern on total expenditure is really different for males and females. This suggests that it might have been more fruitful to consider hypotheses expressed in terms of the parameters of the model y = α M + βM log t + error, for males; y = α F + βF log t + error, for females. Note that the separate compositional invariance hypotheses tested above are the hypotheses βM = 0 and β F = 0 at level 2. All the hypotheses of the lattice can be tested within the standard framework of multivariate linear modelling. We omit the 72 Chapter 3 From theory to practice: some simple apllications details here but show on the lattice the significance probabilities associated with each hypothesis, noting that we can move up the lattice by rejection until level 3 where we would fail to reject the hypothesis α = α F . Before we leave this example we point out that it would be straightforward to introduce some concomitant feature such as age on which pattern may depend and test hypotheses within the associated more general model. 3.4 Testing perturbation hypotheses: Change in cows’ milk The data of Table 1.1.4 are of a before- and after-nature. Each cow has had milk composition determined at the beginning and at the end of the trial and so we have essentially, in standard statistical analysis terms, paired comparisons. The major difference is that we require to use a measure of difference appropriate to compositional change and we have seen this to be perturbation. Thus for each cow we record the set of perturbations below. Control group: compositional change Ident_cow pr mf ch Ca Na K C1 0,1389 0,2278 0,1553 0,1699 0,1400 0,1680 C2 0,1377 0,1661 0,1540 0,2066 0,1549 0,1807 C3 0,1464 0,1525 0,1672 0,1976 0,1572 0,1792 C4 0,1950 0,1564 0,1562 0,1993 0,1163 0,1768 C5 0,1988 0,1423 0,1507 0,1869 0,1470 0,1742 C6 0,1498 0,1979 0,1858 0,1782 0,1237 0,1645 C7 0,1467 0,1552 0,1828 0,1778 0,1564 0,1812 C8 0,1109 0,2690 0,1466 0,2046 0,0996 0,1693 C9 0,1198 0,2005 0,1351 0,1984 0,1654 0,1807 C10 0,2164 0,1624 0,1687 0,1818 0,1111 0,1597 C11 0,1792 0,1585 0,1358 0,1907 0,1645 0,1713 C12 0,1650 0,1836 0,1401 0,1896 0,1477 0,1740 C13 0,1744 0,1999 0,1742 0,1608 0,1310 0,1597 C14 0,1319 0,1689 0,1338 0,2055 0,1713 0,1886 C15 0,1482 0,2426 0,1544 0,1524 0,1476 0,1549 C16 0,1857 0,1891 0,1810 0,1829 0,1042 0,1571 C17 0,1497 0,1552 0,1419 0,2027 0,1623 0,1883 C18 0,1518 0,1703 0,1412 0,1656 0,2064 0,1646 Ident_cow C19 pr 0,1582 mf 0,1437 ch 0,1682 73 Ca 0,1852 Na 0,1733 K 0,1713 Chapter 3 From theory to practice: some simple apllications C20 0,1683 0,1832 0,1618 0,1659 0,1643 0,1565 C21 0,1394 0,2128 0,1999 0,1619 0,1336 0,1523 C22 0,1687 0,1570 0,1399 0,1883 0,1632 0,1829 C23 0,1988 0,1436 0,1529 0,1809 0,1537 0,1700 C24 0,1870 0,1770 0,1561 0,1754 0,1424 0,1622 C25 0,1243 0,2008 0,1520 0,2043 0,1346 0,1840 C26 0,1686 0,2286 0,1465 0,1641 0,1373 0,1549 C27 0,1512 0,1692 0,1658 0,1865 0,1508 0,1766 C28 0,2033 0,2042 0,1676 0,1460 0,1344 0,1445 C29 0,1455 0,1817 0,1783 0,1519 0,1798 0,1628 C30 0,1451 0,2350 0,1886 0,1696 0,1195 0,1422 Treatment group: compositional change Ident_cow pr mf ch Ca Na K T1 0,1753 0,1459 0,1552 0,2122 0,1642 0,1473 T2 0,2090 0,0937 0,1313 0,2341 0,1717 0,1603 T3 0,2387 0,1207 0,1497 0,1832 0,1652 0,1426 T4 0,2398 0,1345 0,1726 0,2310 0,0896 0,1326 T5 0,1173 0,1647 0,1535 0,2482 0,1577 0,1586 T6 0,1701 0,1063 0,1524 0,2508 0,1826 0,1379 T7 0,2018 0,1109 0,1166 0,2800 0,1471 0,1436 T8 0,2142 0,0944 0,1472 0,2488 0,1536 0,1418 T9 0,1890 0,1622 0,2066 0,2182 0,0965 0,1274 T10 0,2097 0,1431 0,1706 0,2082 0,1435 0,1249 T11 0,1562 0,1611 0,1901 0,2452 0,1126 0,1349 T12 0,1292 0,2046 0,1977 0,2104 0,1313 0,1268 T13 0,2538 0,1314 0,1499 0,1665 0,1655 0,1328 T14 0,1959 0,1289 0,1612 0,2324 0,1370 0,1448 T15 0,2154 0,1707 0,1713 0,2456 0,0875 0,1095 T16 0,1748 0,1715 0,1458 0,1959 0,1795 0,1326 T17 0,1446 0,1634 0,1757 0,2265 0,1375 0,1523 T18 0,1690 0,1918 0,1625 0,2510 0,0992 0,1264 T19 0,1791 0,1607 0,1792 0,1990 0,1221 0,1599 T20 0,2149 0,1210 0,1446 0,2589 0,1090 0,1516 T21 0,1799 0,1545 0,1605 0,2272 0,1408 0,1371 T22 0,1723 0,1566 0,1638 0,2460 0,1299 0,1314 T23 0,1778 0,1285 0,1905 0,2468 0,1161 0,1403 T24 0,2045 0,1670 0,1612 0,2124 0,1248 0,1301 T25 0,2063 0,1206 0,1428 0,2287 0,1461 0,1555 T26 0,2709 0,1018 0,1207 0,2491 0,1226 0,1349 T27 0,2099 0,1188 0,1450 0,2617 0,1047 0,1598 T28 0,2046 0,1370 0,1325 0,2779 0,1111 0,1369 T29 0,2808 0,1252 0,1390 0,1813 0,1328 0,1408 T30 0,1245 0,1871 0,1554 0,2084 0,1573 0,1672 74 Chapter 3 From theory to practice: some simple apllications We can address the problems that we face here in three stages by posing three questions. Question 1. Is there any evidence of seasonal change in milk composition. In other words is there any evidence of differences in the milk compositions of the control group between the beginning and end of the trial? Phrased as a compositional hypothesis this is simply a question of whether the centre of the control group perturbations is the identity perturbation. Transformed into logratio terms this is simply asking whether the mean of the alr vectors is a zero vector, a hypothesis easily tested under standard multivariate analysis. The Q-statistic value is 32.5, which when compared with the 95 percentile of χ 2 (15) , namely 25.0, shows significant departure from the identity perturbation. We thus conclude that there is some evidence of a seasonal change which justifies the insistence of having a control group. The centre of the control group perturbations is [pr mf ch Ca Na K]control = [0.1595 0.1835 0.1599 0.1818 0.1458 0.1695]. Question 2. Is there similar evidence of a change in the treatment group? Here the Qstatistic value is even larger, 75.6, again to be compared against the same percentile value, and so we ha ve real evidence of change, with the centre of the treatment group perturbations being [pr mf ch Ca Na K]treat = [0.1928 0.1416 0.1589 0.2309 0.1338 0.1420]. Question 3. The remaining question is to ask whether there are differences between the control and treatment group perturbations and this question can be answered by using a separate sample lattice identical to that for the hongite-kongite comparison of Section 3.1. The three Q-statistics in the same order as for the previous example are 153.7, 45.6 and 212.0 to be compared against 95 percentiles of the chi-squared distribution at 20, 15 and 5 degrees of freedom, all giving significant differences. Thus there is strong evidence of differences between control and treatment changes. 75 Chapter 3 From theory to practice: some simple apllications A good indication of what the nature of this change is can be obtained by computing the perturbation difference between the control and treatment perturbation centres, namely [pr mf ch Ca Na K]treat-control = [0.2015 0.1286 0.1656 0.2117 0.1529 0.1397]. Thus we can see that relatively there is enhancement of protein, carbohydrate and calcium, presumably a successful nutritional result. 3.5 Testing for distributional form 3.5.1 Introduction For compositional problems in which the analysis depends in an assumption of distributional form tests can be applied to assess the multivariate normality of the transformed logratio vectors. For this purpose there is a whole battery of such tests: univariate marginal tests, bivariate angle tests, multivaria te radius tests, with different forms of test statistics: Anderson-Darling, Cramer-von Mises, Watson, with accompanying useful graphical plots. All of these are examined in great detail in Aitchison (1986) and need not divert us here from the main task of presenting principles and practice of compositional data analysis. There has been discussion of whether the choice of divisor in the alr transformation is crucial to the result. So here we present a simple alternative avoiding this problem and which may also provide some measure of the degree of success of the normality assumption. 3.5.2 A useful characterisation of compositional distributional forms The form of the simplicial singular value decomposition suggests a useful way in which to characterise compositional distributions. Suppose that we express a generic D-part composition x in the form x = ξ ⊕ ( u1π 1 ⊗ β1 ) ⊕ . . . ⊕ ( u D−1π D −1βD− 1 ) 76 Chapter 3 From theory to practice: some simple apllications Now if we assume that u = [u1 . . . distribution N D-1 uD-1 ] follows a (D-1)-dimensional normal (0, ID-1 ), then it is simple to establish that x follows a logistic normal distribution with center ξ and centred logratio covariance matrix Γ expressible in terms of Π = diag (π ) and Β as Γ = clr (Β)Π clr ( Β) T , where clr denotes the operation of forming centred logratios from the rows of Β . Similarly a logisticStudent distribution is obtained with the same centre and covariance matrix and with ν degrees of freedom when u follows a St D−1 (ν ,0, I D −1 ) distribution. Further a logistic skew normal distribution for x is obtained if u follows a multivariate skew normal distribution with density function φ D−1 (u | 0, I D−1 ) Φ( uγ T ). We note at this point that this characterisation in terms of the u-distribution is useful for simulation purposes since it requires only simple simulation algorithms for standardised distributions. Simulated data is here useful for testing the effectiveness of the distributional form tests. 3.5.3 Testing procedures The remaining problem is how to exploit these characterisations for the purpose of testing distributional forms. We shall have available a compositional data set, a N × D matrix X whose rows x1 , . . . , x N are N D-part compositions and the first step in testing is to arrive at the appropriate N (D-1)-real vectors u on which to base the tests. This is easily done through the compositional singular value decomposition of X. The first step is to estimate the parameters in the above characterisation. This is easily done from the standard singular value decomposition of the doubly centred matrix Z constructed form the log X. Suppose that the standard singular value decomposition of Z is Z = UPV T , where U and V have zero-sum orthonormal columns and P = diag ( p1 , . . . , p D−1 ) , where p1 , . . . , p D −1 are the singular values in descending order of magnitude. Then it is easy to see that our estimates of the parameters ξ, Π , Β in the power-perturbation characterization are cen(X), P / N − 1 and clr −1 (V T ) . Also, and importantly for our distributional form testing here the vectors for the individual compositions are given as the rows of U N − 1. 