A phylogenetic mixture model for detecting pattern-heterogeneity in gene sequence or character-state data

We describe a general likelihood-based 'mixture model' for inferring phylogenetic trees from gene-sequence or other character-state data. The model accommodates cases in which different sites in the alignment evolve in qualitatively distinct ways, but does not require prior knowledge of these patterns or partitioning of the data. We call this qualitative variability in the pattern of evolution across sites "pattern-heterogeneity" to distinguish it from both a homogenous process of evolution and from one characterized principally by differences in rates of evolution. We present studies to show that the model correctly retrieves the signals of pattern-heterogeneity from simulated gene-sequence data, and we apply the method to protein-coding genes and to a ribosomal 12S data set. The mixture model outperforms conventional partitioning in both these data sets. We implement the mixture model such that it can simultaneously detect rate- and pattern-heterogeneity. The model simplifies to a homogeneous model or a rate- variability model as special cases, and therefore always performs at least as well as these two approaches, and often considerably improves upon them. We make the model available within a Bayesian Markov-chain Monte Carlo framework for phylogenetic inference, as an easy-to-use computer program.

A novel Bayesian approach to drug design with simple and complex enzymes: Use with lens aldehyde dehydrogenase and the glyoxalase pathway

Purpose: Acquiring details of kinetic parameters of enzymes is crucial to biochemical understanding, drug development, and clinical diagnosis in ocular diseases. The correct design of an experiment is critical to collecting data suitable for analysis, modelling and deriving the correct information. As classical design methods are not targeted to the more complex kinetics being frequently studied, attention is needed to estimate parameters of such models with low variance. Methods: We have developed Bayesian utility functions to minimise kinetic parameter variance involving differentiation of model expressions and matrix inversion. These have been applied to the simple kinetics of the enzymes in the glyoxalase pathway (of importance in posttranslational modification of proteins in cataract), and the complex kinetics of lens aldehyde dehydrogenase (also of relevance to cataract). Results: Our successful application of Bayesian statistics has allowed us to identify a set of rules for designing optimum kinetic experiments iteratively. Most importantly, the distribution of points in the range is critical; it is not simply a matter of even or multiple increases. At least 60 % must be below the KM (or plural if more than one dissociation constant) and 40% above. This choice halves the variance found using a simple even spread across the range.With both the glyoxalase system and lens aldehyde dehydrogenase we have significantly improved the variance of kinetic parameter estimation while reducing the number and costs of experiments. Conclusions: We have developed an optimal and iterative method for selecting features of design such as substrate range, number of measurements and choice of intermediate points. Our novel approach minimises parameter error and costs, and maximises experimental efficiency. It is applicable to many areas of ocular drug design, including receptor-ligand binding and immunoglobulin binding, and should be an important tool in ocular drug discovery.

A Bayesian approach to disease gene location using allelic association

A Bayesian approach to analysing data from family-based association studies is developed. This permits direct assessment of the range of possible values of model parameters, such as the recombination frequency and allelic associations, in the light of the data. In addition, sophisticated comparisons of different models may be handled easily, even when such models are not nested. The methodology is developed in such a way as to allow separate inferences to be made about linkage and association by including theta, the recombination fraction between the marker and disease susceptibility locus under study, explicitly in the model. The method is illustrated by application to a previously published data set. The data analysis raises some interesting issues, notably with regard to the weight of evidence necessary to convince us of linkage between a candidate locus and disease.

Detecting recombination in 4-taxa DNA sequence alignments with Bayesian hidden Markov models and Markov chain Monte Carlo

This article presents a statistical method for detecting recombination in DNA sequence alignments, which is based on combining two probabilistic graphical models: (1) a taxon graph (phylogenetic tree) representing the relationship between the taxa, and (2) a site graph (hidden Markov model) representing interactions between different sites in the DNA sequence alignments. We adopt a Bayesian approach and sample the parameters of the model from the posterior distribution with Markov chain Monte Carlo, using a Metropolis-Hastings and Gibbs-within-Gibbs scheme. The proposed method is tested on various synthetic and real-world DNA sequence alignments, and we compare its performance with the established detection methods RECPARS, PLATO, and TOPAL, as well as with two alternative parameter estimation schemes.

