On the Challenge of Fitting Tree Size Distributions in Ecology

Patterns that resemble strongly skewed size distributions are frequently observed in ecology. A typical example represents tree size distributions of stem diameters. Empirical tests of ecological theories predicting their parameters have been conducted, but the results are difficult to interpret because the statistical methods that are applied to fit such decaying size distributions vary. In addition, binning of field data as well as measurement errors might potentially bias parameter estimates. Here, we compare three different methods for parameter estimation – the common maximum likelihood estimation (MLE) and two modified types of MLE correcting for binning of observations or random measurement errors. We test whether three typical frequency distributions, namely the power-law, negative exponential and Weibull distribution can be precisely identified, and how parameter estimates are biased when observations are additionally either binned or contain measurement error. We show that uncorrected MLE already loses the ability to discern functional form and parameters at relatively small levels of uncertainties. The modified MLE methods that consider such uncertainties (either binning or measurement error) are comparatively much more robust. We conclude that it is important to reduce binning of observations, if possible, and to quantify observation accuracy in empirical studies for fitting strongly skewed size distributions. In general, modified MLE methods that correct binning or measurement errors can be applied to ensure reliable results.     

Probability binning and testing agreement between multivariate immunofluorescence histograms: extending the chi-squared test

BACKGROUND: A key problem in immunohistochemistry is assessing when two sample histograms are significantly different. One test that is commonly used for this purpose in the univariate case is the chi-squared test. Comparing multivariate distributions is qualitatively harder, as the "curse of dimensionality" means that the number of bins can grow exponentially. For the chi-squared test to be useful, data-dependent binning methods must be employed. An example of how this can be done is provided by the "probability binning" method of Roederer et al. (1,2,3). METHODS: We derive the theoretical distribution of the probability binning statistic, giving it a more rigorous foundation. We show that the null distribution is a scaled chi-square, and show how it can be related to the standard chi-squared statistic. RESULTS: A small simulation shows how the theoretical results can be used to (a) modify the probability binning statistic to make it more sensitive and (b) suggest variant statistics which, while still exploiting the data-dependent strengths of the probability binning procedure, may be easier to work with. CONCLUSIONS: The probability binning procedure effectively uses adaptive binning to locate structure in high-dimensional data. The derivation of a theoretical basis provides a more detailed interpretation of its behavior and renders the probability binning method more flexible.     

Nonparametric discriminant analysis of phytoplankton species using data from analytical flow cytometry

BACKGROUND: Analytical flow cytometry (AFC) provides rapid and accurate measurement of particles from heterogeneous populations. AFC has been used to classify and identify phytoplankton species, but most methods of discriminant analysis of resulting data have depended on normality assumptions and outcomes have been disappointing. METHODS AND RESULTS: In this study, we consider nonparametric methods based on density estimation. In addition to the familiar kernel method, methods based on wavelets are also implemented. Full five-dimensional wavelet estimation proves to be computationally prohibitive with current workstation power, so we employ projection pursuit for reduction of dimensionality. AFC typically produces very large samples, so we also investigate data simplification through binning. Further modifications to the discrimination strategy are suggested by specific features of phytoplankton data, namely, a hierarchical group structure, the possible presence of many groups, and the likelihood of encountering an aberrant group in a test sample. CONCLUSIONS: We apply all the resultant procedures to appropriate subsets of a very large data set, demonstrate their efficacy, and compare their error rates with those of more conventional methods. We further show that incorporation of the specific features of phytoplankton data into the analysis leads to improved results and provides a general framework for analysis of such data.     

Inexpensive Computation of the Inverse of the Genomic Relationship Matrix in Populations with Small Effective Population Size

Many computations with SNP data including genomic evaluation, parameter estimation, and genome-wide association studies use an inverse of the genomic relationship matrix. The cost of a regular inversion is cubic and is prohibitively expensive for large matrices. Recent studies in cattle demonstrated that the inverse can be computed in almost linear time by recursion on any subset of ∼10,000 individuals. The purpose of this study is to present a theory of why such a recursion works and its implication for other populations. Assume that, because of a small effective population size, the additive information in a genotyped population has a small dimensionality, even with a very large number of SNP markers. That dimensionality is visible as a limited number of effective SNP effects, independent chromosome segments, or the rank of the genomic relationship matrix. Decompose a population arbitrarily into core and noncore individuals, with the number of core individuals equal to that dimensionality. Then, breeding values of noncore individuals can be derived by recursions on breeding values of core individuals, with coefficients of the recursion computed from the genomic relationship matrix. A resulting algorithm for the inversion called “algorithm for proven and young” (APY) has a linear computing and memory cost for noncore animals. Noninfinitesimal genetic architecture can be accommodated through a trait-specific genomic relationship matrix, possibly derived from Bayesian regressions. For populations with small effective population size, the inverse of the genomic relationship matrix can be computed inexpensively for a very large number of genotyped individuals.     

