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1.
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   

2.
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   

3.
Isometric Logratio Transformations for Compositional Data Analysis   总被引：37，自引：0，他引：37  
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   

4.
Multivariate statistical procedures are applied to pisolitic laterite geochemistry in a study of the Golden Grove massive sulphide district. The objective is to optimize identification of geochemical anomalies caused by base metal mineral deposits.The statistical approach used in this paper depends upon geochemical data for appropriate reference groups (or training sets) being available. The target group consists of orientation data from pisolitic laterite about the Gossan Hill Cu-Zn massive sulphide deposit. A group representing background sequence was selected by combining three subareas in a geochemically quiet part of the prospective acid volcano-sedimentary sequence.A multi-element allocation procedure was set up using data from the reference groups. The exploration samples are then allocated, one sample at a time, to either one of the reference group categories, using the probability of group membership. A map showing the relative probability values for each sample site is the final product for interpretation, aided by ancillary use of an index of typicality.The allocation procedures were carried out using different element combinations, these being based on a procedure for subset selection to give maximum separation of reference groups, and on geochemical insight. Whilst many versions of the allocation procedure gave positive identification of the anomaly related to the blind Scuddles Cu-Zn deposit, allocation using only Cu, Pb, Zn and Ag did not. The results emphasize the importance of pathfinder elements in geochemical studies in weathered terrain.The allocation procedure using the most appropriate element combinations provided more positive identification of the main areas of known mineralization than had the previously used empirically derived methods of Smith and Perdrix (1983). The formal allocation procedure has the following additional advantages: results are not markedly affected by a very high value for any single element since robust procedures are incorporated into the analysis; better discrimination appears to be possible for weaker anomalies; separation of target from background can be optimized by formal calculations instead of by trial and error; and better suppression of background variation results.  相似文献   

5.
Geochemical exploration in secondary environments can be viewed as a particular manifestation of indirect geological observation. Geochemical anomalies in complex sample media reflect dispersion signatures, generally much disguised by secondary or higher-order mechanical and physico-chemical processes such as mixing, comminution, dilution, (re)transportation, weathering etc. Such complexities often make a thorough understanding of the origin of any particular sample type difficult ot obtain. The objective of data analysis in this context is to convert the geochemical data into a meaningful “signal”, particularly useful for prospecting, and other, in this case irrelevant, variability or “noise”. The experience of the last decades of practical exploration has clearly shown that statistical as well as geographical geochemical anomaly patterns are multi-element signatures. Using suitable multivariate statistical procedures (in the present case principal components modelling), it is possible to simultaneously define both a background data model and to quantify multivariate geochemical anomalies. This type of data analysis is guided very strongly by geological interaction, in which the emphasis is on modelling the background population(s), coupled with geographic plotting facilities. This outlier-screening facility is critical for many types of geochemical data evaluation. An example of this approach is described below. Another application of indirect multivariate data analysis is represented by PLS (Partial Least Squares) regression, which is a supervised pattern recognition and regression technique. We use it here to predict modal scheelite occurrences from regional stream-sediment data.  相似文献   

6.
Amphibole/liquid partition coefficients for the REE(Damph/liqREE) obtained from natural rocks increase systematically with bulk rock compositional change from basalt to rhyolite and are higher for the middle to heavy REE. Five new experimentally determined sets of DREE (La, Sm, “Eu2+”, Ho, Lu)and 4 published sets are similar to these data and provide values for use in geochemical modelling of magmatic processes involving amphibole, over a range of temperature, pressure and oxygen fugacity. The partition coefficients increase significantly with decreasing temperature, and increase slightly with increasing oxygen fugacity. Pressure does not appear to have a major effect, although one data set suggests increased pressure lowers Damph/liqREE in a basaltic composition.  相似文献   

7.
Measuring Subcompositional Incoherence   总被引：2，自引：0，他引：2  
Subcompositional coherence is a fundamental property of Aitchison’s approach to compositional data analysis and is the principal justification for using ratios of components. We maintain, however, that lack of subcompositional coherence (i.e., incoherence) can be measured in an attempt to evaluate whether any given technique is close enough, for all practical purposes, to being subcompositionally coherent. This opens up the field to alternative methods that might be better suited to cope with problems such as data zeros and outliers while being only slightly incoherent. The measure that we propose is based on the distance measure between components. We show that the two-part subcompositions, which appear to be the most sensitive to subcompositional incoherence, can be used to establish a distance matrix that can be directly compared with the pairwise distances in the full composition. The closeness of these two matrices can be quantified using a stress measure that is common in multidimensional scaling, providing a measure of subcompositional incoherence. The approach is illustrated using power-transformed correspondence analysis, which has already been shown to converge to log-ratio analysis as the power transform tends to zero.  相似文献   

