Local Utility Functions and Local Probability Transformations

This note shows that, under appropriate conditions, preferences may be locally approximated by the linear utility or risk-neutral preference functional associated with a local probability transformation.

Effects of chronic exposure to low-level cadmium on renal tubular function and CYP2A6-mediated coumarin metabolism in healthy human subjects

Relationships between cadmium (Cd) body burden, kidney function and coumarin metabolism were investigated using two groups of 197 and 200 healthy Thais with men and women in nearly equal numbers. A mean age of one group was 30.5 years and it was 39.3 years for the other group. Of 397, 20 subjects (5%) excreted urine Cd between 1.4 mug/g and 3.8 mug/g creatinine and these subjects faced 10-15% increase in the probability of having abnormal urinary excretion of N-acetyl-beta-D-glucosaminidase (NAG-uria). The prevalence of NAG-uria varied with Cd body burden in a dose-dependent manner (chi(2) = 22, P < 0.008). Also NAG-nuria was one of the three kidney effect markers tested that showed the greatest strength of correlation with urine Cd in both men and women (r = 0.48 P < 0.001). In addition, urine Cd excretion of men and women showed a positive correlation (r = 0.46 to 0.54. P < 0.001) with urine 7-hydroxycoumarin (7-OHC) excretion which was used as a marker of liver cytochrome P450 2A6 (CYP2A6) enzyme activity. Urinary CA excretion accounted for 25% of the total variation in urine 7-OHC excretion (P < 0.001). These data suggest that Cd may increase the expression of CYP2A6 in liver, resulting in enhanced coumarin metabolism in subjects with high Cd body burden. (C) 2003 Elsevier Ireland Ltd. All rights reserved.

Wurst: a protein threading server with a structural scoring function, sequence profiles and optimized substitution matrices

Wurst is a protein threading program with an emphasis on high quality sequence to structure alignments (http://www.zbh.uni-hamburg.de/wurst). Submitted sequences are aligned to each of about 3000 templates with a conventional dynamic programming algorithm, but using a score function with sophisticated structure and sequence terms. The structure terms are a log-odds probability of sequence to structure fragment compatibility, obtained from a Bayesian classification procedure. A simplex optimization was used to optimize the sequence-based terms for the goal of alignment and model quality and to balance the sequence and structural contributions against each other. Both sequence and structural terms operate with sequence profiles.

Functional-structural plant modelling using a combination of architectural analysis, L-systems and a canonical model of function

This paper presents a new method for producing a functional-structural plant model that simulates response to different growth conditions, yet does not require detailed knowledge of underlying physiology. The example used to present this method is the modelling of the mountain birch tree. This new functional-structural modelling approach is based on linking an L-system representation of the dynamic structure of the plant with a canonical mathematical model of plant function. Growth indicated by the canonical model is allocated to the structural model according to probabilistic growth rules, such as rules for the placement and length of new shoots, which were derived from an analysis of architectural data. The main advantage of the approach is that it is relatively simple compared to the prevalent process-based functional-structural plant models and does not require a detailed understanding of underlying physiological processes, yet it is able to capture important aspects of plant function and adaptability, unlike simple empirical models. This approach, combining canonical modelling, architectural analysis and L-systems, thus fills the important role of providing an intermediate level of abstraction between the two extremes of deeply mechanistic process-based modelling and purely empirical modelling. We also investigated the relative importance of various aspects of this integrated modelling approach by analysing the sensitivity of the standard birch model to a number of variations in its parameters, functions and algorithms. The results show that using light as the sole factor determining the structural location of new growth gives satisfactory results. Including the influence of additional regulating factors made little difference to global characteristics of the emergent architecture. Changing the form of the probability functions and using alternative methods for choosing the sites of new growth also had little effect. (c) 2004 Elsevier B.V. All rights reserved.

Nonclassicality of a photon-subtracted Gaussian field

We investigate the nonclassicality of a photon-subtracted Gaussian field, which was produced in a recent experiment, using negativity of the Wigner function and the nonexistence of well-behaved positive P function. We obtain the condition to see negativity of the Wigner function for the case including the mixed Gaussian incoming field, the threshold photodetection and the inefficient homodyne measurement. We show how similar the photon-subtracted state is to a superposition of coherent states.

