Estimating and modeling variograms of compositional data with occasional missing variables in R

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1 Estimating and modeling variograms of compositional data with occasional missing variables in R R. Tolosana-Delgado 1, K.G. van den Boogaart 2, V. Pawlowsky-Glahn 3 1 Maritime Engineering Laboratory (LIM), Technical University of Catalonia raimon.tolosana@upc.edu 2 Institute for Stochastics, Technical University Bergakademie, Freiberg, Germany boogaart@math.tu-freiberg.de 3 Dept. Computer Science and Applied Mathematics, University of Girona, Girona, Spain vera.pawlowsky@udg.edu Abstract. Many environmental campaigns typically include regionalized compositional data, showing the relative importance of a set of constituents of samples taken at several locations. Variables considered are not always the same everywhere in the data set, and often are not comparable in their absolute values. On the contrary, log-ratio transformed data are directly homogeneous in these cases, and we can analyse them as usual. Here we propose to characterize their spatial structure by studying the variograms of the set of all pairwise logratios: they are easy to compute, even when some values are missing or with data from different sources; they contain the same information as any set of direct and cross-variograms of a log-ratio transformed composition, and can thus be reexpressed into more classical ways for cokriging; and their model fitting is as easy to understand, visualize and check as in univariate variograms. 1 INTRODUCTION In very extensive geochemical campaigns, in environmental or mineral surveys, it is typical that samples are analysed in different labs, with different techniques and standards, and even sometimes where different subsets of components are observed. These give spatially-referenced compositional data sets with some irregularities: components might be missing in some places, and absolute percentage values might not be comparable among labs, particularly if some data vectors have been closed to sum up to 100%. [3] showed that this constant sum induces a spurious correlation, that spoils all classical statistical techniques, including variogram-based Geostatistics [5]. In the presence of such data gaps, a log-ratio transformation approach to Geostatistics [7] is necessary, as the log-ratio of two components does neither depend on the presence/absence of values on other variables, nor on whether the data were closed or not. However, a log-ratio involving at least a missing variable is not computable. Interpolation in this situation is quite similar to undersampled cokriging [9], where some coordinates of our observation vectors may be missing, and we look for both an interpolation of full vectors at unsampled locations and the completion of the missing variables at the sampled locations. The key issue in this case is the estimation and modelling of (cross-)variograms. Following [6], this contribution presents a way to estimate the covariographic structure of a regionalized compositional data set with irregularities, by using the concept of the variation matrix.

2 2 BASICS OF COMPOSITIONAL DATA ANALYSIS A compositional data set is a data set where each variable shows the relative importance of a part in a whole. The most typical compositional variables in environmental surveillance problems are chemical components, like major oxide and trace element composition of soils, heavy metal composition on trace species (e.g. moss), hydrogeochemical composition of (sub)surface waters, etc. [1] presented compositions as vectors of positive components summing up to a constant (most typically, 1 or 100%), and suggested to transform the data through a set of log-ratio transformations to get rid of the spurious correlation effects induced by the constant sum. It has been later argued [2] that a data set should be regarded as compositional (and log-ratio transformed) as soon as the questions to answer: a) are unrelated with the total sum of the variables, or b) they must be equally meaningful whichever units we use to express the variables (%, mg/l, molarity, etc.). Due to the mentioned spurious correlation effect, classical (geo)statistical concepts must be interpreted with extreme caution when used in compositional data sets. To replace the classical mean vector and covariance matrix, [1] advocates for the use of respectively: the closed geometric mean, ˆm(Z) = 100/(eŷ1 + + eŷd ) [eŷ1,...,eŷd ], where ŷ i = N n=1 log(z ni)/n, and the variation matrix ˆT(Z) = (ˆt ij ), ˆt ij = 1 N N n=1 [ log ( zni z nj ) 2 (ŷ i ŷ j )]. Here, Z = (x ni ) is a compositional data set with D components (i = 1,..., D) and N observations. Note that the variation matrix is the D D symmetric matrix of variances of all pairwise log-ratios. Alternatively, one can choose an isometric log-ratio transformation, ilr(z) = V t log(z) = Z (1) where the log is applied component-wise, and V contains a set of (D 1) orthonormal vectors v i R D orthogonal to 1 = [1,...1]. The result is a new data set without any constraint, which may be treated with any classical statistical technique; geometric results (means, regression intercepts and slopes, principal components, confidence ellipses, etc., symbolized by r ) can be back-transformed to obtain an easier-to-interpret composition ilr 1 (r ) = 100 exp(v r ) 1 t exp(v r ). (2) For instance, the closed geometric mean can be obtained as ˆm(Z) = ilr 1 (E[ilr(Z)]). The variation matrix and the covariance matrix of Z are linked through ˆΣ = Cov[Z ] = 1 2 Vt ˆT V. (3) Note that, due to the orthonormality of the columns of V, i.e. V V t = I, Eq. (3) ensures that ˆΣ and ˆT have the same eigenvectors. Thus, if ˆΣ is positive definite, then ˆT must be negative definite and vice versa. These properties also apply to the theoretical counterparts of the variation matrix and the covariance matrix, though not used here.

