Classification problems of samples of finite mixtures of compositions

A discriminant technique based on mixture models is presented to be applied when observations are a sample of a mixture of compositions with each component following an additive logistic normal distribution on the d-dimensional simplex. The efficiency of this discriminant technique is compared empirically with the efficiency of the standard discriminant technique based on logcontrast. Simulated compositional data and a real dataset are used to carry out these comparisons.     

A Parametric Approach for Dealing with Compositional Rounded Zeros

In this work, a parametric approach for replacing data below the detection limit, also known as rounded zeros, in compositional data sets is proposed. Compositional rounded zeros correspond to small proportions of some whole that cannot be reliably detected by the analytical instruments under given operating conditions. This kind of zeros appear frequently in the data collection process in geosciences. They must be treated in an adequate way before some multivariate analysis can be applied. Our procedure results from a modification of the Expectation-Maximization (EM) algorithm and is based on the additive log-ratio transformation. Its coherence with the nature of compositional data and with basic operations in the simplex sample space is checked. Using real data sets, we find that this approach improves other parametric and non-parametric techniques for compositional rounded zeros.     

BLU Estimators and Compositional Data

One of the principal objections to the logratio approach for the statistical analysis of compositional data has been the absence of unbiasedness and minimum variance properties of some estimators: they seem not to be BLU estimator. Using a geometric approach, we introduce the concept of metric variance and of a compositional unbiased estimator, and we show that the closed geometric mean is a c-BLU estimator (compositional best linear unbiased estimator with respect to the geometry of the simplex) of the center of the distribution of a random composition. Thus, it satisfies analogous properties to the arithmetic mean as a BLU estimator of the expected value in real space. The geometric approach used gives real meaning to the concepts of measure of central tendency and measure of dispersion and opens up a new way of understanding the statistical analysis of compositional data.     

Some aspects of transformations of compositional data and the identification of outliers

The statistical analysis of compositional data is based on determining an appropriate transformation from the simplex to real space. Possible transfonnations and outliers strongly interact: parameters of transformations may be influenced particularly by outliers, and the result of goodness-of-fit tests will reflect their presence. Thus, the identification of outliers in compositional datasets and the selection of an appropriate transformation of the same data, are problems that cannot be separated. A robust method for outlier detection together with the likelihood of transformed data is presented as a first approach to solve those problems when the additive-logratio and multivariate Box-Cox transformations are used. Three examples illustrate the proposed methodology.     

Singularity and Nonnormality in the Classification of Compositional Data

Geologists may want to classify compositional data and express the classification as a map. Regionalized classification is a tool that can be used for this purpose, but it incorporates discriminant analysis, which requires the computation and inversion of a covariance matrix. Covariance matrices of compositional data always will be singular (noninvertible) because of the unit-sum constraint. Fortunately, discriminant analyses can be calculated using a pseudo-inverse of the singular covariance matrix; this is done automatically by some statistical packages such as SAS. Granulometric data from the Darss Sill region of the Baltic Sea is used to explore how the pseudo-inversion procedure influences discriminant analysis results, comparing the algorithm used by SAS to the more conventional Moore–Penrose algorithm. Logratio transforms have been recommended to overcome problems associated with analysis of compositional data, including singularity. A regionalized classification of the Darss Sill data after logratio transformation is different only slightly from one based on raw granulometric data, suggesting that closure problems do not influence severely regionalized classification of compositional data.     

Conditional Spectral Simulation with Phase Identification

Spectral simulation is used widely in electrical engineering to generate random fields with a given covariance spectrum. The algorithms used are fast particularly when based on Fast Fourier Transform (FFT). However, because of lack of phase identification, spectral simulation only generates unconditional realizations. Local data conditioning is obtained typically by adding a simulated kriging residual. This conditioning process requires an additional kriging at each simulated node thus forfeiting the speed advantage of FFT. A new algorithm for conditioning is proposed whereby the phase values are determined iteratively to ensure approximative data reproduction while reproducing the frequency spectrum, that is, the covariance model. A case study is presented to demonstrate the algorithm.     