Regression with input-dependent noise: A Gaussian process treatment

Gaussian processes provide natural non-parametric prior distributions over regression functions. In this paper we consider regression problems where there is noise on the output, and the variance of the noise depends on the inputs. If we assume that the noise is a smooth function of the inputs, then it is natural to model the noise variance using a second Gaussian process, in addition to the Gaussian process governing the noise-free output value. We show that prior uncertainty about the parameters controlling both processes can be handled and that the posterior distribution of the noise rate can be sampled from using Markov chain Monte Carlo methods. Our results on a synthetic data set give a posterior noise variance that well-approximates the true variance.

Flexible Gaussian process wind field models

This technical report builds on previous reports to derive the likelihood and its derivatives for a Gaussian Process with a modified Bessel function based covariance function. The full derivation is shown. The likelihood (with gradient information) can be used in maximum likelihood procedures (i.e. gradient based optimisation) and in Hybrid Monte Carlo sampling (i.e. within a Bayesian framework).

Non-zero mean Gaussian process prior wind field models

This report outlines the derivation and application of a non-zero mean, polynomial-exponential covariance function based Gaussian process which forms the prior wind field model used in 'autonomous' disambiguation. It is principally used since the non-zero mean permits the computation of realistic local wind vector prior probabilities which are required when applying the scaled-likelihood trick, as the marginals of the full wind field prior. As the full prior is multi-variate normal, these marginals are very simple to compute.

First year qualifying report: neural networks for extracting wind vectors from satellite scatterometer data

The ERS-1 Satellite was launched in July 1991 by the European Space Agency into a polar orbit at about km800, carrying a C-band scatterometer. A scatterometer measures the amount of radar back scatter generated by small ripples on the ocean surface induced by instantaneous local winds. Operational methods that extract wind vectors from satellite scatterometer data are based on the local inversion of a forward model, mapping scatterometer observations to wind vectors, by the minimisation of a cost function in the scatterometer measurement space.par This report uses mixture density networks, a principled method for modelling conditional probability density functions, to model the joint probability distribution of the wind vectors given the satellite scatterometer measurements in a single cell (the `inverse' problem). The complexity of the mapping and the structure of the conditional probability density function are investigated by varying the number of units in the hidden layer of the multi-layer perceptron and the number of kernels in the Gaussian mixture model of the mixture density network respectively. The optimal model for networks trained per trace has twenty hidden units and four kernels. Further investigation shows that models trained with incidence angle as an input have results comparable to those models trained by trace. A hybrid mixture density network that incorporates geophysical knowledge of the problem confirms other results that the conditional probability distribution is dominantly bimodal.par The wind retrieval results improve on previous work at Aston, but do not match other neural network techniques that use spatial information in the inputs, which is to be expected given the ambiguity of the inverse problem. Current work uses the local inverse model for autonomous ambiguity removal in a principled Bayesian framework. Future directions in which these models may be improved are given.

General bounds on Bayes errors for regression with Gaussian processes

Based on a simple convexity lemma, we develop bounds for different types of Bayesian prediction errors for regression with Gaussian processes. The basic bounds are formulated for a fixed training set. Simpler expressions are obtained for sampling from an input distribution which equals the weight function of the covariance kernel, yielding asymptotically tight results. The results are compared with numerical experiments.

Gaussian processes:iterative sparse approximations

In recent years there has been an increased interest in applying non-parametric methods to real-world problems. Significant research has been devoted to Gaussian processes (GPs) due to their increased flexibility when compared with parametric models. These methods use Bayesian learning, which generally leads to analytically intractable posteriors. This thesis proposes a two-step solution to construct a probabilistic approximation to the posterior. In the first step we adapt the Bayesian online learning to GPs: the final approximation to the posterior is the result of propagating the first and second moments of intermediate posteriors obtained by combining a new example with the previous approximation. The propagation of em functional forms is solved by showing the existence of a parametrisation to posterior moments that uses combinations of the kernel function at the training points, transforming the Bayesian online learning of functions into a parametric formulation. The drawback is the prohibitive quadratic scaling of the number of parameters with the size of the data, making the method inapplicable to large datasets. The second step solves the problem of the exploding parameter size and makes GPs applicable to arbitrarily large datasets. The approximation is based on a measure of distance between two GPs, the KL-divergence between GPs. This second approximation is with a constrained GP in which only a small subset of the whole training dataset is used to represent the GP. This subset is called the em Basis Vector, or BV set and the resulting GP is a sparse approximation to the true posterior. As this sparsity is based on the KL-minimisation, it is probabilistic and independent of the way the posterior approximation from the first step is obtained. We combine the sparse approximation with an extension to the Bayesian online algorithm that allows multiple iterations for each input and thus approximating a batch solution. The resulting sparse learning algorithm is a generic one: for different problems we only change the likelihood. The algorithm is applied to a variety of problems and we examine its performance both on more classical regression and classification tasks and to the data-assimilation and a simple density estimation problems.

