Decomposing compositional data: minimum chi-squared reduced-rank approximations on the simplex

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1 Decomposin compositional data: minimum chi-squared reduced-rank approximations on the simplex Gert Jan Welte Department of Applied Earth Sciences Delft University of Technoloy PO Box 508 NL-600 GA Delft The Netherlands Introduction The loratio transformation (Aitchison, 1981, 1986 opened the way to statistically riorous analysis of compositional data. Most statistical problems in compositional data analysis can be formulated in terms of loratios, and solved accordinly. However, some problems are more easily described and solved in terms of raw compositional variables. Raw compositions with k components are restricted to a k-1 dimensional subspace of the real space, a so-called simplex, defined by the followin two constraints: k p i i1 1 and p i 0. Compositional variation within suites of rocks and rock-formin minerals is often believed to reflect linear mixin. The correspondin mathematical model, expressed in terms of raw compositional variables may be represented in matrix notation as: P MB E where P is a (n k matrix of observed compositions, believed to represent a set of n mixtures, M is a (n q matrix of mixin coefficients, and B a (q k matrix of end members. The number of end members in the mixture is defined as q. E is a (n k matrix of error terms. Note that M and B are also compositions, so that the rows of E are forced to sum to zero. In many cases of linear mixin, B is assumed to be known exactly. For instance, in a procedure known as normative partitionin, chemical compositions of rocks are expressed as mixtures of perfect model minerals, whose chemical compositions are predicted by theory. However, typical mixin problems in sedimentary eoloy are of an explicit nature, characterised by a near total lack of information about the end members. The nonlinear inverse mixin problem in sedimentary eoloy relies on the existence of an unobservable matrix of perfect mixtures, which is defined as an exact convex linear combination of matrices M and B: X q MB

2 where X q is a (n k matrix of perfect mixtures of exact rank q, implyin that the end members are linearly independent. It thus follows that q min( n, k and: P X q E In other words, X q is a reduced-rank approximation of the compositional data P. The main purpose of this contribution is to present a method for estimatin X q which permits statistically riorous testin of the oodness-of-fit. Current estimation methods The estimation methods developed in the earth sciences (Renner, 1993; Welte, 1997 combine factorisation of P by means of a sinular value decomposition with constrained least squares approximation to ensure that all elements of X q are nonneative. This numerical approach is computationally efficient, because there is as yet no need to estimate M and B (Welte, Instead, use can be made of the unidentifiability (nonuniqueness of model parameters to postulate the existence of a (q q transformation matrix T: 1 Xq MB AT TF AF The estimation problem is now reduced to findin X q AF, where F is a (q k matrix of the first q transposed eienvectors of the data matrix P scaled to unit sum, and A is a (n q matrix of coordinates onto the eienvectors. This problem must be solved for each row of P separately: Minimise T T F a p T Subect to a 1 1 Exact linear equality constraints (unit sum Subect to F T a T h Exact linear inequality constraints (nonneativity/positivity Where h is a k-vector of lower bounds on X values: h 0. It follows that the elements of Xq k are restricted to the interval h ; 1 h. 1 This method of reduced-rank approximation may be used to estimate the number of linearly independent end members q needed to model P independently of the actual end-member compositions B. The weakness of this approach is the lack of a formal oodness-of-fit test for X q. Althouh tests with synthetic data provided support for the use of certain empirical measures (Welte, 1995, 1997; Prins & Welte, 1999a, 1999b; Van der Ark, 1999 this problem was not fully solved. If the number of linearly independent end members has been estimated, a physically realistic mixin model of the data can be produced by solvin the followin set of equations: Minimise MT A Subect to 1T 1 Exact linear equality constraints (unit sum 1 Subect to T B F Exact linear equality constraints (unit sum: nonlinear in T Subect to B 0 Exact linear inequality constraints (nonneativity

