Conditional density approximations with mixtures of polynomials

Mixtures of polynomials (MoPs) are a non-parametric density estimation technique especially designed for hybrid Bayesian networks with continuous and discrete variables. Algorithms to learn one- and multi-dimensional (marginal) MoPs from data have recently been proposed. In this paper we introduce two methods for learning MoP approximations of conditional densities from data. Both approaches are based on learning MoP approximations of the joint density and the marginal density of the conditioning variables, but they differ as to how the MoP approximation of the quotient of the two densities is found. We illustrate and study the methods using data sampled from known parametric distributions, and we demonstrate their applicability by learning models based on real neuroscience data. Finally, we compare the performance of the proposed methods with an approach for learning mixtures of truncated basis functions (MoTBFs). The empirical results show that the proposed methods generally yield models that are comparable to or significantly better than those found using the MoTBF-based method.

Some Marginal Densities of the Wishart Distribution

Евелина Илиева Велева - Разпределението на Уишарт се среща в практиката като разпределението на извадъчната ковариационна матрица за наблюдения над многомерно нормално разпределение. Изведени са някои маргинални плътности, получени чрез интегриране на плътността на Уишарт разпределението. Доказани са необходими и достатъчни условия за положителна определеност на една матрица, които дават нужните граници за интегрирането.

Marginal Densities of the Wishart Distribution

2010 Mathematics Subject Classification: 62H10.

Gaussian anamorphosis in the analysis step of the EnKF: a joint state-variable/observation approach

The analysis step of the (ensemble) Kalman filter is optimal when (1) the distribution of the background is Gaussian, (2) state variables and observations are related via a linear operator, and (3) the observational error is of additive nature and has Gaussian distribution. When these conditions are largely violated, a pre-processing step known as Gaussian anamorphosis (GA) can be applied. The objective of this procedure is to obtain state variables and observations that better fulfil the Gaussianity conditions in some sense. In this work we analyse GA from a joint perspective, paying attention to the effects of transformations in the joint state variable/observation space. First, we study transformations for state variables and observations that are independent from each other. Then, we introduce a targeted joint transformation with the objective to obtain joint Gaussianity in the transformed space. We focus primarily in the univariate case, and briefly comment on the multivariate one. A key point of this paper is that, when (1)-(3) are violated, using the analysis step of the EnKF will not recover the exact posterior density in spite of any transformations one may perform. These transformations, however, provide approximations of different quality to the Bayesian solution of the problem. Using an example in which the Bayesian posterior can be analytically computed, we assess the quality of the analysis distributions generated after applying the EnKF analysis step in conjunction with different GA options. The value of the targeted joint transformation is particularly clear for the case when the prior is Gaussian, the marginal density for the observations is close to Gaussian, and the likelihood is a Gaussian mixture.

Estimadores do tipo núcleo para a densidade de transição de uma cadeia de markov com espaço de estados geral

In this work, we studied the strong consistency for a class of estimates for a transition density of a Markov chain with general state space E ⊂ Rd. The strong ergodicity of the estimates for the density transition is obtained from the strong consistency of the kernel estimates for both the marginal density p(:) of the chain and the joint density q(., .). In this work the Markov chain is supposed to be homogeneous, uniformly ergodic and possessing a stationary density p(.,.)

Genetic Parameters for Reproductive Traits of Crossbred Buffaloes from Brazil, Estimated by Bayesian Inference

