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1.
Like compositions in general, regionalized compositions present the problem of spurious spatial correlation. To avoid this problem, this paper uses the additive-logratio transformation of regionalized compositions, following techniques introduced over the last few years for the statistical analysis of compositional data. It leads to an appropriate definition of a spatial covariance structure to describe spatial dependence between regionalized variables subject to constant-sum constraints in the case of weak stationarity. To illustrate stated problems, simulated data are used. 相似文献
2.
The study of hydrogeochemical data sets frequently calls for statistical dimension reducing techniques. It is well known that
hydrochemical parameters are compositions and, for this type of data, the direct application of classical statistical methods
based on the correlation matrix yield spurious results. But new results on compositional data analysis have identified the
sampling space, the simplex, with an Euclidean space, a fact that allows us to define a simplicial factor analysis strategy,
thus overcoming the problem. For illustration, we use samples from the Llobregat River and its tributaries (NE Spain). Three
unobservable or latent factorial components are extracted, which are identified with pristine waters, potash-mining influence
and urban sewage influence. These three factorial components or compositional factors are plotted in a factorial ternary diagram,
which reflects the relative influence of each one of these factors on each observation. 相似文献
3.
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
4.
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
5.
E. H. Timothy Whitten 《Mathematical Geology》1995,27(6):789-806
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. 相似文献
6.
J. Aitchison 《Mathematical Geology》1990,22(4):487-511
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. 相似文献
7.
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
8.
Isometric Logratio Transformations for Compositional Data Analysis 总被引：37，自引：0，他引：37
J. J. Egozcue V. Pawlowsky-Glahn G. Mateu-Figueras C. Barceló-Vidal 《Mathematical Geology》2003,35(3):279-300
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. 相似文献
9.
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
10.
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
11.
Javier Palarea-Albaladejo Josep A. Martín-Fernández Juan Gómez-García 《Mathematical Geology》2007,39(7):625-645
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. 相似文献
12.
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
13.
Compositional data analysis 总被引：1，自引：0，他引：1
Richard A. Reyment 《地学学报》1989,1(1):29-34
Compositional data occur naturally in the geosciences — tables of chemical analyses, rock-compositions, sedimentary proportions, pollen-analytical tables, etc. The statistical analysis of such data requires special techniques and it is not possible to use standard methods of computing correlation coefficients and carry out multivariate statistical analyses without the risk of incurring grave mistakes. The special property of compositional data, to wit, the fact that the determinations on each specimen sum to a constant, means that the variables involved in the study occur in constrained space defined by the simplex , a restricted part of real space. 相似文献
14.
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
15.
Assessing spatial uncertainty in mapping soil erodibility factor using geostatistical stochastic simulation 总被引：1，自引：0，他引：1
Soil erosion is one of most widespread process of degradation. The erodibility of a soil is a measure of its susceptibility
to erosion and depends on many soil properties. Soil erodibility factor varies greatly over space and is commonly estimated
using the revised universal soil loss equation. Neglecting information about estimation uncertainty may lead to improper decision-making.
One geostatistical approach to spatial analysis is sequential Gaussian simulation, which draws alternative, equally probable,
joint realizations of a regionalised variable. Differences between the realizations provide a measure of spatial uncertainty
and allow us to carry out an error analysis. The objective of this paper was to assess the model output error of soil erodibility
resulting from the uncertainties in the input attributes (texture and organic matter). The study area covers about 30 km2 (Calabria, southern Italy). Topsoil samples were collected at 175 locations within the study area in 2006 and the main chemical
and physical soil properties were determined. As soil textural size fractions are compositional data, the additive-logratio
(alr) transformation was used to remove the non-negativity and constant-sum constraints on compositional variables. A Monte
Carlo analysis was performed, which consisted of drawing a large number (500) of identically distributed input attributes
from the multivariable joint probability distribution function. We incorporated spatial cross-correlation information through
joint sequential Gaussian simulation, because model inputs were spatially correlated. The erodibility model was then estimated
for each set of the 500 joint realisations of the input variables and the ensemble of the model outputs was used to infer
the erodibility probability distribution function. This approach has also allowed for delineating the areas characterised
by greater uncertainty and then to suggest efficient supplementary sampling strategies for further improving the precision
of K value predictions. 相似文献
16.
Perturbation is an operation defined on the simplex and can be used for centering compositional data in a ternary diagram, applying objective criteria. Because a straight line in the original diagram is still astraight line in the perturbed diagram, gridlines or compositional fields defined by straight lines can easily be included in the operation. Simultaneous perturbation of data, gridlines, and/or compositional fields is shown to improve both visualization and graphical interpretation of compositions in ternary diagrams. This is illustrated by some examples using simulated as well as real data. 相似文献
17.
