Response to the letter to the editor by Dr van den Berg

Letters to the editor

On two examples by Iyama and Yoshino

In a recent paper Iyama and Yoshino consider two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal Cohen-Macaulay modules in terms of linear algebra data. In this paper we present two new approaches to these examples. In the first approach we give a relation with cluster categories. In the second approach we use Orlov's result on the graded singularity category. We obtain some new results on the singularity category of isolated singularities which may be interesting in their own right.

Revisió morfológica de smd amb del(5q) aïllada i amb del (5q) + 1

Van den Berghe et al. va descriure el “Sdr. 5q-” com un SMD amb del(5q) aïllada, curs indolent, predomini femení, anèmia macrocítica, amb o sense trombocitosi i megacariocits monolobulats. El SMD amb del(5q) aïllada constitueix un subtipus específic de SMD en la classificació de la OMS. L’objectiu del estudi era revisar mostres de SP i MO de SMD amb del(5q) aïllada o amb del(5q) més una alteració citogenètica. Els resultats mostren la dificultat de classificació dels SMD amb les classificacions actuals, i suggereix que no hi ha diferencies entre els SMD amb del(5q) aïllada i els SMD amb del(5q) +1.

Mixing compositions and other scales

Theory of compositional data analysis is often focused on the composition only. However in practical applications we often treat a composition together with covariableswith some other scale. This contribution systematically gathers and develop statistical tools for this situation. For instance, for the graphical display of the dependenceof a composition with a categorical variable, a colored set of ternary diagrams mightbe a good idea for a first look at the data, but it will fast hide important aspects ifthe composition has many parts, or it takes extreme values. On the other hand colored scatterplots of ilr components could not be very instructive for the analyst, if theconventional, black-box ilr is used.Thinking on terms of the Euclidean structure of the simplex, we suggest to set upappropriate projections, which on one side show the compositional geometry and on theother side are still comprehensible by a non-expert analyst, readable for all locations andscales of the data. This is e.g. done by defining special balance displays with carefully-selected axes. Following this idea, we need to systematically ask how to display, explore,describe, and test the relation to complementary or explanatory data of categorical, real,ratio or again compositional scales.This contribution shows that it is sufficient to use some basic concepts and very fewadvanced tools from multivariate statistics (principal covariances, multivariate linearmodels, trellis or parallel plots, etc.) to build appropriate procedures for all these combinations of scales. This has some fundamental implications in their software implementation, and how might they be taught to analysts not already experts in multivariateanalysis

Statistical treatment of grain-size curves and empirical distributions: densities as compositions?

The preceding two editions of CoDaWork included talks on the possible considerationof densities as infinite compositions: Egozcue and D´ıaz-Barrero (2003) extended theEuclidean structure of the simplex to a Hilbert space structure of the set of densitieswithin a bounded interval, and van den Boogaart (2005) generalized this to the setof densities bounded by an arbitrary reference density. From the many variations ofthe Hilbert structures available, we work with three cases. For bounded variables, abasis derived from Legendre polynomials is used. For variables with a lower bound, westandardize them with respect to an exponential distribution and express their densitiesas coordinates in a basis derived from Laguerre polynomials. Finally, for unboundedvariables, a normal distribution is used as reference, and coordinates are obtained withrespect to a Hermite-polynomials-based basis.To get the coordinates, several approaches can be considered. A numerical accuracyproblem occurs if one estimates the coordinates directly by using discretized scalarproducts. Thus we propose to use a weighted linear regression approach, where all k-order polynomials are used as predictand variables and weights are proportional to thereference density. Finally, for the case of 2-order Hermite polinomials (normal reference)and 1-order Laguerre polinomials (exponential), one can also derive the coordinatesfrom their relationships to the classical mean and variance.Apart of these theoretical issues, this contribution focuses on the application of thistheory to two main problems in sedimentary geology: the comparison of several grainsize distributions, and the comparison among different rocks of the empirical distribution of a property measured on a batch of individual grains from the same rock orsediment, like their composition

Analyzing shapes as compositions of distances

We propose to analyze shapes as “compositions” of distances in Aitchison geometry asan alternate and complementary tool to classical shape analysis, especially when sizeis non-informative.Shapes are typically described by the location of user-chosen landmarks. Howeverthe shape – considered as invariant under scaling, translation, mirroring and rotation– does not uniquely define the location of landmarks. A simple approach is to usedistances of landmarks instead of the locations of landmarks them self. Distances arepositive numbers defined up to joint scaling, a mathematical structure quite similar tocompositions. The shape fixes only ratios of distances. Perturbations correspond torelative changes of the size of subshapes and of aspect ratios. The power transformincreases the expression of the shape by increasing distance ratios. In analogy to thesubcompositional consistency, results should not depend too much on the choice ofdistances, because different subsets of the pairwise distances of landmarks uniquelydefine the shape.Various compositional analysis tools can be applied to sets of distances directly or afterminor modifications concerning the singularity of the covariance matrix and yield resultswith direct interpretations in terms of shape changes. The remaining problem isthat not all sets of distances correspond to a valid shape. Nevertheless interpolated orpredicted shapes can be backtransformated by multidimensional scaling (when all pairwisedistances are used) or free geodetic adjustment (when sufficiently many distancesare used)

