Predicting the species abundance distribution using a model food web

A large number of models of the species abundance distribution (SAD) have been proposed, many of which are generically similar to the log-normal distribution, from which they are often indistinguishable when describing a given data set. Ecological data sets are necessarily incomplete samples of an ecosystem, subject to statistical noise, and cannot readily be combined to yield a closer approximation to the underlying distribution. In this paper, we adopt the Webworld ecosystem model to study the predicted SAD in detail. The Webworld model is complex, and does not allow analytic examination of such features; rather, we use simulation data and an approach similar to that of ecologists analysing empirical data. By examining large sets of fully described data we are able to resolve features which can distinguish between models but which have not been investigated in detail in field data. We find that the power-law normal distribution is superior to both the log-normal and logit-normal distributions, and that the data can improve on even this at the high-population cut-off.     

A simple predictive density based on the p*-formula

The predictive density proposed by Harris (1989) is based onintegrating the density for a new observation with respect tothe estimated sampling distribution of the maximum likelihoodestimator of the unknown parameter. This has good properties,but is rather complicated to compute even for simple models.An approximation to the Harris proposal is considered whichconsists of approximating the sampling distribution of the maximumlikelihood estimator of the unknown parameter by Barndorff-Nielsen's(1983) p*-formula, and then using a Laplace approximation withO(n^–1) correction terms for integrating out the parameter.The result can generally be expressed in terms of standard likelihoodderivatives, and takes a quite simple form for exponential familiesand for location models.     

An approximation method for solving the steady-state probability distribution of probabilistic Boolean networks

MOTIVATION: Probabilistic Boolean networks (PBNs) have been proposed to model genetic regulatory interactions. The steady-state probability distribution of a PBN gives important information about the captured genetic network. The computation of the steady-state probability distribution usually includes construction of the transition probability matrix and computation of the steady-state probability distribution. The size of the transition probability matrix is 2(n)-by-2(n) where n is the number of genes in the genetic network. Therefore, the computational costs of these two steps are very expensive and it is essential to develop a fast approximation method. RESULTS: In this article, we propose an approximation method for computing the steady-state probability distribution of a PBN based on neglecting some Boolean networks (BNs) with very small probabilities during the construction of the transition probability matrix. An error analysis of this approximation method is given and theoretical result on the distribution of BNs in a PBN with at most two Boolean functions for one gene is also presented. These give a foundation and support for the approximation method. Numerical experiments based on a genetic network are given to demonstrate the efficiency of the proposed method.     

Chemical modification studies and the secondary structure of HeLa cell 5.8S rRNA.

Various secondary structure models have been proposed for 5.8 S rRNA. In this paper HeLa cell 5.8 S rRNA is shown to possess several sites that are reactive to carbodiimide at 25 degrees, and other regions that are unreactive. Previous work has established the distribution of reactive and unreactive cytidine residues along the primary structure (11). The secondary structure model of Nazar et al. (7) is fully compatible with the chemical reactivity data whereas other models are partly incompatible. We conclude that the model of Nazar et al. provides the best approximation so far available to the conformation of isolated 5.8 S rRNA. Findings on the effect temperature on the chemical reactivity of different parts of the structure are summarized. The findings described in this paper should provide a basis for examining the specific interaction of 5.8 S rRNA with 28 s rRNA.     

Reduced kinetic models of neutral transport in the tokamak plasma

A method for constructing reduced models of neutrals transport for problems with a reduced dimension has been proposed on the basis of the kinetic equation. The case of a cylindrically symmetric plasma column, which is a good approximation for the tokamak geometry, has been thoroughly analyzed. For this geometry, the kinetic model of neutrals in the isotropic approximation is implemented using an algorithm based on energy grouping of neutrals; this algorithm for atomic hydrogen isotopes is integrated in the ASTRA code.     

Bootstrap Restricted Likelihood Ratio Test for the Detection of Rare Variants

In this paper the detection of rare variants association with continuous phenotypes of interest is investigated via the likelihood-ratio based variance component test under the framework of linear mixed models. The hypothesis testing is challenging and nonstandard, since under the null the variance component is located on the boundary of its parameter space. In this situation the usual asymptotic chisquare distribution of the likelihood ratio statistic does not necessarily hold. To circumvent the derivation of the null distribution we resort to the bootstrap method due to its generic applicability and being easy to implement. Both parametric and nonparametric bootstrap likelihood ratio tests are studied. Numerical studies are implemented to evaluate the performance of the proposed bootstrap likelihood ratio test and compare to some existing methods for the identification of rare variants. To reduce the computational time of the bootstrap likelihood ratio test we propose an effective approximation mixture for the bootstrap null distribution. The GAW17 data is used to illustrate the proposed test.     

