A note on some distributions for nonlinear switching elements with finite dead time

Summary Ricciardi et al. [3] have obtained the output distribution functions of a nonlinear switching element with a Poissonian sequence of impulses at the input. They have studied certain limiting properties of these distribution functions. However, certain limiting properties of these distributions become more obvious if one studies the underlying stochastic processes of these distributions. Further, the methods used by Ricciardi et al. for the derivation of the distribution functions and their limiting properties are quite involved. It might perhaps be useful to derive the main results of their paper by alternative simpler methods and point out the underlying stochastic processes more clearly.     

Poisson adjacency distributions in genome comparison: multichromosomal, circular, signed and unsigned cases

The number of common adjacencies of genetic markers, as a measure of the similarity of two genomes, has been widely used as indicator of evolutionary relatedness and as the basis for inferring phylogenetic relationships. Its probability distribution enables statistical tests in detecting whether significant evolutionary signal remains in the marker order. In this article, we derive the probability distributions of the number of adjacencies for a number of types of genome--signed or unsigned, circular or linear, single-chromosome or multichromosomal. Generating functions are found for singlechromosome cases, from which exact counts can be calculated. Probability approaches are adopted for multichromosomal cases, where we.nd the exact values for expectations and variances. In both cases, the limiting distributions are derived in term of numbers of adjacencies. For all unsigned cases, the limiting distribution is Poisson with parameter 2; for all signed cases, the limiting distribution is Poisson with parameter (1/2).     

Iterated birth and death process as a model of radiation cell survival

The iterated birth and death process is defined as an n-fold iteration of a stochastic process consisting of the combination of instantaneous random killing of individuals in a certain population with a given survival probability s with a Markov birth and death process describing subsequent population dynamics. A long standing problem of computing the distribution of the number of clonogenic tumor cells surviving a fractionated radiation schedule consisting of n equal doses separated by equal time intervals tau is solved within the framework of iterated birth and death processes. For any initial tumor size i, an explicit formula for the distribution of the number M of surviving clonogens at moment tau after the end of treatment is found. It is shown that if i-->infinity and s-->0 so that is(n) tends to a finite positive limit, the distribution of random variable M converges to a probability distribution, and a formula for the latter is obtained. This result generalizes the classical theorem about the Poisson limit of a sequence of binomial distributions. The exact and limiting distributions are also found for the number of surviving clonogens immediately after the nth exposure. In this case, the limiting distribution turns out to be a Poisson distribution.     

LIVES: a new habitat modelling technique for predicting the distribution of species’ occurrences using presence-only data based on limiting factor theory

Predictive modelling techniques using presence-only data have attracted increasing attention because they can provide information on species distributions and their potential habitat for conservation and ecosystem management. However, the existing predictive modelling techniques have several limitations. Here, we propose a novel predictive modelling technique, Limiting Variable and Environmental Suitability (LIVES), for predicting the distributions and potential habitats of species using presence-only data. It is based on limiting factor theory, which postulates that the occurrence of a species is only determined by the factor that most limits its distribution. LIVES predicts the suitability of a candidate grid cell for a species in terms of limiting environmental factor. It also predicts the most limiting factor or the potential limiting factor at the grid cell. The environmental factors can be climatic, geological, biological and any other relevant environmental factors, whether quantitative or qualitative. The predicted habitats consist of the current distribution of the species and the potentially suitable areas for the species where there is currently no record of occurrence. We also compare several properties of LIVES and other predictive modelling techniques. On the basis of 1,000 simulations, the average predictions of LIVES are more accurate than the two other commonly used modelling techniques (BIOCLIM and DOMAIN) for presence-only data.     

