Confronting different models of community structure to species-abundance data: a Bayesian model comparison

Species abundances are undoubtedly the most widely available macroecological data, but can we use them to distinguish among several models of community structure? Here we present a Bayesian analysis of species‐abundance data that yields a full joint probability distribution of each model's parameters plus a relatively parameter‐independent criterion, the posterior Bayes factor, to compare these models. We illustrate our approach by comparing three classical distributions: the zero‐sum multinomial (ZSM) distribution, based on Hubbell's neutral model, the multivariate Poisson lognormal distribution (MPLN), based on niche arguments, and the discrete broken stick (DBS) distribution, based on MacArthur's broken stick model. We give explicit formulas for the probability of observing a particular species‐abundance data set in each model, and argue that conditioning on both sample size and species count is needed to allow comparisons between the two distributions. We apply our approach to two neotropical communities (trees, fish). We find that DBS is largely inferior to ZSM and MPLN for both communities. The tree data do not allow discrimination between ZSM and MPLN, but for the fish data ZSM (neutral model) overwhelmingly outperforms MPLN (niche model), suggesting that dispersal plays a previously underestimated role in structuring tropical freshwater fish communities. We advocate this approach for identifying the relative importance of dispersal and niche‐partitioning in determining diversity of different ecological groups of species under different environmental conditions.     

Bayesian calibration,process modeling and uncertainty quantification in biotechnology

High-throughput experimentation has revolutionized data-driven experimental sciences and opened the door to the application of machine learning techniques. Nevertheless, the quality of any data analysis strongly depends on the quality of the data and specifically the degree to which random effects in the experimental data-generating process are quantified and accounted for. Accordingly calibration, i.e. the quantitative association between observed quantities and measurement responses, is a core element of many workflows in experimental sciences.Particularly in life sciences, univariate calibration, often involving non-linear saturation effects, must be performed to extract quantitative information from measured data. At the same time, the estimation of uncertainty is inseparably connected to quantitative experimentation. Adequate calibration models that describe not only the input/output relationship in a measurement system but also its inherent measurement noise are required. Due to its mathematical nature, statistically robust calibration modeling remains a challenge for many practitioners, at the same time being extremely beneficial for machine learning applications.In this work, we present a bottom-up conceptual and computational approach that solves many problems of understanding and implementing non-linear, empirical calibration modeling for quantification of analytes and process modeling. The methodology is first applied to the optical measurement of biomass concentrations in a high-throughput cultivation system, then to the quantification of glucose by an automated enzymatic assay. We implemented the conceptual framework in two Python packages, calibr8 and murefi, with which we demonstrate how to make uncertainty quantification for various calibration tasks more accessible. Our software packages enable more reproducible and automatable data analysis routines compared to commonly observed workflows in life sciences.Subsequently, we combine the previously established calibration models with a hierarchical Monod-like ordinary differential equation model of microbial growth to describe multiple replicates of Corynebacterium glutamicum batch cultures. Key process model parameters are learned by both maximum likelihood estimation and Bayesian inference, highlighting the flexibility of the statistical and computational framework.     

Factor analysis models for structuring covariance matrices of additive genetic effects: a Bayesian implementation

Multivariate linear models are increasingly important in quantitative genetics. In high dimensional specifications, factor analysis (FA) may provide an avenue for structuring (co)variance matrices, thus reducing the number of parameters needed for describing (co)dispersion. We describe how FA can be used to model genetic effects in the context of a multivariate linear mixed model. An orthogonal common factor structure is used to model genetic effects under Gaussian assumption, so that the marginal likelihood is multivariate normal with a structured genetic (co)variance matrix. Under standard prior assumptions, all fully conditional distributions have closed form, and samples from the joint posterior distribution can be obtained via Gibbs sampling. The model and the algorithm developed for its Bayesian implementation were used to describe five repeated records of milk yield in dairy cattle, and a one common FA model was compared with a standard multiple trait model. The Bayesian Information Criterion favored the FA model.     

Some mathematical aspects of the medical diagnostic process. I: a general mathematical model

Making a medical diagnosis consists of correlating knownpatterns of disease with the various classes of clinical data elicited from the history, physical examination, and batteries of tests relative to the diagnostic dynamics symbolized by atree branching into the various possible diagnostic decisions. In this paper a relational mathematical model of the reasoning aspects of the conventional medical diagnostic process is suggested as a way of extracting a general, formal concept of medical diagnosis. Computer implementation of the model is discussed briefly.     

