An examination of the use of the logratio transformation for the testing of endmember hypotheses

Given a compositional dataset in the absence of any prior information on any mixing process which may have formed it, a complete analysis of mixtures determines three distinct types of estimates in order. These are: (i) the estimate of the number of endmembers or fixed source compositions, of which all the sample compositions of the dataset must be approximate mixtures; (ii) the estimated compositions for each of these chosen number of endmembers; and (iii) the estimated contributions of each of these endmember estimates to each sample. Traditionally, the estimate for the number of endmembers has been assessed either by mapping or by inspection of the coefficients of determination between the observed and estimated variables. Mapping entails the plotting on a map of the region from which the samples were taken, either the contours of the contributions of each endmember to each sample, or some other portrayal of the distribution of endmember abundances. Because it requires the complete analysis, assessment by this method is too elaborate except for final confirmation and display. Alternatively, choosing a number of endmembers, which result in suitability high coefficients of determination for all or most variables, may account for elements which are not part of the conjectured mixing process or, worse, may result in the identification of endmembers which may never in fact have existed. Such an error is similar to overspecifying a multiple regression model. So, the obvious starting point from which to assess the validity, or otherwise choice of endmember numbers, is to examine the matrix of residuals. The differences between the logratio-transformed observed and estimated data form an array of residual logratios. A linear combination of these may be formed for each sample, which, under a random perturbation assumption, should follow a univariate normal distribution. Whether or not this scalar is normal can be readily tested. It can also be examined graphically for such desirable qualities as symmetry when the test for normality may be too severe. This procedure is employed to assess the decompositions of the U.S.G.S. Mid-Pacific data and the Nazca Plate Surface sediments.This paper was presented at the 18th Geochautauqua, Newark, Delaware, 13–14 October 1989.     

A few remarks on the kinetics of static grain growth in rocks

Static grain growth is a relatively simple transformation in which grain size increases under driving forces caused by grain and interphase boundary curvature. Given the relative simplicity of the protocol for grain growth experiments, measurements of grain boundary mobility show surprising variations. Boundary mobilities during grain growth are affected by solute and impurity chemistry, chemical fugacity of trace and major elements, pore size and number, pore fluid chemistry, the presence of melts, and the presence of solid second phases, as well as temperature and pressure. All of these factors may exert influence on grain growth of rocks in natural situations and many are also present during the laboratory experiments. Provided that the necessary kinetics parameters are known, bounds may be placed on the interface mobility when pores, partial melts, or solutes are present. To predict the rate of grain growth in natural situations will require improved laboratory data and careful consideration of the thermodynamic conditions likely to be encountered in nature.     

Endmember graphics

Thek points inp-space corresponding to a specified set ofk (k<p) linearly independent endmember estimates associated with ap-variate (n×p) compositional datasetX, define the vertices of a (k?1) dimensional simplexH. Then estimated mixtures which together account for the systematic variation in the datasetX, are each convex combinations of thek fixed endmember estimates. Accordingly, then-points inp-space which represent these mixtures must be interior points of the simplexH. The purpose of this paper is to describe a simple graphical procedure for examining the positions of the estimated mixtures relative tok=3 or 4 putative endmembers, and for assessing their compliance or otherwise with the convexity constraints. Since the mixture coefficients must be estimated first in order to obtain the mixture estimates, the paper begins with a review of the least-squares partitioning procedure.     

An appraisal of an iterative construction of the endmembers controlling the composition of deep-sea manganese nodules from the Central Indian Ocean Basin

This paper describes an estimation of endmember compositions followed by the assessment of those results by log-ratio variance analysis. As an appraisal, it deals only with the first objective of an endmember analysis namely, to identify endmembers if they exist by estimating their compositions. Following the creation of the endmember estimates, the computation of an array of log-ratio variances was a key innovation in this type of study. Log-ratio variances revealed intrinsic linear associations between the dominant elements on each of the estimated endmember compositions, largely confirming the endmember analysis. The dataset under study contained the concentrations of 16 elements in 93 samples of deep-sea manganese nodules from the Central Indian Ocean Basin. Many previous analyses of these nodules were undertaken to assess the economic potential of the deposits. This study by contrast, quantified the inter-element associations that account for the nodule compositions. Four endmembers were identified. The elements loaded on each were: (1) Mn, Zn, Ni, Cu, Mn-rich, (2) Fe, Ti, P, Co, Fe-rich, (3) Si, Al, Na, K, clay minerals, (3) Mg, ultramafic material, possibly including Mn, Cr, V, Ca, Na. These latter elements were also detected by their log-ratio variances to be associated with Mg on the 4th endmember.     

New 1982 — 1990 photometry of λ Andromedae and its 11-year cycle

We present photoelectric photometry of λ And never before published, obtained between February 1982 and December 1990 at 29 different observatories. Then we combine it with all other photometry available to us (previously published, contained in the I.A.U. Commission 27 Archives, and obtained with the Vanderbilt 16-inch automatic telescope but not yet published), to yield a 14.8-year data base. Analysis reveals a long-term cycle in mean brightness, with a full range of 0^m.15 and a period of 11.4 ± 0.4 years. Because most of our new photometry was concentrated in the 1983-84 observing season, we analyze that one well-defined light curve with a two-spot model. Spot A keeps a 0^m.04 amplitude throughout four rotation cycles whereas the amplitude of spot B diminishes from 0^m.09 down almost to 0^m.03. The spot rotation periods were 55^d.9 ± 0^d.6 and 520^d.8 ± 1^d.0, respectively.     

Use of the Geometric Elements in Numerical Simulations

We derive the transformations to convert the state vector in cartesian coordinates into geometric orbital elements (and conversely the geometric elements into the state vector) for a test particle moving around an oblate planet. These transformations arise from the epicyclic theory and are accurate to second order in eccentricity and inclination. This paper is written to be directly used for computational purposes, such as the numerical study of ring dynamics.     

X-Ray Line Transfer in Plasmas with Large Velocity Gradients

An astrophysically relevant experiment is compared to the output of a multidimensional radiation transfer code in which populations and radiation are self-consistently treated. Experimental Al Ly α spectra obtained with a very high-resolution spectrometer are presented as quantitative evidence of dot plasma non-planar expansion. Analysis of these spectra using the code is performed, in particular examining the effects of velocity gradients in directions other than that of the primary expansion. These calculations are found to be in good agreement with the experimental results. Usage of the Ly α doublet as a planarity diagnostic is discussed.     

Characteristics of pore migration controlled by diffusion through the pore-filling fluid

We analyze drag and drop of pores filled with a fluid phase, e.g., water or melt, in which the constituting elements of the solid matrix are dissolved. Assuming that the diffusion through the fluid-phase dominates bulk transport kinetics, we address the problem of pore motion and calculate the pore mobility and the critical velocity of elongated and lenticular pores on a grain boundary for arbitrary dihedral angle. The found variations in critical velocity and mobility with dihedral angle are modest for given volume of pores with the two considered geometries. For given pore size, however, the dependence on dihedral angle accounts for several orders of magnitude in pore mobility and critical velocity.