Hilbert space on probability density functions with Aitchison geometry

Compositional data analysis motivated the introduction of a complete Euclidean structure in the simplex of D parts. This was based on the early work of J. Aitchison (1986) and completed recently when Aitchinson distance in the simplex was associated with an inner product and orthonormal bases were identified (Aitchison and others, 2002; Egozcue and others, 2003). A partition of the support of a random variable generates a composition by assigning the probability of each interval to a part of the composition. One can imagine that the partition can be refined and the probability density would represent a kind of continuous composition of probabilities in a simplex of infinitely many parts. This intuitive idea would lead to a Hilbert-space of probability densitiesby generalizing the Aitchison geometry for compositions in the simplex into the set probability densities

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 yearsbut at the moment few applications concern the description of natural phenomena withthis type of probability models, as well as the interpretation of their parameters. Theskew–normal distributions family represents an extension of the normal family to whicha parameter (λ) has been added to regulate the skewness. The development of this theoreticalfield has followed the general tendency in Statistics towards more flexible methodsto represent features of the data, as adequately as possible, and to reduce unrealisticassumptions as the normality that underlies most methods of univariate and multivariateanalysis. In this paper an investigation on the shape of the frequency distribution of thelogratio 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 Vulcanoisland (Aeolian archipelago, southern Italy) from 1977 up to now at time intervals ofabout six months. Data of the logratio have been tentatively modeled by evaluating theperformance of the skew–normal model for each well. Values of the λ parameter havebeen 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 ofenvironmental processes affecting the data

Thermodynamics and log–contrast analysis in fluid geochemistry

There are two principal chemical concepts that are important for studying the naturalenvironment. The first one is thermodynamics, which describes whether a system is atequilibrium or can spontaneously change by chemical reactions. The second main conceptis how fast chemical reactions (kinetics or rate of chemical change) take place wheneverthey start. In this work we examine a natural system in which both thermodynamics andkinetic factors are important in determining the abundance of NH+4 , NO−2 and NO−3 insuperficial waters. Samples were collected in the Arno Basin (Tuscany, Italy), a system inwhich natural and antrophic effects both contribute to highly modify the chemical compositionof water. Thermodynamical modelling based on the reduction-oxidation reactionsinvolving the passage NH+4 -& NO−2 -& NO−3 in equilibrium conditions has allowed todetermine 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 expressesthe concentration of H+ in solution, redox potential is used to express the tendency of anenvironment to receive or supply electrons. In this context, oxic environments, as thoseof river systems, are said to have a high redox potential because O2 is available as anelectron acceptor.Principles of thermodynamics and chemical kinetics allow to obtain a model that oftendoes not completely describe the reality of natural systems. Chemical reactions may indeedfail to achieve equilibrium because the products escape from the site of the rectionor because reactions involving the trasformation are very slow, so that non-equilibriumconditions exist for long periods. Moreover, reaction rates can be sensitive to poorly understoodcatalytic effects or to surface effects, while variables as concentration (a largenumber of chemical species can coexist and interact concurrently), temperature and pressurecan have large gradients in natural systems. By taking into account this, data of 91water samples have been modelled by using statistical methodologies for compositionaldata. The application of log–contrast analysis has allowed to obtain statistical parametersto be correlated with the calculated Eh values. In this way, natural conditions in whichchemical equilibrium is hypothesised, as well as underlying fast reactions, are comparedwith those described by a stochastic approach

Automatic classification of breast density

A recent trend in digital mammography is computer-aided diagnosis systems, which are computerised tools designed to assist radiologists. Most of these systems are used for the automatic detection of abnormalities. However, recent studies have shown that their sensitivity is significantly decreased as the density of the breast increases. This dependence is method specific. In this paper we propose a new approach to the classification of mammographic images according to their breast parenchymal density. Our classification uses information extracted from segmentation results and is based on the underlying breast tissue texture. Classification performance was based on a large set of digitised mammograms. Evaluation involves different classifiers and uses a leave-one-out methodology. Results demonstrate the feasibility of estimating breast density using image processing and analysis techniques

Assessing the Precision of Compositional Data in a Stratified Double Stage Cluster Sample: Application to the Swiss Earnings Structure Survey

