Groups of Parts and Their Balances in Compositional Data Analysis

Amalgamation of parts of a composition has been extensively used as a technique of analysis to achieve reduced dimension, as was discussed during the CoDaWork'03 meeting (Girona, Spain, 2003). It was shown to be a non-linear operation in the simplex that does not preserve distances under perturbation. The discussion motivated the introduction in the present paper of concepts such as group of parts, balance between groups, and sequential binary partition, which are intended to provide tools of compositional data analysis for dimension reduction. Key concepts underlying this development are the established tools of subcomposition, coordinates in an orthogonal basis of the simplex, balancing element and, in general, the Aitchison geometry in the simplex. Main new results are: a method to analyze grouped parts of a compositional vector through the adequate coordinates in an ad hoc orthonormal basis; and the study of balances of groups of parts (inter-group analysis) as an orthogonal projection similar to that used in standard subcompositional analysis (intra-group analysis). A simulated example compares results when testing equal centers of two populations using amalgamated parts and balances; it shows that, in certain circumstances, results from both analysis can disagree.     

Subcompositional Patterns in Cenozoic Volcanic Rocks of Hungary

Data selected from an extensive major element database of Cenozoic volcanic rocks (including calc-alkaline andesites, dacites, rhyolites, and alkali basalts) of Hungary are used to illustrate the detection and modeling of subcompositional patterns using a statistical analysis based on the assumption that relative differences between the observed values are more meaningful than absolute ones. In particular, two roughly linear compositional patterns (associated one to the alkaline basalts, the other to the calc-alkaline series) are revealed and evaluated, and it is shown how principal component analysis can be used to obtain the estimated subcomposition of their incidental intersection point.     

川西北甘孜—道孚地区岩浆岩体系的亲缘性研究

不同时期、不同类型的岩浆岩，尽管岩石化学成分的绝对含量不同，但其数据集的内部结构常常存在着某种成因联系。本文应用成分数据统计分析中的变异矩阵分析、子成分分析和主分量分析研究川西北甘孜—道孚地区华力西期、印支期和燕山期不同类型岩浆岩的亲缘性，进而探讨其成因联系。     

Exploring the joint compositional variability of major components and trace elements in the Tellus soil geochemistry survey (Northern Ireland)

The complexity of modern geochemical data sets is increasing in several aspects (number of available samples, number of elements measured, number of matrices analysed, geological-environmental variability covered, etc), hence it is becoming increasingly necessary to apply statistical methods to elucidate their structure. This paper presents an exploratory analysis of one such complex data set, the Tellus geochemical soil survey of Northern Ireland (NI). This exploratory analysis is based on one of the most fundamental exploratory tools, principal component analysis (PCA) and its graphical representation as a biplot, albeit in several variations: the set of elements included (only major oxides vs. all observed elements), the prior transformation applied to the data (none, a standardization or a logratio transformation) and the way the covariance matrix between components is estimated (classical estimation vs. robust estimation). Results show that a log-ratio PCA (robust or classical) of all available elements is the most powerful exploratory setting, providing the following insights: the first two processes controlling the whole geochemical variation in NI soils are peat coverage and a contrast between “mafic” and “felsic” background lithologies; peat covered areas are detected as outliers by a robust analysis, and can be then filtered out if required for further modelling; and peat coverage intensity can be quantified with the %Br in the subcomposition (Br, Rb, Ni).     

Reducing the dimensionality of compositional data sets

The high-dimensionality of many compositional data sets has caused geologists to look for insights into the observed patterns of variability through two dimension-reducing procedures: (i)the selection of a few subcompositions for particular study, and (ii)principal component analysis. After a brief critical review of the unsatisfactory state of current statistical methodology for these two procedures, this paper takes as a starting point for the resolution of persisting difficulties a recent approach to principal component analysis through a new definition of the covariance structure of a composition. This approach is first applied for expository purposes to a small illustrative compositional data set and then to a number of larger published geochemical data sets. The new approach then leads naturally to a method of measuring the extent to which a subcomposition retains the pattern of variability of the whole composition and so provides a criterion for the selection of suitable subcompositions. Such a selection process is illustrated by application to geochemical data sets.     