Efficient and accurate experimental design for enzyme kinetics: Bayesian studies reveal a systematic approach

In areas such as drug development, clinical diagnosis and biotechnology research, acquiring details about the kinetic parameters of enzymes is crucial. The correct design of an experiment is critical to collecting data suitable for analysis, modelling and deriving the correct information. As classical design methods are not targeted to the more complex kinetics being frequently studied, attention is needed to estimate parameters of such models with low variance. We demonstrate that a Bayesian approach (the use of prior knowledge) can produce major gains quantifiable in terms of information, productivity and accuracy of each experiment. Developing the use of Bayesian Utility functions, we have used a systematic method to identify the optimum experimental designs for a number of kinetic model data sets. This has enabled the identification of trends between kinetic model types, sets of design rules and the key conclusion that such designs should be based on some prior knowledge of K-M and/or the kinetic model. We suggest an optimal and iterative method for selecting features of the design such as the substrate range, number of measurements and choice of intermediate points. The final design collects data suitable for accurate modelling and analysis and minimises the error in the parameters estimated. (C) 2003 Elsevier Science B.V. All rights reserved.

Qualitative modelling of sustainable energy scenarios: An application and extension of the Bon Qualitative Input-Output model

Climate change is one of the major challenges facing economic systems at the start of the 21st century. Reducing greenhouse gas emissions will require both restructuring the energy supply system (production) and addressing the efficiency and sufficiency of the social uses of energy (consumption). The energy production system is a complicated supply network of interlinked sectors with 'knock-on' effects throughout the economy. End use energy consumption is governed by complex sets of interdependent cultural, social, psychological and economic variables driven by shifts in consumer preference and technological development trajectories. To date, few models have been developed for exploring alternative joint energy production-consumption systems. The aim of this work is to propose one such model. This is achieved in a methodologically coherent manner through integration of qualitative input-output models of production, with Bayesian belief network models of consumption, at point of final demand. The resulting integrated framework can be applied either (relatively) quickly and qualitatively to explore alternative energy scenarios, or as a fully developed quantitative model to derive or assess specific energy policy options. The qualitative applications are explored here.

Model selection approaches for non-linear system identification: a review

The identification of non-linear systems using only observed finite datasets has become a mature research area over the last two decades. A class of linear-in-the-parameter models with universal approximation capabilities have been intensively studied and widely used due to the availability of many linear-learning algorithms and their inherent convergence conditions. This article presents a systematic overview of basic research on model selection approaches for linear-in-the-parameter models. One of the fundamental problems in non-linear system identification is to find the minimal model with the best model generalisation performance from observational data only. The important concepts in achieving good model generalisation used in various non-linear system-identification algorithms are first reviewed, including Bayesian parameter regularisation and models selective criteria based on the cross validation and experimental design. A significant advance in machine learning has been the development of the support vector machine as a means for identifying kernel models based on the structural risk minimisation principle. The developments on the convex optimisation-based model construction algorithms including the support vector regression algorithms are outlined. Input selection algorithms and on-line system identification algorithms are also included in this review. Finally, some industrial applications of non-linear models are discussed.

A bagging-based correction for the mixture model estimator of population size

Estimation of a population size by means of capture-recapture techniques is an important problem occurring in many areas of life and social sciences. We consider the frequencies of frequencies situation, where a count variable is used to summarize how often a unit has been identified in the target population of interest. The distribution of this count variable is zero-truncated since zero identifications do not occur in the sample. As an application we consider the surveillance of scrapie in Great Britain. In this case study holdings with scrapie that are not identified (zero counts) do not enter the surveillance database. The count variable of interest is the number of scrapie cases per holding. For count distributions a common model is the Poisson distribution and, to adjust for potential heterogeneity, a discrete mixture of Poisson distributions is used. Mixtures of Poissons usually provide an excellent fit as will be demonstrated in the application of interest. However, as it has been recently demonstrated, mixtures also suffer under the so-called boundary problem, resulting in overestimation of population size. It is suggested here to select the mixture model on the basis of the Bayesian Information Criterion. This strategy is further refined by employing a bagging procedure leading to a series of estimates of population size. Using the median of this series, highly influential size estimates are avoided. In limited simulation studies it is shown that the procedure leads to estimates with remarkable small bias.

