On the pointwise limits of bivariate lagrange projectors

A linear algebra proof is given of the fact that the nullspace of a finite-rank linear projector, on polynomials in two complex variables, is an ideal if and only if the projector is the bounded pointwise limit of Lagrange projectors, i.e., projectors whose nullspace is a radical ideal, i.e., the set of all polynomials that vanish on a certain given finite set. A characterization of such projectors is also given in the real case. More generally, a characterization is given of those finite-rank linear projectors, on polynomials in d complex variables, with nullspace an ideal that are the bounded pointwise limit of Lagrange projectors. The characterization is in terms of a certain sequence of d commuting linear maps and so focuses attention on the algebra generated by such sequences.     

The European regional policy and the socio-economic diversity of European regions: A multivariate analysis

There are significant differences among the European Union regions, which have been heightened due to the most recent enlargement in 2004. This paper aims to analyse this diversity and to propose a classification of European regions that is adjusted to the different axes of socio-economic development and, simultaneously, is useful for European regional policy purposes. Multivariate statistical techniques allow the identification of clusters of socio-economic similarity, which are contrasted with the classes considered in the financial proposal of the European Commission (EC) for the period 2007–2013.     

Statistics of extremes for IID data and breakthroughs in the estimation of the extreme value index: Laurens de Haan leading contributions

In the last decades there has been a shift from the parametric statistics of extremes for IID random variables, based on the probabilistic asymptotic results in extreme value theory, towards a semi-parametric approach, where the estimation of the right tail-weight, under a quite general framework, is of major importance. After a brief presentation of classical Gumbel’s block methodology and of later improvements in the parametric framework (multivariate and multi-dimensional extreme value models for largest observations and peaks over threshold approaches), we present a coordinated overview, over the last three decades, of the developments on the estimation of the extreme value index under a semiparametric framework. Laurens de Haan has been one of the leading scientists in the field, (co-)author of many seminal ideas, that he generously shared with dozens (literally) of colleagues and students, thus achieving one of the main goals in a scientist’s life: he gathered around him a bunch of colleagues united in the endeavour of building knowledge. The last section is a personal tribute to Laurens, who fully lives his ideal that co-operation is the heart of Science. To Laurens de Haan, a token of friendship.     

A filtered polynomial approach to density estimation

In this paper, a little known computational approach to density estimation based on filtered polynomial approximation is investigated. It is accompanied by the first online available density estimation computer program based on a filtered polynomial approach. The approximation yields the unknown distribution and density as the product of a monotonic increasing polynomial and a filter. The filter may be considered as a target distribution which gets fixed prior to the estimation. The filtered polynomial approach then provides coefficient estimates for (close) algebraic approximations to (a) the unknown density function and (b) the unknown cumulative distribution function as well as (c) a transformation (e.g., normalization) from the unknown data distribution to the filter. This approach provides a high degree of smoothness in its estimates for univariate as well as for multivariate settings. The nice properties as the high degree of smoothness and the ability to select from different target distributions are suited especially in MCMC simulations. Two applications in Sects. 1 and 7 will show the advantages of the filtered polynomial approach over the commonly used kernel estimation method.      

Three-dimensional voltammetry assisted by parametric signal fitting: A new perspective for the electrochemical evaluation of metal binding in the presence of electrodic adsorption

A new perspective of three-dimensional voltammetry is given for the study of metal complexation in the presence of adsorption phenomena and, eventually, when the electrochemical reversibility of the processes decreases. For this purpose, 3D-voltammograms are obtained as usual, i.e., by combination of chronoamperograms measured at different potential steps, but they are analysed by the new methodology of parametric signal fitting (PSF), which uses the full data matrix to fit a five-parameter equation to every component. The parametric signals can be time-dependent peaks or waves and the optimised parameters inform about the concentrations in both dissolved and adsorbed phases. The analysis can be carried out on both individual and augmented matrices. The preliminary tests on the well-known Cd(II)-glutathione system yield promising results and encourage the combination of 3D-voltammetry and PSF as a complementary tool for the study of metal complexation in complicated systems.     

