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.     

A pseudo-genetic algorithm suited to compositional data

Genetic algorithms can solve least-squares problems where local minima may trap more traditional methods. Although genetic algorithms are applicable to compositional as well as noncompositional data, the standard implementation treats compositional data awkwardly. A need to decode, renormalize, then reincode the fitted parameters to regain a composition is not only computationally costly, but may thwart convergence. A modification to the genetic algorithm, described here, adapts the tools of reproduction, crossover, and mutation to compositional data. The modification consists of replacing crossover with a linear mixture of two parents and replacing mutation with a linear mixture of one of the members of the breeding population and a randomly generated individual. By using continuously evolving populations, rather than discrete generations, reproduction is no longer required. As a test of this new approach, a mixture of four Gaussian functions with given means and variances are deconvolved to recover their mixing proportions.     

Measurement error in compositional data

Compositional data, consisting of vectors of proportions summing to unity such as the geochemical compositions of rocks, have proved difficult to analyze. Recently, the introduction of logistic and logratio transformations between the d-dimensional simplex and Euclidean space has allowed the use of familiar multivariate methods. The problem of how to model and analyze measurement errors in such data is approached through the concept of a perturbation of a composition. Such modeling allows investigation of the role of rescaling, quantification of measurement error, analysis of observor error, and assessment of the effect of measurement error on inferences.     

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.     

Effect of waterwashing on light ends compositional heterogeneity

A number of experiments have been carried out to study in the laboratory the effects of waterwashing on light ends composition. The experimental data were then compared with the results of a simple numerical model. Both approaches confirm the fast removal of the light aromatics by the flushing water and the very slow alteration of the C₁₀₊ compounds. The experimental results also indicate that the normal paraffins are removed faster than the cycloalkanes which is not in agreement with their respective solubility data. the simulations show that, over geological time, compounds such as benzene and toluene should be completely absent in an oil column which is not the case. Therefore, it is likely that in nature oil-water interactions are not as effective as one could imagine, which has implications for oil migration, as well as for the understanding of hydrodynamism phenomena in oil pools.     

End-member modeling of compositional data: Numerical-statistical algorithms for solving the explicit mixing problem

Linear mixing models of compositional data have been developed in various branches of the earth sciences (e.g., geochemistry, petrology, mineralogy, sedimentology) for the purpose of summarizing variation among a series of observations in terms of proportional contributions of (theoretical) end members. Methods of parameter estimation range from relatively straightforward normative partitioning by (nonnegative) least squares, to more sophisticated bilinear inversion techniques. Solving the bilinear mixing problem involves the estimation of both mixing proportions and end-member compositions from the data. Normative partitioning, also known as linear unmixing, thus can be regarded as a special situation of bilinear unmixing with (supposedly) known end members. Previous attempts to model linear mixing processes are reviewed briefly, and a new iterative strategy for solving the bilinear problem is developed. This end-member modeling algorithm is more robust and has better convergence properties than previously proposed numerical schemes. The bilinear unmixing solution is intrinsically nonunique, unless additional constraints on the model parameters are introduced. In situations where no a priori knowledge is available, the concept of an “ optimal ” solution may be used. This concept is based on the trade-off between mathematical and geological feasibility, two seemingly contradictory but equally desirable requirements of the unmixing solution.     

The statistical study of hydraulic geometry: A new direction for compositional data analysis

Hydraulic exponents and unit hydraulic exponents are unit-sum constrained, which requires that they be analyzed by statistical methods designed for compositional data. Though uncertainties remain regarding selection of the best constraining operation and method of handling departures from the unit-sum constraint, neither category of uncertainty should be an impediment to the selection of the appropriate statistical methodology. In a small sample study, the hydraulic geometry of different types of streams were compared: (1) semi-arid: perennial vs. ephemeral; (2) tropical: Puerto Rico vs. West Malaysia; and (3) semi-arid vs. tropical (by pooling the previous data sets). All three comparisons revealed statistically significant differences in either logratio mean vectorsor logratio covariance matrices but not both. All six categories of data had logistic normal distributions. Because the derivatives at a given discharge of curvilinear hydraulic geometry relationships and hydraulic exponents on either side of the breakpoints of piecewise linear relationships are unit-sum constrained, they also can be studied by compositional methods. However, the compositional approach is limited in cases where distributions have large departures from logistic normality and for streams that have negative hydraulic exponents.     

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.     