77 u Chapter 3 From theory to practice: some simple apllications The aim is then to test these uni (i = 1, . . . , D; n = 1, . . . , N) for compositional form. Let us take the specific case of testing the compositional data set for additive logistic normality. The philosophy behind the testing procedure described below is that through the singular value decomposition we have a picture of the dimensionaly of the data set with known proportions of the total variability explained by increasing degrees of approximation. We would surely be reasonably happy if we were sure that for 99 percent of the variability we had satisfied ourselves of additive logistic normality. In this procedure the first column of U$ represents the first order approximations with a proportion p1 explained, the first two columns of U$ the second order approximation with a proportion p2 explained, and so on. Thus in terms of the marginal, bivariate and radius tests as described in Aitchison (1986, Section 7.3) the sequence of testing proceeds as follows. First order tests. Subject the first column of U$ to marginal tests. Second order tests. Subject the second column of U$ to marginal tests; columns 1, 2 to bivariate angle tests; and columns 1, 2 to radius test. Third order tests. Subject the third column of U$ to marginal tests; columns 1,3 and 2, 3 to bivariate angle tests; and columns 1, 2, 3 to radius tests. And so on until the desired degree of approximation is achieved. 3.6 Related types of data 3.6.1 Probability statement data Statisticians will readily recognize that all the above arguments relating to compositional data equally apply to probabilistic statements. It is clear that the standard practice of measuring probabilities on the scale of 0 to 1 is merely a convention and that any meaningful probabilistic statement can be expressed in terms of ratios, equivalently odds. 78 Chapter 3 From theory to practice: some simple apllications For example, subcompositional coherence is simply conditional probability coherence. A clinician may be faced with a differential diagnostic problem among five forms (1, 2, 3, 4, 5) of which 1, 2, 3, are malignant and 4, 5 benign. At a stage in the diagnostic process the clinician, having ruled out the benign forms 4 and 5, ma y wish to make a conditional probabilistic statement involving only the malignant states 1, 2, 3. The process of moving from the full probabilistic statement to the conditional probability statement is exactly analogous to the closure operation of forming a subcomposition from a full composition. Moreover, clearly there is also a principle of conditional coherence, analogous to the subcompositional coherence principle, that must apply here. In relation to probability statements the perturbation operation is a standard process. Bayesians perturb the prior probability assessment x on a finite number D of hypotheses by the likelihood p to obtain the posterior assessment X through the use of Bayes’s formula. Again, in genetic selection, the population composition x of genotypes of one generation is perturbed by differential survival probabilities represented by a perturbation p to obtain the composition X at the next generation, again by the perturbation probabilistic mechanism . 3.6.2 Granulometric data Granulo metric data obtained by sieving techniques are not histograms, as commonly defined, but are weight (or volume)_x_diameter profiles. Mathematically they are third moment distributions of the basic grain diameter distribution, a fact apparently first noted by Hatch (1933); see also Aitchison and Brown (1956) for further details and its relation to the Kolmogorov (1941) breakage model. Thus it could be argued that fitting a probability distribution to such an object is every bit as weird as considering the profile as a composition. Indeed to move from a weight_x_diameter profile to a diameter histogram is nothing more than a perturbation operation. For example if the weight_ x_diameter profile has H diameter intervals I1 , . . . , IH , with centers d1 , . . . , dH and with associated proportional weights p1 , . . . , pH, then on the assumption of uniform specific gravity, the diameter histogram q1 , . . . , qH is approximated by the perturbation [d1 -3 , . . . , dH -3 ] ⊕ [p1 , . . . , pH]. A consequence of the perturbation invariance property of the compositional metric is that the distance between profiles is the same as between histograms, a clearly desirable property. 79 Chapter 3 From theory to practice: some simple apllications Whether grain-size data is considered as grouped ordinal data and some class of univariate distributions is used to characterize each such ‘histogram’ or each histogram is considered a compositional vector is certainly an open question. In situations where the objective is to compare a number of weight_x_diameter profiles, until a satisfactory class of distributions giving good fits to the histogram emerges, the treatment of such data as compositional is certainly viable, with possibilities of inferring the nature of an underlying process through the study of possible differential perturbation processes. 80 Chapter 4 More complex compositional problems Chapter 4 Developing appropriate methodology for more complex compositional problems 4.1 Dimension reducing techniques: logcontrast principal components In unconstrained multivariate analysis principal component analysis is a popular means of investigating the dimension of the variability and of hopefully arriving at linear combinations of variables, which may have some interpretation within the particular discipline. In variation in R D principal components are the natural algebraic form as the inner product of the vector space, namely linear combinations of the components. As we have seen in Section 2.3.3 the inner product takes the form of a logcontrast of the components of the form: a1 log x1 + . . . a D log x D , where a1 + . . . + a D = 0 . The variance of such a logcontrast is aΓa T , where Γ is the centred logratio covariance matrix, and the successive principal logcontrasts are obtained from the eigenvectors (corresponding to the non- zero eigenvalues) of the estimate of Γ , and have the usual properties of orthogonality and with variances simply related to the eigenvalues. Applied to the hongite experience of Table 1.1.1a we have a1 ,..., a5 coefficients as rows of 1st 2nd 3rd 4th logcontrast logcontrast logcontrast logcontrast log A log B log C log D log E ----------------------------------------------0.1945 0.5876 -0.7840 0.0341 -0.0322 -0.0672 0.0867 -0.0112 -0.7069 0.6986 0.7899 -0.5598 -0.2295 -0.0707 0.0701 -0.3656 -0.3658 -0.3640 0.5423 0.5531 with eigenvalues 38.26, 2.186, 0.142, 0.004. The measure of total variability is 1.69 and the first principal logcontrast ‘explains’ 94.2 percent of this variability, the second bringing this to 99.6 percent. Thus we would be justified in regarding the variability 81 Chapter 4 More complex compositional problems of hongite as being largely two-dimensional. Inspection of the first two logcontrasts suggests that the first is involved largely in explaining variability within the (A, B, C) subcomposition and the second variability within the (D, E) subcomposition, and that these variations are orthogonal to each other. The writer can divulge that he had forgotten ho w he had simulated the hongite data set and that this analysis reminded him exactly of the details of the simulation. 4.2 Simplicial singular value decomposition For the record we give the staying- in-the-simplex version of logcontrast principal component analysis. This is by way of the simplicial singular value decomposition. It could reasonably be argued that the major statistical tool in the analysis of multivariate data associated with a metric vector space such as R D , R+D , S D must be the associated singular value decomposition. For a N × D compositional data matrix X with nth composition x n this, as we have already seen in Chapter 2, takes the form x n = ξ ⊕ ( un1π1 ⊗ β1 ) ⊕ . . . ⊕ (u nR π R ⊗ βR ) . It is interesting to apply this to a simple compositional data set such as hongite. The details of the process of estimation will be taken up in the next section. Here we simply record the results. ξ$ = [0.489 0.220 0.099 0.104 0.088], [πˆ 1...πˆ 4 ] = [6.185 1.478 0.377 0.066], ∧ β 1 ∧ β 2 ∧ = β 3 ∧ β 4 0.222 0.169 0.395 0.125 0.329 0198 . 0103 . 0125 . 0.189 0179 . 0.143 0.125 with the N × ( D − 1) set of u coefficients given by 82 0189 . 0.089 0167 . 0.310 0.177 0.364 0.192 , 0.314 Chapter 4 More complex compositional problems Specimen 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 u1 0,1529 0,0177 -0,2512 0,0493 0,2046 0,1314 0,1359 -0,3336 -0,1928 0,4027 -0,0298 0,2084 -0,2796 -0,1612 -0,0858 -0,2951 0,3560 0,1847 -0,1311 -0,1750 -0,0952 0,0614 0,2174 -0,1035 0,0114 u2 u3 0,2871 0,1155 0,0646 -0,0258 0,0752 -0,3607 -0,2561 -0,1047 0,0984 0,0494 0,0803 0,3751 0,2203 -0,3733 0,2479 -0,2955 -0,2078 -0,0418 0,2812 -0,1090 -0,0859 -0,0949 -0,1375 0,1131 0,0848 0,1325 -0,0512 0,0109 0,1437 -0,2816 0,2511 -0,4477 0,5093 0,0224 0,1957 0,1656 -0,0332 -0,1642 -0,2201 0,0340 -0,1939 -0,0156 0,1284 -0,1400 0,2042 -0,0731 0,0350 0,1050 -0,0370 -0,2802 u4 -0,2833 0,0403 0,0705 0,0351 -0,2748 -0,2053 -0,0110 0,0235 0,0122 0,4844 -0,1737 -0,1205 0,3033 0,0101 -0,1349 -0,0771 -0,1994 -0,0841 0,1719 -0,1500 0,1594 0,1608 0,4186 -0,2569 0,0808 The connection between this and the logcontrast principal components is simply that the eigenvalues correspond to the squares of the π ’s, and that the a-coefficients of the logcontrast approach are the clr transforms of the β ’s. The interpretation remains the same. 4.3 Compositional biplots and their interpretation The biplot (Gabriel, 1971, 1981) is a well established graphical aid in other branches of statistical analysis. Its adaptation for compositional data is simple and can prove a useful exploratory and expository tool. For a compositional data matrix X the biplot is based on a singular value decomposition of the doubly centered logratio matrix Z = [zri], where z ri = log{ x ri / g ( xr )} − N −1 N ∑ log{ x ri r =1 83 / g ( x r )}.. Chapter 4 More complex compositional problems Let Z = U diag(k 1 , . . . , kR) V t be the singular value decomposition, where m is the rank of Z, in practice usually m = D – 1, and where the singular values k 1`, . . . , km are in descending order of magnitude. The biplot (Figure 4.3.a) then converts the second order approximation to Z given by the singular value decomposition into a graphical display. Figure 4.3.a consists of an origin O which represents the centre of the compositional data set, a vertex at position (k 1 v i1 , k 2 v i2 )/(N – 1)1/2 for each of the parts, labelled 1, . . . , D, and a case marker at position at (N – 1)1/2 (ur1 , ur2 ) for each of the N cases, labelled c1 , . . . , cN . We term the join of O to a vertex i the ray Oi, and the join of two vertices i and j the link ij. These features constitute a biplot with the following main properties for the interpretation of the compositional variability. Fig. 4.3.a The basic elements of a compositional biplot Links, rays and covariance structure. The links and rays provide information on the covariance structure of the compositional data set. |ij| 2 ≈ var{log( xi / x j )} , | Oi | 2 ≈ var[log{ x i / g ( x)}] , cos( iOj ) = corr [log{ x i / g ( x)}, log{ x i / g ( x)}] . 