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Measurements in sequencing studies are mostly based on counts. There is a lack of theoretical developments for the analysis and modelling of this type of data. Some thoughts in this direction are presented, which might serve as a seed. The main issues addressed are the compositional character of multinomial probabilities and the corresponding representation in orthogonal (isometric) coordinates, and modelling distributions for sequencing data taking into account possible effects of amplification techniques Ministry of Science, Innovation and Universities and European Regional Development Fund [RTI2018-095518-BC21 (C22) (MCIU/AEI/FEDER)]

Bayes theorem (discrete case) is taken as a paradigm of information acquisition. As mentioned by Aitchison, Bayes formula can be identified with perturbation of a prior probability vector and a discrete likelihood function, both vectors being compositional. Considering prior, posterior and likelihood as elements of the simplex, a natural choice of distance between them is the Aitchison distance. Other geometrical features can also be considered using the Aitchison geometry. For instance, orthogonality in the simplex allows to think of orthogonal information, or the perturbation-difference to think of opposite information. The Aitchison norm provides a size of compositional vectors, and is thus a natural scalar measure of the information conveyed by the likelihood or captured by a prior or a posterior. It is called evidence information, or e-information for short. In order to support such e-information theory some principles of e-information are discussed. They essentially coincide with those of compositional data analysis. Also, a comparison of these principles of e-information with the axiomatic Shannon-information theory is performed. Shannoninformation and developments thereof do not satisfy scale invariance and also violate subcompositional coherence. In general, Shannon-information theory follows the philosophy of amalgamation when relating information given by an evidence-vector and some sub-vector, while the dimension reduction for the proposed e-information corresponds to orthogonal projections in the simplex. The result of this preliminary study is a set of properties of e-information that may constitute the basis of an axiomatic theory. A synthetic example is used to motivate the ideas and the subsequent discussion

The discrete case of Bayes’ formula is considered the paradigm of information acquisition. Prior and posterior probability functions, as well as likelihood functions, called evidence functions, are compositions following the Aitchison geometry of the simplex, and have thus vector character. Bayes’ formula becomes a vector addition. The Aitchison norm of an evidence function is introduced as a scalar measurement of information. A fictitious fire scenario serves as illustration. Two different inspections of affected houses are considered. Two questions are addressed: (a) which is the information provided by the outcomes of inspections, and (b) which is the most informative inspection.

The discrete case of Bayes' formula is considered the paradigm of information acquisition. Prior and posterior probability functions, as well as likelihood functions, called evidence functions, are compositions following the Aitchison geometry of the simplex, and have thus vector character. Bayes' formula becomes a vector addition. The Aitchison norm of an evidence function is introduced as a scalar measurement of information. A fictitious fire scenario serves as illustration. Two different inspections of affected houses are considered. Two questions are addressed: (a) which is the information provided by the outcomes of inspections, and (b) which is the most informative inspection.

The Kolmogorov Smirnov test (KS) is a well known test used to asses how a set of observations is significantly different from the probability model specified under the null hypothesis. The KS test statistic quantifies the distance between the empirical distribution function and the hypothetical one. The modification introduced in Monti et al. (2017) consists of computing the mentioned distances as Aitchison distances. In this contribution, we suggest a further modification of the latter test and investigate, by simulation, the asymptotic distribution of the proposed test statistic, checking the appropriateness of a Generalized Extreme Value (GEV) Distribution. The properties of the asymptotic distribution are studied via Monte Carlo simulations.

A modified version of the Kolmogorov-Smirnov (KS) test is presented as a tool to assess whether a specified, although arbitrary, probability model is unsuitable to describe the underlying distribution of a set of observations. The KS test computes distances between points of the sample cumulative distribution function and the hypothetical one as absolute differences between them, and then considering the supreme distance as test statistics. The modification here proposed consists of computing the mentioned distances as Aitchison distances of the probabilities as two part compositions. In this contribution, we investigate by simulation the asymptotic distribution of the proposed test statistic, checking the appropriateness of the Gumbel distribution. The properties of the asymptotic distribution are studied for samples coming from generic distributions such as uniform, normal, lognormal, gamma, beta and exponential with different values of the parameters. A brief Monte Carlo investigation is made of the type I error and power of the test.