Effects of weathering on nickel and vanadyl porphyrins of a Lower Toarcian shale of the Paris basin

The effects of weathering on the organic matter within a narrow sediment interval from surficial to 10 m depth belonging to the Lower Toarcian shales of the Paris basin, were studied at a molecular level. Among the compositional changes observed, the most striking feature is a severe breakdown of the metalloporphyrins (Ni and VO) at the surface, as evidenced by quantitative data. In addition, this degradation of the geoporphyrins is accompanied by substantial modification of their composition with preferential degradation of porphyrins with exocyclic rings over polyalkylporphyrins. These quantitative and qualitative changes are thought to result from exposure of the sediment to weathering conditions. More specifically, abiotic oxidation is likely to be the main weathering process responsible for this superficial alteration of the sedimentary porphyrins. This is particularly suggested by oxidation experiments carried out on a mixture of sedimentary nickel and vanadyl porphyrins where oxidation of the former complexes occurs at a faster rate. Thus, geoporphyrins may be used as molecular indicators to evaluate the degree of oxidation of weathered geological samples containing this kind of molecule.     

Estimation of regionalized compositions: A comparison of three methods

A regionalized composition is a random vector function whose components are positive and sum to a constant at every point of the sampling region. Consequently, the components of a regionalized composition are necessarily spatially correlated. This spatial dependence—induced by the constant sum constraint—is a spurious spatial correlation and may lead to misinterpretations of statistical analyses. Furthermore, the cross-covariance matrices of the regionalized composition are singular, as is the coefficient matrix of the cokriging system of equations. Three methods of performing estimation or prediction of a regionalized composition at unsampled points are discussed: (1) the direct approach of estimating each variable separately; (2) the basis method, which is applicable only when a random function is available that can he regarded as the size of the regionalized composition under study; (3) the logratio approach, using the additive-log-ratio transformation proposed by J. Aitchison, which allows statistical analysis of compositional data. We present a brief theoretical review of these three methods and compare them using compositional data from the Lyons West Oil Field in Kansas (USA). It is shown that, although there are no important numerical differences, the direct approach leads to invalid results, whereas the basis method and the additive-log-ratio approach are comparable.     

Petrogenetic grids for metacarbonate rocks: pressure-temperature phase-diagram projection for mixed-volatile systems

Isothermal or isobaric phase diagram sections as a function of fluid composition (X ^F) are widely used for interpreting the genetic history of metacarbonate rocks. This approach has the disadvantages that: (1) the influence of a key metamorphic variable, either pressure (P) or temperature (T), is obscured; (2) the diagrams are inappropriate for systems that are not fluid-saturated. These problems are avoided by constructing phase-diagram projections in which the volatile composition of the system is projected onto a P-T coordinate frame, i.e., a petrogenetic grid. The univariant curves of such P-T projections trace the conditions of the invariant points of isothermal or isobaric phase-diagram sections, thereby defining the absolute stability of high-variance mineral assemblages, with and without a coexistent fluid phase. Petrogenetic grids for metacarbonate rocks are most useful for the study of regional metamorphism and for systems in which fluid composition has not been externally controlled. A calculated example of a P-T projection for the system CaO−MgO−SiO₂−H₂O−CO₂ suggests that many assemblages (e.g., calcite +tale, enstatite+fluid, magnesite+tremolite, antigorite+diopside+dolomite, and calcite+forsterite+tremolite) in mixed-volatile systems have stability fields that make them useful as P-T indicators. Consideration of the principles governing projection topology demonstrates that the univariant curves around a fluid present invariant point cannot be oriented independently with respect to the direction of compositional variation in the fluid phase. This has the interesting predictive implication that if the direction of compositional variation along one univariant curve around an invariant point is known, then the direction of compositional variation along the remaining curves can be determined solely from topologic constraints. The same constraints can be applied to systems containing simple mineral solutions or melts in order to predict compositional variations.     

An Improved Perturbation Mechanism for Simulated Annealing Simulation

Simulated annealing (SA) is being increasingly used for the generation of stochastic models of spatial phenomena because of its flexibility to integrate data of diverse types and scales. The major shortcoming of SA is the extensive CPU requirements. We present a perturbation mechanism that significantly improves the CPU speed. Two conventional perturbation mechanisms are to (1) randomly select two locations and swap their attribute values, or (2) visit a randomly selected location and draw a new value from the global histogram. The proposed perturbation mechanism is a modification of option 2: each candidate value is drawn from a local conditional distribution built with a template of kriging weights rather than from the global distribution. This results in accepting more perturbations and in perturbations that improve the variogram reproduction for short scale lags. We document the new method, the increased convergence speed, and the improved variogram reproduction. Implementation details of the method such as the size of the local neighborhood are considered.     

