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1.
To avoid spurious spatial correlation when analyzing the spatial covariance structure of regionalized compositions, additive-log-ratio transformation can be used. Here, the additive-log-ratio cokriging estimator, derived in a natural way from this transformation, is shown to be invariant under permutation of components of the untransformed regionalized composition. It leads, as expected, to an exact interpolation. As original data, predicted values of the regionalized composition at unknown points add up to the same constant c and lie between 0 and c.  相似文献   

2.
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   

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.
Three garnet-biotite pairs and eleven garnet-cordierite-biotite triplets from the Steinach aureole (Oberpfalz, North-East Bavaria) were analyzed using an electron probe microanalyzer.The regional metamorphic muscovite-biotite schists contain garnets strongly zoned with Mn-Ca-rich centers and Fe-rich edges, the average composition being almandine 67 — spessartine 4 — pyrope 4 — grossular (+andradite) 25.The first contact garnet that is formed in mica schists of the outermost part of the aureole is small, virtually unzoned, and has an average composition of almandine 52 — spessartine 37 — pyrope 8 — grossular (+andradite) 3. With increasing metamorphic grade, there is a consistent trend to form garnets richer in Fe ending up with a composition almandine 84.5 — spessartine 5.5 — pyrope 7.5 — grossular (+andradite) 2.5. This trend is accompanied by a general increase in grain size and modal amount of garnet. Associated cordierites and biotites also become richer in Fe with increasing grade. While the garnets in the highest grade sillimanite hornfelses are poorly zoned, the transitional andalusite-sillimanite hornfelses contain garnets with distinct but variable zonation profiles.These facts can possibly be explained by the time-temperature relationships in the thermal aureole. In a phase diagram such as the Al-Fe-Mg-Mn tetrahedron, the limiting mineral compositions of a four-phase volume or a three-phase triangle are fixed by T and P (the latter remaining effectively constant within a thermal aureole). Thus, in garnet-cordierite-biotite assemblages, garnet zonation should be controlled by temperature variation rather than by a non-equilibrium depletion process. Taking into account the experimental data of Dahl (1968), a zoned garnet from a transitional andalusite-sillimanite hornfels would reflect a temperature increase of about 40° C during its growth. A hypothetical P-X diagram is proposed to show semi-quantitatively the compositional variation of garnet-cordierite pairs with varying pressures (T constant).  相似文献   

5.
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   

6.
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   

7.
On the Interpretation of Orthonormal Coordinates for Compositional Data   总被引：1，自引：0，他引：1  
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   

8.
Logratio Analysis and Compositional Distance   总被引：10，自引：0，他引：10  
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   

9.
In the linear model of coregionalization (LMC), when applicable to the experimental direct variograms and the experimental cross variogram computed for two random functions, the variability of and relationships between the random functions are modeled with the same basis functions. In particular, structural correlations can be defined from entries of sill matrices (coregionalization matrices) under second-order stationarity. In this article, modified t-tests are proposed for assessing the statistical significance of estimated structural correlations. Their specific aspects and fundamental differences, compared with an existing modified t-test for global correlation analysis with spatial data, are discussed via estimated effective sample sizes, in relation to the superimposition of random structural components, the range of autocorrelation, the presence of correlation at another structure, and the sampling scheme. Accordingly, simulation results are presented for one structure versus two structures (one without and the other with autocorrelation). The performance of tests is shown to be related to the uncertainty associated with the estimation of variogram model parameters (range, sill matrix entries), because these are involved in the test statistic and the degrees of freedom of the associated t-distribution through the estimated effective sample size. Under the second-order stationarity and LMC assumptions, the proposed tests are generally valid.  相似文献   