The objective of the study was to estimate heritability for calving interval (CI) and age at first calving (AFC) and also calculate repeatability for CI in buffaloes using Bayesian inference. The Brazilian Buffaloes Genetic Improvement Program provided the database. Data consists on information from 628 females and four different herds, born between 1980 and 2003. In order to estimate the variance, univariate analyses were performed employing Gibbs sampler procedure included in the MTGSAM software. The model for CI included the random effects direct additive and permanent environment factors, and the fixed effects of contemporary groups and calving orders. The model for AFC included the direct additive random effect and contemporary groups as a fixed effect. The convergence diagnosis was obtained using Geweke that was implemented through the Bayesian Output Analysis package in R software. The estimated averages were 433.2 days and 36.7months for CI and AFC, respectively. The means, medians and modes for the calculated heritability coefficients were similar. The heritability coefficients were 0.10 and 0.42 for CI and AFC respectively, with a posteriori marginal density that follows a normal distribution for both traits. The repeatability for CI was 0.13. The low heritability estimated for CI indicates that the variation in this trait is, to a large extent, influenced by environmental factors such as herd management policies. The age at first calving has clear potential for yield improvement through direct selection in these animals.

Non-Markovian stochastic processes and their applications: from anomalous diffusion to time series analysis

This work provides a forward step in the study and comprehension of the relationships between stochastic processes and a certain class of integral-partial differential equation, which can be used in order to model anomalous diffusion and transport in statistical physics. In the first part, we brought the reader through the fundamental notions of probability and stochastic processes, stochastic integration and stochastic differential equations as well. In particular, within the study of H-sssi processes, we focused on fractional Brownian motion (fBm) and its discrete-time increment process, the fractional Gaussian noise (fGn), which provide examples of non-Markovian Gaussian processes. The fGn, together with stationary FARIMA processes, is widely used in the modeling and estimation of long-memory, or long-range dependence (LRD). Time series manifesting long-range dependence, are often observed in nature especially in physics, meteorology, climatology, but also in hydrology, geophysics, economy and many others. We deepely studied LRD, giving many real data examples, providing statistical analysis and introducing parametric methods of estimation. Then, we introduced the theory of fractional integrals and derivatives, which indeed turns out to be very appropriate for studying and modeling systems with long-memory properties. After having introduced the basics concepts, we provided many examples and applications. For instance, we investigated the relaxation equation with distributed order time-fractional derivatives, which describes models characterized by a strong memory component and can be used to model relaxation in complex systems, which deviates from the classical exponential Debye pattern. Then, we focused in the study of generalizations of the standard diffusion equation, by passing through the preliminary study of the fractional forward drift equation. Such generalizations have been obtained by using fractional integrals and derivatives of distributed orders. In order to find a connection between the anomalous diffusion described by these equations and the long-range dependence, we introduced and studied the generalized grey Brownian motion (ggBm), which is actually a parametric class of H-sssi processes, which have indeed marginal probability density function evolving in time according to a partial integro-differential equation of fractional type. The ggBm is of course Non-Markovian. All around the work, we have remarked many times that, starting from a master equation of a probability density function f(x,t), it is always possible to define an equivalence class of stochastic processes with the same marginal density function f(x,t). All these processes provide suitable stochastic models for the starting equation. Studying the ggBm, we just focused on a subclass made up of processes with stationary increments. The ggBm has been defined canonically in the so called grey noise space. However, we have been able to provide a characterization notwithstanding the underline probability space. We also pointed out that that the generalized grey Brownian motion is a direct generalization of a Gaussian process and in particular it generalizes Brownain motion and fractional Brownain motion as well. Finally, we introduced and analyzed a more general class of diffusion type equations related to certain non-Markovian stochastic processes. We started from the forward drift equation, which have been made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation has been interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time-evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the same memory kernel K(t). We developed several applications and derived the exact solutions. Moreover, we considered different stochastic models for the given equations, providing path simulations.

Marginal cell density in young and old CBA/J and CBA/CAJ mice

The goal of the present study was to compare aging characteristics of the cochlear lateral wall in inbred mouse strains (CBA/J and CBA/CaJ) having very different endocochlear potential (EP)-versus-age profiles to see which anatomic differences might predict their EP differences.