The distribution of soil nutrients with depth: Global patterns and the imprint of plants

To understand the importance of plants in structuring the vertical distributions of soil nutrients, we explored nutrient distributions in the top meter of soil for more than 10,000 profiles across a range of ecological conditions. Hypothesizing that vertical nutrient distributions are dominated by plant cycling relative to leaching, weathering dissolution, and atmospheric deposition, we examined three predictions: (1) that the nutrients that are most limiting for plants would have the shallowest average distributions across ecosystems, (2) that the vertical distribution of a limiting nutrient would be shallower as the nutrient became more scarce, and (3) that along a gradient of soil types with increasing weathering-leaching intensity, limiting nutrients would be relatively more abundant due to preferential cycling by plants. Globally, the ranking of vertical distributions among nutrients was shallowest to deepest in the following order: P > K > Ca > Mg > Na = Cl = SO₄. Nutrients strongly cycled by plants, such as P and K, were more concentrated in the topsoil (upper 20 cm) than were nutrients usually less limiting for plants such as Na and Cl. The topsoil concentrations of all nutrients except Na were higher in the soil profiles where the elements were more scarce. Along a gradient of weathering-leaching intensity (Aridisols to Mollisols to Ultisols), total base saturation decreased but the relative contribution of exchangeable K⁺ to base saturation increased. These patterns are difficult to explain without considering the upward transport of nutrients by plant uptake and cycling. Shallower distributions for P and K, together with negative associations between abundance and topsoil accumulation, support the idea that plant cycling exerts a dominant control on the vertical distribution of the most limiting elements for plants (those required in high amounts in relation to soil supply). Plant characteristics like tissue stoichiometry, biomass cycling rates, above- and belowground allocation, root distributions, and maximum rooting depth may all play an important role in shaping nutrient profiles. Such vertical patterns yield insight into the patterns and processes of nutrient cycling through time.     

Empirical likelihood inference for the area under the ROC curve

For a continuous-scale diagnostic test, the most commonly used summary index of the receiver operating characteristic curve (ROC) is the area under the curve (AUC) that measures the accuracy of the diagnostic test. In this article, we propose an empirical likelihood (EL) approach for the inference on the AUC. First we define an EL ratio for the AUC and show that its limiting distribution is a scaled chi-square distribution. We then obtain an EL-based confidence interval for the AUC using the scaled chi-square distribution. This EL inference for the AUC can be extended to stratified samples, and the resulting limiting distribution is a weighted sum of independent chi-square distributions. Additionally we conduct simulation studies to compare the relative performance of the proposed EL-based interval with the existing normal approximation-based intervals and bootstrap intervals for the AUC.     

On the asymptotic behaviour of the pseudolikelihood ratio test statistic with boundary problems

This paper considers the asymptotic distribution of the likelihood ratio statistic T for testing a subset of parameter of interest θ, θ = (γ, η), H(0) : γ = γ(0), based on the pseudolikelihood L(θ, ??), where ?? is a consistent estimator of ?, the nuisance parameter. We show that the asymptotic distribution of T under H(0) is a weighted sum of independent chi-squared variables. Some sufficient conditions are provided for the limiting distribution to be a chi-squared variable. When the true value of the parameter of interest, θ(0), or the true value of the nuisance parameter, ?(0), lies on the boundary of parameter space, the problem is shown to be asymptotically equivalent to the problem of testing the restricted mean of a multivariate normal distribution based on one observation from a multivariate normal distribution with misspecified covariance matrix, or from a mixture of multivariate normal distributions. A variety of examples are provided for which the limiting distributions of T may be mixtures of chi-squared variables. We conducted simulation studies to examine the performance of the likelihood ratio test statistics in variance component models and teratological experiments.     