3 Subect to M 0 Exact linear inequality constraints (nonneativity A detailed account of the procedure for estimatin M and B is beyond the scope of this contribution (for details see Welte, However, it is important to note that it may not be possible to approximate P as a product of two entirely nonneative matrices M and B. Hence, a set of q linearly independent end members may not exist, even if the fit of X q to P is ood. The implicit estimation of X q thus allows the detection of possible row deeneracy amon the true end members B. An estimation method developed for latent budet analysis (De Leeuw & Van der Heden, 1988, an end-member model used in the social sciences, is based on the EM alorithm for aumentation of incomplete data (Dempster et al., This method employs a maximum likelihood criterion to simultaneously estimate X q, M and B, based on the assumption that P follows a product-multinomial distribution. Accordin to this model, each composition (row of P is considered a random sample from an unknown multinomial distribution. Althouh this approach potentially allows a statistically riorous assessment of the discrepancies between X q and P, it is notoriously slow to convere, and it does not allow testin of the rank of X q without estimatin M and B. In the followin sections a new estimation method will be presented that combines the advantaes of the above approaches: Detection of row deeneracy amon end members by implicit estimation of X q ; Minimisation of a statistically riorous measure of discrepancy between X q and P. Distributional assumptions A statistically riorous measure of discrepancy between P and X q may be formulated by reardin compositions as enumerative data, i.e., counts of discrete units. This is strictly true for many types of eoloical data, such as point counts of mineral rains or map reions. However, other types of compositional data, with units of measurement not correspondin to discrete entities, may also be treated as enumerative. The "count lenth" (sample size N of compositional data can be estimated from the spread of a series of replicate analyses, because samples from a population with fixed composition follow a multinomial distribution if all variation is attributable to random error. Goodness-of-fit statistics for the multinomial distribution have been studied extensively by Cressie & Read (1984 and Read & Cressie (1988, who showed that all commonly used statistics can be rearded as members of a one-parameter family of power-diverence statistics. For lare values of N, as commonly encountered in eoloical data, the behaviour of all power-diverence statistics tends towards that of the well-known Pearson's chi-squared statistic (PCS. Hence, for all practical purposes, minimisation of PCS may be rearded as equivalent to maximum likelihood estimation. Estimation of sample size N from a series of m replicate compositions with k constituents proceeds as follows: m k ( p p m ˆ 1 PCS N, where pˆ p, the vector of arithmetic means. i1 1 pˆ m i1 The fit of the replicates to the model is ood by definition:

4 PCS E (Pr0.5, with (m-1(k-1 derees of freedom. By combinin the above expressions we obtain: (Pr0.5 N m k ( p pˆ i1 1 pˆ Strictly speakin, every matrix of compositional data follows a product-multinomial rather than a multinomial distribution, because of the constant row-sum constraint. However, the oodness-of-fit tests for both types of distributions are alebraically equivalent (Read & Cressie, 1988, provided that the number of derees of freedom is reduced to account for this constraint. Every compositional data matrix thus may be treated as a series of samples from one or more fixed populations. In theory, each composition represents a sample that could have been drawn from a different population, but this need not be the case. The main purpose of end-member modellin is to quantify the relations between these populations. For example, if sediments have been sampled across an entire sedimentary basin fed by multiple sediment sources, each sample can be thouht of as representin the local sediment population (Welte, 001. Each of these local populations in turn represents a mixture of a limited number of end-member populations, whose mixin proportions vary continuously in space. There are no riid assumptions with respect to the distribution of mixin proportions on the simplex spanned by the end members. The Minimum Chi-Squared Decomposition alorithm The basic equation for columnwise scalin of the matrix of compositional data is: 1 W PG where G is a diaonal (k k matrix of nonneative weihts. The obective function we seek to minimise is Pearson's chi-squared statistic (PCS: k n ( w wˆ OPCS N 1 i1 wˆ k n ( p x N x 1 i1 We prefer least-squares estimation to maximum likelihood methods, because the latter are not as computationally efficient (Van der Ark, The method of constrained weihted least squares (CWLS minimises the obective function: O WLS k 1 i1 k n 1 i1 n ( w wˆ ( p x subect to nonneativity and unit-sum constraints on the estimated composition vectors x. By equatin these two obective functions we obtain the followin expression for the minimum chi-squared (MCS scalin factor of the -th column: N n i1 n ( p i1 ( p x x x