84 Chapter 4 More complex compositional problems It is tempting to imagine that this last relation can be used to replace discredited corr(x i, x j) as a measure of the dependence between two components. Unfortunately this measure does not have subcompositional coherence. A more useful result is the following. If links ij and kl intersect in M then cos(iMk) ≈ corr {log( x i / x j ), log( x k / x l )} . A particular case of this is when the two links are at right angles so implying that cos(iMk ) ≈ 0 and there is zero correlation of the two logratios. This is useful in investigation of subcompositions for possible independence. Subcompositional analysis. The center O is the centroid (center of gravity) of the D vertices 1, . . ., D. Since ratios are preserved under formation of subcompositions it follows that the biplot for any subcomposition s is simply formed by selecting the vertices corresponding to the parts of the subcomposition and taking the center Os of the subcompositional biplot as the centroid of these vertices. Coincident vertices. If vertices i and j coincide or nearly so this means that var{log(x i/x j)} is zero or nearly so, so that the ratio x i /x j is constant or nearly so. Collinear vertices. If a subset of vertices, say 1, . . . , C is collinear then we know from our comment on subcompositional analysis that the associated subcomposition has a biplot that is one-dimensional, and then a technical argument leads us to the conclusion that the subcomposition has one-dimensional variability. Technically this one-dimensionality is described by the constancy of C–2 logcontrasts of the components x 1 , . . . , xC . Inspection of these constant logcontrasts may then give further insights into the nature of the compositional variability. Case markers and recovery of data. Such markers have the easily established property that Ocn . ji represents the departure of log(x i /x j) for case cn from the average of this logratio over all the cases. Let P and Pn in Figure 4.3.b denote the projections of the 85 Chapter 4 More complex compositional problems center O and the compositional marker cc on the possibly extended link ji . Then Ocn . ji = ± |PPn | | ji |, where the positive sign is taken if the directions of PPn and ji are the same, otherwise the negative sign is taken. A simple interpretation can be obtained as follows. Consider the extended line ji as divided into positive and negative parts by the point P, the positive part being in the direction of ji from P. If Pn falls on the positive (negative) side of this line then the logratio of log(x ni /x nj) of the nth composition exceeds (falls short of) the average value of this logratio over all cases and the further Pn is from P the greater is this excedance (shortfall); if Pn coincides with P then the compositional logratio coincides with the average. Fig. 4.3.b Interpretation of case markers in a compositional biplot A similar form of interpretation can be obtained from the fact that Ocn .Oi represents the departure of the centered logratio log{x ni /g(xn )} of the nth composition from the average of this centered logratio over all replicates. It must be clear from the above aspects of interpretation that the fundamental elements of a compositional biplot are the links, not the rays as in the case of variation 86 Chapter 4 More complex compositional problems diagrams for unconstrained multivariate data. The complete set of links, by specifying all the relative variances, determines the compositional covariance structure and provides direct information about subcompositional variability and independence. It is also obvious that interpretation of the relative variation diagram is concerned with its internal geometry and would, for example, be unaffected by any rotation or indeed mirror- imaging of the diagram. Another fundamental difference between the practice of biplots for unconstrained and compositional data is in the use of data scaling. For unconstrained data if there are substantial differences in the variances of the components, biplot approximation may concentrate its effort on capturing the nature of the variability of the most variable components and fail to provide any picture of the pattern of variability within the less variable components. Since such differences in variances may simply arise because of scales of measurement a common technique in suc h biplot applications is to apply some form of individual scaling to the components of the unconstrained vectors prior to application of the singular value decomposition. No such individual scaling is necessary for compositional data when the analysis involves logratio transformations. Indeed, since for any set of constants (c1 , . . . , cD), we have cov{log(cix i /cjx j),log(ckx k /clx l)} = corr{log(x i /x j),log(x k /x l)} it is obvious that the covariance structure and therefore the compositional biplot are unchanged by any differential scaling or perturbation of the compositions. This, of course, is simply an aspect of the perturbation invariance of measures of dispersion for compositional data. Only the centering process is affected by such differential scaling. Moreover any attempt at differential scaling of the logratios of the components would be equivalent to applying differential power transformations to the components of the compositions, a distortion which would prevent any compositional interpretatio n from the resulting diagram. 4.4 The Hardy-Weinberg law: an application of compositional biplots and logcontrast principal component analysis 87 Chapter 4 More complex compositional problems In the MN blood group system there are three genotypes, namely MN, MM, NN, and the proportions of these genotypes within a population provide a blood group composition for that population. Table 4.4.1 shows these compositions for 24 native populations; the data are reconstructed from Figure 12 of Gower (1987). Let us suppose that we know nothing about gene tic theory and decide to explore this data set by the construction of a relative variation biplot (Fig. 4.4.a) as described in Section 4.3. For such a 3-part compositional data set the biplot retains all the variability and provides an exact representation of the pattern of variability. The approximate collinearity of the vertices MN, MM, NN indicates that the variability is mainly onedimensional and suggests a logcontrast principal component analysis to determine the form of the constant logcontrast. Suc h a principal component analysis (Aitchison, 1986, Section 8.3) yields the following eigenvalues and logcontrasts λ1 = 2.74 , 0.0031 log MN - 0.7091 log MM + 0.706 log NN; λ2 = 0.079 , 0.816 log MN - 0.406 log MM - 0.411 log NN. Cumulative proportion explained: 0,97 1 5 18 MM 21 MN 23 13 87 16 12 11 9 6 22 14 1715 10 24 3 NN 4 2 1 20 19 = Outliers Fig. 4.4.a Biplot of the MN blood group data The near-constant logcontrast arises from the near-zero second eigenvalue. Moreover the fact that the coefficient 0.816 is approximately twice the coefficients 0.406 and 0.411 suggests that we can obtain a substantial simplification to our interpretation if 88 Chapter 4 More complex compositional problems we consider the constant logcontrast 2 log MN - log MM - log NN = constant We can obtain an estimate 1.348 of the constant from the average value of the logcontrast over the sample of 24 compositions. Moreover the fact that this is approximately log 4 encourages the following conjecture, 2 log MN - log MM - log NN = log 4, a relationship which can be written as 2 MN = 4MM × NN. Thus through examination of the relative variation biplot and its clear indication of the need for a logcontrast principal component analysis we have been led to the rediscovery of the fundamental Hardy-Weinberg equilibrium curve. With this set of 3-part compositions the one-dimensionality of the pattern of variability and the Hardy-Weinberg curve are obvious from the well-known representation of such compositional data sets in a triangular diagram as in Figure 4.4.b. Note that the three cases 1, 5 and 19, circled in the biplot of Figure 4.4.a and having atypicality indices 0.97, 0.97 and 0.99, are the cases which appear to depart most from the Hardy-Weinberg curve in Figure 4.4.b. Omission of any or all of these three cases does not materially affect the form of the Hardy-Weinberg curve. 89 Chapter 4 More complex compositional problems Cum. prop. expl.: MN 0,97 1 Pr. Components: 0,28 0,63 0,14 0,19 0,58 0,18 MM Fig. 4.4.b NN The ternary plot of the MN blood group compositions and the Hardy-Weinberg ‘curve’. We have deliberately used this simple example to demonstrate the effectiveness of logratio analysis and its associated relative variation biplot since there seems to remain some misunderstanding about the transformation involved. For example, Gower (1987, p. 38) mistakenly claims that the logratio transformation fails to cope with the curvature in the data. His confusion lies in not distinguishing between logarithmic and logratio transformations. He correctly points out that a logarithmic transformation, which considers logarithms of components, removes neither the constraint nor the curvature in the data. The logarithmic transformation is, however, not the relevant transformation for compositional data, which provide information only on the relative values or ratios of the components. For successful analysis a logratio transformation involving only ratios of components is required, and as we have seen above this is highly successful not only in taking account of the unit-sum constraint but in modelling the curvature of the Hardy-Weinberg curve. 