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1.
Dealing with Compositional Data: The Freeware CoDaPack 总被引：2，自引：0，他引：2
2.
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
3.
Roger Banks 《Contributions to Mineralogy and Petrology》1979,70(3):237-244
Petrological mixing problems such as modal analysis, magma mixing, and liquid line of descent calculations, can be solved using the methods of linear programming. If estimates of the standard error of the chemical data are introduced as weights into the set of equations, it is possible to assign confidence limits to the solutions which are obtained and to apply formal statistical tests to geological hypotheses based on the mixing model. This approach is applied to petrological data previously analysed by Wright and Doherty (1970) using a combination of linear programming and least squares methods. It is shown that some of the geological inferences which they drew were based on an overoptimistic assessment of the confidence limits on their solutions, and cannot be regarded as proven. 相似文献
4.
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
5.
Ross M. Renner 《Environmental Earth Sciences》2012,67(4):987-997
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. 相似文献
6.
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
7.
Logratio Analysis and Compositional Distance 总被引：10，自引：0，他引：10
J. Aitchison C. Barceló-Vidal J. A. Martín-Fernández V. Pawlowsky-Glahn 《Mathematical Geology》2000,32(3):271-275
The concept of distance between two compositions is important in the statistical analysis of compositional data, particularly in such activities as cluster analysis and multidimensional scaling. This paper exposes the fallacies in a recent criticism of logratio-based distance measures—in particular, the misstatements that logratio methods destroy distance structures and are denominator dependent. Emphasis is on ensuring that compositional data analysis involving distance concepts satisfies certain logically necessary invariance conditions. Logratio analysis and its associated distance measures satisfy these conditions. 相似文献
8.
This paper is part of a special issue of Applied Geochemistry focusing on reliable applications of compositional multivariate statistical methods. This study outlines the application of compositional data analysis (CoDa) to calibration of geochemical data and multivariate statistical modelling of geochemistry and grain-size data from a set of Holocene sedimentary cores from the Ganges-Brahmaputra (G-B) delta. Over the last two decades, understanding near-continuous records of sedimentary sequences has required the use of core-scanning X-ray fluorescence (XRF) spectrometry, for both terrestrial and marine sedimentary sequences. Initial XRF data are generally unusable in ‘raw-format’, requiring data processing in order to remove instrument bias, as well as informed sequence interpretation. The applicability of these conventional calibration equations to core-scanning XRF data are further limited by the constraints posed by unknown measurement geometry and specimen homogeneity, as well as matrix effects. Log-ratio based calibration schemes have been developed and applied to clastic sedimentary sequences focusing mainly on energy dispersive-XRF (ED-XRF) core-scanning. This study has applied high resolution core-scanning XRF to Holocene sedimentary sequences from the tidal-dominated Indian Sundarbans, (Ganges-Brahmaputra delta plain). The Log-Ratio Calibration Equation (LRCE) was applied to a sub-set of core-scan and conventional ED-XRF data to quantify elemental composition. This provides a robust calibration scheme using reduced major axis regression of log-ratio transformed geochemical data. Through partial least squares (PLS) modelling of geochemical and grain-size data, it is possible to derive robust proxy information for the Sundarbans depositional environment. The application of these techniques to Holocene sedimentary data offers an improved methodological framework for unravelling Holocene sedimentation patterns. 相似文献
9.
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). 相似文献
10.
Water chemistry is commonly investigated to determine the suitability of water for various uses. With increased knowledge
of aqueous chemistry, it has become possible to interpret the evolutionary processes that determine water composition and
quality. This paper presents procedures for exploring and modeling the environment using compositional data from water analysis,
utilizing statistical tools in an appropriate sample space. Our procedures build on a methodology based on log-ratios initiated
by John Aitchison in the early 1980's. They are not only useful for interpreting the structure of the data, but also for characterizing
and modeling the influence of geochemical processes acting on the environment. The geochemistry of water samples collected
from wells on Vulcano Island (one of the Aeolian Islands of the Italian province of Sicily) will be used to illustrate the
techniques, although an exhaustive overview would require many different examples. Vulcano island is a quiescent volcanic
area where mobilization of chemical species by weathering of volcanic rocks and input of gaseous components from fumarolic
activity has produced environmental changes expressed in the composition of phreatic waters at the surface and in the shallow
subsurface. Changes in the chemical composition of waters in unconfined aquifers of the northwestern part of the island around
the active crater appear to be useful in understanding the natural processes at work. 相似文献
11.