Compositional Data Analysis of Geological Variability and Process: A Case Study

Developments in the statistical analysis of compositional data over the last two decades have made possible a much deeper exploration of the nature of variability and the possible processes associated with compositional data sets from many disciplines. In this paper, we concentrate on geochemical data. First, we explain how hypotheses of compositional variability may be formulated within the natural sample space, the unit simplex, including useful hypotheses of sub-compositional discrimination and specific perturbational change. Then we develop through standard methodology, such as generalised likelihood ratio tests, statistical tools to allow the systematic investigation of a lattice of such hypotheses. Some of these tests are simple adaptations of existing multivariate tests but others require special construction. We comment on the use of graphical methods in compositional data analysis and on the ordination of specimens. The recent development of the concept of compositional processes is then explained, together with the necessary tools for a staying-in-the-simplex approach, such as the singular value decomposition of a compositional data set. All these statistical techniques are illustrated for a substantial compositional data set, consisting of 209 major oxide and trace element compositions of metamorphosed limestones from the Grampian Highlands of Scotland. Finally, we discuss some unresolved problems in the statistical analysis of compositional processes.     

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.     

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.     

X-ray fluorescence analysis of silicate rocks using fused glass discs and a side-window Rh source tube: accuracy, precision and reproducibility

In this paper, we provide a detailed account of our sample fusion, calibration and instrumentation methods for major-element whole-rock analysis by XRF, and we discuss several aspects of sample preparation and instrument performance that are important considerations for accurate analysis. The fusion procedure involves moderate capital costs and is easy to apply, yielding flat, polished, homogeneous glass discs as cast. The calibration method utilizes a least-squares procedure that rigorously fits data according to both compositional and counting statistical uncertainties. We use a side-window Rh tube for analyzing major elements (including Na) and employ real-time testing for constant count rate to reject spurious results.

The methods result in excellent analytical precision and reproducibility. The standards used for calibration lie within compositional and counting statistical uncertainties of best-fit straight lines. Analyses of replicate discs and repeated analyses of single discs show excellent long-term reproducibility over several months, approaching counting statistical uncertainties in several cases. Comparison with independent measurements made by other laboratories using instrumental neutron activation and X-ray fluorescence analyses shows excellent agreement with our results.

A side-window Rh tube gives increased detection limits for most major elements, but otherwise shows little difference in precision compared to a Cr tube. This means that major and trace elements can be analyzed without changing X-ray sources, which provides saving in terms of time and money, as well as being a convenience to the analyst.     

开放成矿系统元素迁移负异常结构对深部找矿预测的意义

在开放体系成矿过程中,成矿信息应具有时间和空间两方面的属性。空间属性表现为元素在赋矿空间的分布及变化,时间属性则表现为成矿过程中元素的迁移变化。本文以安徽省兆吉口铅锌矿床为例,通过Grant方程进行元素质量迁移研究,建立Na2O负异常结构,并应用成分数据理论的双标图对负异常结构进行验证,结果发现负异常结构携带着时间维度上的成矿信息。成矿元素Pb、Zn的正异常分布于Na2O负异常范围内。Na2O与Pb、Zn、Au负相关,Au与Pb、Zn呈弱负相关,表明该地区可能存在与Pb、Zn同一成矿作用不同成矿阶段形成的Au矿化/体。兆吉口地区的找矿方向应为Pb、Zn、Au三种矿产资源。Na2O负异常在深部并未结束,表明深部仍可能存有矿体。     

On the Interpretation of Orthonormal Coordinates for Compositional Data

The simplex with the Aitchison geometry is a natural sample space for compositional data, that is, observations carrying only relative information (especially proportions, percentages, etc., often occurring in the geosciences). For this reason, standard statistical methods that rely on Euclidean structure of the real space cannot be used directly for statistical analysis. At first, compositional data need to be expressed in coordinates of an orthonormal basis on the simplex (with respect to the Aitchison geometry). The mathematical interpretation of the orthonormal coordinates is derived from the procedure by which they are constructed (called sequential binary partition), and they act as balances between groups of compositional parts. The goal of this paper is to describe the covariance structure of coordinates and, consequently, to provide a complementary interpretation based on log-ratios of parts of the original composition. It must be noted that, in a composition, the ratios themselves contain all the relevant information. The possibilities as well as the limitations of this approach are demonstrated through illustrative examples.     

Mapping Soil Particle Size Fractions Using Compositional Kriging,Cokriging and Additive Log-ratio Cokriging in Two Case Studies

Information on the spatial distribution of soil particle-size fractions (psf) is required for a wide range of applications. Geostatistics is often used to map spatial distribution from point observations; however, for compositional data such as soil psf, conventional multivariate geostatistics are not optimal. Several solutions have been proposed, including compositional kriging and transformation to a composition followed by cokriging. These have been shown to perform differently in different situations, so that there is no procedure to choose an optimal method. To address this, two case studies of soil psf mapping were carried out using compositional kriging, log-ratio cokriging, cokriging, and additive log-ratio cokriging; and the performance of Mahalanobis distance as a criterion for choosing an optimal mapping method was tested. All methods generated very similar results. However, the compositional kriging and cokriging results were slightly more similar to each other than to the other pair, as were log-ratio cokriging and additive log-ratio cokriging. The similar results of the two methods within each pair were due to similarities of the methods themselves, for example, the same variogram models and prediction techniques, and the similar results between the two pairs were due to the mathematical relationship between original and log-ratio transformed data. Mahalanobis distance did not prove to be a good indicator for selecting an optimal method to map soil psf.     