A compositional data analysis package for R providing multiple approaches

”compositions” is a new R-package for the analysis of compositional and positive data.It contains four classes corresponding to the four different types of compositional andpositive geometry (including the Aitchison geometry). It provides means for computation,plotting and high-level multivariate statistical analysis in all four geometries.These geometries are treated in an fully analogous way, based on the principle of workingin coordinates, and the object-oriented programming paradigm of R. In this way,called functions automatically select the most appropriate type of analysis as a functionof the geometry. The graphical capabilities include ternary diagrams and tetrahedrons,various compositional plots (boxplots, barplots, piecharts) and extensive graphical toolsfor principal components. Afterwards, ortion and proportion lines, straight lines andellipses in all geometries can be added to plots. The package is accompanied by ahands-on-introduction, documentation for every function, demos of the graphical capabilitiesand plenty of usage examples. It allows direct and parallel computation inall four vector spaces and provides the beginner with a copy-and-paste style of dataanalysis, while letting advanced users keep the functionality and customizability theydemand of R, as well as all necessary tools to add own analysis routines. A completeexample is included in the appendix

News from compositions, the R package

The R-package “compositions”is a tool for advanced compositional analysis. Its basicfunctionality has seen some conceptual improvement, containing now some facilitiesto work with and represent ilr bases built from balances, and an elaborated subsys-tem for dealing with several kinds of irregular data: (rounded or structural) zeroes,incomplete observations and outliers. The general approach to these irregularities isbased on subcompositions: for an irregular datum, one can distinguish a “regular” sub-composition (where all parts are actually observed and the datum behaves typically)and a “problematic” subcomposition (with those unobserved, zero or rounded parts, orelse where the datum shows an erratic or atypical behaviour). Systematic classificationschemes are proposed for both outliers and missing values (including zeros) focusing onthe nature of irregularities in the datum subcomposition(s).To compute statistics with values missing at random and structural zeros, a projectionapproach is implemented: a given datum contributes to the estimation of the desiredparameters only on the subcompositon where it was observed. For data sets withvalues below the detection limit, two different approaches are provided: the well-knownimputation technique, and also the projection approach.To compute statistics in the presence of outliers, robust statistics are adapted to thecharacteristics of compositional data, based on the minimum covariance determinantapproach. The outlier classification is based on four different models of outlier occur-rence and Monte-Carlo-based tests for their characterization. Furthermore the packageprovides special plots helping to understand the nature of outliers in the dataset.Keywords: coda-dendrogram, lost values, MAR, missing data, MCD estimator,robustness, rounded zeros

Automated Segmentation of Cerebral Vasculature with Aneurysms in 3DRA and TOF-MRA using Geodesic Active Regions: an Evaluation Study

Purpose: To evaluate the suitability of an improved version of an automatic segmentation method based on geodesic active regions (GAR) for segmenting cerebral vasculature with aneurysms from 3D X-ray reconstruc-tion angiography (3DRA) and time of °ight magnetic resonance angiography (TOF-MRA) images available in the clinical routine.Methods: Three aspects of the GAR method have been improved: execution time, robustness to variability in imaging protocols and robustness to variability in image spatial resolutions. The improved GAR was retrospectively evaluated on images from patients containing intracranial aneurysms in the area of the Circle of Willis and imaged with two modalities: 3DRA and TOF-MRA. Images were obtained from two clinical centers, each using di®erent imaging equipment. Evaluation included qualitative and quantitative analyses ofthe segmentation results on 20 images from 10 patients. The gold standard was built from 660 cross-sections (33 per image) of vessels and aneurysms, manually measured by interventional neuroradiologists. GAR has also been compared to an interactive segmentation method: iso-intensity surface extraction (ISE). In addition, since patients had been imaged with the two modalities, we performed an inter-modality agreement analysis with respect to both the manual measurements and each of the two segmentation methods. Results: Both GAR and ISE di®ered from the gold standard within acceptable limits compared to the imaging resolution. GAR (ISE, respectively) had an average accuracy of 0.20 (0.24) mm for 3DRA and 0.27 (0.30) mm for TOF-MRA, and had a repeatability of 0.05 (0.20) mm. Compared to ISE, GAR had a lower qualitative error in the vessel region and a lower quantitative error in the aneurysm region. The repeatabilityof GAR was superior to manual measurements and ISE. The inter-modality agreement was similar between GAR and the manual measurements. Conclusions: The improved GAR method outperformed ISE qualitatively as well as quantitatively and is suitable for segmenting 3DRA and TOF-MRA images from clinical routine.