Likelihood-based inferences under isolation by distance: two-dimensional habitats and confidence intervals

Likelihood-based methods of inference of population parameters from genetic data in structured populations have been implemented but still little tested in large networks of populations. In this work, a previous software implementation of inference in linear habitats is extended to two-dimensional habitats, and the coverage properties of confidence intervals are analyzed in both cases. Both standard likelihood and an efficient approximation are considered. The effects of misspecification of mutation model and dispersal distribution, and of spatial binning of samples, are considered. In the absence of model misspecification, the estimators have low bias, low mean square error, and the coverage properties of confidence intervals are consistent with theoretical expectations. Inferences of dispersal parameters and of the mutation rate are sensitive to misspecification or to approximations inherent to the coalescent algorithms used. In particular, coalescent approximations are not appropriate to infer the shape of the dispersal distribution. However, inferences of the neighborhood parameter (or of the product of population density and mean square dispersal rate) are generally robust with respect to complicating factors, such as misspecification of the mutation process and of the shape of the dispersal distribution, and with respect to spatial binning of samples. Likelihood inferences appear feasible in moderately sized networks of populations (up to 400 populations in this work), and they are more efficient than previous moment-based spatial regression method in realistic conditions.     

Demographic strategies of plant invaders in temporally varying environments

Plant populations may have evolved different demographic strategies to cope with temporal environmental variation. According to the demographic buffering hypothesis, vital rates that are most critical to population persistence are buffered against environmental variation and vary little over time, whereas the demographic lability hypothesis suggests that populations may track and benefit from environmental variation. While the hypotheses of demographic strategies have been widely tested in plant and animal species, they have not been explicitly examined for invasive plants, or in relation to different modelling methods (deterministic vs. stochastic). Here, we tested the demographic buffering and lability hypotheses for 23 populations of eight invasive plant species in relation to life form (woody vs. herbaceous species) and population growth rate using deterministic and stochastic modelling methods, and absolute and relative scales. We found that conclusions of demographic strategies depended on scale, with an absolute scale resulting in stronger negative correlations between the variability and importance of vital rates (i.e., buffering) than a relative scale. Conclusions of demographic strategies were also affected by life form that interacted with method. The populations of woody invaders exhibited buffering regardless of the method used, while for the populations of herbaceous species, deterministic calculations suggested buffering and stochastic calculations suggested lability. Overall, our findings emphasise the role of life form and methodological issues that need to be considered when exploring demographic strategies in fluctuating environments.     

Feature Selection in the Reconstruction of Complex Network Representations of Spectral Data

Complex networks have been extensively used in the last decade to characterize and analyze complex systems, and they have been recently proposed as a novel instrument for the analysis of spectra extracted from biological samples. Yet, the high number of measurements composing spectra, and the consequent high computational cost, make a direct network analysis unfeasible. We here present a comparative analysis of three customary feature selection algorithms, including the binning of spectral data and the use of information theory metrics. Such algorithms are compared by assessing the score obtained in a classification task, where healthy subjects and people suffering from different types of cancers should be discriminated. Results indicate that a feature selection strategy based on Mutual Information outperforms the more classical data binning, while allowing a reduction of the dimensionality of the data set in two orders of magnitude.     