The effect of endodontic materials on the optical density of dyes used in marginal leakage studies

The aim of this study was to determine the effect of the exposure of different endodontic materials to different dye solutions by evaluating the optical density of the dye solutions. Seventy-five plastic tubes were filled with one of the following materials: AH Plus, Sealapex, Portland cement, MTA (Angelus and Pro Root) and fifteen control plastic tubes were not. Each specimen of material and control was immersed in a container with 1 ml of each dye solution. A 0.1 ml-dye solution aliquote was removed before immersion and after 12, 24, 48 and 72 hours of each specimen immersion to record its optical density (OD) in a spectrophotometer. Statistical analysis was performed with ANOVA and Tukey tests (5%). No significant difference was found among any of the solution OD values for AH Plus cement. Portland cement promoted different OD values after 12 hours of immersion. MTA-Angelus cement presented different OD values only for 2% rhodamine B and the MTA-Pro Root cement presented different OD values in all 2% rhodamine B samples. Sealapex cement promoted a reduction in the India Ink OD values. Dye evaluation through OD seems to be an interesting method to select the best dye solution to use in a given marginal leakage study.

Recursive pathways to marginal likelihood estimation with prior-sensitivity analysis

We investigate the utility to computational Bayesian analyses of a particular family of recursive marginal likelihood estimators characterized by the (equivalent) algorithms known as "biased sampling" or "reverse logistic regression" in the statistics literature and "the density of states" in physics. Through a pair of numerical examples (including mixture modeling of the well-known galaxy dataset) we highlight the remarkable diversity of sampling schemes amenable to such recursive normalization, as well as the notable efficiency of the resulting pseudo-mixture distributions for gauging prior-sensitivity in the Bayesian model selection context. Our key theoretical contributions are to introduce a novel heuristic ("thermodynamic integration via importance sampling") for qualifying the role of the bridging sequence in this procedure, and to reveal various connections between these recursive estimators and the nested sampling technique.

A new family of multivariate heavy-tailed distributions with variable marginal amounts of tailweight: Application to robust clustering

We propose a family of multivariate heavy-tailed distributions that allow variable marginal amounts of tailweight. The originality comes from introducing multidimensional instead of univariate scale variables for the mixture of scaled Gaussian family of distributions. In contrast to most existing approaches, the derived distributions can account for a variety of shapes and have a simple tractable form with a closed-form probability density function whatever the dimension. We examine a number of properties of these distributions and illustrate them in the particular case of Pearson type VII and t tails. For these latter cases, we provide maximum likelihood estimation of the parameters and illustrate their modelling flexibility on simulated and real data clustering examples.

Perennial pastures for marginal farming country in southern Queensland. 1. Grass establishment techniques

Efficient ways to re-establish pastures are needed on land that requires a rotation between pastures and crops. We conducted trials in southern inland Queensland with a range of tropical perennial grasses sown into wheat stubble that was modified in various ways. Differing seedbed preparations involved cultivation or herbicide sprays, with or without fertilizer at sowing. Seed was broadcast and sowing time ranged from spring through to autumn on 3 different soil types. Seed quality and post-sowing rainfall were major determinants of the density of sown grass plants in the first year. Light cultivation sometimes enhanced establishment compared with herbicide spraying of standing stubble, most often on harder-setting soils. A nitrogen + phosphorus mixed fertilizer rarely produced any improvement in sown grass establishment and sometimes increased weed competition. The effects were similar for all types of grass seed from hairy fascicles to large, smooth panicoid seeds and minute Eragrostis seeds. There was a strong inverse relationship between the initial density of sown grass established and the level of weed competition.

Modelling of Effect of Non-Uniform Current Density on the Performance of Soluble Lead Redox Flow Batteries

Soluble lead acid redox flow battery (SLRFB) offers a number of advantages. These advantages can be harnessed after problems associated with buildup of active material on. electrodes (residue) are resolved. A mathematical model is developed to understand residue formation in SLRFB. The model incorporates fluid flow, ion transport, electrode reactions, and non-uniform current distribution on electrode surfaces. A number of limiting cases are studied to conclude that ion transport and electrode reaction on anode simultaneously control battery performance. The model fits the reported cell voltage vs. time profiles very well. During the discharge cycle, the model predicts complete dissolution of deposited material from trailing edge side of the electrodes. With time, the active surface area of electrodes decreases rapidly. The corresponding increase in current density leads to precipitous decrease in cell potential before all the deposited material is dissolved. The successive charge-discharge cycles add to the residue. The model correctly captures the marginal effect of flow rate on cell voltage profiles, and identifies flow rate and flow direction as new variables for controlling residue buildup. Simulations carried out with alternating flow direction and a SLRFB with cylindrical electrodes show improved performance with respect to energy efficiency and residue buildup. (C) 2014 The Electrochemical Society. All rights reserved.