Correcting for measurement error in fractional polynomial models using Bayesian modelling and regression calibration,with an application to alcohol and mortality

Exposure measurement error can result in a biased estimate of the association between an exposure and outcome. When the exposure–outcome relationship is linear on the appropriate scale (e.g. linear, logistic) and the measurement error is classical, that is the result of random noise, the result is attenuation of the effect. When the relationship is non‐linear, measurement error distorts the true shape of the association. Regression calibration is a commonly used method for correcting for measurement error, in which each individual's unknown true exposure in the outcome regression model is replaced by its expectation conditional on the error‐prone measure and any fully measured covariates. Regression calibration is simple to execute when the exposure is untransformed in the linear predictor of the outcome regression model, but less straightforward when non‐linear transformations of the exposure are used. We describe a method for applying regression calibration in models in which a non‐linear association is modelled by transforming the exposure using a fractional polynomial model. It is shown that taking a Bayesian estimation approach is advantageous. By use of Markov chain Monte Carlo algorithms, one can sample from the distribution of the true exposure for each individual. Transformations of the sampled values can then be performed directly and used to find the expectation of the transformed exposure required for regression calibration. A simulation study shows that the proposed approach performs well. We apply the method to investigate the relationship between usual alcohol intake and subsequent all‐cause mortality using an error model that adjusts for the episodic nature of alcohol consumption.     

Selection of likelihood parameters for complex models determines the effectiveness of Bayesian calibration

Assessing the parameter uncertainty of complex ecosystem models is a key challenge for improving our understanding of real world abstractions, such as those for explaining carbon and nitrogen cycle at ecosystem scale and associated biosphere-atmosphere-hydrosphere exchange processes. The lack of data about the variance of measurements forces scientists to revisit assumptions used in estimating the parameter distribution of complex ecosystem models.An increasingly used tool for assessing parameter uncertainty of complex ecosystem models is Bayesian calibration. In this paper, we generate two data sets which may represent a seasonal temperature curve or the seasonality of soil carbon dioxide flux and a single high peak put on a low background signal as is e.g. typical for soil nitrous oxide emission. Based on these examples we illustrate that commonly used assumptions for measurement uncertainty can lead to a sampling of wrong areas in the parameter space, incorrect parameter dependencies, and an underestimation of parameter uncertainties. This step needs particular attention by modelers as these issues lead to erroneous model simulations a) in present and future domains, b) misinterpretations of process feedback and functioning of the model, and c) to an underestimation of model uncertainty (e.g. for soil greenhouse gas fluxes).We also test the extension of the Bayesian framework with a model error term to compensate the effects caused by the false assumption of a perfect model and show that this approach can alleviate the observed problems in estimating the model parameter distribution.     

Some mathematical aspects of the diagnostic process. II: A mathematical model of electrocardiographic diagnosis

In a previous paper (Bartholomay, 1971), a general mathematical model of the medical diagnostic process was described. The present paper amounts to a relization of that process in terms of conventional 12-lead electrocardiographic diagnosis as enunciated by Dr. Harold D. Levine (1966) in the course of a collaborative study by Dr. Levine and the present author at the Peter Bent Brigham Hospital of the Harvard Medical School between 1963 and 1966. The main details of the cognitive component of that model are described in detail here. The model has been programmed onto a computer system consisting of an analog-digital converter and general purpose digital computer and amounts to a simulation of Dr. Levine’s electrocardiographic analysis procedure.     

A mathematical model for the leukocyte filtration process

Leukocyte filters are applied clinically to remove leukocytes from blood. In order to optimize leukocyte filters, a mathematical model to describe the leukocyte filtration process was developed by modification of a general theoretical model for depth filtration. The model presented here can be used to predict the time-dependent leukocyte filtration as a function of cell-cell interaction in the filter, filter efficiency, filter capacity, filter dimensions, and leukocyte concentration in the suspension applied to the filter. The results of different leukocyte filtration experiments previously reported in the literature could be well described by the present model. (c) 1995 John Wiley & Sons, Inc.     

Three-dimensional mathematical model of the human spine structure

A method is presented for constructing discrete-parameter type three-dimensional mathematical models and governing equations of motion of the spine structure. The anatomic structure is represented by any combination of rigid bodies, springs, and dashpots in space. These are positioned, orientated, and connected in a manner to represent the true mechanical function of the structure. The rigid bodies are of any shape and have 6 degrees-of-freedom, allowing three-dimensional motion. The springs and dashpots may have up to twenty-one stiffness and damping coefficients respectively to precisely represent the three-dimensional coupled behavior. The method is straightforward and simple to apply. The governing equations are in the matrix form and are easily generated and solved by computer techniques.     