Testing the extreme value domain of attraction for distributions of beneficial fitness effects

In modeling evolutionary genetics, it is often assumed that mutational effects are assigned according to a continuous probability distribution, and multiple distributions have been used with varying degrees of justification. For mutations with beneficial effects, the distribution currently favored is the exponential distribution, in part because it can be justified in terms of extreme value theory, since beneficial mutations should have fitnesses in the extreme right tail of the fitness distribution. While the appeal to extreme value theory seems justified, the exponential distribution is but one of three possible limiting forms for tail distributions, with the other two loosely corresponding to distributions with right-truncated tails and those with heavy tails. We describe a likelihood-ratio framework for analyzing the fitness effects of beneficial mutations, focusing on testing the null hypothesis that the distribution is exponential. We also describe how to account for missing the smallest-effect mutations, which are often difficult to identify experimentally. This technique makes it possible to apply the test to gain-of-function mutations, where the ancestral genotype is unable to grow under the selective conditions. We also describe how to pool data across experiments, since we expect few possible beneficial mutations in any particular experiment.     

A Two Sample CRAMÉR-VON MISES Test for Randomly Censored Data

A CRAMÉR-VON MISES type statistic is introduced for testing the equality of the underlying survival distributions of two populations when observations are subject to arbitrary right censorship. The statistic is appropriate in testing problems where a two-sided alternative is of interest. The asymptotic distribution of the statistic is found; under certain circumstances, the limiting distribution coincides with that of a one sample CRAMÉR-VON MISES type statistic for randomly censored data investigated previously. Approximations to the asymptotic distribution are discussed; an example is given.     

The Distribution of Ocular Chlamydia Prevalence across Tanzanian Communities Where Trachoma Is Declining

BackgroundMathematical models predict an exponential distribution of infection prevalence across communities where a disease is disappearing. Trachoma control programs offer an opportunity to test this hypothesis, as the World Health Organization has targeted trachoma for elimination as a public health concern by the year 2020. Local programs may benefit if a single survey could reveal whether infection was headed towards elimination. Using data from a previously-published 2009 survey, we test the hypothesis that Chlamydia trachomatis prevalence across 75 Tanzanian communities where trachoma had been documented to be disappearing is exponentially distributed.Methods/FindingsWe fit multiple continuous distributions to the Tanzanian data and found the exponential gave the best approximation. Model selection by Akaike Information Criteria (AIC_c) suggested the exponential distribution had the most parsimonious fit to the data. Those distributions which do not include the exponential as a special or limiting case had much lower likelihoods of fitting the observed data. 95% confidence intervals for shape parameter estimates of those distributions which do include the exponential as a special or limiting case were consistent with the exponential. Lastly, goodness-of-fit testing was unable to reject the hypothesis that the prevalence data came from an exponential distribution.ConclusionsModels correctly predict that infection prevalence across communities where a disease is disappearing is best described by an exponential distribution. In Tanzanian communities where local control efforts had reduced the clinical signs of trachoma by 80% over 10 years, an exponential distribution gave the best fit to prevalence data. An exponential distribution has a relatively heavy tail, thus occasional high-prevalence communities are to be expected even when infection is disappearing. A single cross-sectional survey may be able to reveal whether elimination efforts are on-track.     

Prediction of climate warming impacts on plant species could be more complex than expected. Evidence from a case study in the Himalaya

Many studies have investigated the possible impact of climate change on the distributions of plant species. In the present study, we test whether the concept of potential distribution is able to effectively predict the impact of climate warming on plant species.Using spatial simulation models, we related the actual (current species distribution), potential (modelled distribution assuming unlimited dispersal) and predicted (modelled distribution accounting for wind-limited seed dispersal) distributions of two plant species under several warming scenarios in the Sagarmatha National Park (Nepal). We found that the two predicted distributions were, respectively, seven and nine times smaller than the potential ones. Under a +3 °C scenario, both species would likely lose their actual and predicted distributions, while their potential distributions would remain partially safe. Our results emphasize that the predicted distributions of plant species may diverge to a great extent from their potential distributions, particularly in mountain areas, and predictions of species preservation in the face of climate warming based on the potential distributions of plant species are at risk of producing overoptimistic projections.We conclude that the concept of potential distribution is likely to lead to limited or inefficacious conservation of plant species due to its excessively optimistic projections of species preservation. More robust strategies should utilize concepts such as “optimal reintroduction”, which maximizes the benefit–cost ratio of conservation activities by limiting reintroduction efforts to suitable areas that could not otherwise be reached by a species; moreover, such strategies maximize the probability of species establishment by excluding areas that will be endangered under future climate scenarios.     