Isometric Logratio Transformations for Compositional Data Analysis

Geometry in the simplex has been developed in the last 15 years mainly based on the contributions due to J. Aitchison. The main goal was to develop analytical tools for the statistical analysis of compositional data. Our present aim is to get a further insight into some aspects of this geometry in order to clarify the way for more complex statistical approaches. This is done by way of orthonormal bases, which allow for a straightforward handling of geometric elements in the simplex. The transformation into real coordinates preserves all metric properties and is thus called isometric logratio transformation (ilr). An important result is the decomposition of the simplex, as a vector space, into orthogonal subspaces associated with nonoverlapping subcompositions. This gives the key to join compositions with different parts into a single composition by using a balancing element. The relationship between ilr transformations and the centered-logratio (clr) and additive-logratio (alr) transformations is also studied. Exponential growth or decay of mass is used to illustrate compositional linear processes, parallelism and orthogonality in the simplex.     

An internally consistent integration method for critical state models

A new procedure to integrate critical state models including Cam–Clay and modified Cam–Clay is proposed here. The proposed procedure makes use of the linearity of the virgin isotropic compression curve and the parallel anisotropic consolidation lines in e–ln p space which are basic features of the formulation of critical state models. Using this algorithm, a unique final stress state may be found as a function of a single unknown for elastoplastic loading. The key equations are given in this article for the Cam–Clay and modified Cam–Clay models. The use of the Newton–Raphson iterative method to minimize residuals and obtain a converged solution is described here. This new algorithm may be applied using the assumptions of linear elasticity or non‐linear elasticity within a given loading step. The new algorithm proposed here is internally consistent and has computational advantages over the current numerical integration procedures. Numerical examples are presented to show the performance of the algorithm as compared to other integration algorithms. Published in 2005 by John Wiley & Sons, Ltd.     

Sulfur Variations in Glasses from Volcanic Rocks: Effect of Melt Composition on Sulfur Solubility

A worldwide data set of major element and sulfur analyses of undegassed lavas, pumices, and melt inclusions from 14 volcanic locations was selected to investigate the compositional effects on sulfur solubility in magmas. We utilized analyses on calc-alkaline, alkaline, and tholeiitic rocks, with a range of 3400 ppm S variation. There is a strong correlation between chemical composition and the sulfur concentration: the less silicic and the more alkaline the rocks are, the more dissolved sulfur they can carry. Also, sulfur concentration is higher in rocks that represent less polymerized melts. Elemental correlations between FeO and S, well-defined for tholeiites, do not hold for alkaline melts. The compositional effects are at least as important as the better-known pressure, temperature, and f(O₂) dependencies.     

GEOLOGICAL STRUCTURE OF THE SOUTH KHINGAN MANGANESE DEPOSIT AND ESSENTIAL COMPOSITION OF ITS ORES

Development of several different microbeam techniques permits in situ analysis of trace elements in natural and synthetic mineral assemblages, which in turn makes possible determination of two-mineral partitioning behavior for trace elements (D-values). With the experimental approach, control of compositional and physical variables is possible, but equilibrium needs to be carefully assessed, and in some instances compositions and optimal experimental conditions are different from the natural situation. With the natural mineral approach, physical conditions must be assessed independently, accepting the uncertainties involved in geothermobarometric determinations based on coexisting mineral compositions. A potentially complex history of formation must be unravelled as well, with the attendant possibility of non-equilibrium.

Comparison of D-values determined for coexisting amphibole and Ca-clinopyroxene indicates overall good agreement for Sr, Zr, Hf, Y, and REE, but significant discrepancy for Rb, Ba, Nb, and Ta (experimental values much lower than natural values). For coexisting Ca-clinopyroxene and garnet, the spread of data is greater and fewer elements can be compared (Rb, Ba, Nb, and Ta data are not adequate). However Sr, Y, Zr, Hf, and Nd-Lu (of the rare-earth elements) agree reasonably well, but La and Ce experimental values for clinopyroxene/garnet are much lower than the natural values. These differences for both mineral pairs may be attributed in part to critical compositional differences between synthetic and natural minerals, where compositional factors play a key role in controlling accommodation of trace elements in the mineral structure. A second factor is the possibile existence of minute inclusions rich in trace elements trapped in the natural minerals. These inclusions may be submicroscopic and unavoidable, even by microbeam analysis. More work on both experimental and natural systems should resolve the discrepancies and help to realize the full potential for using trace elements to assess petrological processes in the mantle, where knowledge of D-value variation as a function of composition, pressure, and temperature is essential.     