Aitchison Geometry for Probability and Likelihood as a new approach to mathematical statistics

The Aitchison vector space structure for the simplex is generalized to a Hilbert space structure A2(P) for distributions and likelihoods on arbitrary spaces. Centralnotations of statistics, such as Information or Likelihood, can be identified in the algebraical structure of A2(P) and their corresponding notions in compositional data analysis, such as Aitchison distance or centered log ratio transform.In this way very elaborated aspects of mathematical statistics can be understoodeasily in the light of a simple vector space structure and of compositional data analysis. E.g. combination of statistical information such as Bayesian updating,combination of likelihood and robust M-estimation functions are simple additions/perturbations in A2(Pprior). Weighting observations corresponds to a weightedaddition of the corresponding evidence.Likelihood based statistics for general exponential families turns out to have aparticularly easy interpretation in terms of A2(P). Regular exponential families formfinite dimensional linear subspaces of A2(P) and they correspond to finite dimensionalsubspaces formed by their posterior in the dual information space A2(Pprior).The Aitchison norm can identified with mean Fisher information. The closing constant itself is identified with a generalization of the cummulant function and shown to be Kullback Leiblers directed information. Fisher information is the local geometry of the manifold induced by the A2(P) derivative of the Kullback Leibler information and the space A2(P) can therefore be seen as the tangential geometry of statistical inference at the distribution P.The discussion of A2(P) valued random variables, such as estimation functionsor likelihoods, give a further interpretation of Fisher information as the expected squared norm of evidence and a scale free understanding of unbiased reasoning

Processes and consequences in business ethical dilemmas: The oil industry and climate changes

This paper proposes a framework to examine business ethical dilemmas andbusiness attitudes towards such dilemmas. Business ethical dilemmas canbe understood as reflecting a contradiction between a socially detrimentalprocess and a self-interested profitable consequence. This representationallows us to distinguish two forms of behavior differing by whetherpriority is put on consequences or on processes. We argue that theseforms imply very different business attitudes towards society:controversial or competitive for the former and aligned or cooperativefor the latter. These attitudes are then analyzed at the discursive level in order to address the question of good faith in businessargumentation, i.e. to which extent are these attitudes consistent withactual business behaviors. We argue that consequential attitudes mostlyinvolve communication and lobbying actions aiming at eluding the dilemma.Therefore, the question of good faith for consequential attitudes liesin the consistency between beliefs and discourse. On the other hand,procedural attitudes acknowledge the dilemma and claim a change of theprocess of behavior. They thus raise the question of the consistencybetween discourses and actual behavior. We apply this processes/consequencesframework to the case of the oil industry s climate change ethical dilemmawhich comes forth as a dilemma between emitting greenhouse gases and making more profits . And we examine the different attitudes of two oilcorporations-BP Amoco and ExxonMobil-towards the dilemma.

Mean field model for spatially extended systems in the presence of multiplicative noise

We present a mean field model that describes the effect of multiplicative noise in spatially extended systems. The model can be solved analytically. For the case of the phi4 potential it predicts that the phase transition is shifted. This conclusion is supported by numerical simulations of this model in two dimensions.

Reentrant transition induced by multiplictative noise in the Time Dependent Ginzburg-Landau model

An effect of multiplicative noise in the time-dependent Ginzburg-Landau model is reported, namely, that noise at a relatively low intensity induces a phase transition towards an ordered state, whereas strong noise plays a destructive role, driving the system back to its disordered state through a reentrant phase transition. The phase diagram is calculated analytically using a mean-field theory and a more sophisticated approach and is compared with the results from extensive numerical simulations.

Giant acceleration of free diffusion by use of titled periodic potentials

The effective diffusion coefficient for the overdamped Brownian motion in a tilted periodic potential is calculated in closed analytical form. Universality classes and scaling properties for weak thermal noise are identified near the threshold tilt where deterministic running solutions set in. In this regime the diffusion may be greatly enhanced, as compared to free thermal diffusion with, for a realistic experimental setup, an enhancement of up to 14 orders of magnitude.

Difussion in a titled periodic potential: Enhancement, universality, and scaling

An exact analytical expression for the effective diffusion coefficient of an overdamped Brownian particle in a tilted periodic potential is derived for arbitrary potentials and arbitrary strengths of the thermal noise. Near the critical tilt (threshold of deterministic running solutions) a scaling behavior for weak thermal noise is revealed and various universality classes are identified. In comparison with the bare (potential-free) thermal diffusion, the effective diffusion coefficient in a critically tilted periodic potential may be, in principle, arbitrarily enhanced. For a realistic experimental setup, an enhancement by 14 orders of magnitude is predicted so that thermal diffusion should be observable on a macroscopic scale at room temperature.