Intercontinental biotic invasions: what can we learn from native populations and habitats?

The effectiveness of management strategies for invasive species is often hampered by a lack of clear understanding of the factors that limit species distributions. The distribution of exotic species, especially those that are invasive, are often so dynamic that limiting factors are difficult to identify. Comparisons of exotic species between their native ranges, where they are presumably close to equilibrium with controlling factors, and their ranges in areas of introduction can circumvent this difficulty. Such studies would help identify (1) limiting factors for distributions in native ranges, (2) factors associated with a high degree of invasiveness, (3) changes in genetics and morphology since introduction, which also might contribute to invasiveness, and (4) future directions and rates of invasion as a basis for developing detection/warning systems. Findings from such comparative studies would be highly valuable for understanding the dynamics of biological invasions and for improving the effectiveness of management to prevent or control invasives.     

Distribution of age at menopause in two Danish samples

We analyzed the distribution of reported age at natural menopause in two random samples of Danish women (n = 176 and n = 150) to determine the shape of the distribution and to disclose any possible trends in the distribution parameters. It was necessary to correct the frequencies of the reported ages for the effect of differing ages at reporting. The corrected distribution of age at menopause differs from the normal distribution in the same way in both samples. Both distributions could be described by a mixture of two normal distributions. It appears that most of the parameters of the normal distribution mixtures remain unchanged over a 50-year time lag. The position of the distribution, that is, the mean age at menopause, however, increases slightly but significantly.     

Alternatives to the Wright-Fisher model: The robustness of mitochondrial Eve dating

Methods of calculating the distributions of the time to coalescence depend on the underlying model of population demography. In particular, the models assuming deterministic evolution of population size may not be applicable to populations evolving stochastically. Therefore the study of coalescence models involving stochastic demography is important for applications. One interesting approach which includes stochasticity is the O’Connell limit theory of genealogy in branching processes. Our paper explores how many generations are needed for the limiting distributions of O’Connell to become adequate approximations of exact distributions. We perform extensive simulations of slightly supercritical branching processes and compare the results to the O’Connell limits. Coalescent computations under the Wright-Fisher model are compared with limiting O’Connell results and with full genealogy-based predictions. These results are used to estimate the age of the so-called mitochondrial Eve, i.e., the root of the mitochondrial polymorphisms of the modern humans based on the DNA from humans and Neanderthal fossils.     

Exponential pattern of cell age distribution in dividing cells of plant meristems

In order to determine the pattern of cell age distribution in proliferating cells of Allium cepa roots we have measured by cytophotometry two cell size parameters, protein content and surface area projection, in cells that correspond to the entire proliferating population or only to the ana-telophase subpopulation. The size values of ana-telophase cells have been employed to construct theoretical size distributions for the entire proliferating cell population of the root meristem by assuming either a uniform or an exponential cell age distribution. Statistical comparison of theoretical distributions with the experimental one rules out a uniform cell age distribution and strongly favours an exponential age distribution similar to that found in bacteria.     

Beneficial fitness effects are not exponential for two viruses

The distribution of fitness effects for beneficial mutations is of paramount importance in determining the outcome of adaptation. It is generally assumed that fitness effects of beneficial mutations follow an exponential distribution, for example, in theoretical treatments of quantitative genetics, clonal interference, experimental evolution, and the adaptation of DNA sequences. This assumption has been justified by the statistical theory of extreme values, because the fitnesses conferred by beneficial mutations should represent samples from the extreme right tail of the fitness distribution. Yet in extreme value theory, there are three different limiting forms for right tails of distributions, and the exponential describes only those of distributions in the Gumbel domain of attraction. Using beneficial mutations from two viruses, we show for the first time that the Gumbel domain can be rejected in favor of a distribution with a right-truncated tail, thus providing evidence for an upper bound on fitness effects. Our data also violate the common assumption that small-effect beneficial mutations greatly outnumber those of large effect, as they are consistent with a uniform distribution of beneficial effects.     

