Application of discriminant function analysis and change-point problem in dating volcanic ashes

The application of Discriminant function analysis (DFA) is not a new idea in the study of tephrochrology. In this paper, DFA is applied to compositional datasets of two different types of tephras from Mountain Ruapehu in New Zealand and Mountain Rainier in USA. The canonical variables from the analysis are further investigated with a statistical methodology of change-point problems in order to gain a better understanding of the change in compositional pattern over time. Finally, a special case of segmented regression has been proposed to model both the time of change and the change in pattern. This model can be used to estimate the age for the unknown tephras using Bayesian statistical calibration

Some last thoughts on compositional data analysis

One of the disadvantages of old age is that there is more past than future: this, however, may be turned into an advantage if the wealth of experience and, hopefully, wisdom gained in the past can be reflected upon and throw some light on possible future trends. To an extent, then, this talk is necessarily personal, certainly nostalgic, but also self critical and inquisitive about our understanding of the discipline of statistics. A number of almost philosophical themes will run through the talk: search for appropriate modelling in relation to the real problem envisaged, emphasis on sensible balances between simplicity and complexity, the relative roles of theory and practice, the nature of communication of inferential ideas to the statistical layman, the inter-related roles of teaching, consultation and research. A list of keywords might be: identification of sample space and its mathematical structure, choices between transform and stay, the role of parametric modelling, the role of a sample space metric, the underused hypothesis lattice, the nature of compositional change, particularly in relation to the modelling of processes. While the main theme will be relevance to compositional data analysis we shall point to substantial implications for general multivariate analysis arising from experience of the development of compositional data analysis…

Exploration of geological variability and possible processes through the use of compositional data analysis: an example using scottish metamorphosed

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 sets. First we explain how hypotheses of compositional variability may be formulated within the natural sample space, the unit simplex, including useful hypotheses of subcompositional 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 complete 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, namely compositional singular value decompositions. All these statistical techniques are illustrated for a substantial compositional data set, consisting of 209 major-oxide and rare-element compositions of metamorphosed limestones from the Northeast and Central Highlands of Scotland. Finally we point out a number of unresolved problems in the statistical analysis of compositional processes

Compositional hypotheses of subcompositional stability and specific perturbation change and their testing

In standard multivariate statistical analysis common hypotheses of interest concern changes in mean vectors and subvectors. In compositional data analysis it is now well established that compositional change is most readily described in terms of the simplicial operation of perturbation and that subcompositions replace the marginal concept of subvectors. To motivate the statistical developments of this paper we present two challenging compositional problems from food production processes. Against this background the relevance of perturbations and subcompositions can be clearly seen. Moreover we can identify a number of hypotheses of interest involving the specification of particular perturbations or differences between perturbations and also hypotheses of subcompositional stability. We identify the two problems as being the counterpart of the analysis of paired comparison or split plot experiments and of separate sample comparative experiments in the jargon of standard multivariate analysis. We then develop appropriate estimation and testing procedures for a complete lattice of relevant compositional hypotheses

Thermodynamics and log–contrast analysis in fluid geochemistry

There are two principal chemical concepts that are important for studying the natural environment. The first one is thermodynamics, which describes whether a system is at equilibrium or can spontaneously change by chemical reactions. The second main concept is how fast chemical reactions (kinetics or rate of chemical change) take place whenever they start. In this work we examine a natural system in which both thermodynamics and kinetic factors are important in determining the abundance of NH+4 , NO−2 and NO−3 in superficial waters. Samples were collected in the Arno Basin (Tuscany, Italy), a system in which natural and antrophic effects both contribute to highly modify the chemical composition of water. Thermodynamical modelling based on the reduction-oxidation reactions involving the passage NH+4 -> NO−2 -> NO−3 in equilibrium conditions has allowed to determine the Eh redox potential values able to characterise the state of each sample and, consequently, of the fluid environment from which it was drawn. Just as pH expresses the concentration of H+ in solution, redox potential is used to express the tendency of an environment to receive or supply electrons. In this context, oxic environments, as those of river systems, are said to have a high redox potential because O2 is available as an electron acceptor. Principles of thermodynamics and chemical kinetics allow to obtain a model that often does not completely describe the reality of natural systems. Chemical reactions may indeed fail to achieve equilibrium because the products escape from the site of the rection or because reactions involving the trasformation are very slow, so that non-equilibrium conditions exist for long periods. Moreover, reaction rates can be sensitive to poorly understood catalytic effects or to surface effects, while variables as concentration (a large number of chemical species can coexist and interact concurrently), temperature and pressure can have large gradients in natural systems. By taking into account this, data of 91 water samples have been modelled by using statistical methodologies for compositional data. The application of log–contrast analysis has allowed to obtain statistical parameters to be correlated with the calculated Eh values. In this way, natural conditions in which chemical equilibrium is hypothesised, as well as underlying fast reactions, are compared with those described by a stochastic approach