Gaussian quadrature 4D-VAR

A new incremental four-dimensional variational (4D-Var) data assimilation algorithm is introduced. The algorithm does not require the computationally expensive integrations with the nonlinear model in the outer loops. Nonlinearity is accounted for by modifying the linearization trajectory of the observation operator based on integrations with the tangent linear (TL) model. This allows us to update the linearization trajectory of the observation operator in the inner loops at negligible computational cost. As a result the distinction between inner and outer loops is no longer necessary. The key idea on which the proposed 4D-Var method is based is that by using Gaussian quadrature it is possible to get an exact correspondence between the nonlinear time evolution of perturbations and the time evolution in the TL model. It is shown that J-point Gaussian quadrature can be used to derive the exact adjoint-based observation impact equations and furthermore that it is straightforward to account for the effect of multiple outer loops in these equations if the proposed 4D-Var method is used. The method is illustrated using a three-level quasi-geostrophic model and the Lorenz (1996) model.

A new floating model level scheme for the assimilation of boundary layer top inversions: the univariate assimilation of temperature.

The assimilation of observations with a forecast is often heavily influenced by the description of the error covariances associated with the forecast. When a temperature inversion is present at the top of the boundary layer (BL), a significant part of the forecast error may be described as a vertical positional error (as opposed to amplitude error normally dealt with in data assimilation). In these cases, failing to account for positional error explicitly is shown t o r esult in an analysis for which the inversion structure is erroneously weakened and degraded. In this article, a new assimilation scheme is proposed to explicitly include the positional error associated with an inversion. This is done through the introduction of an extra control variable to allow position errors in the a priori to be treated simultaneously with the usual amplitude errors. This new scheme, referred to as the ‘floating BL scheme’, is applied to the one-dimensional (vertical) variational assimilation of temperature. The floating BL scheme is tested with a series of idealised experiments a nd with real data from radiosondes. For each idealised experiment, the floating BL scheme gives an analysis which has the inversion structure and position in agreement with the truth, and outperforms the a ssimilation which accounts only for forecast a mplitude error. When the floating BL scheme is used to assimilate a l arge sample of radiosonde data, its ability to give an analysis with an inversion height in better agreement with that observed is confirmed. However, it is found that the use of Gaussian statistics is an inappropriate description o f t he error statistics o f t he extra c ontrol variable. This problem is alleviated by incorporating a non-Gaussian description of the new control variable in the new scheme. Anticipated challenges in implementing the scheme operationally are discussed towards the end of the article.

Assimilation of RAPID array observations into an ocean model

Experiments assimilating the RAPID dataset of deep temperature and salinity profiles at 26.5°N on the western and eastern Atlantic boundaries into a 1° global NEMO ocean model have been performed. The meridional overturning circulation (MOC) is then assessed against the transports calculated directly from observations. The best initialization found for this short period was obtained by assimilating the EN3 upper-ocean hydrography database prior to 2004, after which different methods of assimilating 5-day average RAPID profiles at the western boundary were tested. The model MOC is strengthened by ∼ 2 Sv giving closer agreement with the RAPID array transports, when the western boundary profiles are assimilated only below 900 m (the approximate depth of the Florida Straits, which are not well resolved) and when the T,S observations are spread meridionally from 10 to 35°N along the deep western boundary. The use of boundary-focused covariances has the largest impact on the assimilation results, otherwise using more conventional Gaussian covariances has a very local impact on the MOC at 26°N with strong adverse impacts on the MOC stream function at higher and lower latitudes. Even using boundary-focused covariances only enables the MOC to be strengthened for ∼ 2 years, after which the increased transport of warm waters leads to a negative feedback on water formation in the subpolar gyre which then reduces the MOC. This negative feedback can be mitigated if EN3 hydrography data continue to be assimilated along with the RAPID array boundary data. Copyright © 2012 Royal Meteorological Society and Crown in the right of Canada.