5 The above expression shows that the reduced-rank estimate X q is required to calculate the MCS scalin factors, implyin that they must be estimated by an iterative process. An initial uess of the MCS scalin factors is based on the fact that the case q = 1 has a simple solution known from chi-squared testin theory (as outlined above for the analysis of replicates. The null hypothesis implies compositional homoeneity of P: H : q 1 0 H1 : q 1 The trivial mixin model with one end member that best fits the data is simply the vector of arithmetic column means. Substitutin the one end-member solution into the eneral expression ives: n p i1 Nn The obective functions are now identical and approximately distributed as χ with (n-1(k-1 derees of freedom. If the null hypothesis is not accepted, more advanced mixin models are needed to account for the variation in P, implyin that q > 1. For the purpose of estimatin X in case q > 1, we define the vector of MCS scalin factors as ~, the vector of diaonal elements of G normalised to unit sum. In other words, we treat ~ as a composition because the oodness-of-fit of the model is solely determined by the relative values of its elements: ~ k 1 The iterative estimation starts by scalin the data usin the current estimate of MCS scalin factors: p w ~ Then the eienvectors V (riht sinular vectors of W are extracted by a sinular value decomposition: T W USV A set of reference vectors F is calculated from the first q columns of V in three steps: First the column sums of vector coefficients are constrained to be nonneative: k 1 v m 0 v m v m 1,,..., k, m 1,,..., q Then a fully nonneative matrix is formed by the linear transformation T qq The mm are calculated from the coefficients of each column of V: VT, where:

6 I v m v 1 mm max v m Where I represents the indicator function: I v0 0 I v0 1 The final step combines scalin and transposition to form F: m fm k ~ 1 m The ~ reference vectors F now obey the followin constraints: F G 1 Linear equality (unit sum F 0 Linear inequality (nonneativity ~ An important result of the above calculations is that X q AFG with unknown coefficients A 1 all of the same order of manitude. It follows that X G ~ AF, implyin that the problem of reduced-rank approximation may be formulated in unweihted form as: Minimise T T F a w T Subect to a 1 1 Linear equality (unit sum Subect to F T a T G ~ 1 h Linear inequality (nonneativity/positivity This is a so-called LSIE problem (least-squares with inequality and equality constraints, which may be solved by applyin the followin transformations: Transformation of a LSIE to a LSI problem by removal of linear equality constraints (LeMaitre, 1979; Transformation of a LSI to a LDP (least-distance prorammin problem of eneral form: Minimise x subect to Ax b (Lawson & Hanson, 1974; Menke, 1989 Transformation of a LDP to a NNLS (nonneative least-squares problem of eneral form: Minimise Cy d subect to y 0 (Lawson & Hanson, 1974; Menke, 1989 The solution to the LSIE problem is obtained by application of the inverse transformations to the NNLS solution. This completes the reduced-rank estimation for the current ~. The vector of MCS scalin coefficients is estimated by Powell's Direction Set Method, a multidimensional minimisation method that does not require calculation of derivatives (Press et al., Multiple random starts may be used to investiate the behaviour of PCS as a function of the composition of ~. Limited experimentation has shown the presence of several local minima, which could be a matter of concern in applications to lare data sets. In the data analysed thus far, multiple minima have comparable PCS values, which merely suests that "valley floors" are not quite flat. The resultin minimum chi-squared estimate is approximately distributed as χ with (n-q(k-q derees of freedom, which allows for riorous oodness-of-fit testin of the reduced rank approximation. The probability of the model X under the null hypothesis as estimated from q