Glassy nuclear fallout debris from near-surface nuclear tests is fundamentally reprocessed earth material. A geochemical approach to analysis of glassy fallout is uniquely suited to determine the means of reprocessing and shed light on the mechanisms of fallout formation. An improved understanding of fallout formation is of interest both for its potential to guide post-detonation nuclear forensic investigations and in the context of possible affinities between glassy debris and other glasses generated by high-energy natural events, such as meteorite impacts and lightning strikes. This study presents a large major-element compositional dataset for glasses within aerodynamic fallout from the Trinity nuclear test (“trinitite”) and a geochemically based analysis of the glass compositional trends. Silica-rich and alkali-rich trinitite glasses show compositions and textures consistent with formation through melting of individual mineral grains—quartz and alkali feldspar, respectively—from the test-site sediment. The volumetrically dominant glass phase—called the CaMgFe glass—shows extreme major-element compositional variability. Compositional trends in the CaMgFe glass are most consistent with formation through volatility-controlled condensation from compositionally heterogeneous plasma. Radioactivity occurs only in CaMgFe glass, indicating that co-condensation of evaporated bulk ground material and trace device material was the main mechanism of radioisotope incorporation into trinitite. CaMgFe trinitite glasses overlap compositionally with basalts, rhyolites, fulgurites, tektites, and microtektites but display greater compositional diversity than all of these naturally formed glasses. Indeed, the most refractory CaMgFe glasses compositionally resemble early solar system condensates—specifically, CAIs. 相似文献
12.
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
13.
Diverse global and local measures of variability appear in the geological literature and, along with methods used to apply them, have been subject to some debate. Measures of variability for three data types—replicate, locational, and compositional—are considered; the source and nature of the variability determine the appropriate type of measure. To illustrate the effects of these measures and expose their inadequacy when improperly applied, the variability of a three-column data set is interpreted under three different suppositions. Geologists need to be aware of the confusion and misleading results that can develop from the use of variance as a measure of variability for locational or compositional data. 相似文献
14.
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
15.
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
16.
The behavior and properties of sediments depend on their compositional characteristics and formation processes, as well as the environmental conditions during their geological history, i.e. post-formation processes. A vertical cut made in a hill in Dhahran, Saudi Arabia, reveals a vivid picture of the inherent heterogeneity of sediments that have been deposited at different geological ages. A review of the geology of the area, as well as laboratory tests, help to determine the possible causes of the variability of soil types and properties in the area. Laboratory tests include basic geotechnical tests, chemical tests, X-ray diffraction analysis, scanning electron microscopy, and thermal analysis. These tests are used to identify different rock types and soils from the face of the cut. The results of this study indicate that the material from this cut varies from clayey shale and limestone rock (Tertiary, lower Eocene) formed some 52 M.Y.
to calcite-cemented sand and pure calcite rock formed in the Quaternary age. 相似文献
17.
Thermal groundwater is currently being exploited for district-scale heating in many locations world-wide. The chemical compositions of these thermal waters reflect the provenance and circulation patterns of the groundwater, which are controlled by recharge, rock type and geological structure. Exploring the provenance of these waters using multivariate statistical analysis (MSA) techniques increases our understanding of the hydrothermal circulation systems, and provides a reliable tool for assessing these resources.Hydrochemical data from thermal springs situated in the Carboniferous Dublin Basin in east-central Ireland were explored using MSA, including hierarchical cluster analysis (HCA) and principal component analysis (PCA), to investigate the source aquifers of the thermal groundwaters. To take into account the compositional nature of the hydrochemical data, compositional data analysis (CoDa) techniques were used to process the data prior to the MSA.The results of the MSA were examined alongside detailed time-lapse temperature measurements from several of the springs, and indicate the influence of three important hydrogeological processes on the hydrochemistry of the thermal waters: 1) salinity and increased water-rock interaction; 2) dissolution of carbonates; and 3) dissolution of sulfides, sulfates and oxides associated with mineral deposits. The use of MSA within the CoDa framework identified subtle temporal variations in the hydrochemistry of the thermal springs, which could not be identified with more traditional graphing methods, or with a standard statistical approach. The MSA was successful in distinguishing different geological settings and different annual behaviours within the group of springs. This study demonstrates the usefulness of the application of MSA within the CoDa framework in order to better understand the underlying controlling processes governing the hydrochemistry of a group of thermal springs in a low-enthalpy setting. 相似文献
18.
Compositional Geometry and Mass Conservation 总被引：1，自引：0，他引：1
Robert F. Shurtz 《Mathematical Geology》2003,35(8):927-937
A geometrical structure is imposed on compositional data by physical and chemical laws, principally mass conservation. Therefore, statistical or mathematical investigation of possible relations between data values and such laws must be consistent with this structure. This demands that geometrical concepts, such as points that specify both mass and composition in linear space, and lines in projective space that specify composition only, be clearly defined and consistent with mass conservation. Mass thus becomes the norm in composition space in place of the Euclidean norm of ordinary space. Coordinate transformations inconsistent with this geometry are accordingly unnatural and misleading. They are also unnecessary because correlation arising from the constant mass presents no unusual difficulty in the analysis of the underlying quadratic form. 相似文献
19.
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
20.
A new approach to null correlations of proportions 总被引：1，自引：0，他引：1
J. Aitchison 《Mathematical Geology》1981,13(2):175-189
Much work on the statistical analysis of compositional data has concentrated on the difficulty of interpreting correlations between proportions with an assortment of tests for nullcorrelations, for independence except for the constraint, F-independence of bounded variables, neutrality in the mean and in the median. This paper questions the appropriateness of characterizing the dependence structure of proportions in terms of such concepts, suggests an alternative method of modeling, develops necessary distribution theory and tests, and illustrates the methodology in applications. 相似文献