Distributions on the simplex

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

Meaning of the λ parameter of skew–normal and log–skew normal distributions in fluid geochemistry

The literature related to skew–normal distributions has grown rapidly in recent years but at the moment few applications concern the description of natural phenomena with this type of probability models, as well as the interpretation of their parameters. The skew–normal distributions family represents an extension of the normal family to which a parameter (λ) has been added to regulate the skewness. The development of this theoretical field has followed the general tendency in Statistics towards more flexible methods to represent features of the data, as adequately as possible, and to reduce unrealistic assumptions as the normality that underlies most methods of univariate and multivariate analysis. In this paper an investigation on the shape of the frequency distribution of the logratio ln(Cl−/Na+) whose components are related to waters composition for 26 wells, has been performed. Samples have been collected around the active center of Vulcano island (Aeolian archipelago, southern Italy) from 1977 up to now at time intervals of about six months. Data of the logratio have been tentatively modeled by evaluating the performance of the skew–normal model for each well. Values of the λ parameter have been compared by considering temperature and spatial position of the sampling points. Preliminary results indicate that changes in λ values can be related to the nature of environmental processes affecting the data

“Unmixing” Tissue Gene Expression Signatures from Tumor Biopsies

Emergent molecular measurement methods, such as DNA microarray, qRTPCR, and many others, offer tremendous promise for the personalized treatment of cancer. These technologies measure the amount of specific proteins, RNA, DNA or other molecular targets from tumor specimens with the goal of “fingerprinting” individual cancers. Tumor specimens are heterogeneous; an individual specimen typically contains unknown amounts of multiple tissues types. Thus, the measured molecular concentrations result from an unknown mixture of tissue types, and must be normalized to account for the composition of the mixture. For example, a breast tumor biopsy may contain normal, dysplastic and cancerous epithelial cells, as well as stromal components (fatty and connective tissue) and blood and lymphatic vessels. Our diagnostic interest focuses solely on the dysplastic and cancerous epithelial cells. The remaining tissue components serve to “contaminate” the signal of interest. The proportion of each of the tissue components changes as a function of patient characteristics (e.g., age), and varies spatially across the tumor region. Because each of the tissue components produces a different molecular signature, and the amount of each tissue type is specimen dependent, we must estimate the tissue composition of the specimen, and adjust the molecular signal for this composition. Using the idea of a chemical mass balance, we consider the total measured concentrations to be a weighted sum of the individual tissue signatures, where weights are determined by the relative amounts of the different tissue types. We develop a compositional source apportionment model to estimate the relative amounts of tissue components in a tumor specimen. We then use these estimates to infer the tissuespecific concentrations of key molecular targets for sub-typing individual tumors. We anticipate these specific measurements will greatly improve our ability to discriminate between different classes of tumors, and allow more precise matching of each patient to the appropriate treatment

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

The preceding two editions of CoDaWork included talks on the possible consideration of densities as infinite compositions: Egozcue and D´ıaz-Barrero (2003) extended the Euclidean structure of the simplex to a Hilbert space structure of the set of densities within a bounded interval, and van den Boogaart (2005) generalized this to the set of densities bounded by an arbitrary reference density. From the many variations of the Hilbert structures available, we work with three cases. For bounded variables, a basis derived from Legendre polynomials is used. For variables with a lower bound, we standardize them with respect to an exponential distribution and express their densities as coordinates in a basis derived from Laguerre polynomials. Finally, for unbounded variables, a normal distribution is used as reference, and coordinates are obtained with respect to a Hermite-polynomials-based basis. To get the coordinates, several approaches can be considered. A numerical accuracy problem occurs if one estimates the coordinates directly by using discretized scalar products. 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 the reference density. Finally, for the case of 2-order Hermite polinomials (normal reference) and 1-order Laguerre polinomials (exponential), one can also derive the coordinates from their relationships to the classical mean and variance. Apart of these theoretical issues, this contribution focuses on the application of this theory to two main problems in sedimentary geology: the comparison of several grain size 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 or sediment, like their composition

Modeling and Classifying Breast Tissue Density in Mammograms

We present a new approach to model and classify breast parenchymal tissue. Given a mammogram, first, we will discover the distribution of the different tissue densities in an unsupervised manner, and second, we will use this tissue distribution to perform the classification. We achieve this using a classifier based on local descriptors and probabilistic Latent Semantic Analysis (pLSA), a generative model from the statistical text literature. We studied the influence of different descriptors like texture and SIFT features at the classification stage showing that textons outperform SIFT in all cases. Moreover we demonstrate that pLSA automatically extracts meaningful latent aspects generating a compact tissue representation based on their densities, useful for discriminating on mammogram classification. We show the results of tissue classification over the MIAS and DDSM datasets. We compare our method with approaches that classified these same datasets showing a better performance of our proposal