Perturbation and powering are two operations in the simplex that define a vector-space structure. Perturbation and powering in the simplex play the same role as the sum and product by scalars in real space. A standard Dirichlet random composition can be shifted by perturbation, and scaled powering by a real scalar. The obtained random composition has a shifted-scaled Dirichlet distribution. The procedure is analogous to standardization of real random variables. The derived distribution is a generalization of the Dirichlet one, and it is studied from a probabilistic point of view. In the simplex, considered as an Euclidean space, the Aitchison measure is the natural (Lebesgue type) measure, which is compatible with its operations and metrics. Therefore, a natural way of describing the generalized (shifted-scaled) Dirichlet probability distributions is using probability densities with respect to the Aitchison measure. This density representation is compared with the traditional probability density with respect to the Lebesgue measure. In particular, the center and variability for both representations are compared.

Observations in daily practice are sometimes registered as positive values larger then a given threshold α. The sample space is in this case the interval (α,+∞), α & 0, which can be structured as a real Euclidean space in different ways. This fact opens the door to alternative statistical models depending not only on the assumed distribution function, but also on the metric which is considered as appropriate, i.e. the way differences are measured, and thus variability

Geochemical data that is derived from the whole or partial analysis of various geologic materials represent a composition of mineralogies or solute species. Minerals are composed of structured relationships between cations and anions which, through atomic and molecular forces, keep the elements bound in specific configurations. The chemical compositions of minerals have specific relationships that are governed by these molecular controls. In the case of olivine, there is a well-defined relationship between Mn-Fe-Mg with Si. Balances between the principal elements defining olivine composition and other significant constituents in the composition (Al, Ti) have been defined, resulting in a near-linear relationship between the logarithmic relative proportion of Si versus (MgMnFe) and Mg versus (MnFe), which is typically described but poorly illustrated in the simplex. The present contribution corresponds to ongoing research, which attempts to relate stoichiometry and geochemical data using compositional geometry. We describe here the approach by which stoichiometric relationships based on mineralogical constraints can be accounted for in the space of simplicial coordinates using olivines as an example. Further examples for other mineral types (plagioclases and more complex minerals such as clays) are needed. Issues that remain to be dealt with include the reduction of a bulk chemical composition of a rock comprised of several minerals from which appropriate balances can be used to describe the composition in a realistic mineralogical framework. The overall objective of our research is to answer the question: In the cases where the mineralogy is unknown, are there suitable proxies that can be substituted? Kew words: Aitchison geometry, balances, mineral composition, oxides Geologische Vereinigung; Institut d’Estadística de Catalunya; International Association for Mathematical Geology; Càtedra Lluís Santaló d’Aplicacions de la Matemàtica; Generalitat de Catalunya, Departament d’Innovació, Universitats i Recerca; Ministerio de Educación y Ciencia; Ingenio 2010.

The simplex, the sample space of compositional data, can be structured as a real Euclidean space. This fact allows to work with the coefficients with respect to an orthonormal basis. Over these coefficients we apply standard real analysis, inparticular, we define two different laws of probability trought the density function and we study their main properties Geologische Vereinigung; Universitat de Barcelona, Equip de Recerca Arqueomètrica; Institut d’Estadística de Catalunya; International Association for Mathematical Geology; Patronat de l’Escola Politècnica Superior de la Universitat de Girona; Fundació privada: Girona, Universitat i Futur.

Hungary lies entirely within the Carpatho-Pannonian Region (CPR), a dominant tectonic unit of eastern Central Europe. The CPR consists of the Pannonian Basin system, and the arc of the Carpathian Mountains surrounding the lowlands in the north, east, and southeast. In the west, the CPR is bounded by the Eastern Alps, whereas in the south, by the Dinaridic belt. (...) Geologische Vereinigung; Universitat de Barcelona, Equip de Recerca Arqueomètrica; Institut d’Estadística de Catalunya; International Association for Mathematical Geology; Patronat de l’Escola Politècnica Superior de la Universitat de Girona; Fundació privada: Girona, Universitat i Futur.