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1.
BLU Estimators and Compositional Data 总被引：5，自引：0，他引：5
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. 相似文献
2.
Large rivers are a major pathway for the erosion products of continents to reach the oceans. The riverine transport of dissolved
and particulate materials is generally related to a large number of interactions involving climate, hydrological, physico-chemical
and biological aspects. Consequently, the investigation of large rivers allows the erosion processes at a global scale to
be addressed, with information about biogeochemical cycles of the elements, weathering rates, physical erosion rates and CO2 consumption by the acid degradation of continental rocks. Today, good databases exist for the major dissolved ions in the
world’s largest rivers. Since concentration of ions in river waters has to be considered in a compositional context, it is
necessary to study the implications of considering the simplex, with its proper geometry, as the natural sample space. Using
the additive (alr) or the isometric (ilr) log-ratio transformations, a composition can be represented as a real vector; but
only in the second case can these coordinates be mapped onto orthogonal axes.
Using data related to the dissolved load of some of the most important rivers in the world, the relationships among the major
ions frequently used in molar ratio mixing diagrams have been investigated with alternative tools. Following the balances
approach, an investigation of the properties of aqueous solutions of electrolytes that are often treated in terms of equilibrium
constants is undertaken. The role played by the source—rain water, weathering of silic, carbonatic and evaporitic rocks, pollution—from
which elements and chemical species can potentially be derived, has been checked through an investigation of a probabilistic
model able to describe the relationships among the different components that contribute to the chemical composition of a river
water sample. 相似文献
3.
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. 相似文献
4.
Hilmar von Eynatten Carles Barceló-Vidal Vera Pawlowsky-Glahn 《Mathematical Geology》2003,35(3):231-251
Perturbation on the simplex is an operation which can be used to numerically describe changes in the composition of, for example, soils, sediments, or rocks. The combination of perturbation and power transformation provides a strong tool for analyzing compositional linear processes in the simplex. When the process is constrained in the sense of a well-known starting (or final) composition, noncentred principal component analysis can be used to estimate the leading perturbation vector of the process. Applying these mathematical tools to chemical major element data from a weathering profile developed on granitoid rocks allows us to model the compositional changes associated with the process of chemical weathering. The comparison of these results with the compositional linear trend defined by erosional products of several of the world's major drainage systems yields close similarities. The latter observation allows for a mathematical formulation of a global mean weathering trend within the system Al2O3–CaO– Na2O– K2O. We further demonstrate the usefulness of the approach for validating processes behind individual trends and for combining the effects of different processes which modify the composition of soils, sediments, and rocks. Alternatives to the Chemical Index of Alteration (CIA) are discussed to obtain a translation-invariant scale for the process of chemical weathering. 相似文献
5.
Simplicial Indicator Kriging 总被引：2，自引：0，他引：2
Indicator kriging (IK) is a spatial interpolation technique devised for estimating a conditional cumulative distribution function at an unsampled location. The result is a discrete approximation, and its corresponding estimated probability density function can be viewed as a composition in the simplex. This fact suggested a compositional approach to IK which, by construction, avoids all its standard drawbacks (negative predictions, not-ordered or larger than one). Here, a simple algorithm to develop the procedure is presented. 相似文献
6.
John Aitchison 《Mathematical Geosciences》1984,16(6):531-564
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. 相似文献
7.
J. Aitchison 《Mathematical Geology》1984,16(6):617-635
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. 相似文献
8.
Application of Compositional Techniques in the Field of Crystal Chemistry: A Case Study of Luzonite, a Sn-Bearing Mineral 总被引：1，自引：1，他引：0
Miria Borgheresi Antonella Buccianti Francesco Di Benedetto David John Vaughan 《Mathematical Geosciences》2013,45(2):183-206
Out-of-equilibrium crystallization often produces complex compositional variability in minerals, generating zoning and other mixing phenomena. The appropriate microchemical characterization of the resulting out-of-equilibrium patterns is of critical importance in understanding the overall physical and chemical properties of the host crystalline phases. In this framework, the modeling of compositional changes assumes a fundamental role. However, when compositional data are used, their management with standard exploratory, statistical, graphical, and numerical tools may give misleading results attributable to the phenomenon of induced correlations. To avoid these problems, methods able to extract compositional data from their constrained space (the simplex) in order to apply standard statistics, have to be adopted. As an alternative, the use of tools having properties able to work in the simplex geometry has to be considered. A luzonite single crystal (ideal composition, Cu3AsS4) exhibiting concentric and sector zoning was studied using electron probe microanalysis in order to understand the mechanisms which give rise to chemical variability and conditions in the developing environment. Compositional variations were determined by collecting data along three different transects. The major and minor elements (Cu, As, S, Fe, Sb, Sn) were analyzed with the aim of characterizing their patterns of association in the crystal and, hence, crystal evolution. The whole covariance structure as well as the chemical relationships between the successive zones was investigated by means of compositional methods, considering both data transformation and the stay in the simplex approach. Results indicate that the crystal grew under quiescent conditions, where chemical control was primarily exercised by the mineral’s surface and only minor effects were due to changes in the composition of the surrounding fluid. Consequently, an oscillatory uptake of chemical components occurred in which a competition between famatinite-like (Cu3SbS4) and kuramite-like (Cu3SnS4) domains characterized the As-poor zones. 相似文献
9.