8.
Chemical reactions in aqueous geochemical systems are driven by nonequilibrium conditions, and their dynamics can be deduced through the distributional analysis (identification of probability laws) of complex compositional indices. In this perspective, compositional data analysis offers the possibility to investigate the behavior of the composition as a whole instead of isolated chemical species, with the awareness that multispecies systems are characterized by the simultaneous interactions among all their parts. We addressed this problem using D???1 isometric log-ratio coordinates describing the D compositional dataset of the river chemistry of the Alpine region (D number of variables), thus working in the \({{\mathbb{R}}^{D - 1}}\) statistical sample space. The D???1 coordinates were chosen using the decreasing variance criterion so that each one could provide information about different space–time properties for the investigated geochemical system. Coordinates dominated by heterogeneity appear to be able to capture regime shifts only on a long-time period and monitor processes on a very wide scale. On the other hand, coordinates characterized by lower variability present multimodality, thus capturing the presence of alternative states in the analyzed spatial domain also for the current time. Further developments are needed to determine the ranges of conditions for which variability and other statistics can be useful indicators of regime shifts on different time–space scales in geochemical systems.  相似文献   

9.
Joint Consistent Mapping of High-Dimensional Geochemical Surveys   总被引：1，自引：0，他引：1  
Geochemical surveys often contain several tens of components, obtained from different horizons and with different analytical techniques. These are used either to obtain elemental concentration maps or to explore links between the variables. The first task involves interpolation, the second task principal component analysis (PCA) or a related technique. Interpolation of all geochemical variables (in wt% or ppm) should guarantee consistent results: At any location, all variables must be positive and sum up to 100 %. This is not ensured by any conventional geostatistical technique. Moreover, the maps should ideally preserve any link present in the data. PCA also presents some problems, derived from the spatial dependence between the observations, and the compositional nature of the data. Log-ratio geostatistical techniques offer a consistent solution to all these problems. Variation-variograms are introduced to capture the spatial dependence structure: These are direct variograms of all possible log ratios of two components. They can be modeled with a function analogous to the linear model of coregionalization (LMC), where for each spatial structure there is an associated variation matrix describing the links between the components. Eigenvalue decompositions of these matrices provide a PCA of that particular spatial scale. The whole data set can then be interpolated by cokriging. Factorial cokriging can also be used to map a certain spatial structure, eventually projected onto those principal components (PCs) of that structure with relevant contribution to the spatial variability. If only one PC is used for a certain structure, the maps obtained represent the spatial variability of a geochemical link between the variables. These procedures and their advantages are illustrated with the horizon C Kola data set, with 25 components and 605 samples covering most of the Kola peninsula (Finland, Norway, Russia).  相似文献   

10.
Do We Really Need Mantle Components to Define Mantle Composition?   总被引：2，自引：0，他引：2  
We discuss the concept of components in the Earth's mantle startingfrom a petrological and geochemical approach, but adopting anew method of projection of geochemical and isotopic data. Thisallows the compositional variability of magmatic associationsto be evaluated in multi-dimensional space, thus simultaneouslyaccounting for a large number of compositional variables. Wedemonstrate that ocean island basalts (OIB) and mid-ocean ridgebasalts (MORB) are derived from a marble-cake mantle, in whichdifferent degrees of partial melting of recycled lithosphere,which are heterogeneous in age and composition, contribute tothe magma genesis. This view is supported by the variabilityin the geochemical and isotopic signatures of OIB that are observedon the scale of a single ocean island as well as on that ofan ocean, mostly varying between two extreme compositions, thatare not strictly related to the commonly accepted mantle components(DMM, EMI, EMII, HIMU). Rather they are a distinctive featureof the mantle source sampled at each ocean island and are stronglydependent on the Pb isotope system. We recommend a change inperspective in studies of MORB–OIB geochemistry from onebased on physically distinct mantle components to a model basedon the existence of a marble-cake-like upper mantle. Althoughresembling the statistical upper mantle, this model impliesthat geochemical homogenization can be attained only withinthe limits of local mantle composition, so that a world-wideuniform depleted reservoir cannot be sampled by simply extendingthe volume of the region undergoing partial melting. KEY WORDS: geochemistry; isotope; mantle; OIB  相似文献   

