Bayesian estimation of g-and-k distributions using MCMC

In this paper we investigate a Bayesian procedure for the estimation of a flexible generalised distribution, notably the MacGillivray adaptation of the g-and-κ distribution. This distribution, described through its inverse cdf or quantile function, generalises the standard normal through extra parameters which together describe skewness and kurtosis. The standard quantile-based methods for estimating the parameters of generalised distributions are often arbitrary and do not rely on computation of the likelihood. MCMC, however, provides a simulation-based alternative for obtaining the maximum likelihood estimates of parameters of these distributions or for deriving posterior estimates of the parameters through a Bayesian framework. In this paper we adopt the latter approach, The proposed methodology is illustrated through an application in which the parameter of interest is slightly skewed.

Patch and whole of surface mass-transfer rate measurements on a stepped rotating cylinder electrode

A recently developed whole of surface electroplating technique was used to obtain mass-transfer rates in the separated flow region of a stepped rotating cylinder electrode. These data are compared with previously reported mass-transfer rates obtained with a patch electrode. It was found that the two methods yield different results, where at lower Reynolds numbers, the mass-transfer rate enhancement was noticeably higher for the whole of the surface electrode than for the patch electrode. The location of the peak mass transfer behind the step, as measured with a patch electrode, was reported to be independent of the Reynolds number in previous studies, whereas the whole of the surface electrode shows a definite Reynolds number dependence. Large eddy simulation results for the recirculating region behind a step are used in this work to show that this difference in behavior is related to the existence of a much thinner fluid layer at the wall for which the velocity is a linear junction of distance from the wall. Consequently, the diffusion layer no longer lies well within a laminar sublayer. It is concluded that the patch electrode responds to the wall shear stress for smooth wall flow as well as for the disturbed flow region behind the step. When the whole of the surface is electro-active, the response is to mass transfer even when this is not a sole function of wall shear stress. The results demonstrate that the choice of the mass-transfer measurement technique in corrosion studies can have a significant effect on the results obtained from empirical data.

A Bayesian approach to imposing curvature on distance functions

The estimated parameters of output distance functions frequently violate the monotonicity, quasi-convexity and convexity constraints implied by economic theory, leading to estimated elasticities and shadow prices that are incorrectly signed, and ultimately to perverse conclusions concerning the effects of input and output changes on productivity growth and relative efficiency levels. We show how a Bayesian approach can be used to impose these constraints on the parameters of a translog output distance function. Implementing the approach involves the use of a Gibbs sampler with data augmentation. A Metropolis-Hastings algorithm is also used within the Gibbs to simulate observations from truncated pdfs. Our methods are developed for the case where panel data is available and technical inefficiency effects are assumed to be time-invariant. Two models-a fixed effects model and a random effects model-are developed and applied to panel data on 17 European railways. We observe significant changes in estimated elasticities and shadow price ratios when regularity restrictions are imposed. (c) 2004 Elsevier B.V. All rights reserved.

Experimental and analytical study of the effect of contact angle on liquid convective heat transfer in microchannels

In this paper we examine the effect of contact angle (or surface wettability) on the convective heat transfer coefficient in microchannels. Slip flow, where the fluid velocity at the wall is non-zero, is most likely to occur in microchannels due to its dependence on shear rate or wall shear stress. We show analytically that for a constant pressure drop, the presence of slip increases the Nusselt number. In a microchannel heat exchanger we modified the surface wettability from a contact angle of 20 degrees-120 degrees using thin film coating technology. Apparent slip flow is implied from pressure and flow rate measurements with a departure from classical laminar friction coefficients above a critical shear rate of approximately 10,000 s(-1). The magnitude of this departure is dependant on the contact angle with higher contact angles surfaces exhibiting larger pressure drop decreases. Similarly, the non-dimensional heat flux is found to decrease relative to laminar non-slip theory, and this decrease is also a function of the contact angle. Depending on the contact angle and the wall shear rate, variations in the heat transfer rate exceeding 10% can be expected. Thus the contact angle is an important consideration in the design of micro, and even more so, nano heat exchangers. (c) 2006 Elsevier Ltd. All rights reserved.