Likelihood-free estimation of model evidence

Statistical methods of inference typically require the likelihood function to be computable in a reasonable amount of time. The class of “likelihood-free” methods termed Approximate Bayesian Computation (ABC) is able to eliminate this requirement, replacing the evaluation of the likelihood with simulation from it. Likelihood-free methods have gained in efficiency and popularity in the past few years, following their integration with Markov Chain Monte Carlo (MCMC) and Sequential Monte Carlo (SMC) in order to better explore the parameter space. They have been applied primarily to estimating the parameters of a given model, but can also be used to compare models. Here we present novel likelihood-free approaches to model comparison, based upon the independent estimation of the evidence of each model under study. Key advantages of these approaches over previous techniques are that they allow the exploitation of MCMC or SMC algorithms for exploring the parameter space, and that they do not require a sampler able to mix between models. We validate the proposed methods using a simple exponential family problem before providing a realistic problem from human population genetics: the comparison of different demographic models based upon genetic data from the Y chromosome.

Bayesian parameter estimation for latent Markov random fields and social networks

Undirected graphical models are widely used in statistics, physics and machine vision. However Bayesian parameter estimation for undirected models is extremely challenging, since evaluation of the posterior typically involves the calculation of an intractable normalising constant. This problem has received much attention, but very little of this has focussed on the important practical case where the data consists of noisy or incomplete observations of the underlying hidden structure. This paper specifically addresses this problem, comparing two alternative methodologies. In the first of these approaches particle Markov chain Monte Carlo (Andrieu et al., 2010) is used to efficiently explore the parameter space, combined with the exchange algorithm (Murray et al., 2006) for avoiding the calculation of the intractable normalising constant (a proof showing that this combination targets the correct distribution in found in a supplementary appendix online). This approach is compared with approximate Bayesian computation (Pritchard et al., 1999). Applications to estimating the parameters of Ising models and exponential random graphs from noisy data are presented. Each algorithm used in the paper targets an approximation to the true posterior due to the use of MCMC to simulate from the latent graphical model, in lieu of being able to do this exactly in general. The supplementary appendix also describes the nature of the resulting approximation.

A unified neurofuzzy model for classification

This work proposes a unified neurofuzzy modelling scheme. To begin with, the initial fuzzy base construction method is based on fuzzy clustering utilising a Gaussian mixture model (GMM) combined with the analysis of covariance (ANOVA) decomposition in order to obtain more compact univariate and bivariate membership functions over the subspaces of the input features. The mean and covariance of the Gaussian membership functions are found by the expectation maximisation (EM) algorithm with the merit of revealing the underlying density distribution of system inputs. The resultant set of membership functions forms the basis of the generalised fuzzy model (GFM) inference engine. The model structure and parameters of this neurofuzzy model are identified via the supervised subspace orthogonal least square (OLS) learning. Finally, instead of providing deterministic class label as model output by convention, a logistic regression model is applied to present the classifier’s output, in which the sigmoid type of logistic transfer function scales the outputs of the neurofuzzy model to the class probability. Experimental validation results are presented to demonstrate the effectiveness of the proposed neurofuzzy modelling scheme.

Settlement patterns in the late Mesolithic of western Scotland: the implications of Bayesian analysis of radiocarbon dates and inter-site technological comparisons

The character of settlement patterns within the late Mesolithic communities of north-west Europe is a topic of substantial debate. An important case study concerns the five shell middens on the island of Oronsay, Inner Hebrides, western Scotland. Two conflicting interpretations have been proposed: the evidence from seasonality indicators and stable isotope analysis of human bones has been used to support a model of year-round settlement on this small island; alternatively, the middens have been interpreted as resulting from short-term intermittent visits to Oronsay within a regionally mobile settlement pattern. We contribute to this debate by describing Storakaig, a newly discovered site on the nearby island of Islay, undertaking a Bayesian chronological analysis and providing evidence for technological continuity between Oronsay and sites elsewhere in the region. While this new evidence remains open to alternative interpretation, we suggest that it makes regional mobility rather than year-round settlement on Oronsay a more viable interpretation for the Oronsay middens. Our analysis also confirms the likely overlap of the late Mesolithic with the earliest Neolithic within western Scotland.