Weighted Logratio Biplots, Correspondence Analysis and Spectral Maps

Starting with logratio biplots for compositional data, which are based on the principle of subcompositional coherence, and then adding weights, as in correspondence analysis, we rediscover Lewi's spectral map and many connections to analyses of two-way tables of non-negative data. Thanks to the weighting, the method also achieves the property of distributional equivalence

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

Bernstein's problem on weighted polynomial approximation

We formulate a necessary and sufficient condition for polynomials to be dense in a space of continuous functions on the real line, with respect to Bernstein's weighted uniform norm. Equivalently, for a positive finite measure [lletra "mu" minúscula de l'alfabet grec] on the real line we give a criterion for density of polynomials in Lp[lletra "mu" minúscula de l'alfabet grec entre parèntesis].

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

Optimum power allocation for parallel Gaussian channels with arbitrary input distributions

The mutual information of independent parallel Gaussian-noise channels is maximized, under an average power constraint, by independent Gaussian inputs whose power is allocated according to the waterfilling policy. In practice, discrete signalling constellations with limited peak-to-average ratios (m-PSK, m-QAM, etc) are used in lieu of the ideal Gaussian signals. This paper gives the power allocation policy that maximizes the mutual information over parallel channels with arbitrary input distributions. Such policy admits a graphical interpretation, referred to as mercury/waterfilling, which generalizes the waterfilling solution and allows retaining some of its intuition. The relationship between mutual information of Gaussian channels and nonlinear minimum mean-square error proves key to solving the power allocation problem.

Social bases of changing income distributions

On the backdrop of very little sociological concern with rising income inequality, this paper examines how key changes in sociodemographic behaviour may help shed additional light on changes in household income distribution and especially on long-term income dynamics and inter-generational mobility. The paper argues that the joint effect of rising marital homogamy in terms of human capital and labour supply contributes generally to widen the income gap between households. Only uner very restrictive conditions, namely when the labour supply of low educated women grows dis-proportionally fast, will women's earnings contribute to more equality. Finally, the paper suggests that women's rising employment commitments contribute positively to equalizing the opportunity structure both via the income effect and if quality care is available, also via more homogenous cultural and cognitive stimulation of children. Mother's work does not generally have adverse effects for children's development.

The world distribution of income (estimated from individual country distributions)

We estimate the world distribution of income by integrating individualincome distributions for 125 countries between 1970 and 1998. Weestimate poverty rates and headcounts by integrating the density functionbelow the $1/day and $2/day poverty lines. We find that poverty ratesdecline substantially over the last twenty years. We compute povertyheadcounts and find that the number of one-dollar poor declined by 235million between 1976 and 1998. The number of $2/day poor declined by 450million over the same period. We analyze poverty across different regionsand countries. Asia is a great success, especially after 1980. LatinAmerica reduced poverty substantially in the 1970s but progress stoppedin the 1980s and 1990s. The worst performer was Africa, where povertyrates increased substantially over the last thirty years: the number of$1/day poor in Africa increased by 175 million between 1970 and 1998,and the number of $2/day poor increased by 227. Africa hosted 11% ofthe world s poor in 1960. It hosted 66% of them in 1998. We estimatenine indexes of income inequality implied by our world distribution ofincome. All of them show substantial reductions in global incomeinequality during the 1980s and 1990s.

Weighted Kernel regression

In the fixed design regression model, additional weights areconsidered for the Nadaraya--Watson and Gasser--M\"uller kernel estimators.We study their asymptotic behavior and the relationships between new andclassical estimators. For a simple family of weights, and considering theIMSE as global loss criterion, we show some possible theoretical advantages.An empirical study illustrates the performance of the weighted estimatorsin finite samples.

Weighted metric multidimensional scaling

This paper establishes a general framework for metric scaling of any distance measure between individuals based on a rectangular individuals-by-variables data matrix. The method allows visualization of both individuals and variables as well as preserving all the good properties of principal axis methods such as principal components and correspondence analysis, based on the singular-value decomposition, including the decomposition of variance into components along principal axes which provide the numerical diagnostics known as contributions. The idea is inspired from the chi-square distance in correspondence analysis which weights each coordinate by an amount calculated from the margins of the data table. In weighted metric multidimensional scaling (WMDS) we allow these weights to be unknown parameters which are estimated from the data to maximize the fit to the original distances. Once this extra weight-estimation step is accomplished, the procedure follows the classical path in decomposing a matrix and displaying its rows and columns in biplots.

Weighted Euclidean biplots

We construct a weighted Euclidean distance that approximates any distance or dissimilarity measure between individuals that is based on a rectangular cases-by-variables data matrix. In contrast to regular multidimensional scaling methods for dissimilarity data, the method leads to biplots of individuals and variables while preserving all the good properties of dimension-reduction methods that are based on the singular-value decomposition. The main benefits are the decomposition of variance into components along principal axes, which provide the numerical diagnostics known as contributions, and the estimation of nonnegative weights for each variable. The idea is inspired by the distance functions used in correspondence analysis and in principal component analysis of standardized data, where the normalizations inherent in the distances can be considered as differential weighting of the variables. In weighted Euclidean biplots we allow these weights to be unknown parameters, which are estimated from the data to maximize the fit to the chosen distances or dissimilarities. These weights are estimated using a majorization algorithm. Once this extra weight-estimation step is accomplished, the procedure follows the classical path in decomposing the matrix and displaying its rows and columns in biplots.

Characterization of intonation in Karṇāṭaka music by parametrizing context-based Svara Distributions

Intonation is a fundamental music concept that has a special relevance in Indian art music. It is characteristic of the rāga and intrinsic to the musical expression of the performer. Describing intonation is of importance to several information retrieval tasks like the development of rāga and artist similarity measures. In our previous work, we proposed a compact representation of intonation based on the parametrization of the pitch histogram of a performance and demonstrated the usefulness of this representation through an explorative rāga recognition task in which we classified 42 vocal performances belonging to 3 rāgas using parameters of a single svara. In this paper, we extend this representation to employ context-based svara distributions, which are obtained with a different approach to find the pitches belonging to each svara. We quantitatively compare this method to our previous one, discuss the advantages, and the necessary melodic analysis to be carried out in future.

Dispersal probability distributions and the wave-front speed problem

The speed and width of front solutions to reaction-dispersal models are analyzed both analytically and numerically. We perform our analysis for Laplace and Gaussian distribution kernels, both for delayed and nondelayed models. The results are discussed in terms of the characteristic parameters of the models