Bayesian Estimates of Transition Probabilities in Seven Small Lithophytic Orchid Populations: Maximizing Data Availability from Many Small Samples

Predicting population dynamics for rare species is of paramount importance in order to evaluate the likelihood of extinction and planning conservation strategies. However, evaluating and predicting population viability can be hindered from a lack of data. Rare species frequently have small populations, so estimates of vital rates are often very uncertain due to lack of data. We evaluated the vital rates of seven small populations from two watersheds with varying light environment of a common epiphytic orchid using Bayesian methods of parameter estimation. From the Lefkovitch matrices we predicted the deterministic population growth rates, elasticities, stable stage distributions and the credible intervals of the statistics. Populations were surveyed on a monthly basis between 18–34 months. In some of the populations few or no transitions in some of the vital rates were observed throughout the sampling period, however, we were able to predict the most likely vital rates using a Bayesian model that incorporated the transitions rates from the other populations. Asymptotic population growth rate varied among the seven orchid populations. There was little difference in population growth rate among watersheds even though it was expected because of physical differences as a result of differing canopy cover and watershed width. Elasticity analyses of Lepanthes rupestris suggest that growth rate is more sensitive to survival followed by growth, shrinking and the reproductive rates. The Bayesian approach helped to estimate transition probabilities that were uncommon or variable in some populations. Moreover, it increased the precision of the parameter estimates as compared to traditional approaches.     

Population dynamics of a threatened cactus species: general assessment and effects of matrix dimensionality

Harrisia portoricensis is an endemic Caribbean cactus currently under threatened status. In this study we used population projection matrices to evaluate the conservation status of this species and we performed a systematic analysis of the effects of matrix dimensionality on the inferred demographic parameters. Results revealed that population growth rates (λ) were 0.946 and 0.961 for the 2007–2008 and 2008–2009 periods respectively, suggesting a declining population with limited persistence ability. Even when the highest elasticity values corresponded to the survival of adults, numerical simulations suggested that increases in either seedling establishment or fecundity could render λ > 1. Our empirical-based analysis using raw demographic data revealed a clear trend for λ values to decrease with increasing matrix dimension. Stasis and fecundity elasticities were also found to decrease whereas retrogression and growth elasticitites increased with increasing matrix dimension. These results are roughly insensitive to the method used to create matrices of different dimensions. For H. portoricensis, large matrices with narrow classifications were required to minimize variations in λ, highlighting the need for large data sets to assess the convergence of results with matrix dimensionality. Our combined results emphasize that under current scenarios the ability of H. portoricensis for population growth is severely limited. Any management strategy designed for the conservation of this species should consider long-term monitoring of populations as well as programs that enhance seedling establishment and adult survival.     

Dynamic adaptive binning: an improved quantification technique for NMR spectroscopic data

The interpretation of nuclear magnetic resonance (NMR) experimental results for metabolomics studies requires intensive signal processing and multivariate data analysis techniques. A key step in this process is the quantification of spectral features, which is commonly accomplished by dividing an NMR spectrum into several hundred integral regions or bins. Binning attempts to minimize effects from variations in peak positions caused by sample pH, ionic strength, and composition, while reducing the dimensionality for multivariate statistical analyses. Herein we develop an improved novel spectral quantification technique, dynamic adaptive binning. With this technique, bin boundaries are determined by optimizing an objective function using a dynamic programming strategy. The objective function measures the quality of a bin configuration based on the number of peaks per bin. This technique shows a significant improvement over both traditional uniform binning and other adaptive binning techniques. This improvement is quantified via synthetic validation sets by analyzing an algorithm’s ability to create bins that do not contain more than a single peak and that maximize the distance from peak to bin boundary. The validation sets are developed by characterizing the salient distributions in experimental NMR spectroscopic data. Further, dynamic adaptive binning is applied to a ¹H NMR-based experiment to monitor rat urinary metabolites to empirically demonstrate improved spectral quantification.     

Demographic model selection using random forests and the site frequency spectrum

Phylogeographic data sets have grown from tens to thousands of loci in recent years, but extant statistical methods do not take full advantage of these large data sets. For example, approximate Bayesian computation (ABC) is a commonly used method for the explicit comparison of alternate demographic histories, but it is limited by the “curse of dimensionality” and issues related to the simulation and summarization of data when applied to next‐generation sequencing (NGS) data sets. We implement here several improvements to overcome these difficulties. We use a Random Forest (RF) classifier for model selection to circumvent the curse of dimensionality and apply a binned representation of the multidimensional site frequency spectrum (mSFS) to address issues related to the simulation and summarization of large SNP data sets. We evaluate the performance of these improvements using simulation and find low overall error rates (~7%). We then apply the approach to data from Haplotrema vancouverense, a land snail endemic to the Pacific Northwest of North America. Fifteen demographic models were compared, and our results support a model of recent dispersal from coastal to inland rainforests. Our results demonstrate that binning is an effective strategy for the construction of a mSFS and imply that the statistical power of RF when applied to demographic model selection is at least comparable to traditional ABC algorithms. Importantly, by combining these strategies, large sets of models with differing numbers of populations can be evaluated.     