4.5 A geological example: interpretation of the biplot of goilite Table 4.5.1 reports a compositional data set which will be new to everyone and so no preconceived ideas will dictate our analysis. It consists of 20 6-part mineral compositions of goilite rocks from a site on the edge of Loch Goil near Carrick Castle. I am told that this is an interesting site so let us see what we can discover 90 Chapter 4 More complex compositional problems about it. Inspection of the variation array of Table 4.5.2 provides little insight into the nature of variability of the goilite compositions of Table 4.5.1. In contrast, the relative variation biplot of Figure 4.5.a, retaining 98.2 per cent of the total compositional variability, allows easy identification of a number of characteristics. For simplicity in our interpretation we shall use only the initial letters to identify the mineral parts. First, we see that the de link is by far the longest indicating the greatest relative variation in the ratios of components is between d and e. Secondly, the near coincidence of the vertices a and c implies that the a and c are in almost constant proportion with the approximate relationship of a/c = 0.55 easily obtained from Table 4.5.1 or from the estimate -0.605 for E{log(a/c)} in the variation array of Table 4.5.2. Note that in the ternary diagram of the abc subcomposition in Figure 4.5.b the representative compositional points lie roughly on a ray through the vertex b. Applying the approximate 95 percent range formula and noting that [ga gb gc gd ge gf ] = [0.157 0.207 0.288 0.102 0.055 0.162] and coefficients of variation for log( e/f) and log(a/e) are -0.716 and -0.214 we obtain the ranges 0.073 < e / f < 1.59; 0.42 < a / c < 0.71. Cumulative proportion explained: 0,89 0,98 0,99 1 1 1 14 8 Broomite 22 Dhuite 13 20 Fyneite 17 10 2 3 7 Eckite 11 4 19 Carronite Arkaigite 12 16 15 5 18 = Outliers Fig. 4.5.a 6 Biplot of goilite mineral compositions 91 21 9 Chapter 4 More complex compositional problems Arkaigite Geometric Mean Broomite Fig. 4.5.b Carronite Ternary diagram for (arkaigite, broomite, carronite) subcompositions showing the near proportionality of arkaigite to carronite Thirdly and most strikingly we see the near-orthogonality of the ab (or cb) link and the links de, df and ef. We can immediately infer that the ratios d/e, d/f and e/f are independent of the ratio of a/b or c/b. Another way of expressing this feature is to state that the subcompositions [c,d,e] and [a,b] are independent. A formal test of this hypothesis of subcompositional independence (Aitchison, 1986, Section 10.3) results in a significance probability 0.27 confirming our conclusion. Fourthly, the collinearity of the three mineral links de, df and ef and the consequent one-dimensionality of the pattern of variability of this (d, e, f)-subcomposition, confirmed by the corresponding subcompositional ternary diagram of Figure 4.5.c, implies some relationship between the proportions of the minerals d, e and f. Direct investigation by logcontrast principal component analysis leads to the following eigenvalues and corresponding logcontrast principal components: λ1 = 12.79, 0.587 log f − 0.785 log e + 0.194 log f ; λ2 = −0.567,− 0.567 log d − 0.225 log e + 0.792 log f . 92 Chapter 4 More complex compositional problems Dhuite Geometric Mean Eckite Fig. 4.5.c Fyneite Ternary diagram of the (dhuite, eckite, fyneite) subcompositions showing the onedimensional pattern of variability of this subcomposition The near-constant logcontrast arises from the near-zero second eigenvalue. Moreover the fact that the coefficients are roughly in the ratios of -2 : -1 : 3 suggests that we can make a substantial simplification to our interpretation if we consider the constant logcontrast -3 log d - log e + 4 log f = constant = 2.46, where the constant value is estimated from the sample average of the logcontrast. This can be simply converted into the approximate relationship; 3 e / f = 0.85 × (f / d) Whether this suggested 'cubic hypothesis' is worth further investigation as a geological finding is a matter for geologists not an ingeolate statistician. As a final comment here we note that any subcomposition can be viewed as a set of logcontrasts (Aitchison, 1984) and so are included in any logcontrast principal component analysis for stud y of the dimensionalty of the pattern of compositional variability. 93 Chapter 4 More complex compositional problems 4.6 Abstract art: the biplot search for understanding Inspection of the variation array of Table 2 provided little insight into the nature of variability of the colour compositions of Table 1.1.5. In contrast, the relative variation biplot of Figure 4.6.a, retaining 98.2 per cent of the total compositional variability, allows easy identification of a number of characteristics. First, we see that the redyellow link is by far the longest indicating that the greatest relative variation in the pictures is between red and yellow. Secondly, the near coincidence of the vertices black and other implies that the artist uses black and the non-primary colour in almost constant proportion with the approximate relationship of other/black = 1.85 easily estimated from Table 1.1.5 or from the estimate 0.605 for E{log(other/black)} in the variation array of Table 2. Thirdly only two compositions, those of paintings 14 and 22, have atypicality indices 0.999 and 0.953 greater than 0.95. From the position of the marker for composition 14 in Figure 4.6.a it is clear that this atypicality is probably due to a combination of its unusually high ratios of yellow to blue with that of white to black, facts easily confirmed from Table 1.1.5. Composition 22 is atypical because of its high blue to yellow, white to black and white to other ratios. Cumulative proportion explained: 0,9 0,98 1 1 1 14 8 1 22 20 13 white 6 3 7 2 blue red yellow 4 other 19 black 12 16 17 10 15 5 11 21 9 18 = Outliers Fig. 4.6.a Biplot of the 22 colour compositions of an abstract artist Fourthly and most strikingly we see the near-orthogonality of the black-white link (or other-white link) and the links blue-red, blue-yellow, red-yellow associated with the 94 Chapter 4 More complex compositional problems primary colours. Thus, we can immediately infer that the ratios in which the artist uses the primary colours are independent of the ratio of black to white, or other to white. Another way of expressing this feature is that the subcompositions (blue, red, yellow) and (black, white) are independent; a formal test of this hypothesis of subcompositional independence (Aitchison, 1986, Section 10.3) results in a significance probability 0.27 confirming our conclusion. Fifthly, the collinearity of the three primary colour links and the consequent one-dimensionality of the pattern of variability of the primary subcomposition, confirmed by the corresponding subcompositional triangular diagram of Figure 4.6.b implies some relationship between the proportions of the primary colours used. Investigation along the lines of the previous example leads to an approximate relationship 3 log(blue) + log(red) - 4 log(yellow) = constant, 3 or, in terms of ratios of colour use, red/yellow ∝ (yellow/blue) . Whether this suggested 'cubic rule' is worth further investigation as an artistic principle is questionable, but such relationships can play an important role in compositional analysis (Aitchison, 1998). blue red Fig. 4.6.b yellow Ternary diagram of (blue,red,yellow)-subcompositions of an abstract artist 95 Chapter 4 More complex compositional problems 4.7 Tektite mineral and major oxide compositions As a further example to illustrate compositional biplot technique and to provide some unusual features which require care in interpretation we consider a data set for 21 tektites (Chao, 1963; Miesch et al, 1966), set out in Table 4.7.1, for which the two compositions are 8-part major-oxide compositions and 8-part mineral compositions. These are subcompositions of the original data set, this reduction being adopted for the sake of simpler exposition. While experimentally these two types of compositions are determined by completely different processes they are obviously chemically related since the minerals are themselves more complicated major oxide compounds. The challenge of the conditional biplot of Figure 4.7, with mineral composition as the response and major-oxide composition as the covariate, is whether it can at least identify these relationships from the compositional data alone, without any additional information about the chemical formulae of the minerals, and hopefully provide other meaningful interpretations of the data. Fig. 4.7 Conditional biplot showing the dependence of the mineral composition on the major oxide compositions for tektite compositions A striking feature of the diagram is that it is indeed successful in identifying which oxides are associated with which minerals. From Table 4.7.2, which provides the 96 Chapter 4 More complex compositional problems chemical association between minerals and major oxides, we see that, apart from SiO 2 , each of the other seven major oxides is associated with only one of the minerals, for example MgO is contained only in enstatite. In the biplot diagram each of these seven major oxide vertices is close to its corresponding mineral vertex. This means that the link associated with any two of these major oxides is nearly parallel to the link of the corresponding minerals and so the mineral logratios are all highly correlated with the corresponding major oxide logratios. It is in this sense that the conditional biplot identifies the chemical relationships. Moreover even SiO 2 , which is a constituent of all eight minerals is nevertheless primarily identified with quartz which is simply its oxide self. Table 4.7.2 Oxides and associated minerals in tektite study _________________________________________________________________ Oxide Mineral Abbreviation Formula _________________________________________________________________ SiO 2 Quartz qu SiO 2 K2O Orthoclase or KAlSi3 O8 Na2 O Albite al NaAlSi3 O8 CaO Anorthite an CaAl2 Si2 O8 MgO Enstatite en MgSiO 3 Fe2 O3 Magnetite ma Fe3 O4 TiO Ilmenite il FeTiO 3 P2 O5 Apatite ap Ca5 (F,Cl)(PO4 )3 __________________________________________________________________ All of this seems splendid until the quality of the approximation is investigated. The proportion of the covariance matrix G which is retained by the biplot is only 0.204. The reason is not too difficult to detect. The singular value decomposition has singular values 1.00, 1.00, 1.00. 0.999, 0.994. 0.868, 0,060 and it would require a fourth order approximation and a four-dimensional biplot to raise the quality to a reasonable 0.911 proportion retained. The reason for this disappointing quality is easily determined. It lies in the fact that within the constraints of compositional data each mineral is almost independently related to its major oxide, in the sense that each mineral logratio is almost perfectly linearly related to the corresponding major-oxide ratio. An analogous situation with unconstrained data would be the assemblage of 97 Chapter 4 More complex compositional problems independent univariate regressions, each with a different response and different covariate, into a multivariate regression. The apparent success of the conditional biplot lies more in the strength of the individual logratio regressions than in the quality of the biplot. It is important here to distinguish between the quality of the biplot and the reliability of the logratio regression of mineral on major oxide composition. The proportion of the mineral variability explained by the regression can be shown to be 0.983. 