Relative variation diagrams for describing patterns of compositional variability

In recognizing that a composition, such as a major oxide or sediment composition, provides information only about the relative, not the absolute, magnitudes of its components, this paper exposes the compositional variation array as the simplest and minimum way of summarizing the pattern of variability within a compositional data set. Such summaries are free of the notorious hazards of the constant-sum constraint and when depicted in relative variation diagrams can often provide substantial insights into the nature of the compositional variability. Concepts and practice are illustrated by reference to a number of real data sets.     

Statistical comparisons of heavy metal pollutants between seven regions of the Polish exclusive economic zone

This paper addresses three intractable difficulties associated with the statistical analysis of compositional data, such as percentages or ppm. These are: (1) that such data do not follow multivariate normal distributions thus rendering inappropriate, standard parametric statistical tests and estimation procedures, (2) the covariance/correlation coefficients between specific pairs of components are determined in whole or in part by the presence or absence of other components, and, (3) the negative bias property. That is, at least one covariance and therefore at least one correlation, must be negative, hence the remaining correlations are prevented from ranging freely between ?1 and +1. It follows that correlation coefficients formed from compositional data are not only not absolute, but also frequently spurious. Standard multivariate procedures based on them are unreliable, and intrinsic associations between components inferred from strong positive correlations in particular, are potentially false. In a recent 2009 paper, it was reported that 59 surface sediment samples from 7 regions in the Polish exclusive economic zone had been chemically analyzed for 16 elements. Enrichment factors together with crude correlation coefficients between selected elements were presented. All these quantities were computed from the initial raw compositional data resulting from the chemical analyses In this paper, a statistical procedure is presented which is distinctly different to the enrichment factor computations based on the same raw compositional data. The procedure generates a log-ratio measure of the abundance of each element in each of the seven regions, thus enabling comparisons of relative levels of pollution between the regions. Although the two techniques are quite unrelated, it is shown that in general, extremely high or low measures of the relative abundances in the regions are associated with correspondingly high or low values of the enrichment factors in the same regions that were reported in the 2009 paper. That is, the statistical analysis confirms the results of the enrichment factor data in the identification of the most to the least polluted regions. In an additional analysis, the residue term was excluded from each sediment sample by rescaling the 16 element concentrations to sum to 100%, thus forming 59 residue-free sub-compositions. Crude correlation coefficients were computed for pairs of elements of this sub-compositional data. These revealed that certain correlations based on the initial raw data that were reported in the 2009 paper for the same pairs of elements, were not only inconsistent, but sometimes also contradictory. Such contradictions imply that intrinsic geochemical element associations inferred in that paper from such correlations were false.     

The statistical analysis of geochemical compositions

The analysis and interpretation of compositional data, such as major oxide compositions of rocks, has been traditionally plagued by the so-called constant-sum or closure problem. Particular difficulties have been the lack of a satisfactory, interpretable covariance structure and of rich, tractable, parametric classes of distributions on the simplex sample space. Consideration of logistic and logratio transformations between the simplex and Euclidan space has allowed the introduction of new concepts of covariance structure and of classes of logistic-normal distributions which have now opened up a substantial and meaningful array of statistical methodology for compositional data. From the motivation of a wide variety of practical geological problems we examine the range of possibilities with this new approach to the constant-sum problem.     

The bimodal composition of carbonatites: Reality or misconception?

The concept of compositional bimodality in carbonatites has become widely accepted and has been used to impose restrictions on the composition of carbonatite magmas. We agree that mineralogical bimodality exists in carbonatites (most are either calcitic or dolomitic/ankeritic), but we argue that there is no compositional bimodality. The idea of bimodality is based on the interpretation of a variety of element distribution diagrams which were compiled only from chemical analyses in which SiO₂ is < 10 wt.%. All others were rejected. Even with such a restricted data set the case for compositional bimodality is extremely weak, but the inclusion of analyses with higher SiO₂ content destroys it completely. Yet these more siliceous compositions must be included, for many carbonatites contain substantial amounts of Fe–Mg silicates which are an essential part of the magmatic mineralogy of the rocks. They account for much of the Mg in carbonatites that are otherwise calcitic. Many such carbonatites contain well in excess of 10 wt.% SiO₂. Supporters of the bimodality concept argue that liquids having compositions between calcite and dolomite can precipitate neither calcite nor dolomite because the minimum on the solid solution loops in the system calcite–dolomite permits only a carbonate of intermediate composition. Therefore, it is argued, liquids of such intermediate composition cannot be parental to calcitic and dolomitic carbonatites; their parent magmas must be calcitic and dolomitic. This deduction is incorrect. It is well established that dolomitic liquids have calcite as the liquidus phase over substantial temperature intervals, and that this is followed by dolomite precipitation. Mixed calcite–dolomite carbonatites are explicable in this way. Therefore, dolomitic liquids can be parental to calcitic carbonatites. However, dolomitic carbonatites cannot crystallize from a calcitic liquid. We suggest that intermediate composition carbonatite magmas are probably common. Bimodality in carbonatites is solely mineralogical, not compositional.     