A method of evaluating the stationary rate of gene migration (a cartographic approach)

A method of geographic mapping of the stationary (limiting) gene migration rate has been developed. The method is based on approximation of the empirical distribution of gene frequencies by a theoretical steady-state distribution. The maximum likelihood method and the chi 2 minimization method are used to obtain consistent estimations of the gene migration rate as a parameter of the steady-state distribution. The new method makes it possible to determine the geographical distribution of the ratio between the properties of the population migration structure described by the stepping-stone and island models and to construct a geographical map of chi 2 values. This map approximately reflects the distribution of natural selection pressure on the gene pool if genetic processes are quasisteady.     

3D Magnetic Holes in Collisionless Plasmas

Recent multispacecraft observations in the Earth’s magnetosphere have revealed an abundance of magnetic holes—localized magnetic field depressions. These magnetic holes are characterized by the plasma pressure enhancement and strongly localized currents flowing around the hole boundaries. There are several numerical and analytical models describing 2D configurations of magnetic holes, but the 3D distribution of magnetic fields and electric currents is studied poorly. Such a 3D magnetic field configuration is important for accurate investigation of charged particle dynamics within magnetic holes. Moreover, the 3D distribution of currents can be used for distant probing of magnetic holes in the magnetosphere. In this study, a 3D magnetic hole model using the single-fluid approximation and a spatial scale hierarchy with the distinct separation of gradients is developed. It is shown that such 3D holes can be obtained as a generalization of 1D models with the plasma pressure distribution adopted from the kinetic approach. The proposed model contains two magnetic field components and field-aligned currents. The magnetic field line configuration resembles the magnetic trap where hot charged particles bounce between mirror points. However, the approximation of isotropic pressure results in a constant plasma pressure along magnetic field lines, and the proposed magnetic hole model does not confine plasma along the field direction.     

A version of the EM algorithm for proportional hazard model with random effects

Proportional hazard models with multivariate random effects (frailties) acting multiplicatively on the baseline hazard have recently become a topic of an intensive research. One of the main practical problems related to the models is the estimation of parameters. To this aim, several approaches based on the EM algorithm have been proposed. The major difference between these approaches is the method of the computation of conditional expectations required at the E-step. In this paper an alternative implementation of the EM algorithm is proposed, in which the expected values are computed with the use of the Laplace approximation. The method is computationally less demanding than the approaches developed previously. Its performance is assessed based on a simulation study and compared to a non-EM based estimation approach proposed by Ripatti and Palmgren (2000).     

Survival and hazard functions for progressive diseases using saddlepoint approximations

Saddlepoint approximations for the computation of survival and hazard functions are introduced in the context of parametric survival analysis. Although these approximations are computationally fast, accurate, and relatively straightforward to implement, their use in survival analysis has been lacking. We approximate survival functions using the Lugannani and Rice saddlepoint approximation to the distribution function or by numerically integrating the saddlepoint density approximation. The hazard function is approximated using the saddlepoint density and distribution functions. The approximations are especially useful for consideration of survival and hazard functions for waiting times in complicated models. Examples include total or partial waiting times for a disease that progresses through various stages (convolutions of distributions).     

Predicting the impacts of climate change on the distribution of species: are bioclimate envelope models useful?

Modelling strategies for predicting the potential impacts of climate change on the natural distribution of species have often focused on the characterization of a species’ bioclimate envelope. A number of recent critiques have questioned the validity of this approach by pointing to the many factors other than climate that play an important part in determining species distributions and the dynamics of distribution changes. Such factors include biotic interactions, evolutionary change and dispersal ability. This paper reviews and evaluates criticisms of bioclimate envelope models and discusses the implications of these criticisms for the different modelling strategies employed. It is proposed that, although the complexity of the natural system presents fundamental limits to predictive modelling, the bioclimate envelope approach can provide a useful first approximation as to the potentially dramatic impact of climate change on biodiversity. However, it is stressed that the spatial scale at which these models are applied is of fundamental importance, and that model results should not be interpreted without due consideration of the limitations involved. A hierarchical modelling framework is proposed through which some of these limitations can be addressed within a broader, scale‐dependent context.     

Models for the logarithmic species abundance distributions

Three models, developed by Karlin, McGregor and Ewens to describe evolving populations of selectively neutral genotypes, are shown to lead to various versions of Fisher's logarithmic series distribution for species abundance. Statistical inference procedures and measures of diversity which have been developed in one of the two contexts are therefore also applicable in the other context, and the paper reviews and extends these links. Some work of Fisher, Good and Rao is shown to be based on a faulty version of the logarithmic distribution, which, nevertheless, is a good approximation to a consistent version.     