10.
Chemical components such as SiO 2,TiO 2,MnO, P 2 O 5,and especially Fe 2 O 3 of the iron ores of Bicholim Mine, Northern Goa, have been determined for lateral and vertical sections of the mine at equal intervals of 3 and 1 m, respectively, so as to form the spatial (time) series. Univariate stationary models of the type Autoregressive moving average—ARMA (p, q)—were established for each series on the basis of statistical analyses of their auto (acf) and partial auto (pacf) correlation functions. These models were used for forecasting assay values at different lead distances from any pivot. Principles of parsimony simplified all of the candidate ARMA (p, q) models to pure AR (p) models, and the univariate forecasts were significantly improved by multivariate stochastic forecasts.  相似文献   

11.
Quantification of discrete pressure–temperature domains in deformed chlorite + white mica‐bearing metapelites was undertaken on mineral compositions derived by two‐dimensional microprobe compositional mapping of selected areas of rock thin sections. In order to achieve compositional information at sufficient analytical precision, spatial resolution and sample coverage within a typical analysis time of 1 day, an optimization of measurement methods was necessary. The method presented here allows collection of raw counts for eight different element concentrations at an analytical precision of ～1–2 wt%. X‐ray intensity multiplane maps (one map per measured chemical element) are translated into concentration multiplane maps, utilizing selected conventionally measured spot analyses combined with the Castaing approximation for each mineral. As this step requires identification of the different minerals present in the mapped area, a statistical clustering technique to identify different groups of composition was developed, guided by simple petrographic inspection of the thin section, to delineate the important minerals in the mapped area. Finally, the compositions of each pixel are translated into a mineral structural formula thus yielding a new kind of image with a high content of petrological information. The reliability of the mineral composition images was emphasized by carrying out precision tests on the analytical data. The possible use of chemical maps to infer the P–T–deformation history of metamorphic rocks is illustrated with two samples from the Spitzbergen and the Sambagawa blueschist facies belts. In both samples, a strong correlation between structures and chemistry is observed. Qualitative estimates of P–T conditions from the Si‐content of mica and chlorite are in good agreement with their location in microstructures that formed at different times. Therefore, the combination of chemical maps with microstructural observations is a very powerful approach to understand both the evolution of complex metamorphic rocks and the control by deformation of mineral reactivity.  相似文献   

12.
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   

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.
The dominant feature distinguishing one method of principal components analysis from another is the manner in which the original data are transformed prior to the other computations. The only other distinguishing feature of any importance is whether the eigenvectors of the inner product-moment of the transformed data matrix are taken directly as the Q-mode scores or scaled by the square roots of their associated eigenvalues and called the R-mode loadings. If the eigenvectors are extracted from the product-moment correlation matrix, the variables, in effect, were transformed by column standardization (zero means and unit variances), and the sum of the p-largest eigenvalues divided by the sum of all the eigenvalues indicates the degree to which a model containing pcomponents will account for the total variance in the original data. However, if the data were transformed in any manner other than column standardization, the eigenvalues cannot be used in this manner, but can only be used to determine the degree to which the model will account for the transformed data. Regardless of the type of principal components analysis that is performed—even whether it is Ror Q-mode—the goodness-of-fit of the model to the original data is given better by the eigenvalues of the correlation matrix than by those of the matrix that was actually factored.  相似文献   

15.
含水层渗透系数K的空间变异性研究   总被引：17，自引：0，他引：17  
朝伦巴根  和泰 《地质学报》1994,68(4):358-367
本文结合桥梁实例对公路双曲拱桥出现的一些主要病害进行了初步分析，并进而对桥梁状况提出评定意见和处理建议，仅供参考。  相似文献   