Leading coordinate analysis of reaction pathways in proton chain transfer: Application to a two-proton transfer model for the green fluorescent protein

The ‘leading coordinate’ approach to computing an approximate reaction pathway, with subsequent determination of the true minimum energy profile, is applied to a two-proton chain transfer model based on the chromophore and its surrounding moieties within the green fluorescent protein (GFP). Using an ab initio quantum chemical method, a number of different relaxed energy profiles are found for several plausible guesses at leading coordinates. The results obtained for different trial leading coordinates are rationalized through the calculation of a two-dimensional relaxed potential energy surface (PES) for the system. Analysis of the 2-D relaxed PES reveals that two of the trial pathways are entirely spurious, while two others contain useful information and can be used to furnish starting points for successful saddle-point searches. Implications for selection of trial leading coordinates in this class of proton chain transfer reactions are discussed, and a simple diagnostic function is proposed for revealing whether or not a relaxed pathway based on a trial leading coordinate is likely to furnish useful information.

Multidimensional modelling to investigate interspecies hydrogen transfer in anaerobic biofilms

Anaerobic digestion is a multistep process, mediated by a functionally and phylogenetically diverse microbial population. One of the crucial steps is oxidation of organic acids, with electron transfer via hydrogen or formate from acetogenic bacteria to methanogens. This syntrophic microbiological process is strongly restricted by a thermodynamic limitation on the allowable hydrogen or formate concentration. In order to study this process in more detail, we developed an individual-based biofilm model which enables to describe the processes at a microbial resolution. The biochemical model is the ADM1, implemented in a multidimensional domain. With this model, we evaluated three important issues for the syntrophic relationship: (i) is there a fundamental difference in using hydrogen or formate as electron carrier? (ii) Does a thermodynamic-based inhibition function produced substantially different results from an empirical function? and; (iii) Does the physical colocation of acetogens and methanogens follow directly from a general model. Hydrogen or formate as electron carrier had no substantial impact on model results. Standard inhibition functions or thermodynamic inhibition function gave similar results at larger substrate field grid sizes (> 10 mu m), but at smaller grid sizes, the thermodynamic-based function reduced the number of cells with long interspecies distances (> 2.5 mu m). Therefore, a very fine grid resolution is needed to reflect differences between the thermodynamic function, and a more generic inhibition form. The co-location of syntrophic bacteria was well predicted without a need to assume a microbiological based mechanism (e.g., through chemotaxis) of biofilm formation.

In vivo vitamin C supplementation increases phosphoinositol transfer protein expression in peripheral blood mononuclear cells from healthy individuals

Ascorbate can act as both a reducing and oxidising agent in vitro depending on its environment. It can modulate the intracellular redox environment of cells and therefore is predicted to modulate thiol-dependent cell signalling and gene expression pathways. Using proteomic analysis of vitamin C-treated T cells in vitro, we have previously reported changes in expression of five functional protein groups associated with signalling, carbohydrate metabolism, apoptosis, transcription and immune function. The increased expression of the signalling molecule phosphatidylinositol transfer protein (PITP) was also confirmed using Western blotting. Herein, we have compared protein changes elicited by ascorbate in vitro, with the effect of ascorbate on plasma potassium levels, on peripheral blood mononuclear cell (PBMC) apoptosis and PITP expression, in patients supplemented with vitamin C (0-2 g/d) for up to 10 weeks to investigate whether in vitro model systems are predictive of in vivo effects. PITP varied in expression widely between subjects at all time-points analysed but was increased by supplementation with 2 g ascorbate/d after 5 and 10 weeks. No effects on plasma potassium levels were observed in supplemented subjects despite a reduction of K+ channel proteins in ascorbate-treated T cells in vitro. Similarly, no effect of vitamin C supplementation on PBMC apoptosis was observed, whilst ascorbate decreased expression of caspase 3 recruitment domain protein in vitro. These data provide one of the first demonstrations that proteomics may be valuable in developing predictive markers of nutrient effects in vivo and may identify novel pathways for studying mechanisms of action in vivo.

Modelling of non-stationary processes using radial basis function networks

This paper reports preliminary progress on a principled approach to modelling nonstationary phenomena using neural networks. We are concerned with both parameter and model order complexity estimation. The basic methodology assumes a Bayesian foundation. However to allow the construction of pragmatic models, successive approximations have to be made to permit computational tractibility. The lowest order corresponds to the (Extended) Kalman filter approach to parameter estimation which has already been applied to neural networks. We illustrate some of the deficiencies of the existing approaches and discuss our preliminary generalisations, by considering the application to nonstationary time series.

Bayesian invariant measurements of generalisation for continuous distributions

A family of measurements of generalisation is proposed for estimators of continuous distributions. In particular, they apply to neural network learning rules associated with continuous neural networks. The optimal estimators (learning rules) in this sense are Bayesian decision methods with information divergence as loss function. The Bayesian framework guarantees internal coherence of such measurements, while the information geometric loss function guarantees invariance. The theoretical solution for the optimal estimator is derived by a variational method. It is applied to the family of Gaussian distributions and the implications are discussed. This is one in a series of technical reports on this topic; it generalises the results of ¸iteZhu95:prob.discrete to continuous distributions and serve as a concrete example of a larger picture ¸iteZhu95:generalisation.