Quantifying and testing differences among means of compositional data suites

Commonly, geological studies compare mean values of two or more compositional data suites in order to determine if, how, and by how much they differ. Simple approaches for evaluating and statistically testing differences in mean values for open data fail for compositional (closed) data. A new parameter, an f-value, therefore has been developed, which correctly quantifies the differences among compositional mean values and allows testing those differences for statistical significance. In general, this parameter quantifies only therelative factor by which compositional variables differ across data suites; however for situations where, arguably, at least one component has neither increased nor decreased, anabsolute f-value can be computed. In situations where the compositional variables have undergone many perturbations, arguments based upon thef-values and the central limit theorem indicate that logratios of compositional variables should be normally distributed.     

Quantitative determination of metamorphic reaction history: mass balance relations between groundmass and mineral inclusion assemblages in metamorphic rocks

Qualitative and quantitative information about metamorphic reaction history and PT paths may be obtained from mineral inclusions in garnet by comparing the mineralogy, distribution, and compositions of paragenetically-related inclusions with minerals in the groundmass assemblage. Using the algebraic technique of singular value decomposition (SVD), we document mass balance relations between inclusion and groundmass assemblages in metapelitic rocks from two metamorphic terranes that experienced different peak metamorphic conditions, and whose transition from inclusion to groundmass assemblage records different PT path segments relative to peak conditions. We calculate mass balances relating an inclusion assemblage consisting in part of armored relics of chloritoid to groundmass mineral assemblages in a kyanite-staurolite mica schist from the Solitude Range, British Columbia, and an inclusion assemblage of kyanite, staurolite, and rutile to groundmass minerals in a sillimanite-cordierite gneiss from the Skagit Gneiss, North Cascade Range, Washington. Mass balances for each rock are consistent with reaction histories inferred from petrographic observations. In the Solitude Range schist, the results of mass balance calculations are consistent with the growth of staurolite and garnet at the expense of chloritoid during prograde metamorphism and suggest that chlorite, although not preserved as an inclusion, was involved in initial staurolite growth. In the Skagit sillimanite gneiss, mass balance relations exist between the inclusion suite, which formed during high pressure metamorphism, and the associated groundmass assemblage, which equilibrated at high temperature but much lower pressure. Mass balance does not exist between the groundmass of the Skagit sillimanite gneiss and the groundmass of a nearby kyanite-staurolite schist that has been proposed as a possible lower-grade equivalent of the sillimanite-bearing rocks. These results indicate that, although compositional modification and selective preservation of minerals must be taken into account, mineral inclusion suites may nevertheless preserve enough compositional information to allow reconstruction of complete or nearly complete pre-existing assemblages. This information may not be retrievable from any other source if no lower-grade equivalents of the rocks of interest are exposed.     

Chemical gradient at diffusive interfaces in magma chambers

The results of high pressure experiments on diffusion and Soret separation in natural silicate melts show that the diffusive behaviour between natural silicic and mafic magmas can be approximately modelled as if the system were a binary mixture of SiO₂ and other components such as MgO+FeO+CaO. Steady state compositional profiles across a diffusive interface between silicic and mafic magma layers are calculated on the basis of phenomenological relationships for the fluxes of chemical species and heat in the binary mixtures, using the experimental data of diffusion and Soret coefficients in natural silicate melts. The compositional profiles show a curvature with a minimum SiO₂ value within the interface due to the Soret effect and temperature dependence of diffusion coefficient. The compositional gradient at the lower half of the diffusive interface is similar to that resulting from the Soret separation of a mafic melt regardless of the composition of the silicic magmas. These results suggest that picritic magma can be formed in the interfacial region between the mafic and silicic magma layers. The compositional gradient explains chemical variation of mafic to picritic inclusions in a mixed andesite of the Abu Volcano Group, Japan.     