Extremal lattices and the construction of lattice rules

After introducing the concept of extremal lattices we show how these can be used to construct lattice rules of a high trigonometric degree.     

Multicriteria variable selection for classification of production batches

In many industrial processes hundreds of noisy and correlated process variables are collected for monitoring and control purposes. The goal is often to correctly classify production batches into classes, such as good or failed, based on the process variables. We propose a method for selecting the best process variables for classification of process batches using multiple criteria including classification performance measures (i.e., sensitivity and specificity) and the measurement cost. The method applies Partial Least Squares (PLS) regression on the training set to derive an importance index for each variable. Then an iterative classification/elimination procedure using k-Nearest Neighbor is carried out. Finally, Pareto analysis is used to select the best set of variables and avoid excessive retention of variables. The method proposed here consistently selects process variables important for classification, regardless of the batches included in the training data. Further, we demonstrate the advantages of the proposed method using six industrial datasets.     

ALTERNATIVE MEASURES OF THE EXPLANATORY POWER OF GENERAL MULTIVARIATE REGRESSION MODELS

In this paper R ²-type measures of the explanatory power of multivariate linear and categorical probit models proposed in the literature are reviewed and their deficiencies discussed. It is argued that a measure of the explanatory power should take into account the components which are explicitly modelled when a regression model is estimated while it should be indifferent to components not explicitly modelled. Based on this view three different measures for multivariate probit models are proposed. Results of a simulation study are presented, designed to compare two measures in various situations, to evaluate the BC _a bootstrap technique for testing the hypothesis that the corresponding measure is zero, and to calculate approximate confidence intervals. The BC _a bootstrap technique turned out to work quite well for a wide range of situations, but may lead to misleading results if the true values of the corresponding measure are close to zero.     

Mathematical modeling of dispersion in single interface flow analysis

This work describes the optimization of the recently proposed fluid management methodology single interface flow analysis (SIFA) using chemometrics modelling. The influence of the most important physical and hydrodynamic flow parameters of SIFA systems on the axial dispersion coefficients estimated with the axially dispersed plug-flow model, was evaluated with chemometrics linear (multivariate linear regression) and non-linear (simple multiplicative and feed-forward neural networks) models. A D-optimal experimental design built with three reaction coil properties (length, configuration and internal diameter), flow-cell volume and flow rate, was adopted to generate the experimental data. Bromocresol green was used as the dye solution and the analytical signals were monitored by spectrophotometric detection at 614 nm. Results demonstrate that, independent of the model type, the statistically relevant parameters were the reactor coil length and internal diameter and the flow rate. The linear and non-linear multiplicative models were able to estimate the axial dispersion coefficient with validation r² = 0.86. Artificial neural networks estimated the same parameter with an increased accuracy (r² = 0.93), demonstrating that relations between the physical parameters and the dispersion phenomena are highly non-linear. The analysis of the response surface control charts simulated with the developed models allowed the interpretation of the relationships between the physical parameters and the dispersion processes.     

A differentially weighted Monte Carlo method for two-component coagulation

The direct simulation Monte Carlo (DSMC) method for population balance modeling is capable of retaining the history of each simulation particle and is thus able to deal with multivariate properties in a simple and straightforward manner. As opposed to conventional DSMC approaches that track equally weighted simulation particles, a differentially weighted Monte Carlo method is extended to simulate two-component coagulation processes and is thereby able to simulate the micromixing of the components. A new feature of the method for this bivariate population balance modeling is that it is possible to specify how the simulation particles are distributed over the compositional axis. This allows us to obtain information about particles in those regions of the size and composition distribution functions where the non-weighted MC methods place insufficient simulation particles to obtain an inaccurate solution. The new feature results in lower statistical noise for simulating two-component coagulation, which is validated by using two-component coagulation cases for which analytical solutions exist (a discrete process with sum kernel for initial monodisperse populations and a process with constant kernel for initial polydisperse populations).