A Novel Breast Tissue Density Classification Methodology

It has been shown that the accuracy of mammographic abnormality detection methods is strongly dependent on the breast tissue characteristics, where a dense breast drastically reduces detection sensitivity. In addition, breast tissue density is widely accepted to be an important risk indicator for the development of breast cancer. Here, we describe the development of an automatic breast tissue classification methodology, which can be summarized in a number of distinct steps: 1) the segmentation of the breast area into fatty versus dense mammographic tissue; 2) the extraction of morphological and texture features from the segmented breast areas; and 3) the use of a Bayesian combination of a number of classifiers. The evaluation, based on a large number of cases from two different mammographic data sets, shows a strong correlation ( and 0.67 for the two data sets) between automatic and expert-based Breast Imaging Reporting and Data System mammographic density assessment

A new approach to the classification of mammographic masses and normal breast tissue

A new approach to mammographic mass detection is presented in this paper. Although different algorithms have been proposed for such a task, most of them are application dependent. In contrast, our approach makes use of a kindred topic in computer vision adapted to our particular problem. In this sense, we translate the eigenfaces approach for face detection/classification problems to a mass detection. Two different databases were used to show the robustness of the approach. The first one consisted on a set of 160 regions of interest (RoIs) extracted from the MIAS database, being 40 of them with confirmed masses and the rest normal tissue. The second set of RoIs was extracted from the DDSM database, and contained 196 RoIs containing masses and 392 with normal, but suspicious regions. Initial results demonstrate the feasibility of using such approach with performances comparable to other algorithms, with the advantage of being a more general, simple and cost-effective approach

The mapping of the local contributions of Fermi and Coulomb correlation into intracule and extracule density distributions

The contributions of the correlated and uncorrelated components of the electron-pair density to atomic and molecular intracule I(r) and extracule E(R) densities and its Laplacian functions ∇2I(r) and ∇2E(R) are analyzed at the Hartree-Fock (HF) and configuration interaction (CI) levels of theory. The topologies of the uncorrelated components of these functions can be rationalized in terms of the corresponding one-electron densities. In contrast, by analyzing the correlated components of I(r) and E(R), namely, IC(r) and EC(R), the effect of electron Fermi and Coulomb correlation can be assessed at the HF and CI levels of theory. Moreover, the contribution of Coulomb correlation can be isolated by means of difference maps between IC(r) and EC(R) distributions calculated at the two levels of theory. As application examples, the He, Ne, and Ar atomic series, the C2-2, N2, O2+2 molecular series, and the C2H4 molecule have been investigated. For these atoms and molecules, it is found that Fermi correlation accounts for the main characteristics of IC(r) and EC(R), with Coulomb correlation increasing slightly the locality of these functions at the CI level of theory. Furthermore, IC(r), EC(R), and the associated Laplacian functions, reveal the short-ranged nature and high isotropy of Fermi and Coulomb correlation in atoms and molecules

The relevance of the Laplacian of intracule and extracule density distributions for analyzing electron-electron interactions in molecules

A topological analysis of intracule and extracule densities and their Laplacians computed within the Hartree-Fock approximation is presented. The analysis of the density distributions reveals that among all possible electron-electron interactions in atoms and between atoms in molecules only very few are located rigorously as local maxima. In contrast, they are clearly identified as local minima in the topology of Laplacian maps. The conceptually different interpretation of intracule and extracule maps is also discussed in detail. An application example to the C2H2, C2H4, and C2H6 series of molecules is presented

A comparative analysis by means of quantum molecular similarity measures of density distributions derived from conventional ab initio and density functional methods

A procedure based on quantum molecular similarity measures (QMSM) has been used to compare electron densities obtained from conventional ab initio and density functional methodologies at their respective optimized geometries. This method has been applied to a series of small molecules which have experimentally known properties and molecular bonds of diverse degrees of ionicity and covalency. Results show that in most cases the electron densities obtained from density functional methodologies are of a similar quality than post-Hartree-Fock generalized densities. For molecules where Hartree-Fock methodology yields erroneous results, the density functional methodology is shown to yield usually more accurate densities than those provided by the second order Møller-Plesset perturbation theory