Qualitative behavior of a selectively neutral allelic model

A model of Fleming and Viot for describing frequency distributions for selectively neutral allelic populations and for multidimensional numerical genetic traits is studied, with specific attention given to two qualitative aspects that are derived: the coherence of the random wandering distribution and the tendency to cluster at microscopic scales. A new quantity, called the clustering or occupation index, is introduced to measure relative patchiness of the distribution in terms of a limiting subcell occupation frequency. Results of one- and two-dimensional computer-generated simulations are presented which provide an estimate of the theoretical expected distribution of allelic types. Evidence of the robustness of the model, in the sense that it arises from a variety of assumptions about the reproductive mechanism, is presented in the Appendix.     

The distribution of individual heterozygosity in natural populations

Estimation of the distribution of the level of individual heterozygosity within natural populations is explored with both Monte-Carlo simulation studies and data from natural populations. Simulations indicate that heterozygosities estimated from as few as a dozen randomly chosen loci may, to some degree, reflect (r = 0.35) heterozygosity determined by 100 independent loci. The shape of the expected distribution of heterozygosity is heavily dependent upon levels of heterozygosity at the loci. Complete genetic data for 12 loci from 997 Fundulus heteroclitus are used to describe the distributions of heterozygosity for different localities, for age classes and for sexes. The distributions deviate from normality. Distributions from different localities are not different, but the distributions are heterogeneous among age classes at one of two localities and are heterogeneous between the sexes.     

Stochastic models for aggregation processes

Three models are presented, which describe the aggregation of objects into groups and the distributions of groups sizes and group numbers within habitats. The processes regarded are pure accumulation processes which involve only formation and invasion of groups. Invasion represents the special case of fusion when only single objects - and not groups - join a group of certain size. The basic model is derived by a single parameter, the formation probability q, which represents the probability of an object to form a new group. A novel, discrete and finite distribution that results for the group sizes is deduced from this aggregation process and it is shown that it converges to a geometric distribution if the number of objects tends to infinity. Two extensions of this model, which both converge to the Waring distribution, are added: the model can be extended either with a beta distributed formation probability or with the assumption that the invasion probability depends on the group size. Relationships between the limiting distributions involved are discussed.     

On the distributions of bootstrap support and posterior distributions for a star tree

Several authors have recently noted that when data are generated from a star topology, posterior probabilities can often be very large, even with arbitrarily large sequence lengths. This is counter to intuition, which suggests convergence to the limit of equal probability for each topology. Here the limiting distributions of bootstrap support and posterior probabilities are obtained for a four-taxon star tree. Theoretical results are given, providing confirmation that this counterintuitive phenomenon holds for both posterior probabilities and bootstrap support. For large samples the limiting results for posterior probabilities are the same regardless of the prior. With equal-length terminal edges, the limiting distribution is similar but not the same across different choices for the lengths of the edges. In contrast to previous results, the case of unequal lengths of terminal edges is considered. With two long edges, the posterior probability of the tree with long edges together tends to be much larger. Using the neighbor-joining algorithm, with equal edge lengths, the distribution of bootstrap support tends to be qualitatively comparable to posterior probabilities. As with posterior probabilities, when two of the edges are long, bootstrap support for the tree with long branches together tends to be large. The bias is less pronounced, however, as the distribution of bootstrap support gets close to uniform for this tree, whereas posterior probabilities are much more likely to be large. Our findings for maximum likelihood estimation are based entirely on simulation and in contrast suggest that bootstrap support tends to be fairly constant across edge-length choices.