A tale of two cycles – distinguishing quasi-cycles and limit cycles in finite predator–prey populations

Periodic predator – prey dynamics in constant environments are usually taken as indicative of deterministic limit cycles. It is known, however, that demographic stochasticity in finite populations can also give rise to regular population cycles, even when the corresponding deterministic models predict a stable equilibrium. Specifically, such quasi-cycles are expected in stochastic versions of deterministic models exhibiting equilibrium dynamics with weakly damped oscillations. The existence of quasi-cycles substantially expands the scope for natural patterns of periodic population oscillations caused by ecological interactions, thereby complicating the conclusive interpretation of such patterns. Here we show how to distinguish between quasi-cycles and noisy limit cycles based on observing changing population sizes in predator – prey populations. We start by confirming that both types of cycle can occur in the individual-based version of a widely used class of deterministic predator – prey model. We then show that it is feasible and straightforward to accurately distinguish between the two types of cycle through the combined analysis of autocorrelations and marginal distributions of population sizes. Finally, by confronting these results with real ecological time series, we demonstrate that by using our methods even short and imperfect time series allow quasi-cycles and limit cycles to be distinguished reliably.     

植物种群生存力分析研究进展

对十多年来国外植物PVA的研究进行了综合评述;具体分析了影响植物种群生存力的各种随机性因子及确定性因子;总结了植物PVA研究的方法步骤及采用的模拟模型;探讨了植物PVA的难点,PVA对管理措施的评价效果;并提出对今后植物PVA的研究展望,认为PVA是研究濒危植物种群灭绝及评价管理或保护措施的有力工具;发展描述复杂种间关系的多种种的PVA模型以及包含多个影响因素的PVA应用模型是未来植物PVA的研究方向。     

Unsupervised discovery of microbial population structure within metagenomes using nucleotide base composition

An approach to infer the unknown microbial population structure within a metagenome is to cluster nucleotide sequences based on common patterns in base composition, otherwise referred to as binning. When functional roles are assigned to the identified populations, a deeper understanding of microbial communities can be attained, more so than gene-centric approaches that explore overall functionality. In this study, we propose an unsupervised, model-based binning method with two clustering tiers, which uses a novel transformation of the oligonucleotide frequency-derived error gradient and GC content to generate coarse groups at the first tier of clustering; and tetranucleotide frequency to refine these groups at the secondary clustering tier. The proposed method has a demonstrated improvement over PhyloPythia, S-GSOM, TACOA and TaxSOM on all three benchmarks that were used for evaluation in this study. The proposed method is then applied to a pyrosequenced metagenomic library of mud volcano sediment sampled in southwestern Taiwan, with the inferred population structure validated against complementary sequencing of 16S ribosomal RNA marker genes. Finally, the proposed method was further validated against four publicly available metagenomes, including a highly complex Antarctic whale-fall bone sample, which was previously assumed to be too complex for binning prior to functional analysis.     

Estimation of Plant Demographic Parameters from Stage‐Structured Censuses

Summary This article presents some statistical methods for estimating the parameters of a population dynamics model for annual plants. The model takes account of reproduction, immigration, seed survival in a seed bank, and plant growth. The data consist of the number of plants in several developmental stages that were measured in a number of populations for a few consecutive years; they are incomplete since seeds could not be counted. It is assumed that there are no measurement errors or that measurement errors are binomial and not frequent. Some statistical methods are developed within the framework of estimating equations or Bayesian inference. These methods are applied to oilseed rape data.     