Conditioning Geostatistical Operations to Nonlinear Volume Averages

The all-important process of data integration calls for algorithms that can handle secondary data often defined as nonlinear averages of the primary (hard) data over specific areas or volumes. It is suggested to approximate these nonlinear averages by linear averages of a nonlinear transform of the primary variable. Kriging of such nonlinear transforms, followed by the inverse transform, allows exact reproduction of all original data, both of point support and nonlinear volume averages. In a simulation mode, the previous cokriging provides the mean and variance of a conditional distribution from which to draw a simulated value, which is then backtransformed into a simulated value of the primary variable. The nonlinear averaged data values are then reproduced exactly. The direct sequential simulation algorithm adopted does not call for using any Gaussian distribution.     

Titanite dates crystallization: Slow Pb diffusion during super‐solidus re‐equilibration

Titanite can be found in rocks of wide compositional range, is reactive, growing or regrowing during metamorphic and hydrothermal events, and is generally amenable to U–Pb geochronology. Experimental evidence suggest that titanite has a closure temperature for Pb ranging from 550 to 650°C, and thus titanite dates are commonly interpreted as cooling ages. However, this view has been challenged in recent years by evidence from natural titanite which suggests the closure temperature may be significantly higher (up to 800°C). Here, we investigate titanite in an enclave of migmatitic gneiss included within a granite intrusion. The titanite crystals exhibit textural features characteristic of fluid‐mediated mass transfer processes on length scales of <100 µm. These textural features are associated with variation in both Pb concentrations and distinct U–Pb isotopic compositions. Zr‐in‐titanite thermometry indicates that modification of the titanite occurred at temperatures in excess of 840°C, in the presence of a high‐T silicate melt. The Pb concentration gradients preserved in these titanite crystals are used to determine the diffusivity of Pb in titanite under high‐T conditions. We estimate diffusivities ranging from 2 × 10^?22 to 5 × 10^?25 m²/s. These results are significantly lower than experimental data predict yet are consistent with other empirical data on natural titanites, suggesting that Pb diffusivity is similar to that of Sr. Thus our data challenge the wide‐held assumption that U–Pb titanite dates only reflect cooling ages.     

Progressive evolution of whole‐rock composition during metamorphism revealed by multivariate statistical analyses

The geochemical evolution of metamorphic rocks during subduction‐related metamorphism is described on the basis of multivariate statistical analyses. The studied data set comprises a series of mapped metamorphic rocks collected from the Sanbagawa metamorphic belt in central Shikoku, Japan, where metamorphic conditions range from the pumpellyite–actinolite to epidote–amphibolite facies. Recent progress in computational and information science provides a number of algorithms capable of revealing structures in large data sets. This study applies k‐means cluster analysis (KCA) and non‐negative matrix factorization (NMF) to a series of metapelites, which is the main lithotype of the Sanbagawa metamorphic belt. KCA describes the structures of the high‐dimensional data, while NMF provides end‐member decomposition which can be useful for evaluating the spatial distribution of continuous compositional trends. The analysed data set, derived from previously published work, contains 296 samples for which 14 elements (Si, Ti, Al, Fe, Mn, Mg, Ca, Na, K, P, Rb, Sr, Zr and Ba) have been analysed. The KCA and NMF analyses indicate five clusters and four end‐members, respectively, successfully explaining compositional variations within the data set. KCA indicates that the chemical compositions of metapelite samples from the western (Besshi) part of the sampled area differ significantly from those in the east (Asemigawa). In the west, clusters show a good correlation with the metamorphic grade. With increasing metamorphic grade, there are decreases in SiO₂ and Na₂O and increases in other components. However, the compositional change with metamorphic grade is less obvious in the eastern area. End‐member decomposition using NMF revealed that the evolutional change of whole‐rock composition, as correlated with metamorphic grade, approximates a stoichiometric increase of a garnet‐like component in the whole‐rock composition, possibly due to the precipitation of garnet and effusion of other components during progressive dehydration. Thermodynamic modelling of the evolution of the whole‐rock composition yielded the following results: (1) the whole‐rock composition at lower metamorphic grade favours the preferential crystallization of garnet under the conditions of the garnet zone, with biotite becoming stable together with garnet in higher‐grade rock compositions under the same P–T conditions; (2) with higher‐grade whole‐rock compositions, more H₂O is retained. These results provide insight into the mechanism suppressing dehydration under high‐P metamorphic conditions. This mechanism should be considered in forward modelling of the fluid cycle in subduction zones, although such a quantitative model has yet to be developed.