Slopes of Avian Species-Area Relationships, Human Population Density, and Environmental Factors

There is increasing interest in how humans influence spatial patterns in biodiversity. One of the most frequently noted and marked of these patterns is the increase in species richness with area, the species-area relationship (SAR). SARs are used for a number of conservation purposes, including predicting extinction rates, setting conservation targets, and identifying biodiversity hotspots. Such applications can be improved by a detailed understanding of the factors promoting spatial variation in the slope of SARs, which is currently the subject of a vigorous debate. Moreover, very few studies have considered the anthropogenic influences on the slopes of SARs; this is particularly surprising given that in much of the world areas with high human population density are typically those with a high number of species, which generates conservation conflicts. Here we determine correlates of spatial variation in the slopes of species-area relationships, using the British avifauna as a case study. Whilst we focus on human population density, a widely used index of human activities, we also take into account (1) the rate of increase in habitat heterogeneity with increasing area, which is frequently proposed to drive SARs, (2) environmental energy availability, which may influence SARs by affecting species occupancy patterns, and (3) species richness. We consider environmental variables measured at both local (10 km x 10 km) and regional (290 km x 290 km) spatial grains, but find that the former consistently provides a better fit to the data. In our case study, the effect of species richness on the slope SARs appears to be scale dependent, being negative at local scales but positive at regional scales. In univariate tests, the slope of the SAR correlates negatively with human population density and environmental energy availability, and positively with the rate of increase in habitat heterogeneity. We conducted two sets of multiple regression analyses, with and without species richness as a predictor. When species richness is included it exerts a dominant effect, but when it is excluded temperature has the dominant effect on the slope of the SAR, and the effects of other predictors are marginal.

Slopes of Avian Species-Area Relationships, Human Population Density, and Environmental Factors

There is increasing interest in how humans influence spatial patterns in biodiversity. One of the most frequently noted and marked of these patterns is the increase in species richness with area, the species–area relationship (SAR). SARs are used for a number of conservation purposes, including predicting extinction rates, setting conservation targets, and identifying biodiversity hotspots. Such applications can be improved by a detailed understanding of the factors promoting spatial variation in the slope of SARs, which is currently the subject of a vigorous debate. Moreover, very few studies have considered the anthropogenic influences on the slopes of SARs; this is particularly surprising given that in much of the world areas with high human population density are typically those with a high number of species, which generates conservation conflicts. Here we determine correlates of spatial variation in the slopes of species–area relationships, using the British avifauna as a case study. Whilst we focus on human population density, a widely used index of human activities, we also take into account (1) the rate of increase in habitat heterogeneity with increasing area, which is frequently proposed to drive SARs, (2) environmental energy availability, which may influence SARs by affecting species occupancy patterns, and (3) species richness. We consider environmental variables measured at both local (10 km × 10 km) and regional (290 km × 290 km) spatial grains, but find that the former consistently provides a better fit to the data. In our case study, the effect of species richness on the slope SARs appears to be scale dependent, being negative at local scales but positive at regional scales. In univariate tests, the slope of the SAR correlates negatively with human population density and environmental energy availability, and positively with the rate of increase in habitat heterogeneity. We conducted two sets of multiple regression analyses, with and without species richness as a predictor. When species richness is included it exerts a dominant effect, but when it is excluded temperature has the dominant effect on the slope of the SAR, and the effects of other predictors are marginal.