Random field models and priors on wind

This report seeks to make concrete some of the ideas we have been discussing about sensible priors for winds over the ocean. In particular, random field models are reviewed, as are permissible covariance functions. The criteria which these covariance functions must satisfy in order that vorticity and divergence exist and are continuous are defined. The use of Helmholtz theorem is discussed, and possible choices for the covariances are suggested.

Random graph coloring:Statistical physics approach

The problem of vertex coloring in random graphs is studied using methods of statistical physics and probability. Our analytical results are compared to those obtained by exact enumeration and Monte Carlo simulations. We critically discuss the merits and shortcomings of the various methods, and interpret the results obtained. We present an exact analytical expression for the two-coloring problem as well as general replica symmetric approximated solutions for the thermodynamics of the graph coloring problem with p colors and K-body edges. ©2002 The American Physical Society.

Coloring random graphs and maximizing local diversity

We study a variation of the graph coloring problem on random graphs of finite average connectivity. Given the number of colors, we aim to maximize the number of different colors at neighboring vertices (i.e. one edge distance) of any vertex. Two efficient algorithms, belief propagation and Walksat are adapted to carry out this task. We present experimental results based on two types of random graphs for different system sizes and identify the critical value of the connectivity for the algorithms to find a perfect solution. The problem and the suggested algorithms have practical relevance since various applications, such as distributed storage, can be mapped onto this problem.

Measuring the spatial arrangement patterns of pathological lesions in histological sections of brain tissue

The development of abnormal protein aggregates in the form of extracellular plaques and intracellular inclusions is a characteristic feature of many neurodegenerative diseases such as Alzheimer's disease (AD), Creutzfeldt-Jakob disease (CJD) and the fronto-temporal dementias (FTD). An important aspect of a pathological protein aggregate is its spatial topography in the tissue. Lesions may not be randomly distributed within a histological section but exhibit spatial pattern, a departure from randomness either towards regularity or clustering. Information on the spatial pattern of a lesion may be useful in elucidating its pathogenesis and in studying the relationships between different lesions. This article reviews the methods that have been used to study the spatial topography of lesions. These include simple tests of whether the distribution of a lesion departs significantly from random using randomized points or sample fields, and more complex methods that employ grids or transects of contiguous fields and which can detect the intensity of aggregation and the sizes, distribution and spacing of the clusters. The usefulness of these methods in elucidating the pathogenesis of protein aggregates in neurodegenerative disease is discussed.

The spacing and timing of [a chic] critique

This rejoinder reflects an important step, for me, in a preoccupation with methodology that has provided me with many hours of enjoyable reading, not to mention anxiety. For me the ‘reality’ of the incommensurable nature of paradigms and acceptance of the legitimacy of a range of conceptual and philosophical traditions came late. As a constructionist I find myself on the ‘anything goes’ end of methodology choice. This paper and my main paper ought not to be read as a critique of ‘middle range’ theory, but as a critique of an important and necessary aspect of the way we all seek to inscribe facts and structure our writing. What follows is a reflection of the influence Bruno Latour’s writings have had on my ways of seeing and perhaps an unhealthy emphasis on the small things that combine to produce convincing arguments and ‘facts’.

Emulation of random output simulators

Computer models, or simulators, are widely used in a range of scientific fields to aid understanding of the processes involved and make predictions. Such simulators are often computationally demanding and are thus not amenable to statistical analysis. Emulators provide a statistical approximation, or surrogate, for the simulators accounting for the additional approximation uncertainty. This thesis develops a novel sequential screening method to reduce the set of simulator variables considered during emulation. This screening method is shown to require fewer simulator evaluations than existing approaches. Utilising the lower dimensional active variable set simplifies subsequent emulation analysis. For random output, or stochastic, simulators the output dispersion, and thus variance, is typically a function of the inputs. This work extends the emulator framework to account for such heteroscedasticity by constructing two new heteroscedastic Gaussian process representations and proposes an experimental design technique to optimally learn the model parameters. The design criterion is an extension of Fisher information to heteroscedastic variance models. Replicated observations are efficiently handled in both the design and model inference stages. Through a series of simulation experiments on both synthetic and real world simulators, the emulators inferred on optimal designs with replicated observations are shown to outperform equivalent models inferred on space-filling replicate-free designs in terms of both model parameter uncertainty and predictive variance.

