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.

Second-order processes driven by dichotomous noise

We study free second-order processes driven by dichotomous noise. We obtain an exact differential equation for the marginal density p(x,t) of the position. It is also found that both the velocity ¿(t) and the position X(t) are Gaussian random variables for large t.

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.

Rotifer abundance, biomass, and secondary production after the recovery of hydrologic connectivity between a river and two marginal lakes (Sao Paulo, Brazil)

Rotifera density, biomass, and secondary production on two marginal lakes of Paranapanema River were compared after the recovery of hydrologic connectivity with the river (Sao Paulo State, Brazil). Daily samplings were performed in limnetic zone of both lakes during the rainy season immediately after lateral inflow of water and, in the dry period, six months after hydrologic connectivity recovery. In order to identify the factors that affect rotifer population dynamics, lake water level, volume, depth, temperature, transparency, dissolved oxygen, pH, alkalinity, conductivity, suspended solids, nutrients, and chlorophyll-a were determined. Variations of water physical and chemical factors that affect rotifer population were related to the lake-river degree of connection and to water level rising after drought. The water lateral inflow from the river resulted in an increase in lake water volume, depth, and transparency and a decrease in water pH, alkalinity, and suspended solids. The lake with the wider river connection, more frequent biota exchange, and larger amount of particulate and dissolved materials was richer and more diverse, while rotifer density, biomass, and productivity were lower in both periods studied. Density, biomass, and secondary production were higher in the lake with the smaller river connection and the higher physical and chemical stability. Our results show that the connectivity affects the limnological stability, associated to seasonality. Stable conditions, caused by low connectivity in dry periods, were related with high density, biomass and secondary production. Conversely, instability conditions in rainy periods were associated to elevated richness and diversity values, caused by exchange biota due to higher connectivity. (C) 2008 Elsevier GmbH. All rights reserved.

Composite pre-heating: Effects on marginal adaptation, degree of conversion and mechanical properties

Objectives. This study evaluated the effect of composite pre-polymerization temperature and energy density on the marginal adaptation (MA), degree of conversion (DC), flexural strength (FS), and polymer cross-linking (PCL) of a resin composite (Filtek Z350, 3M/ESPE). Methods. For MA, class V cavities (4mmx2mmx2mm) were prepared in 40 bovine incisors. The adhesive system Adper Single Bond 2 (3M/ESPE) was applied. Before being placed in the cavities, the resin composite was either kept at room-temperature (25 degrees C) or previously pre-heated to 68 degrees C in the Calset (TM) device (AdDent Inc., Danbury, CT, USA). The composite was then light polymerized for 20 or 40s at 600mW/cm(2) (12 or 24 J/cm(2), respectively). The percentage of gaps was analyzed by scanning electron microscopy, after sectioning the restorations and preparing epoxy resin replicas. DC (n = 3) was obtained by FT-Raman spectroscopy on irradiated and non-irradiated composite surfaces. FS (n = 10) was measured by the three-point-bending test. KHN (n = 6) was measured after 24h dry storage and again after immersion in 100% ethanol solution for 24 h, to calculate PCL density. Data were analyzed by appropriate statistical analyses. Results. The pre-heated composite showed better MA than the room-temperature groups. A higher number of gaps were observed in the room-temperature groups, irrespective of the energy density, mainly in the axial wall (p < 0.05). Composite pre-heating and energy density did not affect the DC, FS and PCL (p > 0.05). Significance. Pre-heating the composite prior to light polymerization similar in a clinical situation did not alter the mechanical properties and monomer conversion of the composite, but provided enhanced composite adaptation to cavity walls. (C) 2010 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved.

Fine-scale genetic structure and marginal processes in an expanding population of Biscutella laevigata L. (Brassicaceae).