7 PCS is known to deviate from the theoretical χ in cases with low expected frequencies (Read & Cressie, 1988; Greenwood & Nikulin, 1996, which is not uncommon for eoloical data. In such cases an exact chi-squared test should be used (Romesbur et al., References Aitchison. J. (1981 A new approach to null correlations of proportions. Journal of Mathematical Geoloy, 13, Aitchison, J. (1986 The statistical analysis of compositional data. Chapman and Hall, London. Cressie, N. & Read, T.R.C. (1984 Multinomial oodness-of-fit tests. Journal of the Royal Statistical Society, Series B, 46, De Leeuw, J., and Van der Heden, P.G.M. (1988 The analysis of time budets with a latent time-budet model. In: E. Diday et al., Eds., Data analysis and informatics 5. North Holland Publishers, Amsterdam, p Dempster, A.P., Laird, N.M. & Rubin, D.B. (1977 Maximum Likelihood from incomplete data via the EM alorithm. Journal of the Royal Statistical Society, Series B, 39, Greenwood, P.E. & Nikulin, M.S. (1996 A uide to Chi-Squared testin. Wiley & Sons, New York. Lawson, C.L., and Hanson, R.J. (1974 Solvin least squares problems. Prentice-Hall, Enlewood Cliffs. LeMaitre, R.W. (1979 A new eneralised petroloical mixin model. Contributions to Mineraloy and Petroloy, 71, Menke, W. (1989 Geophysical data analysis: discrete inverse theory, revised edition. Academic Press, San Dieo. Press, W.H., Teukolsky, S.A., Vetterlin, W.T. & Flannery, B.P. (1994 Numerical recipes in Fortran: The art of scientific computin, second edition. University Press, Cambride. Prins, M.A. & Welte, G.J. (1999a End-member modelin of siliciclastic rain-size distributions: The late Quaternary record of eolian and fluvial sediment supply to the Arabian Sea and its paleoclimatic sinificance. In: Numerical Experiments in Stratiraphy: Recent Advances in Stratiraphic / Sedimentoloic Computer Simulations (Ed. by J.W. Harbauh, L. Watney, et al., SEPM Special Publication 6, p Prins, M.A. & Welte, G.J. (1999b End-member modellin of rain-size distributions of sediment mixtures. In: Prins, M.A. (1999 Pelaic, hemipelaic and turbidite deposition in the Arabian Sea durin the late Quaternary [PhD Thesis Utrecht University]. Geoloica Ultraiectina 168, p Read, T.R.C. & Cressie, N. (1988 Goodness-of-fit statistics for discrete multivariate data. Spriner Verla, New York. Renner, R.M. (1993 The resolution of a compositional data set into mixtures of fixed source compositions. Applied Statistics, Journal of the Royal Statistical Society, Series C, 4, Romesbur, H.C., Marshall, K. & Mauk, T.P. (1981 FITEST: a computer proram for "exact chi-square" oodness-of-fit sinificance tests. Computers and Geosciences, 7, Van der Ark, L.A. (1999 Contributions to latent budet analysis: a tool for the analysis of compositional data [Ph.D. Thesis Utrecht University], DSWO Press, Leiden. Welte, G.J. (1995 Unravellin mixed provenance of coastal sands: The Po delta and adacent beaches of the northern Adriatic sea as a test case. In: Geoloy of deltas (M.A. Oti & G. Postma, Eds., Balkema, Rotterdam, p

8 Welte, G.J. (1997 End-member modellin of compositional data: Numerical-statistical alorithms for solvin the explicit mixin problem. Journal of Mathematical Geoloy, 9, Welte, G.J. (001 Quantitative analysis of detrital modes: statistically riorous confidence reions in ternary diarams and their use in sedimentary petroloy. Earth-Science Reviews, in press.