Numerical comparison of several approximations of the word count distribution in random sequences.

The exact distribution of word counts in random sequences and several approximations have been proposed in the past few years. The exact distribution has no theoretical limit but may require prohibitive computation time. On the other hand, approximate distributions can be rapidly calculated but, in practice, are only accurate under specific conditions. After making a survey of these distributions, we compare them according to both their accuracy and computational cost. Rules are suggested for choosing between Gaussian approximations, compound Poisson approximation, and exact distribution. This work is illustrated with the detection of exceptional words in the phage Lambda genome.     

Linear noise approximation method for calculating nonequilibrium fluctuations of the occupation numbers in radiative-collisional average ion models

A novel method is proposed for calculating nonequilibrium fluctuations of the mean occupation numbers of the electron shells in the radiative-collisional average-ion models of multicharged plasma kinetics. For the class of Slater ionic models, equations are derived for the mean occupation numbers of the electron shells and their fluctuations in the Fokker-Planck approximation. To calculate the fluctuations, the Fokker-Planck equation is linearized in the vicinity of the steady-state nonequilibrium solution to the kinetic equations (linear noise approximation). The method proposed allows one to take into account both the nonequilibrium correlations of the occupation-number fluctuations and the thermodynamically equilibrium statistical correlations related to the Coulomb interaction among bound electrons. The relation among the coefficients in the Fokker-Planck equation for the occupation-number fluctuations of the electron shells is discussed based on the fluctuation-dissipative theorem.     

A Measure to Evaluate the Closeness of Satterthwaite's Approximation

Satterthwaite's approximation of the distribution of a nonnegative linear combination of independent mean squares is addressed in this article. A measure is developed to quantify the closeness of this approximation by using certain optimization results given by THIBAUDEAU and STYAN (1985). The main advantage of the proposed measure is to provide a theoretical framework for determining conditions under which Satterthwaite's approximation may be inadequate. This is demonstrated in three examples portraying frequently encountered problems in analysis of variance.     

Equations of heat distribution within the body

A vector integral equation describing heat distribution within the body has been derived. The factors considered are heat conduction, forced convection via the circulatory system, environmental exchange, metabolic heat production, and change in heat content. The vector partial differential equation and alternative forms incorporating boundary conditions were also developed. A difference equation based on a first-order approximation to the fundamental equations was derived to form the basis of a model for heat distribution within the body. It has been shown that factors involving conduction and convection must be considered independently unless the temperature of the blood flowing from a region of the body is equal to the average temperature of the tissue in that region. If this relation between tissue and blood temperature does exist, only a single temperature from each eleeent is needed to describe the heat distribution. In this latter case, models which ascribe all heat transfer to “equivalent” conduction or to convection can give valid predictions.     

Semiclassical approximations of stochastic epidemiological processes towards parameter estimation using as prime example the SIS system with import

In this paper we investigate several schemes to approximate the stationary distribution of the stochastic SIS system with import. We begin by presenting the model and analytically computing its stationary distribution. We then approximate this distribution using Kramers–Moyal approximation, van Kampen's system size expansion, and a semiclassical scheme, also called WKB or eikonal approximation depending on its different applications in physics. For the semiclassical scheme, done in the context of the Hamilton–Jacobi formalism, two approaches are taken. In the first approach we assume a semiclassical ansatz for the generating function, while in the second the solution of the master equation is approximated directly. The different schemes are compared and the semiclassical approximation, which performs better, is then used to analyse the time dependent solution of stochastic systems for which no analytical expression is known. Stochastic epidemiological models are studied in order to investigate how far such semiclassical approximations can be used for parameter estimation.     

Some Approximations of the Borel-Tanner or the Generalized Poisson Distribution

This paper considers some approximations for the Borel-Tanner (Generalized Poisson) sums by using (i) Gram-Charlier Poisson expansion, (ii) Mixture of two Poisson distributions, (iii) Variance stabilizing technique, and (iv) negative binomial distribution. It has been found that the approximation obtained by using the negative binomial distribution seems to be more efficient than the other approximation.     

Function approximation approach to the inference of reduced NGnet models of genetic networks

Background  

The inference of a genetic network is a problem in which mutual interactions among genes are deduced using time-series of gene expression patterns. While a number of models have been proposed to describe genetic regulatory networks, this study focuses on a set of differential equations since it has the ability to model dynamic behavior of gene expression. When we use a set of differential equations to describe genetic networks, the inference problem can be defined as a function approximation problem. On the basis of this problem definition, we propose in this study a new method to infer reduced NGnet models of genetic networks.