4.8 Subcompositional analysis A common problem in compositional data analysis appears to be marginal analysis in the sense of locating subcompositions of greatest or of least variability. For this purpose the measure of total variation provides for any subcomposition s of a full compositions x the estimate of the ratio trace{Γ ( s )} / trace{Γ ( x )} as the proportion of the total variation explained by the subcomposition. In such forms of analysis it should be noted that a (1, . . . , C–1)-subcomposition is a set of C–1 particular logcontrasts and so the variability explained by a C-part subcomposition can also be compared with that achieved by the first C–1 principal logcontrasts. We can illustrate this simply for the hongite experience of Table 1.1.1a. For example for 3-part subcompositions we have the 10 possible subcompositions in ascending order of variability (where 1=A. . . . , 5=E): 98 Chapter 4 More complex compositional problems Subcompositions A A A B C A A B B A D B B D D C C C C B E D E E E E D E D C Proportion of variability explained 0.08 0.17 0.20 0.27 0.44 0.51 0.53 0.90 0.91 0.94 We may note here that the (A,B,C)–subcomposition is the most variable, in concurrence with our interpretation of the first logcontrast principal component of Section 4.1. We may also note that this proportion 0.94 is comparable to that obtained by the first principal logcontrast component. 4.9 Compositions in an explanatory role Another interesting form of subcompositional analysis is where the composition plays the role of regressor, for example in categorical regression, where we wish to examine the extent to which, for example, type of rock depends on full major oxide composition or some subcomposition. For binary regression a sensible approach is to set the conditional model of type t, say 0 and 1, for given composition x as follows: pr (t = 1| x ) = 1 − pr (t = 0 | x) = F (α 0 + ∑i=1α i log x i ) , where D ∑ D i =1 αi = 0 . Hypotheses that the categorization depends only on a subcomposition, for example on the subcomposition formed from parts 1, . . . , C is then simply specified by α C + 1 = . . . = α D = 0 , and so the whole lattice of subcompositional hypotheses can be readily and systematically investigated. A striking example of the use of this technique is to be found in discriminating 99 Chapter 4 More complex compositional problems between two types of limestone. Thomas and Aitchison (1998) show that of the 17part (major-oxide, trace element) composition a simple major-oxide subcomposition (CaO,Fe2 O3 ,MgO) provides excellent discrimination, equal to that of the full composition. Figures 4.9.a and 4.9.b show the separation in logratio and ternary diagram space, respectively. Dufftown Inchory 10log(Fe2O3/MgO) -1 -2 -3 1' Fig. 4.9.a 2' 3' 4' log(CaO/MgO) 5' Scattergram of logratios log(CaO / MgO) and log(Fe2O3/ MgO) for Scottish limestones CaO Fe2O3 Fig. 4.9.b Dufftown Inchory MgO Ternary diagram of ‘centre perturbed’ (CaO, Fe2O3, MgO) subcompositions of Scottish limestones 100 Chapter 4 More complex compositional problems 4.10 Experiments with mixtures Another range of problems where compositional data play a role as comcomitants is in experiments with mixtures. Here the usual aim is to determine whether and in way a quantitative response depends on the composition of a mixture of ingredients. A simple and typical example is where the experiment is aimed at determining how the microhardness (kg/mm2 ) of glass depends on the relative proportions of Ge, Sb, Se used in its manufacture. Such problems are quite common in many disciplines. There is no reason why the response should be univariate. Aitchison and Bacon-Shone (1984) give an example of an investigation into how the brilliance and vorticity of girandole fireworks may depend on a 5-part mixture of light producing, propellent and binding components. Indeed the response may be a composition. The simplest model for such investigations is clearly when the expected response is a logcontrast of the ingredients and it is clear from the discussion of the previous section how investigation of subcompositional hypotheses would proceed. It is, however, possible to have a more general model involving second power terms in logratios, together with a hierarchy of hypotheses of inactivity of parts, of partition additivity , completely additive. For full details on the motivation for such definitions, for the practical meaning of the hypotheses and for implementation of a testing lattice, see Aitchison and Bacon-Shone (1984) and Aitchison (1986, Sections 12.4-5). 4.11 Forms of independence Because of the constant sum constraint, equivalently because of the nature of the simplex sample space, independence hypotheses must clearly take radically different forms from those associated with R D . For example, the analogue of complete independence of components in unconstrained space is for compositional data complete subcompositional independence, in which any subset of non-overlapping subcompositions is independent. These, of course can be specified in terms of associated logratios and in fact result in a particular parameterisation of the 101 Chapter 4 More complex compositional problems covariance structure. Tests of such hypotheses are readily available; see, for example, Aitchison (1986, Chapter 10). We use the time budgets of Table 1.1.6 to provide a very simple example, and examine the hypothesis that the work and leisure subcompositions are independent. This is almost clear in the biplot of Figure 4.11, in which the links of the working parts are roughly at right angles to the links of the leisure parts, indicating lack of correlation. The formal test involves testing whether the correlations between work logrations and leisure logratios are all zero. This is easily assessed and results in a significance probability of 0.56, so that we cannot reject the hypothesis of independence of work and leisure parts of the statistician’s day. Cumulative proportion explained: 0,42 0,65 0,82 0,95 1 18 1 Administration Consultation 14 5 Other 13 19 11 15 8 12 9 3 20 Sleep 10 7 4 16 Research 6 2 Teaching 17 = Outliers Fig. 4.11 Biplot of the time budgets of the statistician’s day 102 Chapter 5 Compositional processes Chapter 5 Compositional processes: a statistical search for understanding 5.1 Introduction Most scientists are interested in the nature of the process which has led to the data they observe, not least geologists in their search for explanations of how our planet has developed geologically. Unfortunately they are seldom in the fortunate position of observing a closed system where fundamental principles such as conservation of mass and energy apply. Commonly the only data available take the form of compositional data providing information only on relative magnitudes of the constituents of the specimens. In some disciplines there is a wide variety of terminology associated with such realised or hypothetical compositional processes. For example, geological langua ge contains many terms to describe a whole variety of envisaged geochemical processes, such as denudation, diagenesis, erosion, gravity transport, metasomatism, metamorphism, orogenesis, polymetamorphism, sedimentation, transportation, weathering. Often the data for the study of such possible processes consist of variable compositional vectors, such as major oxide compositions, major and minor element compositions, granulometric weight by diameter profiles such as (sand, silt, clay) sediments or palaeontological compositions such as foraminifera abundances. It is our purpose here to study a variety of ways in which statistical analysis of the variability in such data sets may be directed towards quantification of such processes and also, where there may be rival hypotheses as to the nature of the geological process, the extent to which the nature of the variability may be used to distinguish between the hypotheses. 5.2 Differential perturbation processes Many of the terms used by geologists to describe the processes they study appear to envisage some kind of differential change in the components of the composition – denudation, erosion, sedimentation, metamorphism, weathering. Since differential 103 Chapter 5 Compositional processes change in compositions is simply characterised by the simplex operation of perturbation this seems the sensible tool for the mathematical statistical study of such processes. The fundamentals for such a study were set out in Aitchison and Thomas (1998). Briefly the argument went as follows. Consider a process whic h results in an observable D-part composition x(t) = [x 1 (t) , . . . , x D(t)] which varies with some ordered variable such as time t. Since processes are commonly assumed to take place continuously over time we can attempt to describe such a process in a time-differential way by relating the composition x(t + dt) at time t + dt to the composition x(t) at previous time t in terms of a small perturbation. Since such an infinitesimal perturbation will be a slight departure from the identity perturbation (1/D, . . . , 1/D ) the process can be set out as x ( t + dt) = x ( t ) ⊕ (1 / D )[1 + δ1 ( t ) dt , . . . ,1 + δD ( t ) dt] Sometimes it is convenient to assume that such a perturbation is in the D-simplex but since the perturbation operation is invariant with respect to scale there is strictly no need for such a requirement. The original development then moved to a set of differential equation in logratios for which the solution is x (t ) − x( t 0 ) ⊕ [exp {∫ δ (u)du}(i = 1,..., D)], t t0 i where x ( t 0 ) is the known or assumed composition at time t 0 . With differentiation now defined on the simplex we note that an alternative expression of the process is in terms of the simple differential equation Dx (t ) = C[exp( δ i ( t ) :i = 1,..., D] with the known value at t 0 being the ‘boundary conditio n’. An interesting and important special case is where δi (t ) = γ i h(t ) , when the relationship takes the form of a simple compositional regression in a powerperturbation form as x (t) = x (t0 ) ⊕ H (t ) ⊗ β , 104 Chapter 5 Compositional processes where H (t ) = ∫ h( t ) dt and β = C[(exp( γ i ) :i = 1,..., D]. t t0 With actual compositional data the regression either in logratio terms or in staying in the simplex mode is easily accomplished. The important feature here is the possibility of alternative approaches to interpretation. 5.3 A simple example: Arctic lake sediments We continue the example used by Aitchison and Thomas (1998) to illustrate various ways of describing the process by which the variability of Arctic lake sediments may depend on depth. The previous study arrived at logratio regression equations log{sand(t) / clay(t)} = 9.70 -2.74 log t; log{silt(t) / clay(t)} = 4.80 - 1.10 log t, and differential perturbation relationship x ( t + dt ) = x ( t ) ⊕ 13 [1 − 146 . / t ), 1 + 0,18 / t , 1 + 1.28 / t ]. The stay-in-the-simplex versions of these are the compositional regression x ( t ) = [ 0.9928 0.0071 0.0001] ⊕ log t ⊗ [0.046 0.238 0.716] and the compositional differential equation Dx( t ) = (1 / t ) ⊗ [ 0.046 0.238 0.716] All provide the same interpretation of the process: as depth increases sand gives way to silt and more so to clay with differential effects decreasing with depth. Which characterisation of the process is chosen may well depend on personal choice. 