安徽省兆吉口铅锌矿床成矿地球化学机制研究 总被引：1，自引：0，他引：1
自20世纪30年代起, 勘查地球化学就在矿产资源勘查领域发挥着重要作用。目前, 矿业界对勘查地球化学回归到基础勘查理论研究有着明确的需要。多维异常体系应用基础理论的提出是我国学者在该领域的积极响应。多维异常体系定义为“在特定的成矿地质时期, 成矿系统中存在的空间有序共存、成因机理各异、成矿指示递进的多属性异常体系”。其中, 多属性异常的形成机制及其在成矿空间中的结构关系, 是探讨矿床成矿地球化学机制和指导矿产勘查的基础, 同时也是勘查地球化学研究的前沿方向。本论文以位于安徽省东至县的兆吉口浅成低温热液型铅锌矿床为研究对象, 通过元素质量迁移定量计算, 研究典型剖面上元素活动规律, 构建矿致异常结构模式, 揭示矿床成矿地球化学机制; 利用图示方法展现不同水平断面上元素异常分布形态, 为深部成矿预测指明方向; 利用分形模型和基于成分数据理论的主成分分析、因子分析等方法, 研究地表岩屑中元素分布特征及影响因素, 指导研究区外围矿床地球化学勘查。 相似文献
18.
Compositional data are very common in the earth sciences. Nevertheless, little attention has been paid to the spatial interpolation of these data sets. Most interpolators do not necessarily satisfy the constant sum and nonnegativity constraints of compositional data, nor take spatial structure into account. Therefore, compositional kriging is introduced as a straightforward extension of ordinary kriging that complies with these constraints. In two case studies, the performance of compositional kriging is compared with that of the additive logratio-transform. In the first case study, compositional kriging yielded significantly more accurate predictions than the additive logratio-transform, while in the second case study the performances were comparable. 相似文献
19.
A Critical Approach to Probability Laws in Geochemistry 总被引：2，自引：0，他引：2
Probability laws in geochemistry have been a major issue of concern over the last decades. The lognormal on the positive real
line or the additive logistic normal on the simplex are two classical laws of probability to model geochemical data sets due
to their association with a relative measure of difference. This fact is not fully exploited in the classical approach when
viewing both the positive real line and the simplex as subsets of real space with the induced geometry. But it can be taken
into account considering them as real linear vector spaces with their own structure. This approach implies using a particular
geometry and a measure different from the usual ones. Therefore, we can work with the coordinates with respect to an orthonormal
basis. It could be shown that the two mentioned laws are associated with a normal distribution on the coordinates. In this
contribution both approaches are compared, and a real data set is used to illustrate similarities and differences. 相似文献
20.
月球表面的元素和物质成分分布是理解月球成岩与地质演化历史的重要线索。嫦娥一号干涉成像光谱仪(IIM)是我国首台月球探测成像光谱仪器,其获得的大量月球高光谱数据已成为我国未来探测月球成分与地质演化研究的宝贵基础数据。本文利用探月工程地面应用系统发布的IIM B版本2C级数据,开发出一套数据再定标流程,获得了较为可靠的月表相对反射率数据。我们在新校正数据的基础上开展月球表面FeO、TiO_2的反演建模,获得了全月FeO和TiO_2分布图,这些图件是进行月球地质填图的基础。校正数据反演的FeO和TiO_2分布与前人对Clementine UVVIS数据的反演结果相近,表明干涉成像光谱仪数据具有较大的应用潜力。高地的低铁岩石成分(一般小于8%)佐证了月球月壳形成的过程中的岩浆洋分异假说,而月海玄武岩的TiO_2成分变化范围较大(0~13%)则表明月海玄武岩来源于不同的月幔源区。根据嫦娥一号干涉成像光谱仪全月FeO分布图,可将月球表面物质类型总体划分为高地斜长岩和月海玄武岩,而根据TiO_2分布可以进一步将月海玄武岩划分为5种不同钛含量的玄武岩岩石类型。FeO和TiO_2在全月范围内的分布表明Apollo和Luna返回的月球样品不能够代表全月范围内的矿物成分多样性,月球岩浆演化历史比前人认为的要复杂。未来月球样品返回任务(如嫦娥五号)如能赴这些特殊地区进行取样,将很有可能返回重要的月球科学研究发现和成果。 相似文献