Correction to: Multivariate cluster weighted models using skewed distributions

Advances in Data Analysis and Classification - In the original publication of the article, the line after equation (5) has been published incorrectly     

On well-posedness of semilinear parabolic and elliptic problems in the hyperbolic space

We investigate existence and uniqueness of solutions to semilinear parabolic and elliptic equations in bounded domains of the n-dimensional hyperbolic space (n?3). Lp→Lq estimates for the semigroup generated by the Laplace-Beltrami operator are obtained and then used to prove existence and uniqueness results for parabolic problems. Moreover, under proper assumptions on the nonlinear function, we establish uniqueness of positive classical solutions and nonuniqueness of singular solutions of the elliptic problem; furthermore, for the corresponding semilinear parabolic problem, nonuniqueness of weak solutions is stated.     

Uniqueness and non-uniqueness of bounded solutions to singular nonlinear parabolic equations

We investigate well-posedness of initial-boundary value problems for a class of nonlinear parabolic equations with variable density. At some part of the boundary, called singular boundary, the density can either vanish or diverge or not need to have a limit. We provide simple conditions for uniqueness or non-uniqueness of bounded solutions, depending on the behaviour of the density near the singular boundary.     

Clustering Multivariate Longitudinal Observations: The Contaminated Gaussian Hidden Markov Model

The Gaussian hidden Markov model (HMM) is widely considered for the analysis of heterogenous continuous multivariate longitudinal data. To robustify this approach with respect to possible elliptical heavy-tailed departures from normality, due to the presence of outliers, spurious points, or noise (collectively referred to as bad points herein), the contaminated Gaussian HMM is here introduced. The contaminated Gaussian distribution represents an elliptical generalization of the Gaussian distribution and allows for automatic detection of bad points in the same natural way as observations are typically assigned to the latent states in the HMM context. Once the model is fitted, each observation has a posterior probability of belonging to a particular state and, inside each state, of being a bad point or not. In addition to the parameters of the classical Gaussian HMM, for each state we have two more parameters, both with a specific and useful interpretation: one controls the proportion of bad points and one specifies their degree of atypicality. A sufficient condition for the identifiability of the model is given, an expectation-conditional maximization algorithm is outlined for parameter estimation and various operational issues are discussed. Using a large-scale simulation study, but also an illustrative artificial dataset, we demonstrate the effectiveness of the proposed model in comparison with HMMs of different elliptical distributions, and we also evaluate the performance of some well-known information criteria in selecting the true number of latent states. The model is finally used to fit data on criminal activities in Italian provinces. Supplementary materials for this article are available online     

Interfacial free energy driven nanophase separation in poly(3-hexylthiophene)/[6,6]-phenyl-C61-butyric acid methyl ester thin films

The nanostructure of thermally annealed thin films of poly(3-hexylthiophene) (P3HT) and [6,6]-phenyl-C61-butyric acid methyl ester (PCBM) blends on hydrophobic and hydrophilic substrates was studied to unravel the relationship between the substrate properties and the phase structure of polymer blends in confined geometry. Indeed, the nature of the employed substrates was found to affect the extent of phase separation, the PCBM aggregation state and the texture of the whole system. In particular, annealing below the melting temperature of the polymer yielded the formation of PCBM nanometric crystallites on the hydrophobic substrates, while mostly amorphous microscopic aggregates were formed on the hydrophilic ones. Moreover, while an enhanced in-plane orientation of P3HT lamellae was promoted on hydrophobic substrates, a markedly tilted geometry was produced on the hydrophilic ones. The observed effects were interpreted in terms of a simple model connecting the interface free energy for the blend films to the different polymer chain mobility and diffusion velocity of PCBM molecules on the different substrates.     

Finite mixtures of unimodal beta and gamma densities and the k-bumps algorithm

This paper addresses the problem of estimating a density, with either a compact support or a support bounded at only one end, exploiting a general and natural form of a finite mixture of distributions. Due to the importance of the concept of multimodality in the mixture framework, unimodal beta and gamma densities are used as mixture components, leading to a flexible modeling approach. Accordingly, a mode-based parameterization of the components is provided. A partitional clustering method, named $k$ -bumps, is also proposed; it is used as an ad hoc initialization strategy in the EM algorithm to obtain the maximum likelihood estimation of the mixture parameters. The performance of the $k$ -bumps algorithm as an initialization tool, in comparison to other common initialization strategies, is evaluated through some simulation experiments. Finally, two real applications are presented.     

On well-posedness of the semilinear heat equation on the sphere

We are concerned with existence, uniqueness and nonuniqueness of nonnegative solutions to the semilinear heat equation in open subsets of the n-dimensional sphere. Existence and uniqueness results are obtained using L ^p ?? L ^q estimates for the semigroup generated by the Laplace?CBeltrami operator. Moreover, under proper assumptions on the nonlinear function, we establish nonuniqueness of weak solutions, when n??? 3; to do this, we shall prove uniqueness of positive classical solutions and nonuniqueness of singular solutions of the corresponding semilinear elliptic problem.     

Experimental and in silico characterization of a biologically active inosose

Inositols have been recently reported to show a biological activity as inhibitors of both glycosidase and amyloid-β protein. After having harvested good crystals suitable for single crystal X-ray diffraction, we performed a comparison with the data inferred by means of a molecular dynamics simulation, based on the use of an appropriate Force Field coupled to the most performing charging scheme. This approach allowed a detailed analysis extended to ultra-fine details, such as atomic displacement parameters. It confirmed the good validity of a robust approach already tested by us in previous studies. A NMR analysis of the molecule in solution was also carried out, to compare the structural findings suggested by the X-ray analysis with the ones in solution and avoid confining them to the solid-state. In this framework, we investigated the above-mentioned inhibiting activity of a class of inososes, by means of a molecular docking investigation, which proved the suggested validity of the studied compound as inhibitor of the α-glucosidase.     

Testing Serial Independence via Density-Based Measures of Divergence

This article reviews some nonparametric serial independence tests based on measures of divergence between densities. Among others, the well-known Kullback–Leibler, Hellinger, Tsallis, and Rosenblatt divergences are analyzed. Moreover, their copula-based version is taken into account. Via a wide simulation study, the performances of the considered serial independence tests are compared under different settings. Both single-lag and multiple-lag testing procedures are investigated to find out the best “omnibus” solution.