11.
Biodegradation of crude oil causes volumetrically important compositional changes, which lead to significant deterioration in quality, in particular during the early stages of alteration. To better understand these effects we focussed on a detailed assessment of light to moderate levels of alteration. We investigated a suite of 40 crude oil samples from five different petroleum systems to evaluate the extent of alteration occurring in reservoirs. Based on a comprehensive geochemical characterization, five individual crude oil sequences were defined, where compositional variability is mainly due to microbial activity in the reservoir. In particular, samples from the Gullfaks field (offshore Norway) and from a petroleum system offshore Angola illustrate that conventional molecular biodegradation parameters, such as the Pr/n-C17 and Ph/n-C18 alkane ratios are not suitable for defining the extent of biodegradation in petroleum reservoirs. Here, we suggest a new molecular biodegradation parameter, the degradative loss, that can be used to quantify depletion in individual crude oil constituents. The approach allows improved assessment of the extent of biodegradation in crude oil samples by means of the mean degradative loss. It is demonstrated that crude oil quality, as assessed from API gravity, can be predicted directly from the molecular composition of crude oils. Our data clearly indicate that the degradation patterns of light hydrocarbons and n-alkanes differ in different petroleum systems. This suggests that microbial communities are different and therefore generate different molecular degradation patterns which have to be evaluated individually for each system.  相似文献   

12.
Commonly used methods for calculating component scores are reviewed. Means, variances, and the covariance structures of the resulting sets of scores are examined both by calculations based on a large set of electron microprobe analyses of melilite (supplied by D. Velde)and by a survey of recent geological applications of principal component analysis. Most of the procedures used to project raw data into the new vector space yield uncorrelated scores. In exceptions so far encountered, correlations between scores seem to have been occasioned by the use of unstandardized variables with components calculated from a correlation matrix. In a number of cases substantive interpretations of such correlations have been proposed. A different set of correlations results for the same data if scores are computed from standardized variables and components based on the covariance matrix. If unscaled components are rotated by the varimax procedure, the result is a return to the original space. In the work reported here, nevertheless, scores calculated from varimax-rotated scaled vectors are uncorrelated.  相似文献   

13.
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   

14.
Compositional data analysis requires selecting an orthonormal basis with which to work on coordinates. In most cases this selection is based on a data driven criterion. Principal component analysis provides bases that are, in general, functions of all the original parts, each with a different weight hindering their interpretation. For interpretative purposes, it would be better to have each basis component as a ratio or balance of the geometric means of two groups of parts, leaving irrelevant parts with a zero weight. This is the role of principal balances, defined as a sequence of orthonormal balances which successively maximize the explained variance in a data set. The new algorithm to compute principal balances requires an exhaustive search along all the possible sets of orthonormal balances. To reduce computational time, the sets of possible partitions for up to 15 parts are stored. Two other suboptimal, but feasible, algorithms are also introduced: (i) a new search for balances following a constrained principal component approach and (ii) the hierarchical cluster analysis of variables. The latter is a new approach based on the relation between the variation matrix and the Aitchison distance. The properties and performance of these three algorithms are illustrated using a typical data set of geochemical compositions and a simulation exercise.  相似文献   

15.
Isothermal or isobaric phase diagram sections as a function of fluid composition (X F) are widely used for interpreting the genetic history of metacarbonate rocks. This approach has the disadvantages that: (1) the influence of a key metamorphic variable, either pressure (P) or temperature (T), is obscured; (2) the diagrams are inappropriate for systems that are not fluid-saturated. These problems are avoided by constructing phase-diagram projections in which the volatile composition of the system is projected onto a P-T coordinate frame, i.e., a petrogenetic grid. The univariant curves of such P-T projections trace the conditions of the invariant points of isothermal or isobaric phase-diagram sections, thereby defining the absolute stability of high-variance mineral assemblages, with and without a coexistent fluid phase. Petrogenetic grids for metacarbonate rocks are most useful for the study of regional metamorphism and for systems in which fluid composition has not been externally controlled. A calculated example of a P-T projection for the system CaO−MgO−SiO2−H2O−CO2 suggests that many assemblages (e.g., calcite +tale, enstatite+fluid, magnesite+tremolite, antigorite+diopside+dolomite, and calcite+forsterite+tremolite) in mixed-volatile systems have stability fields that make them useful as P-T indicators. Consideration of the principles governing projection topology demonstrates that the univariant curves around a fluid present invariant point cannot be oriented independently with respect to the direction of compositional variation in the fluid phase. This has the interesting predictive implication that if the direction of compositional variation along one univariant curve around an invariant point is known, then the direction of compositional variation along the remaining curves can be determined solely from topologic constraints. The same constraints can be applied to systems containing simple mineral solutions or melts in order to predict compositional variations.  相似文献   