3 3 STRUCTURAL ANALYSIS FOR COMPOSITIONS For a regionalized composition Z( x), we can follow the idea of the variation matrix and work with Γ = (γ ij ) the matrix of direct variograms of all log-ratios of any two variables (i, j), estimated by ˆγ ij ( h) = 1 2N( h) n,m N( h) ( log z ni log z ) 2 mi (4) z nj z mj and called the intrinsic variation matrix. Surprising as it might be, [7] show that this matrix of D 2 direct variograms contains the same information as the array of all (symmetric) cross-covariances σ ij kl ( [ ( h) = Cov log (Z i ( x)/z k ( x)),log Z j ( x + h)/z l ( x + )] h) between any two possible pair-wise log-ratios, containing D 4 (cross-)covariance functions, or of any set Ψ = (ψ ij ) of auto- and cross-variograms ψ ij = Cov [ v t i log(z),vt j log(z)] of an ilr-transformed (Eq. 1) data set ilr(z( x)) = Z ( x). Note that Z ( x) is an unbounded regionalized variable, thus its intrinsic covariance structure Ψ must be conditionally negative definite. Then, because this variogram system and the intrinsic variation matrix are related through Ψ = 0.5V t Γ V, we deduce that Γ must be conditionally positive definite. In the modelling chapter, we may take Ψ( h) as a linear model of corregionalization, i.e. Ψ( h) = K k=1 C k (1 ρ k ( h)), a linear combination of some (chosen) positive definite correlograms ρ k ( h) with positive (semi-)definite C k covariance matrices (to estimate). Then, just by the linearity of (3), we may model the experimental intrinsic variation matrix by Γ( K h) = B k (1 ρ k ( h)) k=1 where B k = 0.5V t C k V are negative semi-definite matrices to estimate. Thus, there is no need to devise new routines to fit these models: we can just modify the routines checking definiteness to force the C k to be negative semi-definite. Moreover, (4) transparently admits missing values: wherever a log-ratio involves a missing, one has one sampled point less to estimate the variogram at the lags involving that location, i.e. the number of data pairs N ij ( h) will now depend on the two variable indices (i, j). The resulting variogram matrix may not be valid itself, but this is of limited importance, because the experimental variograms just guide the fitting of a valid model. 4 R PROGRAMMING For our purposes, an important limitation of existing geostatistical R packages is the lack of a truly vectorial approach to multivariate geostatistics: geor is mostly univariate (it does not even allow the computation of a cross-variogram); and gstat, a multivariate geostatistics package, must be given the variables (e.g., the several parts of the composition, or their ilr coefficients, or the set of all pairwise log-ratios) one by one. Additionally, none of the packages work transparently with missing values during variogram fitting. Therefore the proposed algorithm of computing all pairwise log-ratio variograms and