105 Chapter 5 Compositional processes 5.4 Exploration for possible differential processes Given a compositional data set X = [ x1 , ;...; x N ] forming some possible process but with no obvious driving variable such as time, temperature or pressure, it is of interest to explore the possibility that there may be some unknown process at work. A suitable tool for such an investigation is the simplicial singular value decomposition (Aitchison et al, 2002). With each x n expressible in power-perturbation form: x n = ξ ⊕ ( un1 s1 ⊗ β1 ) ⊕ . . . ⊕ (u n, D−1 s D−1 ⊗ βD−1 ) where ξ is the centre of the data set. Here the hope is that the singular values are decreasing so rapidly that the variability will be described by a low order truncation of the power-perturbation combination. Suppose that for the Arctic lake sediments we were unaware of the possibility of depth as a process variable. Then the application of the singular value decomposition gives a representation x n = ξ ⊕ ( un1s1 ⊗ β1 ) ⊕ ( un2 s2 ⊗ β2 ), where s1 = 9,51, s2 = 1..85;ξ = [0.1780.5640.258]; β1 = [0.1360.3040.560],β 2 = [ 0.1960.6290.175]. The second order approximation explains 96.3 of the total variability. This should encourage the search for a possible driving variable. If depth is considered and plotted against the resulting u’s a log- like scatter of points is obtained confirming the nature of the earlier regression analysis. The expression of a process in terms of a power-perturbation combination is in simplicial terms a range space description. It should be realised that for any range space description say range(Β ) there is available a corresponding null space 106 Chapter 5 Compositional processes description null( Β⊥ ) . This is convenient if the objective is to produce some law- like description of the process. This was the situation in Aitchison and Thomas (1998) in the study of olivines. There, for example for kimberlitess, the range space approach would have resulted in [ Fe, Mg , Si ] = ξ ⊕ ( u ⊗ [0.128 0.461 0.411]) corresponding to the null space law- like description 0.065 log Fe + 0.67222 log Mg − 0.738 log Si = constant or equivalently in equilibrium form Fe Si 0. 089 Mg Si 0. 011 = 1.13. As a further illustrative analysis of the 25 5-part hongite compositions of Aitchison (1986) provides an interesting insight into the variability. The simplicial singular value decomposition gives the following results s = [ 6.185 1.478 0.377 0.066] ξ = [ 0.489 0.220 0.099 0104 . 0.088] β1 β2 β3 β3 = [ 0.222 = [ 0.170 = [ 0.395 = [ 0.125 0.329 0.198 0.103 0.125 0.084 0.189 0.177] 0.179 0.089 0.364] 0.143 0.167 0.192] 0.125 0.311 0.314] The consequence is that the second order power-perturbation approximation explains 94 percent of the total variability. Moreover the nature of β1 and β2 indicate that the first order approximation is associated with stability of the (4,5)-subcomposition with the second order complementing this with a subprocess involving the stability of the (1, 2, 3)-subcomposition. 107 Chapter 5 Compositional processes 5.5 Convex linear mixing processes Another popular way of studying compositional data is in terms of convex linear modelling processes. Such an approach is based on some such assumption as conservation of mass There is, of course, no way that compositional data can be used to support such a mass conservation hypothesis since compositions carry no information about mass. Compositions can, however, be analyzed within models which assume conservation of mass. All these models assume that there are source compositions, say ξ1 , . . . , ξC , from which a generic observed composition x arises as a convex linear combination x = π1ξ1 + . . . + π C ξC where π = [π1 , . . . , π C ] ∈ S C is the vector of mixing proportions. The form of modelling obviously depends on the extent of the information about the number of sources and the source compositions. At the ‘ignorance end’ neither the number of sources nor their compositions are known – the so-called endmember problem as presented, for example, in Renner (1993) and Weltje (197). At the opposite extreme the problem may be to test a hypothesis that the sources are specified ompositions ξ1 , . . . , ξC . Many intermediate situations can be visualised: an example is the pollution problem analysed by Aitchison and Bacon-Shone (1999), where there are not only samples from the target set but also sampled compositions from the source. The additive nature of such modelling does not mean that basic principles of compositional data analysis are thereby neglected. For example an approach to the so-called endmember problem where a set of say C endmember compositions ξ1 , . . . , ξC is sought such that each composition x n (n = 1, . . . , N) of the data set can be expressed as a convex linear combination Π n of ξ1 , . . . , ξC , uses as criterion of success the magnitude of N ∑ ∆ (x 2 n n =1 108 ,ξn ) Chapter 5 Compositional processes while monitoring the magnitude of ∑∆ 2 (ξ b , ξ c ). b< c All that are required to implement such a procedure are good algorithms for the minimisation of functions over a product of simplices. Our own, still at the testing stage, are based on an iterative search program where each step involves perturbations of the attained position, For the hongite data set for example on the supposition that there are three endme mbers these turn out to be ξ1 = [ 0.358 0.052 0.265 0.059 0.266] ξ2 = [ 0.507 0.351 0.004 0.085 0.053] . ξ3 = [ 0.374 0.055 0.392 0146 . 0.033] 5.6 Distinguishing between alternative hypotheses Each of the processes – differential perturbation and convex linear mixing – will result in fitted compositions, say x nP and x nC , for each of the observed compositions x n . The goodness of fit G P and G C of each of these processes may then be reasonably judged in terms some such measures as N N G = ∑ ∆ ( x n , xn ),G = ∑ ∆ ( x n , xn ). P 2 P C n =1 2 C n =1 In such a comparison, of course, we would be comparing processes of the same order of complexity. We do not attempt here to develop any formal statistical test for such a comparison. That would certainly involve many assumptions about the nature of the residual variability and possibly lead to more argument than any simple sensible comparison of the goodness of fit measures. For the hongite data set we can compare these goodness of fit measures at various orders of approximation: 109 Chapter 5 Compositional processes Order 2 3 4 5 Differential perturbation GP 2.332 0.146 0.004 0 Convex lineal mixing GC 3.731 1.851 0.402 0 It is fairly clear that for this data set the differential perturbation model has the edge over the convex linear model. This is in concurrence with the known method by which the data set was originally simulated. 110 Postlude Pockets of resistance and confusion Postlude Pockets of resistance and confusion There are a number of well-defined categories of response to the problems of compositional data analysis. I hope readers do not recognize their position in any of the categories. The wishful thinkers No problem exists (Gower,1987) or, at worst, it is some esoteric mathematical statistical curiosity which has not worried our predecessors and so should not worry us. Let us continue to calculate and interpret correlations of raw components. After all if we omit one of the parts the constant-sum constraint no longer applies. Someday, somehow, what we are doing will be shown by someone to have been correct all the time. The describers As long as we are just describing a compositional data set we can use any characteristics. In describing compositional data we can use arithmetic means, covariance matrices of raw components and indeed any linear methods such as principal components of the raw components. After all we are simply describing the data set in summary form, not analyzing it (Le Maitre, 1982). The openers The fact that most compositions are recorded by first arriving experimentally at an 'open vector' of quantities of the D parts constituting some whole and then forming a 'closed vector', the composition, seems to have led to a particular form of wishful thinking. All will be resolved if we can reopen the closed vector in some ideal way and then perform some statistical analysis on the open vectors to reveal the inner secrets of the compositions. The notion that there is some magic powder which can be sprinkled on closed data to make them open and unconstrained dies hard. Most recently Whitten (1995) takes as closed vectors major-oxide compositions of rocks 111 Postlude Pockets of resistance and confusion expressed as percentages by weight, scales by whole rock specific gravities to obtain 'open vectors' recorded in g/100cc. His argument depends on attempts to establish that whole rock specific gravity is independent of the composition of the rock (To someone with virtually no knowledge of geology a seemingly naive concept) by a series of regression studies in which whole rock specific gravities are regressed against at most two of the constituent major oxides. Percentages of explanation of over 50 per cent are cavalierly regarded as indications of independence. And why we may ask was not a regression on the complete set of major oxides considered. These would certainly have led to even higher percentages of explanation. Apart from this statistical criticism the consequent open vectors are peculiarly placed geometrically, being only minor displacements from a different constraining hyperplane. If only such openers would realize that in any opened composition the ratios of components are the same as in the closed composition so that any scale invariant procedure applied to the opened composition will be identical to that procedure applied to the closed composition. Opening compositions is indeed superfluous folly. The null correlationists Pearson was the originator of this school. The idea developed from a study of the composition (shape) of Plymouth shrimps; see Aitchison (1986, Chapter 3) for an account of his ingenious early bootstrap experiment. Others, in particular Chayes and Kruskal (1966) and Darroch and Ratcliff (1970, 1978) have attempted this approach. The basic idea here is related to the openers’ ideas. Because of the ‘negative bias’ in correlations of raw components of compositions, zero correlation obviously does not have its usual meaning in relation to independence. There must be some non- zero value of such a correlation, called the null correlation, which corresponds to ‘independence’. Usually the null correlation is surmised by some opening out procedure, as for example the oft-quoted Chayes-Kruskall method. The concept of null correlation is spurious and indeed unnecessary. All meaningful concepts of compositional dependence and independence can be studied within the simplex and in relation to the logratio covariance structures already specified. The pathologists A study of the compositional literature suggests that much of compositional data 112 Postlude Pockets of resistance and confusion analysis in the period 1965-85 was directed at trying to find some inspiration from calculation of crude correlations and other linear methods. Those who were aware that things go wrong with crude correlations attempted to describe the