16.
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   

17.
A new method has been developed to separate the compositional variations in ocean island basalts into those that result from variations in source composition and from the melting process itself. The approach depends on correlations between isotope ratios, which can only come from source inhomogeneities, and elemental concentrations. Analysis of three data sets shows that the inhomogeneities beneath Theistareykir, in NE Iceland, Kilauea and Pitcairn can be produced by subduction of oceanic islands and volcanic ridges. The thicknesses of the lithosphere on which such islands were constructed and potential temperatures of the plumes that produced them can be estimated from the geochemical observations. Model ages are harder to determine, though simple assumptions give about 400 Ma for the Theistareykir source and 1.2 Ga for Kilauea. The model may also provide a physical explanation for the commonly used isotopic classification of ocean island basalts, with the isotopic composition changing from HIMU through EMII to EMI as the melt fraction increases. These results have been obtained from a small number of data sets obtained from ocean island basalts erupted in small areas during short time intervals. More such observations are needed to discover whether geochemical observations from other islands are consistent with the same model.  相似文献   

18.
The activity of a given mineral component in a silicate melt can be calculated from the compositions of coexisting melt and crystals, provided that 1) the component is an independently variable component of the crystal, and 2) appropriate thermodynamic data for the component are known. This approach is used to calibrate the compositional dependence of the activities of forsterite, fayalite, anorthite, and albite from experimental data on natural mafic-to-intermediate melts. The natural logarithms of the activities of forsterite and anorthite can be closely approximated as second-degree polynomial functions of the melt composition (r 2=0.99 and 0.97, respectively); corresponding fits for fayalite and albite are significantly poorer (r 2=0.81 and 0.87, respectively). The shapes of the fitted activity surfaces yield information about speciation in silicate melts. The activity models for forsterite and anorthite provide excellent geothermometers with standard deviations of temperature residuals of approximately 10° C. These geothermometers, when combined with the activity models for fayalite and albite, can be used to predict the temperature at which olivine or plagioclase will crystallize from a melt, along with the composition of the crystals.  相似文献   

19.
Geochemical data are typical compositional data which should be opened prior to univariate and multivariate data analysis. In this study, a frequency-based method (robust principal component analysis, RPCA) and a frequency-space-based method (spectrum–area fractal model, S–A) are applied to explore the effects of the data closure problem and to study the integrated geochemical anomalies associated with polymetallic Cu mineralization using a stream sediment geochemical dataset collected from the Zhongteng district, Fujian Province (China). The results show that: (1) geochemical data should be opened prior to RPCA to avoid spurious correlation between variables; (2) geochemical pattern is a superimposition of multi-processes and should be decomposed; and (3) the S–A fractal model is a powerful tool for decomposing the mixed geochemical pattern.  相似文献   

20.
One of the most important observations that can be obtained from the study of an aquifer system dominated by mixing is the contribution of each end-member water to the chemical composition of every water parcel in the aquifer. Once the first-order effect of mixing has been taken into account via the mixing proportions, water–rock interaction can be used to explain the remaining variability. There are many sources of uncertainty that can prevent the accurate calculation of the mixing proportions of a mixing-dominated system, but the type and intensity of the chemical reactions that have taken place as a consequence of mixing is one of the most critical. Here the uncertainty in the computed mixing proportions of samples from a “synthetic” aquifer system derived from the actuation of different chemical reactions are assessed (always remembering that the chemical reactions are a second-order effect). These uncertainties are explored using two different geochemical codes in order to infer the limits of both methodological approaches: PHREEQC, as an example of a standard geochemical code; and M3, as an example of a Principal Component-based geochemical code. Several synthetic water samples are created with the direct approach of PHREEQC, both by pure mixing and including different types of chemical reactions. Together with the chemical information of the end-member waters, these samples are then fed into PHREEQC (inverse modelling) and M3 and the mixing proportions and mineral mass transfers are computed. PHREEQC calculations give a reasonable estimate of the real mixing proportions and the chemistry of the groundwaters. However, similar mixing proportions and mass transfers can be obtained using different sets of reactions, indicating a source of uncertainty that should be overcome with additional chemical information. For M3, where synthetic samples have been included in a real data set of groundwater samples from the Scandinavian Shield, mixing proportions are only mildly affected either by the number of compositional variables or the number of samples used for the Principal Component Analysis (PCA). However, the robustness of the output is quite sensitive to whether only conservative compositional variables are used or both conservative and non-conservative compositional variables. Mass balance calculations in M3 are much more sensitive to non-conservative compositional variables and the recommendation here is not to use non-conservative variables with PCA-based codes if any information about reactions is to be obtained.  相似文献   

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