Bayesian calibration of the Unified budburst model in six temperate tree species

Numerous phenology models developed to predict the budburst date of trees have been merged into one Unified model (Chuine, 2000, J. Theor. Biol. 207, 337–347). In this study, we tested a simplified version of the Unified model (Unichill model) on six woody species. Budburst and temperature data were available for five sites across Belgium from 1957 to 1995. We calibrated the Unichill model using a Bayesian calibration procedure, which reduced the uncertainty of the parameter coefficients and quantified the prediction uncertainty. The model performance differed among species. For two species (chestnut and black locust), the model showed good performance when tested against independent data not used for calibration. For the four other species (beech, oak, birch, ash), the model performed poorly. Model performance improved substantially for most species when using site-specific parameter coefficients instead of across-site parameter coefficients. This suggested that budburst is influenced by local environment and/or genetic differences among populations. Chestnut, black locust and birch were found to be temperature-driven species, and we therefore analyzed the sensitivity of budburst date to forcing temperature in those three species. Model results showed that budburst advanced with increasing temperature for 1–3 days °C⁻¹, which agreed with the observed trends. In synthesis, our results suggest that the Unichill model can be successfully applied to chestnut and black locust (with both across-site and site-specific calibration) and to birch (with site-specific calibration). For other species, temperature is not the only determinant of budburst and additional influencing factors will need to be included in the model.     

An efficient approach to automate the manual trial and error calibration of activated sludge models

An efficient approach is introduced to help automate the rather tedious manual trial and error way of model calibration currently used in activated sludge modeling practice. To this end, we have evaluated a Monte Carlo based calibration approach consisting of four steps: (i) parameter subset selection, (ii) defining parameter space, (iii) parameter sampling for Monte Carlo simulations and (iv) selecting the best Monte Carlo simulation thereby providing the calibrated parameter values. The approach was evaluated on a formerly calibrated full-scale ASM2d model for a domestic plant (located in The Netherlands), using in total 3 months of dynamic oxygen, ammonia and nitrate sensor data. The Monte Carlo calibrated model was validated successfully using ammonia, oxygen and nitrate data collected at high measurement frequency. Statistical analysis of the residuals using mean absolute error (MAE), root mean square error (RMSE) and Janus coefficient showed that the calibrated model was able to provide statistically accurate and valid predictions for ammonium, oxygen and nitrate. This shows that this pragmatic approach can perform the task of model calibration and therefore be used in practice to save the valuable time of modelers spent on this step of activated sludge modeling. The high computational demand is a downside of this approach but this can be overcome by using distributed computing. Overall we expect that the use of such systems analysis tools in the application of activated sludge models will improve the quality of model predictions and their use in decision making.     

The role of mathematical models in helminth population biology

This brief review aims to illustrate how theory can aid in our understanding of the factors that determine the regulation and stability of parasite abundance, and influence the impact of control measures. The current generation of models are obviously crude, and ignore much biological detail, but they are often able to capture qualitative trends observed in real communities. As such, their analysis and investigation can provide important conceptional insights or, in some circumstances, they can be of value in a predictive role (e.g. the impact of chemotherapy in human communities).This field of research, however, is still in its infancy and much remains to be done to improve biological realism in model formulation and to extent the methods of analysis and interpretation. In the latter context, for example, the current analytical methods for the study of the dynamical properties of non-linear systems of differential and partial differential equations are inadequate for many areas of biological application. Future advances in applied mathematics will, therefore, be of great importance. As far as biological realism is concerned, three areas require urgent attention. The first concerns the treatment of heterogeneity in worm loads within host communities. The generative factors of parasite aggregation are many and varied and little is understood at present of how these processes influence a parasite's population response to perturbation induced, for example, by control measures. Stochastic models are required to examine this problem but current work in this area is very limited.The second area concerns immunity to parasitic infection. Few models take account of the substantive body of experimental work which attests to the significance of host responses (both specific and non-specific) to parasite invasion as determinants of parasite abundance within both an individual host and in the community at large. A start has been made in the investigation of models which mimic acquired immunity and immunological “memory” but much refinement and elaboration is needed (Anderson &; May, 1985a). In particular, the next generation of models should address the details of antibody-antigen and cell-antigen interactions in individual hosts as well as the broader questions concerning herd immunity. Heterogeneity in immunological responsiveness as a consequence of host nutritional status or genetic background must also be condsidered.The final topic is that of population genetics. Geneticists invariably consider changes in gene frequencies without reference to changes in parasite or host abundance, ecologists and epidemiologists have tended to study changes in abundance without reference to changes in genetic structure while immunologists have focused on the mechanisms of resistance to parasitic infection without reference to population or genetic changes. It is becoming increasingly apparent that host genetic background and genetic heterogeneity within parasite populations (e.g. the malarial parasites of man) are important determinants of observed population events (Medley &; Anderson, 1985). Future research must attempt to meld the areas of genetics, population dynamics and immunology. Such an integration presents a fascinating challenge.     