A variety of approaches to the testing of distributional forms for compositional data has appeared in the literature, all based on logratio or Box–Cox transformation techniques and to a degree dependent on the divisor chosen in the formation of ratios for these transformations. This paper, recognizing the special algebraic–geometric structure of the standard simplex sample space for compositional problems, the use of the fundamental simplicial singular value decomposition, and an associated power-perturbation characterization of compositional variability, attempts to provide a definitive approach to such distributional testing problems. Our main consideration is the characterization and testing of additive logistic–normal form, but we also indicate possible applications to logistic skew normal forms once a full range of multivariate tests emerges. The testing strategy is illustrated with both simulated data and applications to some real geological compositional data sets. 相似文献
10.
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。 相似文献
11.
Dealing with Compositional Data: The Freeware CoDaPack 总被引：2，自引：0，他引：2
12.
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. 相似文献
13.
Dealing with Zeros and Missing Values in Compositional Data Sets Using Nonparametric Imputation 总被引：5，自引：0，他引：5
The statistical analysis of compositional data based on logratios of parts is not suitable when zeros are present in a data set. Nevertheless, if there is interest in using this modeling approach, several strategies have been published in the specialized literature which can be used. In particular, substitution or imputation strategies are available for rounded zeros. In this paper, existing nonparametric imputation methods—both for the additive and the multiplicative approach—are revised and essential properties of the last method are given. For missing values a generalization of the multiplicative approach is proposed. 相似文献
14.
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. 相似文献
15.
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. 相似文献
16.
《地学前缘(英文版)》2020,11(5):1681-1693
From a compilation of geochemical data for the discrimination of the tectonic settings of mid-ocean ridge(MOR;3730 samples) and oceanic plateau(OP;3656 samples),we present two new discriminant functions and diagrams obtained from censored multivariate discordant outlier-free datasets.Ten different sets of data(original concentrations as well as isometric log-ratio transformed(ilr) variables;all 10 major(M) elements as well as all 10 major and 6 trace elements MT) were used to evaluate the quality of discrimination from linear discriminant analysis(LDA) and canonical analysis.Two selected multidimensional models ilrM(9 ilr transformed variables of multi-normally distributed 10 major elements) and ilrMT(15 ilr transformed variables of multi-normally distributed combined 10 major and 6 trace elements),considered as the best or most representative(total of5650 samples for ilr_M and 2858 for ilrMT),provided percent success values,respectively,of 80.9% for the MOR and 81.1% for the OP(ilr_M) and 88.5% for the MOR and 90.1% for the OP(ilrMT).Both processes(log-ratio transformation and multi-normality) rendered the percent success values similar for both groups(MOR and OP).The respective discriminant functions were successfully used for four tests from known tectonic settings and four application cases(two for ophiolites and two for Precambrian rocks),documenting thus the utility of the new discrimination procedure for the MOR and OP tectonic settings.Furthermore,we showed that our multidimensional procedure is robust against analytical errors or uncertainties,as well as post-emplacement compositional changes caused by element mobility from both low or high temperature alteration.The robustness against the gain or loss of a single element at a time was also documented,from which the ilr_(MT) model was evaluated as more robust than the ilr_M model.A new online computer program MOROPdisc was written in Java Framework ZK,which is freely available for use at our web portal http://tlaloc.ier.unam.mx. 相似文献
17.
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. 相似文献
18.
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. 相似文献
19.
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. 相似文献
20.
Progressive evolution of whole‐rock composition during metamorphism revealed by multivariate statistical analyses 
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下载免费PDF全文 Kenta Yoshida Tatsu Kuwatani Takao Hirajima Hikaru Iwamori Shotaro Akaho 《Journal of Metamorphic Geology》2018,36(1):41-54
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. 相似文献