The Effect of nutritional supplementation on subjective and objective measures of visual and retinal function:a random controlled trial

In industrialised countries age-related macular disease (ARMD) is the leading cause of visual loss in older people. Because oxidative stress is purported to be associated with an increased risk of disease development the role of antioxidant supplementation is of interest. Lutein is a carotenoid antioxidant that accumulates within the retina and is thought to filter blue light. Increased levels of lutein have been associated with reduced risk of developing ARMD and improvements in visual and retinal function in eyes with ARMD. The aim of this randomised controlled trial (RCT) was to investigate the effect of a lutein-based nutritional supplement on subjective and objective measures of visual function in healthy eyes and in eyes with age-related maculopathy (ARM) – an early form of ARMD. Supplement withdrawal effects were also investigated. A sample size of 66 healthy older (HO), healthy younger (HY), and ARM eyes were randomly allocated to receive a lutein-based supplement or no treatment for 40 weeks. The supplemented group then stopped supplementation to look at the effects of withdrawal over a further 20 weeks. The primary outcome measure was multifocal electroretinogram (mfERG) N1P1 amplitude. Secondary outcome measures were mfERG N1, P1 and N2 latency, contrast sensitivity (CS), Visual acuity (VA) and macular pigment optical density (MPOD). Sample sizes were sufficient for the RCT to have an 80% power to detect a significant clinical effect at the 5% significance level for all outcome measures when the healthy eye groups were combined, and CS, VA and mfERG in the ARM group. This RCT demonstrates significant improvements in MPOD in HY and HO supplemented eyes. When HY and HO supplemented groups were combined, MPOD improvements were maintained, and mfERG ring 2 P1 latency became shorter. On withdrawal of the supplement mfERG ring 1 N1P1 amplitude reduced in HO eyes. When HO and HY groups were combined, mfERG ring 1 and ring 2 N1P1 amplitudes were reduced. In ARM eyes, ring 3 N2 latency and ring 4 P1 latency became longer. These statistically significant changes may not be clinically significant. The finding that a lutein-based supplement increases MPOD in healthy eyes, but does not increase mfERG amplitudes contrasts with the CARMIS study and contributes to the debate on the use of nutritional supplementation in ARM.

Spectral pattern, grouping, and the pitches of complex tones and their components

Harmonically related components are typically heard as a unified entity with a rich timbre and a pitch corresponding to the fundamental frequency. Mistuning a component generally has four consequences: (i) the global pitch of the complex shifts in the same direction as the mistuning; (ii) the component makes a reduced contribution to global pitch; (iii) the component is heard out as a separate sound with a pure timbre; (iv) its pitch differs from that of a pure tone of equal frequency in a small but systematic way. Local interactions between neighbouring components cannot explain these effects; instead they are usually explained in terms of the global operation of a single harmonic-template mechanism. However, several observations indicate that separate mechanisms govern the selection of spectral components for perceptual fusion and for the computation of global pitch. First, an increase in mistuning causes a harmonic to be heard out before it begins to be excluded from the computation of global pitch. Second, a single even harmonic added to an odd-harmonic complex is typically more salient than its odd neighbours. Third, the mistuning of a component in frequency-shifted stimuli, or stimuli with a moderate spectral stretch, results in changes in salience and component pitch like those seen for harmonic stimuli. Fourth, the global pitch of frequency-shifted stimuli is predicted well by the weighted fit of a harmonic template, but, with the exception of the lowest component, the fusion of individual partials for shifted stimuli is best predicted by the common pattern of spectral spacing. Fifth, our sensitivity to spectral pattern is surprisingly resistant to random variations in component spacing induced by applying mistunings to several harmonics at once. These findings are evaluated in the context of an autocorrelogram model of the proposed pitch/grouping dissociation. © S. Hirzel Verlag · EAA.

Molecular phase space transport in water:non-stationary random walk model

Molecular transport in phase space is crucial for chemical reactions because it defines how pre-reactive molecular configurations are found during the time evolution of the system. Using Molecular Dynamics (MD) simulated atomistic trajectories we test the assumption of the normal diffusion in the phase space for bulk water at ambient conditions by checking the equivalence of the transport to the random walk model. Contrary to common expectations we have found that some statistical features of the transport in the phase space differ from those of the normal diffusion models. This implies a non-random character of the path search process by the reacting complexes in water solutions. Our further numerical experiments show that a significant long period of non-stationarity in the transition probabilities of the segments of molecular trajectories can account for the observed non-uniform filling of the phase space. Surprisingly, the characteristic periods in the model non-stationarity constitute hundreds of nanoseconds, that is much longer time scales compared to typical lifetime of known liquid water molecular structures (several picoseconds).