Alternative ways to estimate change points in multinomial sequences. An application to an authorship attribution problem

The statistical analysis of literary style is the part of stylometry that compares measurable characteristics in a text that are rarely controlled by the author, with those in other texts. When the goal is to settle authorship questions, these characteristics should relate to the author’s style and not to the genre, epoch or editor, and they should be such that their variation between authors is larger than the variation within comparable texts from the same author. For an overview of the literature on stylometry and some of the techniques involved, see for example Mosteller and Wallace (1964, 82), Herdan (1964), Morton (1978), Holmes (1985), Oakes (1998) or Lebart, Salem and Berry (1998). Tirant lo Blanc, a chivalry book, is the main work in catalan literature and it was hailed to be “the best book of its kind in the world” by Cervantes in Don Quixote. Considered by writters like Vargas Llosa or Damaso Alonso to be the first modern novel in Europe, it has been translated several times into Spanish, Italian and French, with modern English translations by Rosenthal (1996) and La Fontaine (1993). The main body of this book was written between 1460 and 1465, but it was not printed until 1490. There is an intense and long lasting debate around its authorship sprouting from its first edition, where its introduction states that the whole book is the work of Martorell (1413?-1468), while at the end it is stated that the last one fourth of the book is by Galba (?-1490), after the death of Martorell. Some of the authors that support the theory of single authorship are Riquer (1990), Chiner (1993) and Badia (1993), while some of those supporting the double authorship are Riquer (1947), Coromines (1956) and Ferrando (1995). For an overview of this debate, see Riquer (1990). Neither of the two candidate authors left any text comparable to the one under study, and therefore discriminant analysis can not be used to help classify chapters by author. By using sample texts encompassing about ten percent of the book, and looking at word length and at the use of 44 conjunctions, prepositions and articles, Ginebra and Cabos (1998) detect heterogeneities that might indicate the existence of two authors. By analyzing the diversity of the vocabulary, Riba and Ginebra (2000) estimates that stylistic boundary to be near chapter 383. Following the lead of the extensive literature, this paper looks into word length, the use of the most frequent words and into the use of vowels in each chapter of the book. Given that the features selected are categorical, that leads to three contingency tables of ordered rows and therefore to three sequences of multinomial observations. Section 2 explores these sequences graphically, observing a clear shift in their distribution. Section 3 describes the problem of the estimation of a suden change-point in those sequences, in the following sections we propose various ways to estimate change-points in multinomial sequences; the method in section 4 involves fitting models for polytomous data, the one in Section 5 fits gamma models onto the sequence of Chi-square distances between each row profiles and the average profile, the one in Section 6 fits models onto the sequence of values taken by the first component of the correspondence analysis as well as onto sequences of other summary measures like the average word length. In Section 7 we fit models onto the marginal binomial sequences to identify the features that distinguish the chapters before and after that boundary. Most methods rely heavily on the use of generalized linear models

Dynamic graphics of parametrically linked multivariate methods used in compositional data analysis

Many multivariate methods that are apparently distinct can be linked by introducing one or more parameters in their definition. Methods that can be linked in this way are correspondence analysis, unweighted or weighted logratio analysis (the latter also known as "spectral mapping"), nonsymmetric correspondence analysis, principal component analysis (with and without logarithmic transformation of the data) and multidimensional scaling. In this presentation I will show how several of these methods, which are frequently used in compositional data analysis, may be linked through parametrizations such as power transformations, linear transformations and convex linear combinations. Since the methods of interest here all lead to visual maps of data, a "movie" can be made where where the linking parameter is allowed to vary in small steps: the results are recalculated "frame by frame" and one can see the smooth change from one method to another. Several of these "movies" will be shown, giving a deeper insight into the similarities and differences between these methods

How does basis set superposition error change the potential surfaces for hydrogen-bonded dimers?

We describe a simple method to automate the geometric optimization of molecular orbital calculations of supermolecules on potential surfaces that are corrected for basis set superposition error using the counterpoise (CP) method. This method is applied to the H-bonding complexes HF/HCN, HF/H2O, and HCCH/H2O using the 6-31G(d,p) and D95 + + (d,p) basis sets at both the Hartree-Fock and second-order Møller-Plesset levels. We report the interaction energies, geometries, and vibrational frequencies of these complexes on the CP-optimized surfaces; and compare them with similar values calculated using traditional methods, including the (more traditional) single point CP correction. Upon optimization on the CP-corrected surface, the interaction energies become more negative (before vibrational corrections) and the H-bonding stretching vibrations decrease in all cases. The extent of the effects vary from extremely small to quite large depending on the complex and the calculational method. The relative magnitudes of the vibrational corrections cannot be predicted from the H-bond stretching frequencies alone