4 fitting an LMC-variogram model with negative semidefinite matrices was newly implemented in R within the compositions software package [10]. Since variances are always positive and their variablity is typically proportional to their mean value, the optimal fitting is done on a log scale, and weighted with the number of pairs in each distance class of the empirical variogram. For a given multivariate variogram model Γ(h; θ) = (γ ij (h : θ)) depending on a vector of parameters θ, the corresponding objective function (goodness-of-fit) is given by gof(θ) = p p h Bins i=1 j=1,j i ln (γ ij (h; θ)) ln (ˆγ ij (h)) 2 For reasonable starting values, it is possible to automatically minimize this function with the non-linear optimization procedure nlm of basic R [8]. 5 EXAMPLE To illustrate this variogram fitting proposal, we use a geochemical data set of 601 samples from river and stream sediments from the Grazer Paläozoikum (Styria, Austria) analysed for 34 compositional parts (9 major oxides, 25 trace elements), kindly provided by J. C. Davis. This region is mostly covered by shales, limestones and dolomites, with some crystalline basement outcrops and Tertiary clastic sediments [11]. In this contribution, we take 7 major oxides (K, Na, Ca, P, Fe, Mg, Mn), randomly removed 400 values (to simulate the loss), reclose the remaining to sum up to 100%, and compute and fit the log-variograms. We fitted a pure spherical model, Γ(h) = C 0 (1 δ 0h ) + C s sph (h/r) with positive definite matrices C 0 for the nugget and C s for the partial sill, and sph( ) a standard univariate spherical variogram with unit sill, with a range parameter r to fit. Figure 1 visualizes the achieved fit, for the original data set and the same data set with randomly created missing values. It also provides the corresponding commands needed to compute such intrinsic variation matrix with the package. A further example with the classical reference Jura data set [4] is provided in the package [10]. 6 CONCLUSIONS The array of all log-ratio variograms (the intrinsic variation matrix) allows to fit multivariate variogram models semiautomatically. The fit can be visualized in a matrix of variograms, where each panel can be interpreted separately, like a univariate variogram: it is not necessary to understand the particular aspects of cross-variogram fitting, when checking the fit. Since these variograms are strictly positive, it is also possible to use a relative-scale fitting procedure, weighting variogram values according to their (inverse) expected value. Thus, the fitting is focused on the more important small variogram values to short distances, instead of being dominated by the higher variogram values. Moreover, this method can be applied even in the presence of missing values. The complete procedure has been implemented in the compositions package for R.

5 K Na semi variogram Ca P Fe Mg Mn lag distance (km) Figure 1: Empirical intrinsic variation matrix (symbols) compared with the fitted theoretical model (lines), with a single spherical structure of fitted range 8.8km plus a nugget effect. Black (circles, thick line) is used for the original data set, red (cross, thin line) for the data set with artificial missing values. Substantial departures of the model from the empirical version are in both cases only visible beyond the range. The fit can be obtained by the following sequence of commands in R: > library(compositions)... loading data > empvar <- logratiovariogram(comp,x) > vgmodel <- CompLinModCoReg(~nugget()+sph(5),comp) > fitted <- vgmfit2lrv(empvar,vgmodel,iterlim = 1000)

6 ACKNOWDLEDGEMENT This research was funded by the spanish Ministry of Science and Innovation though a Juan de la Cierva subprogram, supported by the European Social Fund (ESF-FSE), and through the Research project MTM REFERENCES [1] J. Aitchison. The Statistical Analysis of Compositional Data. Chapman & Hall Ltd., London, [2] C. Barceló-Vidal. Fundamentación matemática del análisis de datos composicionales. Technical Report IMA RR, Departament d Informática i Matemática Aplicada, Universitat de Girona, Spain, [3] F. Chayes. On correlation between variables of constant sum. Journal of Geophysical Research, 65: , [4] P. Goovaerts. Geostatistics for Natural Resources Evaluation. Oxford University Press, New York, [5] V. Pawlowsky-Glahn. On spurious spatial covariance between variables of constant sum. Science de la Terre, Sér. Informatique, 21: , [6] V. Pawlowsky-Glahn and H. Burger. Spatial structure analysis of regionalized compositions. Mathematical Geology, 24: , [7] V. Pawlowsky-Glahn and R.A. Olea. Geostatistical Analysis of Compositional Data. Oxford University Press, [8] R Development Core Team. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, [9] R. Tolosana-Delgado, J.J. Egozcue, and V. Pawlowsky-Glahn. Cokriging of compositions: Log-ratios and unbiasedness. In J.M. Ortiz and X. Emery, editors, Geostatistics Chile 2008, pages Gecamin Ltd., Santiago de Chile, [10] K. G. van den Boogaart, R. Tolosana, and M. Bren. compositions: Compositional Data Analysis. R package version , [11] L. Weber and J.C. Davis. Multivariate statistical analysis of stream-sediment geochemistry in the grazer paläozoikum, austria. Mineralium Deposita, 25: , 1990.