A Bayesian formulation of search, control and the exploration/exploitation trade-off

A new approach to optimisation is introduced based on a precise probabilistic statement of what is ideally required of an optimisation method. It is convenient to express the formalism in terms of the control of a stationary environment. This leads to an objective function for the controller which unifies the objectives of exploration and exploitation, thereby providing a quantitative principle for managing this trade-off. This is demonstrated using a variant of the multi-armed bandit problem. This approach opens new possibilities for optimisation algorithms, particularly by using neural network or other adaptive methods for the adaptive controller. It also opens possibilities for deepening understanding of existing methods. The realisation of these possibilities requires research into practical approximations of the exact formalism.

Bayesian regression filter and the issue of priors

We propose a Bayesian framework for regression problems, which covers areas which are usually dealt with by function approximation. An online learning algorithm is derived which solves regression problems with a Kalman filter. Its solution always improves with increasing model complexity, without the risk of over-fitting. In the infinite dimension limit it approaches the true Bayesian posterior. The issues of prior selection and over-fitting are also discussed, showing that some of the commonly held beliefs are misleading. The practical implementation is summarised. Simulations using 13 popular publicly available data sets are used to demonstrate the method and highlight important issues concerning the choice of priors.

Bayesian training of mixture density networks

Mixture Density Networks (MDNs) are a well-established method for modelling the conditional probability density which is useful for complex multi-valued functions where regression methods (such as MLPs) fail. In this paper we extend earlier research of a regularisation method for a special case of MDNs to the general case using evidence based regularisation and we show how the Hessian of the MDN error function can be evaluated using R-propagation. The method is tested on two data sets and compared with early stopping.

Regression with input-dependent noise: A Bayesian treatment

In most treatments of the regression problem it is assumed that the distribution of target data can be described by a deterministic function of the inputs, together with additive Gaussian noise having constant variance. The use of maximum likelihood to train such models then corresponds to the minimization of a sum-of-squares error function. In many applications a more realistic model would allow the noise variance itself to depend on the input variables. However, the use of maximum likelihood to train such models would give highly biased results. In this paper we show how a Bayesian treatment can allow for an input-dependent variance while overcoming the bias of maximum likelihood.

Bayesian classification with Gaussian processes

We consider the problem of assigning an input vector bfx to one of m classes by predicting P(c|bfx) for c = 1, ldots, m. For a two-class problem, the probability of class 1 given bfx is estimated by s(y(bfx)), where s(y) = 1/(1 + e^-y). A Gaussian process prior is placed on y(bfx), and is combined with the training data to obtain predictions for new bfx points. We provide a Bayesian treatment, integrating over uncertainty in y and in the parameters that control the Gaussian process prior; the necessary integration over y is carried out using Laplace's approximation. The method is generalized to multi-class problems (m >2) using the softmax function. We demonstrate the effectiveness of the method on a number of datasets.

Approximate Bayesian techniques for inference in stochastic dynamical systems

This thesis is concerned with approximate inference in dynamical systems, from a variational Bayesian perspective. When modelling real world dynamical systems, stochastic differential equations appear as a natural choice, mainly because of their ability to model the noise of the system by adding a variant of some stochastic process to the deterministic dynamics. Hence, inference in such processes has drawn much attention. Here two new extended frameworks are derived and presented that are based on basis function expansions and local polynomial approximations of a recently proposed variational Bayesian algorithm. It is shown that the new extensions converge to the original variational algorithm and can be used for state estimation (smoothing). However, the main focus is on estimating the (hyper-) parameters of these systems (i.e. drift parameters and diffusion coefficients). The new methods are numerically validated on a range of different systems which vary in dimensionality and non-linearity. These are the Ornstein-Uhlenbeck process, for which the exact likelihood can be computed analytically, the univariate and highly non-linear, stochastic double well and the multivariate chaotic stochastic Lorenz '63 (3-dimensional model). The algorithms are also applied to the 40 dimensional stochastic Lorenz '96 system. In this investigation these new approaches are compared with a variety of other well known methods such as the ensemble Kalman filter / smoother, a hybrid Monte Carlo sampler, the dual unscented Kalman filter (for jointly estimating the systems states and model parameters) and full weak-constraint 4D-Var. Empirical analysis of their asymptotic behaviour as a function of observation density or length of time window increases is provided.