Advanced mean field methods:theory and practice

A major problem in modern probabilistic modeling is the huge computational complexity involved in typical calculations with multivariate probability distributions when the number of random variables is large. Because exact computations are infeasible in such cases and Monte Carlo sampling techniques may reach their limits, there is a need for methods that allow for efficient approximate computations. One of the simplest approximations is based on the mean field method, which has a long history in statistical physics. The method is widely used, particularly in the growing field of graphical models. Researchers from disciplines such as statistical physics, computer science, and mathematical statistics are studying ways to improve this and related methods and are exploring novel application areas. Leading approaches include the variational approach, which goes beyond factorizable distributions to achieve systematic improvements; the TAP (Thouless-Anderson-Palmer) approach, which incorporates correlations by including effective reaction terms in the mean field theory; and the more general methods of graphical models. Bringing together ideas and techniques from these diverse disciplines, this book covers the theoretical foundations of advanced mean field methods, explores the relation between the different approaches, examines the quality of the approximation obtained, and demonstrates their application to various areas of probabilistic modeling.

UncertML: an XML schema for exchanging uncertainty

Authors from Burrough (1992) to Heuvelink et al. (2007) have highlighted the importance of GIS frameworks which can handle incomplete knowledge in data inputs, in decision rules and in the geometries and attributes modelled. It is particularly important for this uncertainty to be characterised and quantified when GI data is used for spatial decision making. Despite a substantial and valuable literature on means of representing and encoding uncertainty and its propagation in GI (e.g.,Hunter and Goodchild 1993; Duckham et al. 2001; Couclelis 2003), no framework yet exists to describe and communicate uncertainty in an interoperable way. This limits the usability of Internet resources of geospatial data, which are ever-increasing, based on specifications that provide frameworks for the ‘GeoWeb’ (Botts and Robin 2007; Cox 2006). In this paper we present UncertML, an XML schema which provides a framework for describing uncertainty as it propagates through many applications, including online risk management chains. This uncertainty description ranges from simple summary statistics (e.g., mean and variance) to complex representations such as parametric, multivariate distributions at each point of a regular grid. The philosophy adopted in UncertML is that all data values are inherently uncertain, (i.e., they are random variables, rather than values with defined quality metadata).

Web based expert elicitation of uncertainties in environmental model inputs

When constructing and using environmental models, it is typical that many of the inputs to the models will not be known perfectly. In some cases, it will be possible to make observations, or occasionally physics-based uncertainty propagation, to ascertain the uncertainty on these inputs. However, such observations are often either not available or even possible, and another approach to characterising the uncertainty on the inputs must be sought. Even when observations are available, if the analysis is being carried out within a Bayesian framework then prior distributions will have to be specified. One option for gathering or at least estimating this information is to employ expert elicitation. Expert elicitation is well studied within statistics and psychology and involves the assessment of the beliefs of a group of experts about an uncertain quantity, (for example an input / parameter within a model), typically in terms of obtaining a probability distribution. One of the challenges in expert elicitation is to minimise the biases that might enter into the judgements made by the individual experts, and then to come to a consensus decision within the group of experts. Effort is made in the elicitation exercise to prevent biases clouding the judgements through well-devised questioning schemes. It is also important that, when reaching a consensus, the experts are exposed to the knowledge of the others in the group. Within the FP7 UncertWeb project (http://www.uncertweb.org/), there is a requirement to build a Webbased tool for expert elicitation. In this paper, we discuss some of the issues of building a Web-based elicitation system - both the technological aspects and the statistical and scientific issues. In particular, we demonstrate two tools: a Web-based system for the elicitation of continuous random variables and a system designed to elicit uncertainty about categorical random variables in the setting of landcover classification uncertainty. The first of these examples is a generic tool developed to elicit uncertainty about univariate continuous random variables. It is designed to be used within an application context and extends the existing SHELF method, adding a web interface and access to metadata. The tool is developed so that it can be readily integrated with environmental models exposed as web services. The second example was developed for the TREES-3 initiative which monitors tropical landcover change through ground-truthing at confluence points. It allows experts to validate the accuracy of automated landcover classifications using site-specific imagery and local knowledge. Experts may provide uncertainty information at various levels: from a general rating of their confidence in a site validation to a numerical ranking of the possible landcover types within a segment. A key challenge in the web based setting is the design of the user interface and the method of interacting between the problem owner and the problem experts. We show the workflow of the elicitation tool, and show how we can represent the final elicited distributions and confusion matrices using UncertML, ready for integration into uncertainty enabled workflows.We also show how the metadata associated with the elicitation exercise is captured and can be referenced from the elicited result, providing crucial lineage information and thus traceability in the decision making process.

