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Univariate regression models have rich literature for counting data. However, this is not the case for multivariate count data. Therefore, we present the Multivariate Generalized Linear Mixed Models framework that deals with a multivariate set of responses, measuring the correlation between them through random effects that follows a multivariate normal distribution. This model is based on a GLMM with a random intercept and the estimation process remains the same as a standard GLMM with random effects integrated out via Laplace approximation. We efficiently implemented this model through the TMB package available in R. We used Poisson, negative binomial (NB), and COM-Poisson distributions. To assess the estimator properties, we conducted a simulation study considering four different sample sizes and three different correlation values for each distribution. We achieved unbiased and consistent estimators for Poisson and NB distributions; for COM-Poisson estimators were consistent, but biased, especially for dispersion, variance, and correlation parameter estimators. These models were applied to two datasets. The first concerns a sample from 30 different sites collected in Australia where the number of times each one of the 41 different ant species was registered; which results in an impressive 820 variance-covariance and 41 dispersion parameters estimated simultaneously, let alone the regression parameters. The second is from the Australia Health Survey with 5 response variables and 5190 respondents. These datasets can be considered overdispersed by the generalized dispersion index. The COM-Poisson model overcame the other two competitors considering three goodness-of-fit indexes. Therefore, the proposed model is capable of dealing with multivariate count data, and measuring any kind of correlation between them taking into account the effects of the covariates., Comment: 22pages, 12 figures, 5 tables

In the analysis of count data often the equidispersion assumption is not suitable, hence the Poisson regression model is inappropriate. As a generalization of the Poisson distribution, the COM-Poisson distribution can deal with under-, equi- and overdispersed count data. It is a member of the exponential family of distributions and has well known special cases. In spite of the nice properties of the COM-Poisson distribution, its location parameter does not correspond to the expectation, which complicates the interpretation of regression models. In this paper, we propose a straightforward reparametrization of the COM-Poisson distribution based on an approximation to the expectation of this distribution. The main advantage of our new parametrization is the straightforward interpretation of the regression coefficients in terms of the expectation, as usual in the context of generalized linear models. Furthermore, the estimation and inference for the new COM-Poisson regression model can be done based on the likelihood paradigm. We carried out simulation studies to verify the finite sample properties of the maximum likelihood estimators. The results from our simulation study show that the maximum likelihood estimators are unbiased and consistent for both regression and dispersion parameters. We observed that the empirical correlation between the regression and dispersion parameter estimators is close to zero, which suggests that these parameters are orthogonal. We illustrate the application of the proposed model through the analysis of three data sets with over-, under- and equidispersed count data. The study of distribution properties through a consideration of dispersion, zero-inflated and heavy tail indexes, together with the results of data analysis show the flexibility over standard approaches., Comment: 23 pages, 12 figures and 7 tables. Submitted article

Tweedie regression models provide a flexible family of distributions to deal with non-negative highly right-skewed data as well as symmetric and heavy tailed data and can handle continuous data with probability mass at zero. The estimation and inference of Tweedie regression models based on the maximum likelihood method are challenged by the presence of an infinity sum in the probability function and non-trivial restrictions on the power parameter space. In this paper, we propose two approaches for fitting Tweedie regression models, namely, quasi- and pseudo-likelihood. We discuss the asymptotic properties of the two approaches and perform simulation studies to compare our methods with the maximum likelihood method. In particular, we show that the quasi-likelihood method provides asymptotically efficient estimation for regression parameters. The computational implementation of the alternative methods is faster and easier than the orthodox maximum likelihood, relying on a simple Newton scoring algorithm. Simulation studies showed that the quasi- and pseudo-likelihood approaches present estimates, standard errors and coverage rates similar to the maximum likelihood method. Furthermore, the second-moment assumptions required by the quasi- and pseudo-likelihood methods enables us to extend the Tweedie regression models to the class of quasi-Tweedie regression models in the Wedderburn's style. Moreover, it allows to eliminate the non-trivial restriction on the power parameter space, and thus provides a flexible regression model to deal with continuous data. We provide \texttt{R} implementation and illustrate the application of Tweedie regression models using three data sets., Comment: 34 pages, 8 figures

We propose a new class of discrete generalized linear models based on the class of Poisson-Tweedie factorial dispersion models with variance of the form $\mu + \phi\mu^p$, where $\mu$ is the mean, $\phi$ and $p$ are the dispersion and Tweedie power parameters, respectively. The models are fitted by using an estimating function approach obtained by combining the quasi-score and Pearson estimating functions for estimation of the regression and dispersion parameters, respectively. This provides a flexible and efficient regression methodology for a comprehensive family of count models including Hermite, Neyman Type A, P\'olya-Aeppli, negative binomial and Poisson-inverse Gaussian. The estimating function approach allows us to extend the Poisson-Tweedie distributions to deal with underdispersed count data by allowing negative values for the dispersion parameter $\phi$. Furthermore, the Poisson-Tweedie family can automatically adapt to highly skewed count data with excessive zeros, without the need to introduce zero-inflated or hurdle components, by the simple estimation of the power parameter. Thus, the proposed models offer a unified framework to deal with under, equi, overdispersed, zero-inflated and heavy-tailed count data. The computational implementation of the proposed models is fast, relying only on a simple Newton scoring algorithm. Simulation studies showed that the estimating function approach provides unbiased and consistent estimators for both regression and dispersion parameters. We highlight the ability of the Poisson-Tweedie distributions to deal with count data through a consideration of dispersion, zero-inflated and heavy tail indices, and illustrate its application with four data analyses. We provide an \texttt{R} implementation and the data sets as supplementary materials., Comment: 31 pages, 8 figures

