Markov chain montecarlo method applied to rounding zeros of compositional data: first approach

As stated in Aitchison (1986), a proper study of relative variation in a compositional data set should be based on logratios, and dealing with logratios excludes dealing with zeros. Nevertheless, it is clear that zero observations might be present in real data sets, either because the corresponding part is completely absent –essential zeros– or because it is below detection limit –rounded zeros. Because the second kind of zeros is usually understood as “a trace too small to measure”, it seems reasonable to replace them by a suitable small value, and this has been the traditional approach. As stated, e.g. by Tauber (1999) and by Martín-Fernández, Barceló-Vidal, and Pawlowsky-Glahn (2000), the principal problem in compositional data analysis is related to rounded zeros. One should be careful to use a replacement strategy that does not seriously distort the general structure of the data. In particular, the covariance structure of the involved parts –and thus the metric properties– should be preserved, as otherwise further analysis on subpopulations could be misleading. Following this point of view, a non-parametric imputation method is introduced in Martín-Fernández, Barceló-Vidal, and Pawlowsky-Glahn (2000). This method is analyzed in depth by Martín-Fernández, Barceló-Vidal, and Pawlowsky-Glahn (2003) where it is shown that the theoretical drawbacks of the additive zero replacement method proposed in Aitchison (1986) can be overcome using a new multiplicative approach on the non-zero parts of a composition. The new approach has reasonable properties from a compositional point of view. In particular, it is “natural” in the sense that it recovers the “true” composition if replacement values are identical to the missing values, and it is coherent with the basic operations on the simplex. This coherence implies that the covariance structure of subcompositions with no zeros is preserved. As a generalization of the multiplicative replacement, in the same paper a substitution method for missing values on compositional data sets is introduced

Compositional Data Analysis with R

R from http://www.r-project.org/ is ‘GNU S’ – a language and environment for statistical computing and graphics. The environment in which many classical and modern statistical techniques have been implemented, but many are supplied as packages. There are 8 standard packages and many more are available through the cran family of Internet sites http://cran.r-project.org . We started to develop a library of functions in R to support the analysis of mixtures and our goal is a MixeR package for compositional data analysis that provides support for operations on compositions: perturbation and power multiplication, subcomposition with or without residuals, centering of the data, computing Aitchison’s, Euclidean, Bhattacharyya distances, compositional Kullback-Leibler divergence etc. graphical presentation of compositions in ternary diagrams and tetrahedrons with additional features: barycenter, geometric mean of the data set, the percentiles lines, marking and coloring of subsets of the data set, theirs geometric means, notation of individual data in the set . . . dealing with zeros and missing values in compositional data sets with R procedures for simple and multiplicative replacement strategy, the time series analysis of compositional data. We’ll present the current status of MixeR development and illustrate its use on selected data sets

Searching for consensus in ahp-group decision making. A bayesian perspective

This paper sets out to identify the initial positions of the different decision makers who intervene in a group decision making process with a reduced number of actors, and to establish possible consensus paths between these actors. As a methodological support, it employs one of the most widely-known multicriteria decision techniques, namely, the Analytic Hierarchy Process (AHP). Assuming that the judgements elicited by the decision makers follow the so-called multiplicative model (Crawford and Williams, 1985; Altuzarra et al., 1997; Laininen and Hämäläinen, 2003) with log-normal errors and unknown variance, a Bayesian approach is used in the estimation of the relative priorities of the alternatives being compared. These priorities, estimated by way of the median of the posterior distribution and normalised in a distributive manner (priorities add up to one), are a clear example of compositional data that will be used in the search for consensus between the actors involved in the resolution of the problem through the use of Multidimensional Scaling tools

Bayesian tools for zero counts in compositional data

The log-ratio methodology makes available powerful tools for analyzing compositional data. Nevertheless, the use of this methodology is only possible for those data sets without null values. Consequently, in those data sets where the zeros are present, a previous treatment becomes necessary. Last advances in the treatment of compositional zeros have been centered especially in the zeros of structural nature and in the rounded zeros. These tools do not contemplate the particular case of count compositional data sets with null values. In this work we deal with \count zeros" and we introduce a treatment based on a mixed Bayesian-multiplicative estimation. We use the Dirichlet probability distribution as a prior and we estimate the posterior probabilities. Then we apply a multiplicative modi¯cation for the non-zero values. We present a case study where this new methodology is applied. Key words: count data, multiplicative replacement, composition, log-ratio analysis

Contrapuntos de la invención

Un instrumento musical implica la presencia de un registro sonoro que afecta tanto a la organización de los sonidos, silencios y ruidos, como a la disposición corporal que con él se va forjando. Desde esta consideración, la organización de la música llevada a cabo con las tecnologías eléctricas y electrónicas supone una profunda modificación de ambos aspectos. La llegada de la electricidad implica una triple dislocación: respecto a la transmisión de lo sonoro, a su posibilidad de reproducción y a la escucha. Estas dislocaciones son puestas en relación con invenciones que, desde el órgano de Ctesibios hasta el clavecín ocular de Castel, nos dibujan un marco en el que música, técnica, sensibilidad y sistema económico-social, tejen sus nexos. A lo largo de este recorrido se trazan lo que se ha denominado contrapuntos de la invención, que pueden tomar en las figuras de J.S. Bach y de J. Cage sus ejemplos más prominentes

Measurement Quality in Indicators of Compositions: a Compositional Multitrait-Multimethod Approach

Compositional data, also called multiplicative ipsative data, are common in survey research instruments in areas such as time use, budget expenditure and social networks. Compositional data are usually expressed as proportions of a total, whose sum can only be 1. Owing to their constrained nature, statistical analysis in general, and estimation of measurement quality with a confirmatory factor analysis model for multitrait-multimethod (MTMM) designs in particular are challenging tasks. Compositional data are highly non-normal, as they range within the 0-1 interval. One component can only increase if some other(s) decrease, which results in spurious negative correlations among components which cannot be accounted for by the MTMM model parameters. In this article we show how researchers can use the correlated uniqueness model for MTMM designs in order to evaluate measurement quality of compositional indicators. We suggest using the additive log ratio transformation of the data, discuss several approaches to deal with zero components and explain how the interpretation of MTMM designs di ers from the application to standard unconstrained data. We show an illustration of the method on data of social network composition expressed in percentages of partner, family, friends and other members in which we conclude that the faceto-face collection mode is generally superior to the telephone mode, although primacy e ects are higher in the face-to-face mode. Compositions of strong ties (such as partner) are measured with higher quality than those of weaker ties (such as other network members)