Water-rock interactions and chemical compositional variations during ductile deformation of the NW-striking shear zone in the Jiapigou gold belt,China

Two kinds of mylonite series rocks, felsic and mafic, have been recognized in the NW-striking shear zone of the Jiapigou gold belt. During ductile deformation, a large amount of fluid interacted intensively with the mylonite series rocks: plagioclases were sericitized and theAn values declined rapidly, finally all of them were transformed to albites; dark minerals were gradually replaced by chlorites (mostly ripidolite). Meanwhile, large-scale and extensive carbonation also took place, and the carbonatization minerals varied from calcite to dolomite and ankerite with the development of deformation. The δ¹³C values of the carbonates are −3.0‰ – −5.6‰ suggesting a deep source of carbon. The ductile deformation is nearly an iso-volume one (f _v≈1). With the enhancement of shear deformation, SiO₂ in the two mylonite series rocks was depleted, while volatile components suchs as CO₂ and H₂O, and some ore-forming elements such as Au and S were obviously enriched. But it is noted that the enrichment of Au in both the mylonite series rocks did not reach the paygrade of gold. The released SiO₂ from water-rock interactions occurred in the form of colloids and absorbed gold in the fluid. When brittle structures were formed locally in the ductile shear zone, the ore-forming fluids migrated to the structures along microfractures, and preciptated auriferous quartz because of reduction of pressure and temperature. Fluid inclusion study shows that the temperature and pressure of the ore-forming fluids are 245–292°C and 95.4–131.7 MPa respectively; the salinity is 12.88–16.33wt% NaCl; the fluid-phase is rich in Ca²⁺, K⁺, Na⁺, Mg²⁺, F⁻ and Cl⁻, while the gaseous phases are rich in CO₂ and CH₄. The δD and δ¹⁸O, values of the ore-forming fluid are −84.48‰ – −91.73‰ and −0.247‰ – +2.715‰ respectively, suggesting that the fluid is composed predominantly of meteoric water. This project is financially supported by the National Natural Science Foundation of China (No. 9488010).     

On criteria for measures of compositional difference

Simple perceptions about the nature of compositions lead through logical necessity to certain forms of analysis of compositional data. In this paper the consequences of essential requirements of scale, perturbation and permutation invariance, together with that of subcompositional dominance, are applied to the problem of characterizing change and measures of difference between two compositions. It will be shown that one strongly advocated scalar measure of difference fails these tests of logical necessity, and that one particular form of scalar measure of difference (the sum of the squares of all possible logratio differences in the components of the two compositions), although not unique, emerges as the simplest and most tractable satisfying the criteria.     

形态相关系数及其在地球化学数据分析中的应用

地球化学数据元素间的相关分析具有重要意义。地球化学数据是一种不遵循正态分布的成分数据,其封闭特征存在挖掘和分析的困难和障碍,因此许多传统的统计方法不适合使用。主要通过元素值的趋势分析了地球化学元素之间的相关关系,并提出了形态相关系数的概念。该方法不需要数据服从正态分布,可以忽略闭包特征的影响。实验表明,该方法简单、稳定、准确,可以显示数据元素之间的关系。此外,方法计算过程消除了回溯,因此适用于大数据的实时和动态分析。     

成分数据理论和无监督聚类K- means方法提取背景和异常信息——以安徽省兆吉口铅锌矿床为例

地球化学数据是应用地球化学研究的重要组成部分，是化学勘查工作的基础成果。勘查地球化学数据基本上以元素的质量百分浓度 (简称浓度)的形式表达，是典型的成分数据。其表达的是“组分/总体”相对质量贡献信息，而不是绝对的质量变化信息。浓度数据分布在单纯形空间，而不是整个欧式空间。对成分数据进行处理之前，进行适当的对数比值转换处理可以提高其信息表达。本文以安徽省兆吉口铅锌矿床土壤中Pb数据为示范案例，通过对数比值转换方法优化浓度数据的结构以提高相对信息的表达，并利用无监督学习K- means聚类方法根据对数比值转换数据分布空间质心的距离识别背景和异常信息，最后对K- means聚类方法识别的背景和异常与迭代2倍标准差法和浓度- 面积分形分析法进行比较以衡量其表现。结果表明：浓度数据表达的是相对质量信息，而不是绝对质量关系，不同样品间不能通过比较浓度高低推断出质量的多寡关系。对数比值方法可以有效地提高浓度数据的结构和信息表达，K- means方法能够准确识别对数比值转换数据的背景和异常信息，其效果类似浓度- 面积分形分析方法，比迭代2倍标准差法好。     

Open and closed compositional data in petrology

The nature of chemical and modal closed and open compositional data for igneous petrology is reviewed. Chemical analyses of Ben Nevis (Ontario) Archaean volcanic rocks support earlier assertions that major- and trace-element g/100 cc igneous-rock data usually comprise open compositional variables which, unlike traditional closed constant-sum weight percentages, are not constrained by closure.     