稀土元素在韧性剪切带体积亏损研究中的应用：以胶南造山带构造岩为例

稀土元素在韧性剪切带中发生有规律的成分变异,随着变形程度的增强,单个稀土元素,轻稀土,重稀土,稀土元素的总量均明显增加,而稀土配分模式不变。通过花岗质构造岩变形前后的等比分析,稀土元素的成分变异主要是由于较大的体积亏损而引起,并根据成分变异与体积亏损之间的函数关系,进一步确定了构造岩变形前后的体积亏损率。     

岩石地球化学数据挖掘及弱异常识别--以新疆阿舍勒铜矿为例

勘查地球化学找矿工作的重点在于正确解译地球化学数据,以便从冗杂的地质信息中精准提取与成矿有关的异常信息,指导找矿研究。然而,地球化学数据属于成分数据,具有闭合效应,只有对数据进行正确的预处理才能应用多元统计分析方法,还原元素真实的空间分布。本文在阿舍勒铜锌矿区外围南侧区域共收集1009件地表原生晕样品,对样品中的13种微量元素进行测试,并对原始数据、对数及ilr变换后的数据进行EDA分析,对比数据空间分布及内部结构特征。运用(稳健)主成分分析,结合成分数据双标图及第一主成分点位图,解译三类数据指示的元素组合与成矿信息之间的关联。随后运用多重分形滤波技术,对以ilr变换为基础的稳健主成分得分数据分解元素组合异常和背景分布特征。结果表明:①经过对数及ilr变换后的数据相比原始数据空间尺度更均匀,数据近似正态分布;②三类数据双标图表明,仅ilr变换后的数据消除了“闭合效应”,且其第一主成分元素分组揭示了研究区铜矿化与铅锌多金属矿化组合;以对数变换与ilr变换为基础的第一主成分点位图表明,后者主成分得分异常能够较好指示研究区地质找矿信息;③结合研究区地质找矿信息、元素组合异常及背景空间分布特征,最终圈定3个有利找矿靶区。     

Reassessment of the garnet-clinopyroxene Fe−Mg exchange thermometer: I. Evaluation of the Pattison and Newton (1989) experiments

This is Part 1 of a two-part study on the garnet(Grt)-clinopyroxene(Cpx) Fe−Mg exchange equilibrium widely used in geothermometry of amphibolites, granulites and eclogites. The experimental data set previously published by Pattison and Newton (PN) on this equilibrium comprises potentially the largest source of data used for extraction of thermodynamic properties for Grt and Cpx and for formulation of a new thermometer expression (Part 2 of this study). However, the experiments suggested unusual Grt and Cpx solution properties and resulted in a geothermometer expression that generally returns low temperatures for granulites. To verify the reliability of the experiments, and to test the experimental technique of Grt buffering used in the PN study, 69 of the 125 PN runs were reanalysed using wavelength dispersive spectroscopic analysis guided by backscattered electron images. Special attention was given to analysis of compositional changes in Grt in addition to Cpx. With minor exceptions, the reanalysis supports the analytical results of the original PN study. The mechanism of equilibration in the runs was a multicomponent solution-precipitation process involving dissolution of metastable Grt and Cpx starting materials and precipitation of newly formed Cpx and Grt, the latter assumed to represent equilibrim compositions. The experiments cannot be regarded either as reversed or as strictly bracketing. Nevertheless, Cpx compositional parameters, including Mg#, Na, Ca and Al, show the same smooth variations with temperature, pressure and composition as found by PN, suggestive of a close approach to equilibrium. Grt in run products retained the moderate heterogeneity of the starting Grt, although there are subtle compositional trends indicating small changes in Grt Mg# consistent with mass balance constraints. These results uphold the essential validity of the technique of Grt buffering used in the experiments. Nevertheless, it is possible that the final Grt compositional may not coincide with the nominal starting Grt compositions, requiring assignment of compositional uncertainties to Grt for purposes of thermodynamic modelling of the data. Cpx analyses also require assignment of non-zero compositional uncertainties.