Evolutionary processes acting at the expanding margins of a species' range are still poorly understood. Genetic drift is considered prevalent in marginal populations, and the maintenance of genetic diversity during recolonization might seem puzzling. To investigate such processes, a fine-scale investigation of 219 individuals was performed within a population of Biscutella laevigata (Brassicaceae), located at the leading edge of its range. The survey used amplified fragment length polymorphisms (AFLPs). As commonly reported across the whole species distribution range, individual density and genetic diversity decreased along the local axis of recolonization of this expanding population, highlighting the enduring effect of the historical colonization on present-day diversity. The self-incompatibility system of the plant may have prevented local inbreeding in newly found patches and sustained genetic diversity by ensuring gene flow from established populations. Within the more continuously populated region, spatial analysis of genetic structure revealed restricted gene flow among individuals. The distribution of genotypes formed a mosaic of relatively homogenous patches within the continuous population. This pattern could be explained by a history of expansion by long-distance dispersal followed by fine-scale diffusion (that is, a stratified dispersal combination). The secondary contact among expanding patches apparently led to admixture among differentiated genotypes where they met (that is, a reshuffling effect). This type of dynamics could explain the maintenance of genetic diversity during recolonization.

Are there vascular density gradients along myocardial laser channels?

BACKGROUND: Clinical studies suggest that transmyocardial laser revascularization may improve regional blood flow of the subendocardial layer. The vascular growth pattern of laser channels was analyzed. METHODS: Twenty pigs were randomized to undergo ligation of left marginal arteries (n = 5), to undergo transmyocardial laser revascularization of the left lateral wall (n = 5), to undergo both procedures (n = 5) or to a control group (n = 5). All the animals were sacrificed after 1 month. Computed morphometric analysis of vascular density of the involved area was expressed as number of vascular structures per square millimeter (+/-1 standard deviation). RESULTS: The vascular density of the scar tissue of the laser channel was significantly increased in comparison with myocardial infarction alone: 49.6+/-12.8/mm2 versus 25.5+/-8.6/mm2 (p < 0.0001). The vascular densities of subendocardial and subepicardial channel areas were similar: 52.9+/-16.8/mm2 versus 46.3+/-13.6/mm2 (p = 0.41). The area immediately adjacent to the channels showed a vascular density similar to that of normal tissue: 6.02+/-1.7/mm2 versus 5.2+/-1.9/mm2 (p = 0.08). In the infarction + transmyocardial laser revascularization group, the channels were indistinguishable from infarction scar. CONCLUSIONS: Scars of transmyocardial laser revascularization channels exhibit an increased vascular density in comparison with scar tissue of myocardial infarction, which does not extend into their immediate vicinity. There was no vascular density gradient along the longitudinal axis of the channels.

Immunohistochemical characterization of the chick marginal retina

The retina is a highly differentiated tissue with a complex layered structure that has been extensively characterized. However, most of the previous studies focused on the histology of the central retina while little is known about the cellular composition, organization and function of the marginal retina. Recent research has identified a subpopulation of multipotential progenitor cells in the marginal regions of the retina, closest to the ciliary body ("ciliary marginal zone"). These cells are capable of differentiation in response to an appropriate stimulus. Thus, it is possible that the structure and composition of the marginal retina are distinct from those of the central retina to accommodate the potential addition of newly formed neurons. To characterize the cellular profile of the chick marginal retina, we labeled it immunohistochemically for markers whose staining pattern is well established in the central retina: calbindin, calretinin, protein kinase C, and choline acetyltransferase. Calbindin was present at very low levels in the marginal retina putative photoreceptor layer. Calretinin-positive horizontal cells were also sparse close to the ciliary marginal zone. The bipolar cells in the marginal outer plexiform layer were positive for anti-protein kinase C antibodies, but the density of labeling was also decreased in relation to the central retina. In contrast, the marginal starburst cholinergic amacrine cell pattern was very similar to the central retina. From these data we conclude that the structure of the marginal retina is significantly different from that of the central retina. In particular, the expression of late retina markers in the marginal retina decreased in comparison to the central retina.