16.
Compositional data analysis   总被引：1，自引：0，他引：1  
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   

17.
The petrochemistry of kimberlites from Yakutia and Lesotho has been studied using a silicate melt model with the SiO2, CO2 and H2O derivatives as the main anions.A model has been developed, according to which the dissolution of H2O in an ultramafic melt results in orthosilicates (H2SiC 4 -2 , H3SiO 4 – , H4SiO4 etc.) rather than metasilicates, while the dissolution of CO2 produces additional hydrocarbonate complexes. It suggests that at high PCO 2 1 , and where the orthosilicic calcium salt clusters are likely to be present in the magma, the kimberlite melt can break down into carbonate and silicate liquids. Therefore, the composition of kimberlite magma will be determined by the H2O/CO2 ratio under the relatively constant fluid pressure. This can be seen from the distinct fluidrs trend in the H2O-CO2-SiO2 diagram for the Yakutia and Lesotho diamond-bearing kimberlites. The H2O/CO2 ratio changes with the liquidus temperature along this trend (Perchuk and Vaganov 1977) which suggests that liquid immiscibility predominates over the simple CO2 solubility in the melts of kimberlite composition. The well-known Boyd's diagrams for the equilibrium PT-conditions in peridotites have been applied along with new experimental data to natural Cpx and Opx, and the PT-parameters were correlated for peridotite inclusions in kimberlite pipes in Yakutia and Lesotho. The liquidus temperatures for the extrapolated area of these correlations gave depths (pressures) at which kimberlite magmas are formed (200–250 km).The hypothesis on SiO2 partitioning between the melt and the fluid was used to calculate the composition of dry initial kimberlite which characterised the average mantle composition: SiO2 — 45.12; TiO2 — 2.49; Al2O3 — 3.58; Cr2O3 — 0.12; FeO — 9.32; MnO — 0.16; CoO — 0.11; MgO — 23.47; CaO — 13.44; Na2O — 0.20; K2O — 1.12; P2O5 — 0.69; S — 0.18; sum — 100 wt.%. This kimberlite is close to wehrlite in composition.  相似文献   

18.
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   

19.
Criteria to Compare Estimation Methods of Regionalized Compositions   总被引：1，自引：0，他引：1  
The additive logratio (alr) transformation has been used in several case studies to predict regionalized compositions using standard geostatistical estimation methods such as ordinary kriging and ordinary cokriging. It is a simple method that allows application to transformed data all the body of knowledge available for geostatistical analysis of coregionalizations without a constant sum constraint. To compare the performance of methods, it is customary to use a univariate crossvalidation approach based on the leaving-one-out technique to evaluate the performance for each attribute separately. For multivariate observations this approach is difficult to interpret in terms of overall performance. Therefore, we propose using appropriate distances in real space and in the simplex, to improve the crossvalidation approach and, going a step forward, to adapt the concept of stress from multidimensional scaling to obtain a global measure of performance for each method. The Lyons West oil field of Kansas is used to illustrate the impactof using different distances in the performance of ordinary kriging versus ordinary cokriging.  相似文献   

20.
《Chemical Geology》2006,225(1-2):1-15
Microprobe monazite dating has been increasingly used to constrain the timing of deformation and metamorphism because of the potential to date very small monazite domains (down to 5 μm or less) in structural and petrologic context. This paper presents an analytical strategy, presentation format, and error considerations for microprobe monazite dating. The strategy involves high-resolution compositional mapping to delineate compositional domains within monazite crystals. Then for each compositional domain, a series of Th, U and Pb analyses are made, and a single date and error are calculated. The number of analyses in each domain is determined by the desired statistical precision of the date. Results from several monazite grains are typically combined and, along with textural relationships, are used to build an argument that the dates constrain the age of a deformation or metamorphic event. The total error involves three components: short-term random error (dominated by counting statistical uncertainty), short-term systematic error (uncertainty in background correction, conductive coating variation, and calibration), and long-term systematic error (uncertainty in standard composition, mass absorption factors, decay constants, etc.). In homogeneous compositional domains, short-term random errors (2σ) of less than 10 m.y. can be obtained from five to ten analyses. However, short-term systematic error, mainly background estimation uncertainty, would typically result in a doubling of the magnitude of random error. Microprobe dates are presented as a single Gaussian probability distribution for each domain, along with representative compositional maps. It is recommended that a consistency standard be analyzed during each analytical session and the results be reported along with those from the unknown. This proposed strategy and format are compatible with those of other geochronological techniques; they incorporate analytical limitations associated with trace, as opposed to major element, microprobe analysis, and will allow better comparisons to be made between labs and between different geochronological techniques.  相似文献   

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