A covariance structure model for the admixture of binary genetic variation

I derive a covariance structure model for pairwise linkage disequilibrium (LD) between binary markers in a recently admixed population and use a generalized least-squares method to fit the model to two different data sets. Both linked and unlinked marker pairs are incorporated in the model. Under the model, a pairwise LD matrix is decomposed into two component matrices, one containing LD attributable to admixture, and another containing, in an aggregate form, LD specific to the populations forming the mixture. I use population genetics theory to show that the latter matrix has block-diagonal structure. For the data sets considered here, I show that the number of source populations can be determined by statistical inference on the canonical correlations of the sample LD matrix.     

The diffusion process approach to one-compartmental stochastic models: A mathematical note

We consider the one-dimension (one-compartment) exponential model using a diffusion process approach. In particular, we summarize the known results in the case where the stochastic component of the model is a Gaussian white noise process with mean zero and variance σ². Finally, we briefly illustrate a number of cases where similar forms of model arise.     

Bayesian hierarchical error model for analysis of gene expression data

MOTIVATION: Analysis of genome-wide microarray data requires the estimation of a large number of genetic parameters for individual genes and their interaction expression patterns under multiple biological conditions. The sources of microarray error variability comprises various biological and experimental factors, such as biological and individual replication, sample preparation, hybridization and image processing. Moreover, the same gene often shows quite heterogeneous error variability under different biological and experimental conditions, which must be estimated separately for evaluating the statistical significance of differential expression patterns. Widely used linear modeling approaches are limited because they do not allow simultaneous modeling and inference on the large number of these genetic parameters and heterogeneous error components on different genes, different biological and experimental conditions, and varying intensity ranges in microarray data. RESULTS: We propose a Bayesian hierarchical error model (HEM) to overcome the above restrictions. HEM accounts for heterogeneous error variability in an oligonucleotide microarray experiment. The error variability is decomposed into two components (experimental and biological errors) when both biological and experimental replicates are available. Our HEM inference is based on Markov chain Monte Carlo to estimate a large number of parameters from a single-likelihood function for all genes. An F-like summary statistic is proposed to identify differentially expressed genes under multiple conditions based on the HEM estimation. The performance of HEM and its F-like statistic was examined with simulated data and two published microarray datasets-primate brain data and mouse B-cell development data. HEM was also compared with ANOVA using simulated data. AVAILABILITY: The software for the HEM is available from the authors upon request.     

The effect of interfering substances on the disinfection process: a mathematical model

AIMS: To gain a greater understanding of the effect of interfering substances on the efficacy of disinfection. METHODS AND RESULTS: Current kinetic disinfection models were augmented by a term designed to quantify the deleterious effect of soils such as milk on the disinfection process of suspended organisms. The model was based on the assumption that inactivation by added soil occurred at a much faster rate than microbial inactivation. The new model, the fat-soil model, was also able to quantify the effect of changing the initial inoculum size (1 x 10(7)-5 x 10(7) ml(-1) of Staphylococcus aureus) on the outcome of the suspension tests. Addition of catalase to the disinfection of Escherichia coli by hydrogen peroxide, resulted in changes to the shape of the log survivor/time plots. These changes were modelled on the basis of changing biocide concentration commensurate with microbial inactivation. CONCLUSIONS: The reduction in efficacy of a disinfectant in the presence of an interfering substance can be quantified through the use of adaptations to current disinfection models. SIGNIFICANCE AND IMPACT OF THE STUDY: Understanding the effect of soil on disinfection efficacy allows us to understand the limitations of disinfectants and disinfection procedures. It also gives us a mechanism with which to investigate the soil tolerance of new biocides and formulations.     

Bayesian calibration, validation, and uncertainty quantification of diffuse interface models of tumor growth

The idea that one can possibly develop computational models that predict the emergence, growth, or decline of tumors in living tissue is enormously intriguing as such predictions could revolutionize medicine and bring a new paradigm into the treatment and prevention of a class of the deadliest maladies affecting humankind. But at the heart of this subject is the notion of predictability itself, the ambiguity involved in selecting and implementing effective models, and the acquisition of relevant data, all factors that contribute to the difficulty of predicting such complex events as tumor growth with quantifiable uncertainty. In this work, we attempt to lay out a framework, based on Bayesian probability, for systematically addressing the questions of Validation, the process of investigating the accuracy with which a mathematical model is able to reproduce particular physical events, and Uncertainty quantification, developing measures of the degree of confidence with which a computer model predicts particular quantities of interest. For illustrative purposes, we exercise the process using virtual data for models of tumor growth based on diffuse-interface theories of mixtures utilizing virtual data.