Precision of released figures is not only an important quality feature of official statistics,it is also essential for a good understanding of the data. In this paper we show a casestudy of how precision could be conveyed if the multivariate nature of data has to betaken into account. In the official release of the Swiss earnings structure survey, the totalsalary is broken down into several wage components. We follow Aitchison's approachfor the analysis of compositional data, which is based on logratios of components. Wefirst present diferent multivariate analyses of the compositional data whereby the wagecomponents are broken down by economic activity classes. Then we propose a numberof ways to assess precision

Stability of cascade networks via fluid models

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Etravirine Concentrations in Cerebrospinal Fluid in HIV-Infected Patients

Cerebrospinal fluid Etravirine concentrations were measured in 12 asymptomatic HIV-infected patients. Median ETR concentration in plasma was 611.5 ng/mL (148-991) and median CSF ETR concentration was 7.24 ng/ml (3.5-17.9). In all cases Etravirine levels were above the IC50 range(0.39-2.4ng/ml) and CSF viral load was &40 copies/ml in all patients with undetectable plasma viral load. Our data suggest that ETR achieves concentrations several times above the IC50 range in CSF. All patients with undetectable plasma viral load were virologically suppressed in CSF while receiving an ETR-containing regimen. ETR may help in controlling HIV-1 in CNS.

A comparison of the alr and ilr transformations for kernel density estimation of compositional data

In a seminal paper, Aitchison and Lauder (1985) introduced classical kernel densityestimation techniques in the context of compositional data analysis. Indeed, they gavetwo options for the choice of the kernel to be used in the kernel estimator. One ofthese kernels is based on the use the alr transformation on the simplex SD jointly withthe normal distribution on RD-1. However, these authors themselves recognized thatthis method has some deficiencies. A method for overcoming these dificulties based onrecent developments for compositional data analysis and multivariate kernel estimationtheory, combining the ilr transformation with the use of the normal density with a fullbandwidth matrix, was recently proposed in Martín-Fernández, Chacón and Mateu-Figueras (2006). Here we present an extensive simulation study that compares bothmethods in practice, thus exploring the finite-sample behaviour of both estimators

Chaos and order in the contemporary city. The impact of urban spatial structure on population density and commuting distance in Barcelona, 1986-2001.

One of the criticisms leveled at the model of dispersed city found all over the world is its unarticulated, random, and undifferentiated nature. To check this idea in the Barcelona Metropolitan Region, we estimated the impact of the urban spatial structure (CBD, subcenters and transportation infrastructures) over the population density and commuting distance. The results are unfavorable to the hypothesis of the increasing destructuring of cities given that the explanatory capacity of both functions improves over time, both when other control variables are not included and when they are included.

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

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

Nuclear magnetic resonance chemical shifts with the statistical average of orbital-dependent model potentials in Kohn-Sham density functional theory

The performance of the SAOP potential for the calculation of NMR chemical shifts was evaluated. SAOP results show considerable improvement with respect to previous potentials, like VWN or BP86, at least for the carbon, nitrogen, oxygen, and fluorine chemical shifts. Furthermore, a few NMR calculations carried out on third period atoms (S, P, and Cl) improved when using the SAOP potential

Effect of basis set superposition error on the electron density of molecular complexes

The effect of basis set superposition error (BSSE) on molecular complexes is analyzed. The BSSE causes artificial delocalizations which modify the first order electron density. The mechanism of this effect is assessed for the hydrogen fluoride dimer with several basis sets. The BSSE-corrected first-order electron density is obtained using the chemical Hamiltonian approach versions of the Roothaan and Kohn-Sham equations. The corrected densities are compared to uncorrected densities based on the charge density critical points. Contour difference maps between BSSE-corrected and uncorrected densities on the molecular plane are also plotted to gain insight into the effects of BSSE correction on the electron density

A quantum molecular similarity analysis of changes in molecular electron density caused by basis set flotation and electric field application

Quantum molecular similarity (QMS) techniques are used to assess the response of the electron density of various small molecules to application of a static, uniform electric field. Likewise, QMS is used to analyze the changes in electron density generated by the process of floating a basis set. The results obtained show an interrelation between the floating process, the optimum geometry, and the presence of an external field. Cases involving the Le Chatelier principle are discussed, and an insight on the changes of bond critical point properties, self-similarity values and density differences is performed