ESTIMATION IN COMPOSITIONAL DATA ANALYSIS

Compositional data arise naturally in several branches of science,including chemistry,geology,biology,medicine,ecology and manufacturing design.In chemistry,these constrained data seem to occur typicallywhen raw data are normalized or when output is obtained from a constrained estimation procedure,suchas might be used in a source apportionment problem.It is important not only for chemists to be awarethat the usual multivariate statistical techniques are not applicable to constrained data,but also to haveaccess to appropriate techniques as they become available.The currently available methodology is dueprincipally to Aitchison and is based on log-normal models.This paper suggests new parametric andnon-parametric approaches to significantly improve the existing methodology.In the parametric setting,some recent work of Rayens and Srinivasan is extended and a practical regression model is proposed.In the development of the non-parametric approach,minimum distance methods coupled withmultivariate bootstrap techniques are used to obtain point and region estimators.     

A petrographic,chemical, and isotopic study of calcium‐aluminum‐rich inclusions and aluminum‐rich chondrules from the Axtell (CV3) chondrite

Abstract— Petrographic, compositional, and isotopic characteristics were studied for three calcium‐aluminum‐rich inclusions (CAIs) and four plagioclase‐bearing chondrules (three of them Al‐rich) from the Axtell (CV3) chondrite. All seven objects have analogues in Allende (CV3) and other primitive chondrites, yet Axtell, like most other chondrites, contains a distinctive suite of CAIs and chondrules. In common with Allende CAIs, CAIs in Axtell exhibit initial ²⁶Al/²⁷Al ratios ((²⁶Al/²⁷Al)₀) ranging from ～5 × 10^?5 to <1.1 × 10^?5, and plagioclase‐bearing chondrules have (²⁶Al/²⁷Al)₀ ratios of ～3 × 10^?6 and lower. One type‐A CAI has the characteristics of a FUN inclusion. The Al‐Mg data imply that the plagioclase‐bearing chondrules began to form >2 Ma after the first CAIs. As in other CV3 chondrites, some objects in Axtell show evidence of isotopic disturbance. Axtell has experienced only mild thermal metamorphism (<600 °C), probably not enough to disturb the Al‐Mg systematics. Its CAIs and chondrules have suffered extensive metasomatism, probably prior to final accretion. These data indicate that CAIs and chondrules in Axtell (and other meteorites) had an extended history of several million years before their incorporation into the Axtell parent body. These long time periods appear to require a mechanism in the early solar system to prevent CAIs and chondrules from falling into the Sun via gas drag for several million years before final accretion. We also examined the compositional relationships among the four plagioclase‐bearing chondrules (two with large anorthite laths and two barred‐olivine chondrules) and between the chondrules and CAIs. Three processes were examined: (1) igneous differentiation, (2) assimilation of a CAI by average nebular material, and (3) evaporation of volatile elements from average nebular material. We find no evidence that igneous differentiation played a role in producing the chondrule compositions, although the barred olivine compositions can be related by addition or subtraction of olivine. Methods (2) and (3) could have produced the composition of one chondrule, AXCH‐1471, but neither process explains the other compositions. Our study indicates that plagioclase‐bearing objects originated through a variety of processes.     

BOX-COX TRANSFORMATIONS IN THE ANALYSIS OF COMPOSITIONAL DATA

The statistical analysis of compositional data is of fundamental importance to practitioners in generaland to chemists in particular.The existing methodology is principally due to Aitchison,who effectivelyuses two transformations,a ratio followed by the logarithmic,to create a useful,coherent theory thatin principle allows the plethora of normal-based multivariate techniques to be used on the transformeddata.This paper suggests that the well-known class of Box-Cox transformations can be employed inplace of the logarithmic to significantly improve the existing methodology.This is supported in part byshowing that one of the most basic problems that Aitchison managed to overcome,namely thespecification of an interpretable covariance structure for compositional data,can be resolved,or nearlyresolved,once the ratio transformation has been applied.Hence the resolution is not directly dependenton the logarithmic transformation.It is then verified that access to the general Box-Cox family will allowa more accurate use of the normal-based multivariate techniques,simply because better fits to normalitycan be achieved.Finally,maximum likelihood estimation and some associated asymptotics are employedto construct confidence intervals for ratios of the true,unknown compositional constituents.Heretoforethis had not been done even in the context of the logarithmic transformation.Applications to real dataare presented.