Solvable model for distribution networks on random graphs

We propose a simple model that captures the salient properties of distribution networks, and study the possible occurrence of blackouts, i.e., sudden failings of large portions of such networks. The model is defined on a random graph of finite connectivity. The nodes of the graph represent hubs of the network, while the edges of the graph represent the links of the distribution network. Both, the nodes and the edges carry dynamical two state variables representing the functioning or dysfunctional state of the node or link in question. We describe a dynamical process in which the breakdown of a link or node is triggered when the level of maintenance it receives falls below a given threshold. This form of dynamics can lead to situations of catastrophic breakdown, if levels of maintenance are themselves dependent on the functioning of the net, once maintenance levels locally fall below a critical threshold due to fluctuations. We formulate conditions under which such systems can be analyzed in terms of thermodynamic equilibrium techniques, and under these conditions derive a phase diagram characterizing the collective behavior of the system, given its model parameters. The phase diagram is confirmed qualitatively and quantitatively by simulations on explicit realizations of the graph, thus confirming the validity of our approach. © 2007 The American Physical Society.

Typical kernel size and number of sparse random matrices over Galois fields: a statistical physics approach

Using methods of statistical physics, we study the average number and kernel size of general sparse random matrices over GF(q), with a given connectivity profile, in the thermodynamical limit of large matrices. We introduce a mapping of GF(q) matrices onto spin systems using the representation of the cyclic group of order q as the q-th complex roots of unity. This representation facilitates the derivation of the average kernel size of random matrices using the replica approach, under the replica symmetric ansatz, resulting in saddle point equations for general connectivity distributions. Numerical solutions are then obtained for particular cases by population dynamics. Similar techniques also allow us to obtain an expression for the exact and average number of random matrices for any general connectivity profile. We present numerical results for particular distributions.

Properties of sparse random matrices over finite fields

Typical properties of sparse random matrices over finite (Galois) fields are studied, in the limit of large matrices, using techniques from the physics of disordered systems. For the case of a finite field GF(q) with prime order q, we present results for the average kernel dimension, average dimension of the eigenvector spaces and the distribution of the eigenvalues. The number of matrices for a given distribution of entries is also calculated for the general case. The significance of these results to error-correcting codes and random graphs is also discussed.

Noisy random Boolean formulae:a statistical physics perspective

Properties of computing Boolean circuits composed of noisy logical gates are studied using the statistical physics methodology. A formula-growth model that gives rise to random Boolean functions is mapped onto a spin system, which facilitates the study of their typical behavior in the presence of noise. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding macroscopic phase transitions. The framework is employed for deriving results on error-rates at various function-depths and function sensitivity, and their dependence on the gate-type and noise model used. These are difficult to obtain via the traditional methods used in this field.

Analysis of population dispersion in histological sections

Stereology and other image analysis methods have enabled rapid and objective quantitative measurements to be made on histological sections. These mesurements may include total volumes, surfaces, lengths and numbers of cells and blood vessels or pathological lesions. Histological features, however, may not be randomly distributed across a section but exhibit 'dispersion', a departure from randomness either towards regularity or aggregation. Information of population dispersion may be valuable not only in understanding the two-or three-dimensional structure but also in elucidating the pathogenesis of lesions in pathological conditions. This article reviews some of the statistical methods available for studying dispersion. These range from simple tests of whether the distribution of a histological faeture departs significantly from random to more complex methods which can detect the intensity of aggregation and the sizes, distribution and spacing of the clusters.

The usefulness of spatial pattern analysis in understanding the pathogenesis of neurodegenerative disorders, with particular reference to plaque formation in Alzheimer's disease

Discrete, microscopic lesions are developed in the brain in a number of neurodegenerative diseases. These lesions may not be randomly distributed in the tissue but exhibit a spatial pattern, i.e., a departure from randomness towards regularlity or clustering. The spatial pattern of a lesion may reflect its development in relation to other brain lesions or to neuroanatomical structures. Hence, a study of spatial pattern may help to elucidate the pathogenesis of a lesion. A number of statistical methods can be used to study the spatial patterns of brain lesions. They range from simple tests of whether the distribution of a lesion departs from random to more complex methods which can detect clustering and the size, distribution and spacing of clusters. This paper reviews the uses and limitations of these methods as applied to neurodegenerative disorders, and in particular to senile plaque formation in Alzheimer's disease.