nature of the disease instead of trying to find a cure. Thus we have many papers with titles such as ‘An effect of closure on the structure of principal component’ (Chayes and Trochimczyk, 1978) and ‘The effect of closure on the measure of similarity between samples’ (Butler, 1979). The non-transformists Despite his warning about the spuriousness of correlations of crude proportions, Pearson would have been unhappy about the solution through logratio transformations. He had bitter arguments (Pearson, 1905, 1906) with some of the rediscoverers (for example, Kapteyn, 1903 ) of the lognormal distribution. This lay in his distrust of transformations: what can possibly be the meaning of the logarithm of weight? I had hoped that we were now sufficiently convinced, particularly in geology, that the lognormal distribution has a central role to play in many geological applications. But the mention of a logratio of components still brings forth that same resistance. What is the meaning of such a logratio is a question posed by Fisher in the discussion of Aitchison (1982) and even more recently by Whitten (1995). We hope that the analogy with the lognormal distribution and the comments earlier that every piece of compositional statistical analysis can be carried out within the simplex may mean that this resistance will soon collapse. The sphericists There have been various attempts to escape from the unit simplex to what are thought to be simpler or more familiar sample spaces. One popular idea (Atkinson and Stephens in the discussion of Aitchison (1982), and Stephens(1982)) is to move from the unit simplex S D to the positive orthant of the unit hypersphere by the transformation zi = ui (i = 1, . . . , D) and then to use established theory of distributions on the hypersphere. There are two insuperable difficulties about such a transformation. First, the transformation is only onto part of the hypersphere and so established distributional theory, associated as it is with the whole hypersphere, does 113 Postlude Pockets of resistance and confusion not apply. There is clearly no way round this since the simplex and hypersphere are topologically different: there is no way of transforming a triangle to the surface of a two-dimensional sphere. As serious a difficulty is the impossibility of representing the fundamental operation of perturbation on the simplex as something tractable on the hypersphere. This is not surprising since the fundamental algebraic operation on the hypersphere is rotation and this bears no relationship to the structure of perturbation. The additional step of Stanley (1990) in transforming z to spherical polar coordinates further complicates such issues. Although the angles involved are scale invariant functions of the composition their relationship to the composition is bewilderingly complicated. Moreover there would be no subcompositional coherence since in terms of our previous discussion scientist B would be transforming onto a hypersphere of lower dimension with impossibly complicated relationships between the angles used by scientist A and B. The Dirichlet extenders Many statisticians are attempting to extend the Dirichlet class of distributions on the simplex in the hope that greater generality will bring greater realism than the simple Dirichlet class. Unfortunately I think they are likely to fail, since even the simple Dirichlet class with all its elegant mathematical properties does not have any exact perturbation properties. Conclusion The only sensible conclusion, it seems to me, is to reiterate my advice to my students. Recognize your sample space for what it is. Pay attention to its properties and follow through any logical necessities arising from these properties. The solution here to the apparent awkwardness of the sample space is not so difficult. 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Buccianti, Nardi, G. and Potenza, R., eds., Proceedings of IAMG'98 - The Fourth Annual Conference of the International Association for Mathematical Geology: De Frede Editore, Napoli, p. 555-8.. 119 120 Appendix Tables Appendix Tables Table 1.1.1a Compositions of 25 specimens of hongite ___________________________________________ Specimen Percentages by weight of minerals no A B C D E ___________________________________________ 1 2 3 4 5 48.8 48.2 37.0 50.9 44.2 31.7 23.8 9.1 23.8 38.3 3.8 9.0 34.2 7.2 2.9 6.4 9.2 9.5 10.1 7.7 9.3 9.8 10.2 8.0 6.9 6 7 8 9 10 52.3 44.6 34.6 41.2 42.6 26.2 33.0 5.2 11.7 46.6 4.2 4.6 42.9 26.7 0.7 12.5 12.2 9.6 9.6 5.6 4.8 5.6 7.7 10.8 4.5 11 12 13 14 15 49.9 45.2 32.7 41.4 46.2 19.5 37.3 8.5 12.9 17.5 11.4 2.7 38.9 23.4 15.8 9.5 5.5 8.0 15.8 8.3 9.7 9.3 11.9 6.5 12.2 16 17 18 19 20 32.3 43.2 49.5 42.3 44.6 7.3 44.3 32.3 15.8 11.5 40.9 1.0 3.1 20.4 23.8 12.9 7.8 8.7 8.3 11.6 6.6 3.7 6.3 13.2 8.5 21 45.8 16.6 16.8 12.0 8.8 22 49.9 25.0 6.8 10.9 7.4 23 48.6 34.0 2.5 9.4 5.5 24 45.5 16.6 17.6 9.6 10.7 25 45.9 24.9 9.7 9.8 9.7 ___________________________________________ 121 Appendix Tables Table 1.1.1b Compositions of 25 specimens of kongite _____________________________________________ Specimen Percentages by weight of minerals no A B C D E _____________________________________________ 1 2 3 4 5 33.5 47.6 52.7 44.5 42.3 6.1 14.9 23.9 24.2 47.6 41.3 16.1 6.0 10.7 0.6 7.1 14.8 8.7 11.9 4.1 12.0 6.6 8.7 8.7 5.4 6 7 8 9 10 51.8 47.9 51.2 19.3 46.1 33.2 21.5 23.6 2.3 23.4 1.9 10.7 6.2 65.8 10.4 7.0 9.5 13.3 5.8 11.5 6.1 10.4 5.7 6.8 8.6 11 12 13 14 15 30.6 49.7 49.4 38.4 41.6 6.7 28.1 24.3 9.5 19.0 43.0 5.1 7.6 30.6 17.3 6.3 8.0 8.5 14.8 13.8 13.4 9.1 10.2 6.7 8.3 16 17 18 19 20 42.3 45.7 45.5 52.1 46.2 43.3 23.9 20.3 17.9 14.3 1.6 10.3 13.6 10.7 18.5 5.9 11.6 10.9 7.9 12.2 6.9 8.5 9.7 11.4 8.8 21 47.2 30.9 4.6 6.3 11.0 22 45.4 33.3 4.0 11.9 5.4 23 48.6 23.4 8.7 10.7 8.6 24 31.2 4.5 47.0 10.2 7.1 25 44.3 15.0 19.4 10.5 10.8 ___________________________________________ 122 Appendix Table 1.1.2 Tables (sand, silt, clay) compositions (percentages by weight) and water depth (m) of 39 Arctic lake sediments _____________________________________________ Sediment Percentages by weight Water no sand silt clay depth _____________________________________________ 1 77.5 19.5 3.0 10.4 2 71.9 24.9 3.2 11.7 3 50.7 36.1 13.2 12.8 4 52.2 40.9 6.6 13.0 5 70.0 26.5 3.5 15.7 6 66.5 32.2 1.3 16.3 7 43.1 55.3 1.6 18.0 8 53.4 36.8 9.8 18.7 9 15.5 54.4 30.1 20.7 10 31.7 41.5 26.8 22.1 11 12 13 14 15 16 17 18 19 20 65.7 70.4 17.4 10.6 38.2 10.8 18.4 4.6 15.6 31.9 27.8 29.0 53.6 69.8 43.1 52.7 50.7 47.4 50.4 45.1 6.5 0.6 29.0 19.6 18.7 36.5 30.9 48.0 34.0 23.0 22.4 24.4 25.8 32.5 33.6 36.8 37.8 36.9 42.2 47.0 21 22 23 24 25 26 27 28 29 30 9.5 17.1 10.5 4.8 2.6 11.4 6.7 6.9 4.0 7.4 53.5 48.0 55.4 54.7 45.2 52.7 46.9 49.7 44.9 51.6 37.0 34.9 34.1 41.0 52.2 35.9 46.4 43.4 51.1 40.9 47.1 48.4 49.4 49.5 59.2 60.1 61.7 62.4 69.3 73.6 31 4.8 49.5 45.7 74.4 32 4.5 48.5 47.0 78.5 33 6.6 52.1 41.3 82.9 34 6.7 47.3 45.9 87.7 35 7.4 45.6 46.9 88.1 36 6.0 48.9 45.1 90.4 37 6.3 53.8 39.9 90.6 38 2.5 48.0 49.5 97.7 39 2.0 47.8 50.2 103.7 _____________________________________________ 123 Appendix Tables Table 1.1.3 Household expenditures (HK$) on four commodity groups of 20 single men (M) and 20 single women (W) _________________________________________________________________ Commodity group Commodity group ID 1 2 3 4 ID 1 2 3 4 no no _________________________________________________________________ M1 M2 M3 M4 M5 497 591 153 839 942 302 789 1308 668 892 842 287 1585 781 2476 291 365 584 395 1740 W1 W2 W3 W4 W5 820 184 921 488 721 114 74 66 80 83 183 6 1686 103 176 154 20 455 115 104 M6 755 M7 388 M8 617 M9 248 M10 1641 764 655 879 438 440 428 153 757 22 6471 438 233 719 65 2063 W6 W7 W8 W9 W10 614 801 396 864 845 55 56 59 65 64 441 357 61 1618 1935 193 214 80 352 414 M11 1180 M12 619 M13 253 M14 661 M15 1981 1243 768 684 99 422 15 739 71 869 1489 813 204 48 188 1032 W11 404 W12 781 W13 457 W14 1029 W15 1047 97 47 103 71 90 33 1906 136 244 653 47 452 108 189 298 M16 1746 746 2662 1594 W16 552 91 185 158 M17 1865 915 5184 1767 W17 718 104 583 304 M18 238 552 29 75 W18 495 114 65 74 M19 1199 1095 261 344 W19 382 77 230 147 M20 1524 964 1739 1410 W20 1090 59 313 177 _________________________________________________________________ 1 2 3 4 Housing, including fuel and light Foodstuffs, including alcohol and tobacco Other goods, including clothing, footwear and durable goods Services, including transport and vehicles 124 Appendix Tables Table 1.1.4 Dietary compositions of the milk of 60, thirty in the control group and 30 in the treatment group (pr = protein, mf = milk fat, ch = carbohydrate) Control group before ______________________________________________ pr mf ch Ca Na K 10-2 x 10-2 x 10-2 x ____________________________________________ 0.3098 0.2237 0.4410 0.0103 0.0025 0.0127 0.2679 0.3687 0.3377 0.0084 0.0030 0.0144 0.2583 0.3392 0.3747 0.0074 0.0047 0.0157 0.2450 0.2614 0.4617 0.0090 0.0090 0.0140 0.3715 0.1477 0.4514 0.0098 0.0032 0.0163 0.2451 0.2987 0.4263 0.0104 0.0032 0.0163 0.3797 0.2268 0.3660 0.0064 0.0080 0.0131 0.2286 0.2723 0.4709 0.0097 0.0026 0.0159 0.2381 0.2182 0.5199 0.0100 0.0016 0.0122 0.3731 0.1937 0.4051 0.0109 0.0020 0.0153 0.1988 0.4113 0.3632 0.0056 0.0080 0.0131 0.3178 0.1908 0.4678 0.0058 0.0067 0.0111 0.2446 0.2976 0.4272 0.0114 0.0018 0.0175 0.2680 0.2357 0.4731 0.0041 0.0085 0.0106 0.3448 0.2428 0.3840 0.0098 0.0040 0.0148 0.2107 0.4630 0.2955 0.0154 0.0016 0.0138 0.2767 0.1796 0.5177 0.0040 0.0089 0.0130 0.3286 0.2883 0.3584 0.0065 0.0038 0.0143 0.2168 0.3149 0.4421 0.0083 0.0043 0.0136 0.2325 0.2858 0.4544 0.0049 0.0066 0.0157 0.3140 0.1600 0.4967 0.0092 0.0053 0.0149 0.3007 0.2313 0.4451 0.0084 0.0016 0.0131 0.1966 0.3840 0.3933 0.0101 0.0031 0.0128 0.1207 0.5170 0.3328 0.0075 0.0042 0.0179 0.1728 0.4103 0.3892 0.0112 0.0015 0.0150 0.1655 0.5171 0.2841 0.0094 0.0066 0.0173 0.3257 0.1735 0.4761 0.0059 0.0044 0.0142 0.2177 0.3711 0.3788 0.0147 0.0021 0.0155 0.2628 0.3019 0.4022 0.0131 0.0035 0.0164 0.3754 0.1718 0.4256 0.0112 0.0009 0.0150 ______________________________________________ 125 Appendix Table 1.1.4 Tables (continued) Treatment group before ______________________________________________ pr mf ch Ca Na K 10-2 x 10-2 x 10-2 x ____________________________________________ 0.3270 0.1956 0.4500 0.0068 0.0083 0.0123 0.3758 0.1720 0.4267 0.0071 0.0057 0.0125 0.2473 0.3304 0.3924 0.0086 0.0059 0.0156 0.2624 0.2719 0.4344 0.0090 0.0054 0.0169 0.2811 0.2700 0.4226 0.0042 0.0108 0.0112 0.3456 0.2318 0.4003 0.0039 0.0069 0.0115 0.4216 0.1417 0.4138 0.0080 0.0024 0.0125 0.2465 0.3286 0.3980 0.0087 0.0046 0.0135 0.2468 0.3266 0.3945 0.0092 0.0052 0.0178 0.3486 0.1670 0.4575 0.0118 0.0015 0.0135 0.3217 0.2407 0.4055 0.0069 0.0126 0.0128 0.2165 0.3268 0.4260 0.0111 0.0035 0.0161 0.3296 0.2173 0.4197 0.0092 0.0110 0.0133 0.2324 0.3370 0.4026 0.0086 0.0022 0.0172 0.2252 0.3160 0.4245 0.0099 0.0072 0.0171 0.1756 0.4177 0.3797 0.0091 0.0037 0.0143 0.3169 0.2167 0.4373 0.0051 0.0116 0.0125 0.2226 0.3809 