Alex Woronow

Quantifying and testing differences among means of compositional data suites

Commonly, geological studies compare mean values of two or more compositional data suites in order to determine if, how, and by how much they differ. Simple approaches for evaluating and statistically testing differences in mean values for open data fail for compositional (closed) data. A new parameter, an “f-value,” therefore has been developed, which correctly quantifies the differences among compositional mean values and allows testing those differences for statistical significance. In general, this parameter quantifies only therelative factor by which compositional variables differ across data suites; however for situations where, arguably, at least one component has neither increased nor decreased, anabsolute f-value can be computed. In situations where the compositional variables have undergone many perturbations, arguments based upon thef-values and the central limit theorem indicate that logratios of compositional variables should be normally distributed.

Chemical changes induced in aragonite using treatments for the destruction of organic material

Abstract Treatments used to destroy or remove organic material prior to chemical analyses of carbonate material alter the elemental composition of aragonite. To assess chemical changes in aragonite caused by various treatments, an abiogenic aragonite crystal was powdered and split into five groups. One group was left untreated as the control group. The four remaining groups were treated as follows: (1) soaking in a 5% sodium hypochlorite solution for 24 hr.; (2) boiling for 3 min. in a 5 N sodium hydroxide solution; (3) soaking in a 30% hydrogen peroxide solution for 24 hr.; and (4) heating for 2 hr. at 400°C. Each group was split subsequently and analyzed for Ca, Sr, Mg, Mn, Fe, Na and K content. Using a statistical technique designed specifically for compositional data, analyses indicate that all of the treatments except heating caused significant changes in the composition of the aragonite. Although the extent of the chemical alteration differs according to the particular treatment, broadly similar changes occurred in all chemically treated samples. Most notable are decreases in Fe and Mg, possibly related to their occurrence in the aragonite as adsorbed ions, as interstitial ions, or in structural defect sites. The significant changes caused by these chemical treatments can lead to misinterpretations of laboratory-altered chemical compositions as products of natural processes.

Complete subcompositional independence testing of closed arrays

Abstract The problem of testing for correlations in closed data, where all components sum to a constant, has undermined confidence in many inferences concerning geologic data. By their nature, much data of geologic interest are closed. Fortunately, significant progress has occurred recently in the rigorous treatment of closed data. The program documented here exploits these advances to provide a test for complete subcompositional independence. If the hypothesis of complete subcompositional independence can be demonstrated, then no nonoverlapping subvectors of the data display correlations beyond those induced solely through closure.

Generating random numbers on a simplex

Algorithms for three types of random compositional vectors are provided in this article. The first is intuitive. Random basis (“open”) vectors are transformed into compositional vectors by dividing each vectors components by the sum of its components. The second algorithm uses a coordinate transformation to map uniform random vectors in a d-dimensional hypertetrahedron into uniformly distributed compositional variables on the d + 1 dimensional simplex. The third algorithm operates upon a logratio population covariance matrix and vector of means to produce random compositional vectors drawn from the same distribution. The statistical attributes of the crude compositional data also are honored by this method.

Endmember unmixing of compositional data

Abstract Manipulation of the n -endmember mixing equations, appropriate to compositional data, leads to a general method (1) to test hypotheses about endmember compositions and (2) to recover the concentrations of all compositional components in an endmember when the concentrations of n − 1 components are known or measured in that endmember. Additionally, Pearce element-ratio analysis is shown to be a special (and not efficiently posed) two-endmember case of n -endmember unmixing. Information available on Pearce element-ratio diagrams as values of the slopes of lines is contained on Harker diagrams as values of points.

Methods for quantifying, statistically testing, and graphically displaying shifts in compositional abundances across data suites

Abstract Because of the constant-sum problem, compositional variables cannot be evaluated as can unconstrained variables. The concentration of every component may shift in response to the changing concentration of each and every other component. However, a multiplicative factor, “ f -value,” can be defined that both maps one suite of compositional concentrations into another and has a simple and useful interpretation of being a location difference. For instance, an f -value of 1.0 for a compositional component indicates that any observed, systematic shifts in its concentration resulted entirely from the changing concentrations of other components. The computer program given here computes these factors and determines if they differ significantly from one. An auxiliary program graphically displays the results.

Mercurian crater-filling classes constrain the emplacement process of the intercrater plains material

Abstract A multivariate analysis of crater-filling classes for craters larger than 10 km diameter on Mercury constrains the process of intercrater plains emplacement to have been one that affected both the intercrater plains and the densely cratered terrain in a similar manner. Any emplacement process considerably restricted in time, such as ejecta from a single basin or an episode of volcanic eruptions that was brief compared to the time span of the late heavy bombardment, violates the observations.

Power and robustness of Aitchison's test for complete subcompositional independence in closed arrays

This paper examines some aspects of the power and robustness of the test for complete subcompositional independence proposed by Aitchison (1982). Although the computed test statistics commonly do not approach being χ2 distributed throughout their range, the upper tail of their distribution does mimic the χ2 distribution sufficiently to yield a quite robust test when variates are drawn from identical distributions with different distribution parameters or even when variates are drawn from different distributions. But the magnitude of correlations among the variables and the proportion of correlated to independent variables that compose the closed data vectors affect the power of the test.

On the internet with a PC

Abstract Using a PC to access the Internet and all its services is not difficult—once basic computer hardware, software, and network connections have been installed and configured. Many Internet networking problems can be avoided by contacting network experts before selecting the hardware and network software. This article will introduce some of the hardware, software, and Internet networking options available to the would-be Internet Surfer, and some of the pitfalls that one might encounter. Once the network hardware and software are installed successfully, one can explore the Internet through a number of different software packages that allow the sharing of computer files and printers, as well as logging onto remote host-computers, searching World Wide Web, posting and reading notices on virtually any topic, and sending and receiving electronic mail. This article will describe the functions of several programs that can access these Internet services and how to obtain them.

Identifying minerals controlling the chemical evolution of igneous rocks: Beyond Pearce element-ratio diagrams

Abstract A new method for identifying mineral controls on the chemical evolution of magmas has statistical power and interpretative reliability exceeding those of the conserved-denominator method. Conserved denominator (or “Pearce element-ratio”) analysis compares two chemical variables that have each been divided by the same third chemical variable. The common divisor introduces “induced correlations” that tend to reduce statistical power—the method tends to assert the presence of mineral controls where none exist. No available method can remove or even quantify the effects of these induced correlations. By recasting chemical data into a new form, standard regression analysis serves to evaluate the same geochemical hypotheses, evaluated by common-denominator analysis. This new approach devolves from simple geometric relationships among compositional variables, is consistent with a linear mixing paradigm, and does not introduce induced correlations. Furthermore, tests for co-control by minerals with common cations (e.g., olivine and orthopyroxene) are easily formulated with this new method. Case studies involving olivine and olivine + plagioclase control, as well as synthetic-data studies, illustrate and verify this new method.