珍珠的成分特点研究

运用ICP-MS、IC、XRD、IR等分析方法,对我国不同成因、不同产地的珍珠及其贝壳的成分进行了系统的研究,结果表明:淡水养殖珍珠中的P、Mn、Ba的含量相对较高,而海水珍珠中的Sr、Zn、Cr、Ni的含量相对较高;与淡水养殖珍珠相比,海水养殖珍珠中的阴、阳离子种类多,含量高,氨基酸总含量高而稳定;海水养殖珍珠及其贝壳或淡水养殖珍珠及其贝壳在成分上的变化基本一致,而海水养殖珍珠的珍珠层与珍珠核之间差异较大。该结论对于珍珠的鉴定及其理论药理研究具有重要的意义。     

Effect of various transformations on the analysis of percentage data

Proper analysis of transformed data arrays (such as percentages) requires paying special attention to the effects of the transformation process itself. Effects of several commonly used transformations (including percentage formation, row and column normalization, and the square root transformation) have been examined with emphasis placed on changes in the statistical and geometrical properties of column vectors that accompany the application of the transformation. Even though many transformations, including taking the square root, open up the percentage array, this does not allow one to ignore the fact that percentage formation may have considerably modified the statistical and geometrical properties of the columns of the matrix. In preparing to analyze percentages one should give serious consideration to using the row normalized form of the data matrix. The individual elements in such a matrix are the direction cosines of the vector in M-dimensional space, the row vectors are of unit length, and the row normalized matrix computed from the closed array is equal to the row normalized, open matrix that is unobservable. Application of a column transformation (such as range restriction and proportion of the maximum) destroys the equality of the open and percentage row normalized matrices. Despite repeated claims to the contrary, one can not deduce the statistical and geometrical properties of the open matrix given only the statistical and geometrical properties of the closed matrix.     

Recognition of geochemical footprints of mineral systems in the regolith at regional to continental scales

Understanding the character of Australia's extensive regolith cover is crucial to the continuing success of mineral exploration. We hypothesise that the regolith contains geochemical fingerprints of processes related to the development and preservation of mineral systems at a range of scales. We test this hypothesis by analysing the composition of surface sediments within greenfield regional-scale (southern Thomson Orogen) and continental-scale (Australia) study areas. In the southern Thomson Orogen area, the first principal component (PC1) derived in our study [Ca, Sr, Cu, Mg, Au and Mo at one end; rare earth elements (REEs) and Th at the other] is very similar to the empirical vector used by a local company (enrichment in Sr, Ca and Au concomitant with depletion in REEs) to successfully site exploration drill holes for Cu–Au mineralisation. Mapping of the spatial distribution of PC1 in the region reveals several areas of elevated values and possible mineralisation potential. One of the strongest targets in the PC1 map is located between Brewarrina and Bourke in northern New South Wales. Here, exploration drilling has intersected porphyry Cu–Au mineralisation with up to 1 wt% Cu, 0.1 g/t Au, and 717 ppm Zn. The analysis of a comparable geochemical dataset at the continental scale yields a compositionally similar PC1 (Ca, Sr, Mg, Cu, Au and Mo at one end; REEs and Th at the other) to that of the regional study. Mapping PC1 at the continental scale shows patterns that (1) are spatially compatible with the regional study and (2) reveal several geological regions of elevated values, possibly suggesting an enhanced potential for porphyry Cu–Au mineralisation. These include well-endowed mineral provinces such as the Curnamona and Capricorn regions, but also some greenfield regions such as the Albany-Fraser/western Eucla, western Murray and Eromanga geological regions. We conclude that the geochemical composition of Australia's regolith may hold critical information pertaining to mineralisation within/beneath it.     

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.     

The analysis of compositional effects as exemplified by the study of elections

Some people are influenced by other people who live near them. They vote for the same political party as their neighbours and they go to church on Sunday because that is what most other people in the area do. This phenomenon is known as the compositional effect of the environment. This article offers a survey of the history of the analysis of such effects, of the explanations that can be given for them and of the research methods that can be used to establish them. The most important conclusions regarding political preference are given, and a direction in which research into compositional effects should develop is also indicated.