0.3668 0.0064 0.0088 0.0145 0.2820 0.2373 0.4514 0.0085 0.0040 0.0168 0.2180 0.3414 0.4138 0.0066 0.0042 0.0161 0.3460 0.2307 0.3926 0.0106 0.0046 0.0155 0.3065 0.2337 0.4336 0.0125 0.0014 0.0122 0.2522 0.2965 0.4227 0.0141 0.0016 0.0130 0.3312 0.1541 0.4896 0.0073 0.0048 0.0130 0.2800 0.2365 0.4562 0.0115 0.0015 0.0144 0.2704 0.2809 0.4256 0.0119 0.0009 0.0104 0.5041 0.0875 0.3808 0.0104 0.0027 0.0146 0.3187 0.2490 0.4041 0.0111 0.0037 0.0134 0.2396 0.3502 0.3793 0.0106 0.0033 0.0170 0.2424 0.2725 0.4592 0.0117 0.0015 0.0127 _______________________________________________ 126 Appendix Table 1.1.4 Tables (continued) Control group after ______________________________________________ pr mf ch Ca Na K 10-2 x 10-2 x 10-2 x ____________________________________________ 0.2582 0.3057 0.4107 0.0105 0.0021 0.0128 0.2381 0.3954 0.3356 0.0112 0.0030 0.0168 0.2405 0.3291 0.3985 0.0093 0.0047 0.0179 0.2877 0.2461 0.4342 0.0108 0.0063 0.0149 0.4395 0.1251 0.4049 0.0109 0.0028 0.0169 0.2040 0.3285 0.4400 0.0103 0.0022 0.0149 0.3427 0.2165 0.4115 0.0070 0.0077 0.0146 0.1469 0.4245 0.4000 0.0115 0.0015 0.0156 0.1941 0.2976 0.4779 0.0135 0.0018 0.0150 0.4360 0.1699 0.3690 0.0107 0.0012 0.0132 0.2302 0.4212 0.3186 0.0069 0.0085 0.0145 0.3338 0.2230 0.4174 0.0070 0.0063 0.0123 0.2351 0.3279 0.4102 0.0101 0.0013 0.0154 0.2475 0.2789 0.4435 0.0059 0.0102 0.0140 0.2942 0.3392 0.3415 0.0086 0.0034 0.0132 0.2112 0.4724 0.2886 0.0152 0.0009 0.0117 0.2809 0.1890 0.4981 0.0055 0.0098 0.0166 0.3244 0.3192 0.3291 0.0070 0.0051 0.0153 0.2164 0.2855 0.4692 0.0097 0.0047 0.0147 0.2310 0.3091 0.4341 0.0048 0.0064 0.0145 0.2411 0.1875 0.5468 0.0082 0.0039 0.0125 0.3304 0.2364 0.4056 0.0103 0.0017 0.0156 0.2461 0.3472 0.3786 0.0115 0.0030 0.0137 0.1321 0.5356 0.3041 0.0077 0.0035 0.0170 0.1276 0.4896 0.3516 0.0136 0.0012 0.0164 0.1447 0.6130 0.2158 0.0080 0.0047 0.0139 0.3044 0.1814 0.4878 0.0068 0.0041 0.0155 0.2352 0.4027 0.3373 0.0114 0.0015 0.0119 0.2248 0.3225 0.4217 0.0117 0.0037 0.0157 0.3039 0.2252 0.4477 0.0106 0.0006 0.0119 _______________________________________________ 127 Appendix Table 1.1.4 Tables (continued) Treatment group after ______________________________________________ pr mf ch Ca Na K 10-2 x 10-2 x 10-2 x ____________________________________________ 0.3575 0.1780 0.4357 0.0090 0.0085 0.0113 0.5056 0.1038 0.3607 0.0107 0.0063 0.0129 0.3635 0.2455 0.3616 0.0097 0.0060 0.0137 0.3510 0.2040 0.4182 0.0116 0.0027 0.0125 0.2246 0.3028 0.4419 0.0071 0.0116 0.0121 0.3966 0.1662 0.4115 0.0066 0.0085 0.0107 0.5544 0.1024 0.3145 0.0146 0.0023 0.0117 0.3587 0.2107 0.3980 0.0147 0.0048 0.0130 0.2509 0.2850 0.4385 0.0108 0.0027 0.0122 0.4076 0.1332 0.4351 0.0137 0.0012 0.0094 0.2939 0.2268 0.4510 0.0099 0.0083 0.0101 0.1521 0.3636 0.4580 0.0127 0.0025 0.0111 0.4641 0.1584 0.3491 0.0085 0.0101 0.0098 0.2870 0.2738 0.4091 0.0126 0.0019 0.0157 0.2693 0.2995 0.4037 0.0135 0.0035 0.0104 0.1894 0.4421 0.3416 0.0110 0.0041 0.0117 0.2816 0.2176 0.4722 0.0071 0.0098 0.0117 0.2154 0.4184 0.3414 0.0092 0.0050 0.0105 0.2896 0.2187 0.4638 0.0097 0.0028 0.0154 0.3070 0.2707 0.3921 0.0112 0.0030 0.0160 0.3749 0.2146 0.3794 0.0145 0.0039 0.0128 0.3195 0.2214 0.4297 0.0186 0.0011 0.0097 0.2654 0.2255 0.4766 0.0206 0.0011 0.0108 0.3843 0.1460 0.4478 0.0088 0.0034 0.0096 0.3690 0.1822 0.4162 0.0168 0.0014 0.0143 0.4646 0.1813 0.3257 0.0188 0.0007 0.0089 0.5987 0.0588 0.3123 0.0154 0.0016 0.0132 0.4122 0.2157 0.3385 0.0195 0.0026 0.0116 0.3991 0.2600 0.3126 0.0114 0.0026 0.0143 ______________________________________________ 128 Appendix Tables Table 1.1.5 Six-part colour compositions of 22 paintings by an abstract artist ________________________________________________________ Painting Proportions of total area assigned to colours No black white blue red yellow other ________________________________________________________ 1 2 3 4 5 0.125 0.143 0.147 0.164 0.197 0.243 0.224 0.231 0.209 0.151 0.153 0.111 0.058 0.120 0.132 0.031 0.051 0.129 0.047 0.033 0.181 0.159 0.133 0.178 0.188 0.266 0.313 0.303 0.282 0.299 6 7 8 9 10 0.157 0.153 0.115 0.178 0.164 0.256 0.232 0.249 0.167 0.183 0.072 0.101 0.176 0.048 0.158 0.116 0.062 0.025 0.143 0.027 0.153 0.170 0.176 0.118 0.186 0.246 0.282 0.259 0.347 0.281 11 12 13 14 15 0.175 0.168 0.155 0.126 0.199 0.211 0.192 0.251 0.273 0.170 0.070 0.120 0.091 0.045 0.080 0.104 0.044 0.085 0.156 0.076 0.157 0.171 0.161 0.131 0.158 0.283 0.305 0.257 0.269 0.318 16 17 18 19 20 0.163 0.136 0.184 0.169 0.146 0.196 0.185 0.152 0.207 0.240 0.107 0.162 0.110 0.111 0.141 0.054 0.020 0.039 0.057 0.038 0.144 0.193 0.165 0.156 0.184 0.335 0.304 0.350 0.300 0.250 21 0.200 0.172 0.059 0.120 0.136 0.313 22 0.135 0.225 0.217 0.019 0.187 0.217 _________________________________________________________ 129 Appendix Tables Table 1.1.6 Activity patterns of a statistician for 20 days ______________________________________________ Day Proportion of day in activity No te co ad re ot sl ______________________________________________ 1 0.144 0.091 0.179 0.107 0.263 0.217 2 0.162 0.079 0.107 0.132 0.265 0.254 3 0.153 0.101 0.131 0.138 0.209 0.267 4 0.177 0.087 0.140 0.132 0.155 0.310 5 0.158 0.110 0.139 0.116 0.258 0.219 6 0.165 0.079 0.113 0.113 0.275 0.255 7 0.159 0.084 0.117 0.094 0.225 0.321 8 0.161 0.105 0.123 0.110 0.267 0.234 9 0.163 0.126 0.105 0.106 0.227 0.273 10 0.169 0.102 0.104 0.104 0.235 0.286 11 0.149 0.113 0.123 0.115 0.256 0.244 12 0.118 0.100 0.145 0.096 0.192 0.349 13 0.106 0.112 0.135 0.104 0.205 0.338 14 0.163 0.142 0.109 0.115 0.260 0.211 15 0.151 0.122 0.126 0.121 0.235 0.245 16 0.163 0.101 0.126 0.142 0.232 0.237 17 0.176 0.084 0.094 0.098 0.213 0.335 18 0.104 0.093 0.148 0.090 0.269 0.295 19 0.111 0.111 0.118 0.086 0.216 0.358 20 0.105 0.090 0.135 0.117 0.168 0.385 ________________________________________________ Notes: te = teaching; co = consultation; ad = administration; re = research; ot = other wakeful activities; sl = sleep 130 Appendix Tables Table 1.1.7 Typical river and fishing location pollutant compositions Pollutant a River 1 River 2 River 3 Location A Location B Location C b c d 0.6541 0.1553 0.1129 0.0777 0.5420 0.3497 0.0349 0.0734 0.2450 0.2924 0.2450 0.2176 0.2503 0.0420 0.5571 0.1506 0.3334 0.1704 0.2026 0.2936 0.4332 0.1409 0.1352 0.2907 0.4014 0.1864 0.2619 0.1503 0.3820 0.1169 0.3480 0.1531 0.4033 0.2300 0.2168 0.1498 0.4706 0.2207 0.1594 0.1493 0.3140 0.1060 0.3896 0.1904 0.2460 0.2278 0.3488 0.1774 131 Appendix Tables Table 4.5.1 Six-part mineral compositions of 22 specimens of goilite a b c d e f ______________________________________________________________ 1 0.125 0.353 0.266 0.163 0.031 0.181 2 0.143 0.224 0.313 0.111 0.051 0.159 3 0.147 0.231 0.303 0.058 0.129 0.133 4 0.164 0.209 0.282 0.120 0.047 0.178 5 0.197 0.151 0.299 0.132 0.033 0.188 6 0.157 0.256 0.246 0.072 0.116 0.153 7 0.153 0.232 0.282 0.101 0.062 0.170 8 0.115 0.249 0.259 0.176 0.025 0.176 9 0.178 0.167 0.347 0.048 0.143 0.118 10 0.164 0.183 0.281 0.158 0.027 0.186 11 0.175 0.211 0.283 0.070 0.104 0.157 12 0.168 0.192 0.305 0.120 0.044 0.171 13 0.155 0.251 0.257 0.091 0.085 0.161 14 0.126 0.273 0.269 0.045 0.156 0.131 15 0.199 0.170 0.318 0.080 0.076 0.158 16 0.163 0.196 0.335 0.107 0.054 0.144 17 0.136 0.185 0.304 0.162 0.020 0.193 18 0.184 0.152 0.350 0.110 0.039 0.165 19 0.169 0.207 0.300 0.111 0.057 0.156 20 0.146 0.240 0.250 0.141 0.038 0.184 21 0.200 0.172 0.313 0.059 0.120 0.136 22 0.135 0.225 0.217 0.217 0.019 0.187 ______________________________________________________________ a: arkaigite d: dhuite b: broomite e: eckite c: carronite f: fyneite Table 4.5.2 Variation array for goilite compositional data set Column j a b c d e f _________________________________________ Row i a b c d e f 0 -0.275 -0.605 0.432 1.047 -0.027 0.307 0 -0.330 0.706 1.322 0.247 0.129 0.270 0 1.037 1.652 0.578 0.502 0.617 0.465 0.646 0.486 0.628 0 1.071 0.615 0 -0.459 -1.074 Estimates below the diagonal are of E(log(xj /xi ) and above the diagonal of 132 0.225 0.221 0.213 0.314 0.769 0 var{log( xi / x j )} Appendix Tables Table 4.7.1 Major-oxide and mineral compositions of 21 tektites Major oxide compositions Case SiO2 K2 O Na 2 O CaO MgO Fe2 O3 TiO P2 O5 _____________________________________________________ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 70.83 80.10 80.17 78.40 78.37 77.21 78.19 76.11 76.68 76.09 76.25 76.23 75.59 75.58 75.38 75.51 75.13 74.94 73.36 72.70 71.89 1.86 1.99 2.24 1.90 2.43 2.42 2.23 2.38 1.81 2.04 2.22 2.03 2.42 2.40 2.21 2.25 1.84 1.84 1.93 1.63 1.60 1.20 0.52 1.37 0.49 1.53 0.56 1.36 0.55 1.84 0.78 1.80 0.96 1.71 0.65 1.59 0.81 1.27 0.59 1.60 0.67 1.63 0.74 1.50 0.51 1.72 0.79 1.84 0.79 1.77 0.79 1.61 0.74 1.42 0.54 1.50 0.66 1.44 0.61 1.43 0.41 1.28 0.045 0.46 0.030 0.65 0.42 0.110 0.66 0.37 0.180 0.60 0.59 0.050 0.69 0.70 0.050 0.59 0.50 0.060 0.62 0.73 0.230 0.74 0.59 0.220 0.74 0.56 0.005 0.83 0.54 0.230 0.80 0.74 0.270 0.74 0.58 0.330 0.77 0.66 0.200 0.73 0.95 0.210 0.71 0.95 0.320 0.78 0.67 0.350 0.75 0.61 0.170 0.90 0.43 0.130 0.86 0.75 0.310 0.87 0.70 0.320 0.99 0.78 0.270 1.05 0.005 0.020 0.030 0.010 0.020 0.060 0.040 0.040 0.010 0.040 0.050 0.050 0.050 0.050 0.060 0.050 0.050 0.040 0.030 0.070 0.040 Minerals compositions Case qu or al an en ma il ap ______________________________________________________ 1 62.02 10.99 10.15 2.58 1.15 0.040 1.23 0.010 2 61.13 11.76 11.59 2.30 1.05 0.160 1.25 0.050 3 59.17 13.25 12.94 2.58 0.92 0.260 1.14 0.070 4 58.93 11.23 11.50 2.66 1.47 0.070 1.31 0.020 5 53.79 14.36 15.56 3.74 1.74 0.070 1.12 0.050 6 52.54 14.30 15.22 4.37 1.24 0.090 1.18 0.140 7 55.20 13.17 14.46 2.96 1.82 0.330 1.41 0.090 8 52.78 14.06 13.45 3.76 1.47 0.320 1.41 0.090 9 57.90 10.69 10.74 2.86 1.39 0.010 1.58 0.020 10 54.19 12.05 13.53 3.06 1.34 0.330 1.52 0.090 11 53.22 13.12 13.79 3.34 1.84 0.390 1.41 0.120 12 55.38 11.99 12.69 2.20 1.44 0.480 1.46 0.120 13 51.24 14.30 14.55 3.59 1.64 0.290 1.39 0.120 14 50.15 14.18 15.56 3.59 2.37 0.300 1.35 0.120 15 50.97 13.06 14.97 3.53 2.37 0.460 1.48 0.140 16 52.39 13.29 13.62 3.34 1.67 0.510 1.42 0.120 17 54.92 10.87 12.01 2.35 1.52 0.250 1.71 0.120 18 54.01 10.87 12.69 3.01 1.07 0.190 1.63 0.090 19 51.99 11.40 12.18 2.83 1.87 0.450 1.65 0.070 20 52.95 9.63 12.09 1.58 1.74 0.460 1.88 0.170 21 52.79 9.45 10.83 1.97 1.94 0.390 1.99 0.090 ______________________________________________________ qu: quartz en: enstatite or: orthoclase ma: magnetite al: albite il: ilmenite 133 an: anorthite ap: apatite Appendix Tables Table 4.7.2 Oxides and associated minerals in tektite study _________________________________________________________________ Oxide Mineral Abbreviation Formula _________________________________________________________________ SiO 2 Quartz qu SiO 2 K2O Orthoclase or KAlSi3 O8 Na2 O Albite al NaAlSi3 O8 CaO Anorthite an CaAl2 Si2 O8 MgO Enstatite en MgSiO 3 Fe2 O3 Magnetite ma Fe3 O4 TiO Ilmenite il FeTiO 3 P2 O5 Apatite ap Ca5 (F,Cl)(PO4 )3 __________________________________________________________________ 134
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