Evolutionary dynamics as a component of stage-structured matrix models: an example using Trillium grandiflorum

Evolution by natural selection improves fitness and may therefore influence population trajectories. Demographic matrix models are often employed in conservation studies to project population dynamics, but such analyses have not incorporated evolutionary dynamics. We project evolutionarily informed population trajectories for a population of the perennial plant Trillium grandiflorum, which is declining due to high levels of herbivory by white-tailed deer. Individuals with later flowering times are less often consumed, so there is selection on this trait. We first incorporated selection analyses into a deterministic matrix model in three ways (corresponding to different methods that have been used for analyzing evolution in structured populations). Because it is not clear which of these methods works best for stage-structured models, we compared each with a more realistic, individual-based model. Deterministic models using fitness averaged over the phenotypic distribution gave trajectories that were similar to those of the individual-based model, whereas the deterministic model using fitness at the mean phenotype gave a much faster rate of evolution than that which was observed. This illustrates that subtle differences in the way in which one splices evolution into demographic models can have a large effect on expected outcomes. This study demonstrates that, by combining demographic and selection analyses, one can gauge the potential relevance of evolution to population dynamics and persistence.     

Short prokaryotic DNA fragment binning using a hierarchical classifier based on linear discriminant analysis and principal component analysis

Metagenomics is an emerging field in which the power of genomic analysis is applied to an entire microbial community, bypassing the need to isolate and culture individual microbial species. Assembling of metagenomic DNA fragments is very much like the overlap-layout-consensus procedure for assembling isolated genomes, but is augmented by an additional binning step to differentiate scaffolds, contigs and unassembled reads into various taxonomic groups. In this paper, we employed n-mer oligonucleotide frequencies as the features and developed a hierarchical classifier (PCAHIER) for binning short (≤ 1,000 bps) metagenomic fragments. The principal component analysis was used to reduce the high dimensionality of the feature space. The hierarchical classifier consists of four layers of local classifiers that are implemented based on the linear discriminant analysis. These local classifiers are responsible for binning prokaryotic DNA fragments into superkingdoms, of the same superkingdom into phyla, of the same phylum into genera, and of the same genus into species, respectively. We evaluated the performance of the PCAHIER by using our own simulated data sets as well as the widely used simHC synthetic metagenome data set from the IMG/M system. The effectiveness of the PCAHIER was demonstrated through comparisons against a non-hierarchical classifier, and two existing binning algorithms (TETRA and Phylopythia).     

Moving forward in circles: challenges and opportunities in modelling population cycles

Population cycling is a widespread phenomenon, observed across a multitude of taxa in both laboratory and natural conditions. Historically, the theory associated with population cycles was tightly linked to pairwise consumer–resource interactions and studied via deterministic models, but current empirical and theoretical research reveals a much richer basis for ecological cycles. Stochasticity and seasonality can modulate or create cyclic behaviour in non‐intuitive ways, the high‐dimensionality in ecological systems can profoundly influence cycling, and so can demographic structure and eco‐evolutionary dynamics. An inclusive theory for population cycles, ranging from ecosystem‐level to demographic modelling, grounded in observational or experimental data, is therefore necessary to better understand observed cyclical patterns. In turn, by gaining better insight into the drivers of population cycles, we can begin to understand the causes of cycle gain and loss, how biodiversity interacts with population cycling, and how to effectively manage wildly fluctuating populations, all of which are growing domains of ecological research.     

Models for spatially distributed populations: The effect of within-patch variability

This paper studies population models which have the following three ingredients: populations are divided into local subpopulations, local population dynamics are nonlinear and random events occur locally in space. In this setting local stochastic phenomena have a systematic effect on average population density and this effect does not disappear in large populations. This result is an outcome of the interaction of the three ingredients in the models and it says that stochastic models of systems of patches can be expected to give results for average population density that differ systematically from those of deterministic models. The magnitude of these differences is related to the degree of nonlinearity of local dynamics and the magnitude of local variability. These results explain those obtained from a number of previously published models which give conclusions that differ from those of deterministic models. Results are also obtained that show how stochastic models of systems of patches may be simplified to facilitate their study.     

Competition and adaptation in a community of migrating fish populations

Within a discrete scheme, the process of population migration is set by some nonnegative Markov matrix. In studying an appropriate class of competition models, nonlinear methods of convex analysis (monotone operator theory) prove to be highly effective. For special matrices (cyclic and perron ones), conditions of steady coexistence in and of competitor displacement from the community have been found. Model mechanisms of migration route adaptation for a separate population and for a family of populations from one vertical trophic chain have been proposed. The major characteristic of a migration route turns out to be the relative time of population dwelling in one or another region. Specific (perron) vectors of migration matrices correspond to these populations. It is revealed that in the course of co-adaptation the perron vectors of predator and prey migration matrixes practically coincide.