Probability distribution modelling to improve stability in nonlinear MIMO control

We consider the direct adaptive inverse control of nonlinear multivariable systems with different delays between every input-output pair. In direct adaptive inverse control, the inverse mapping is learned from examples of input-output pairs. This makes the obtained controller sub optimal, since the network may have to learn the response of the plant over a larger operational range than necessary. Moreover, in certain applications, the control problem can be redundant, implying that the inverse problem is ill posed. In this paper we propose a new algorithm which allows estimating and exploiting uncertainty in nonlinear multivariable control systems. This approach allows us to model strongly non-Gaussian distribution of control signals as well as processes with hysteresis. The proposed algorithm circumvents the dynamic programming problem by using the predicted neural network uncertainty to localise the possible control solutions to consider.

Stochastic volatility and the goodness-of-fit of the Heston model

Recently, Drǎgulescu and Yakovenko proposed an analytical formula for computing the probability density function of stock log returns, based on the Heston model, which they tested empirically. Their research design inadvertently favourably biased the fit of the data to the Heston model, thus overstating their empirical results. Furthermore, Drǎgulescu and Yakovenko did not perform any goodness-of-fit statistical tests. This study employs a research design that facilitates statistical tests of the goodness-of-fit of the Heston model to empirical returns. Robustness checks are also performed. In brief, the Heston model outperformed the Gaussian model only at high frequencies and even so does not provide a statistically acceptable fit to the data. The Gaussian model performed (marginally) better at medium and low frequencies, at which points the extra parameters of the Heston model have adverse impacts on the test statistics. © 2005 Taylor & Francis Group Ltd.

A basis function approach to Bayesian inference in diffusion processes

In this paper, we present a framework for Bayesian inference in continuous-time diffusion processes. The new method is directly related to the recently proposed variational Gaussian Process approximation (VGPA) approach to Bayesian smoothing of partially observed diffusions. By adopting a basis function expansion (BF-VGPA), both the time-dependent control parameters of the approximate GP process and its moment equations are projected onto a lower-dimensional subspace. This allows us both to reduce the computational complexity and to eliminate the time discretisation used in the previous algorithm. The new algorithm is tested on an Ornstein-Uhlenbeck process. Our preliminary results show that BF-VGPA algorithm provides a reasonably accurate state estimation using a small number of basis functions.

Area summation in human vision at and above detection threshold

The initial image-processing stages of visual cortex are well suited to a local (patchwise) analysis of the viewed scene. But the world's structures extend over space as textures and surfaces, suggesting the need for spatial integration. Most models of contrast vision fall shy of this process because (i) the weak area summation at detection threshold is attributed to probability summation (PS) and (ii) there is little or no advantage of area well above threshold. Both of these views are challenged here. First, it is shown that results at threshold are consistent with linear summation of contrast following retinal inhomogeneity, spatial filtering, nonlinear contrast transduction and multiple sources of additive Gaussian noise. We suggest that the suprathreshold loss of the area advantage in previous studies is due to a concomitant increase in suppression from the pedestal. To overcome this confound, a novel stimulus class is designed where: (i) the observer operates on a constant retinal area, (ii) the target area is controlled within this summation field, and (iii) the pedestal is fixed in size. Using this arrangement, substantial summation is found along the entire masking function, including the region of facilitation. Our analysis shows that PS and uncertainty cannot account for the results, and that suprathreshold summation of contrast extends over at least seven target cycles of grating. © 2007 The Royal Society.