Beta regression models are a suitable choice for continuous response variables on the unity interval. Random effects add further flexibility to the models and accommodate data structures such as hierarchical, repeated measures and longitudinal, which typically induce extra variability and/or dependence. Closed expressions cannot be obtained for parameter estimation and numerical methods are required and possibly combined with sampling algorithms. We focus on likelihood inference and related algorithms for the analysis of beta mixed models motivated by two real problems with grouped data structures. The first is a study on a life quality index of industry workers with data collected according to an hierarchical sampling scheme. The second is a study with a nested and longitudinal data structure assessing the impact of hydroelectric power plants upon measures of water quality indexes up, downstream and at the reservoirs of the dammed rivers. Relevant scientific hypothesis are investigated by comparing alternative models. The analysis uses different algorithms including data-cloning, an alternative to numerical approximations which also assess identifiability. Confidence intervals based on profiled likelihoods are compared to those obtained by asymptotic quadratic approximations, showing relevant differences for parameters related to the random effects.

Minkowski valuations provide a systematic framework for quantifying different aspects of morphology. In this paper we apply vector- and tensor-valued Minkowski valuations to neuronal cells from the cat's retina in order to describe their morphological structure in a comprehensive way. We introduce the framework of Minkowski valuations, discuss their implementation for neuronal cells and show how they can discriminate between cells of different types., Comment: 14 pages, 18 postscript figures

The two-dimensional interpolating readout concept of the Virtual-Pixel detector (ViP detector) goes along with an enormous reduction of electronic channels compared to pure pixel devices. However, the special concept of the readout structure demands for adequate position reconstruction methods. Theoretical considerations reflecting the choice and the combination of linear algorithms with emphasis on correct position reconstruction and spatial resolution are presented. A subsequent two-dimensional coordinate transformation further improves the image response of the system. Measured images show how far the theoretical predictions of the simulations can be verified and to what extent they can be used to improve the position reconstruction., Comment: 25 pages, 18 figures, Submitted to Nucl. Instr. and Meth. A

We investigate the gas gain behaviour of a triple-GEM configuration in gas mixtures of argon, krypton and xenon with ten and thirty percent of carbon dioxide at pressures between 1 and 3 bar. Since the signal widths affect the dead time behaviour of the detector we present signal length measurements to evaluate the use of the triple-GEM in time-resolved X-ray imaging., Comment: 19 pages, 21 figures, revised version, accepted for publication in Nucl. Instr. and Meth. A

A MicroCAT (Micro Compteur A Trous) structure which is used for avalanche charge multiplication in gas filled radiation detectors has been optimised with respect to maximum electron transparency and minimum ion feedback. We report on the charge transfer behaviour and the achievable gas gain of this device. A three-dimensional electron and ion transfer simulation is compared to results derived from electric current measurements. Similarly, we present studies of the charge transfer behaviour of a GEM (Gas Electron Multiplier) by current measurements and simulations. Finally, we investigate the combination of the MicroCAT and the GEM by measurements with respect to the performance at different voltage settings, gas mixtures and gas pressures., Comment: 26 pages, 32 figures

In a follow-up study to a previous analysis of the IRAS 1.2Jy catalogue, we quantify the morphological fluctuations in the PSCz survey. We use a variety of measures, among them the family of scalar Minkowski functionals. We confirm the existence of significant fluctuations that are discernible in volume-limited samples out to 200Mpc/h. In contrast to earlier findings, comparisons with cosmological N-body simulations reveal that the observed fluctuations roughly agree with the cosmic variance found in corresponding mock samples. While two-point measures, e.g. the variance of count-in-cells, fluctuate only mildly, the fluctuations in the morphology on large scales indicate the presence of coherent structures that are at least as large as the sample., Comment: aa style, 13 pages, 9 figures, matching accepted version

Minkowski functionals constitute a family of order parameters which discriminate spatial patterns according to size, shape and connectivity. Here we point out, that these scalar descriptors can be complemented by vector-valued curvature measures also known as Quermass vectors. Using examples of galaxy clusters, we demonstrate that the Quermass vectors provide additional morphological information on directional features and symmetries displayed by spatial data., Comment: 14 pages, Latex, to appear in Physica A

We determine the Minkowski functionals for a sample of Abell/ACO clusters, 401 with measured and 16 with estimated redshifts. The four Minkowski functionals (including the void probability function and the mean genus) deliver a global description of the spatial distribution of clusters on scales from $10$ to $60\hMpc$ with a clear geometric interpretation. Comparisons with mock catalogues of N--body simulations using different variants of the CDM model demonstrate the discriminative power of the description. The standard CDM model and the model with tilted perturbation spectrum cannot generate the Minkowski functionals of the cluster data, while a model with a cosmological constant and a model with breaking of the scale invariance of perturbations (BSI) yield compatible results., Comment: 10 pages, 13 Postscript figures, uses epsf.sty and mn.sty (included), submitted to MNRAS

We propose a novel method for the description of spatial patterns formed by a coverage of point sets representing galaxy samples. This method is based on a complete family of morphological measures known as Minkowski functionals, which includes the topological Euler characteristic and geometric descriptors to specify the content, shape and connectivity of spatial sets. The method is numerically robust even for small samples, independent of statistical assumptions, and yields global as well as local morphological information. We illustrate the method by applying it to a Poisson process, a `double-Poisson' process, and to the Abell catalogue of galaxy clusters., Comment: submitted to A & A, Main Journal. TeX, 17 pages excl.figures; contact T. Buchert: TOB@ibma.ipp-garching.mpg.de