Describing and communicating uncertainty within the semantic web

The Semantic Web relies on carefully structured, well defined, data to allow machines to communicate and understand one another. In many domains (e.g. geospatial) the data being described contains some uncertainty, often due to incomplete knowledge; meaningful processing of this data requires these uncertainties to be carefully analysed and integrated into the process chain. Currently, within the SemanticWeb there is no standard mechanism for interoperable description and exchange of uncertain information, which renders the automated processing of such information implausible, particularly where error must be considered and captured as it propagates through a processing sequence. In particular we adopt a Bayesian perspective and focus on the case where the inputs / outputs are naturally treated as random variables. This paper discusses a solution to the problem in the form of the Uncertainty Markup Language (UncertML). UncertML is a conceptual model, realised as an XML schema, that allows uncertainty to be quantified in a variety of ways i.e. realisations, statistics and probability distributions. UncertML is based upon a soft-typed XML schema design that provides a generic framework from which any statistic or distribution may be created. Making extensive use of Geography Markup Language (GML) dictionaries, UncertML provides a collection of definitions for common uncertainty types. Containing both written descriptions and mathematical functions, encoded as MathML, the definitions within these dictionaries provide a robust mechanism for defining any statistic or distribution and can be easily extended. Universal Resource Identifiers (URIs) are used to introduce semantics to the soft-typed elements by linking to these dictionary definitions. The INTAMAP (INTeroperability and Automated MAPping) project provides a use case for UncertML. This paper demonstrates how observation errors can be quantified using UncertML and wrapped within an Observations & Measurements (O&M) Observation. The interpolation service uses the information within these observations to influence the prediction outcome. The output uncertainties may be encoded in a variety of UncertML types, e.g. a series of marginal Gaussian distributions, a set of statistics, such as the first three marginal moments, or a set of realisations from a Monte Carlo treatment. Quantifying and propagating uncertainty in this way allows such interpolation results to be consumed by other services. This could form part of a risk management chain or a decision support system, and ultimately paves the way for complex data processing chains in the Semantic Web.

Research on reliability evaluation of reconfigurable manufacturing system

It is a crucial task to evaluate the reliability of manufacturing process in product development process. Process reliability is a measurement of production ability of reconfigurable manufacturing system (RMS), which serves as an integrated performance indicator of the production process under specified technical constraints, including time, cost and quality. An integration framework of manufacturing process reliability evaluation is presented together with product development process. A mathematical model and algorithm based on universal generating function (UGF) is developed for calculating the reliability of manufacturing process with respect to task intensity and process capacity, which are both independent random variables. The rework strategies of RMS are analyzed under different task intensity based on process reliability is presented, and the optimization of rework strategies based on process reliability is discussed afterwards.

Raman fiber lasers with a random distributed feedback based on Rayleigh scattering

We demonstrate lasing based on a random distributed feedback due to the Raman amplified Rayleigh backscattering in different types of cavities with and without conventional point-action reflectors. Quasistationary generation of a narrowband spectrum is achieved despite the random nature of the feedback. The generated spectrum is localized at the reflection or gain spectral maxima in schemes with and without point reflectors, respectively. The length limit for a conventional cavity and the minimal pump power required for the lasing based purely on a random distributed feedback are determined. © 2010 The American Physical Society.

CW lasing in a telecom fibre due to the random distributed feedback via Rayleigh scattering

We demonstrate a fibre laser with a mirrorless cavity that operates via Rayleigh scattering amplified through the Raman effect. The properties of such random distributed feedback laser appear different from those of both traditional random lasers and conventional fibre lasers. ©2010 IEEE.

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.

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.

Ultra-long raman laser with a feedback based on the Rayleigh scattering

This paper reports the Rayleigh scattering effects in ultra-long Raman fibre laser. It has been found that in a long fibre cavity (-100 km) the distributed feedback due to Rayleigh back scattering at propagation of light between fibre Bragg grating reflectors may be comparable with the lumped feedback provided by the FBG itself. As a result, Raman lasing in the fibre span limited by lumped (FBG) reflector at one side only appears possible due to significant reflection from the RS-based "random" distributed mirror at the other side. Thus, it concludes that a distributed Rayleigh scattering "random" mirror can form a cavity together with a single FBG spliced to the opposite cavity end.

Random distributed feedback fibre laser

The concept of random lasers making use of multiple scattering in amplifying disordered media to generate coherent light has attracted a great deal of attention in recent years. Here, we demonstrate a fibre laser with a mirrorless open cavity that operates via Rayleigh scattering, amplified through the Raman effect. The fibre waveguide geometry provides transverse confinement and effectively one-dimensional random distributed feedback, leading to the generation of a stationary near-Gaussian beam with a narrow spectrum, and with efficiency and performance comparable to regular lasers. Rayleigh scattering due to inhomogeneities within the glass structure of the fibre is extremely weak, making the operation and properties of the proposed random distributed feedback lasers profoundly different from those of both traditional random lasers and conventional fibre lasers.

Dual-wavelength, ultralong Raman laser with Rayleigh-scattering feedback

We present experimental demonstration of a 200-km-long, dual-wavelength Raman laser utilizing two slightly different-wavelength fiber Bragg gratings, one on each side of the fiber span. The obtained results clearly prove the generation of two independent Raman lasers with a distributed “random” Rayleigh scattering mirror forming a cavity together with each of the individual fiber Bragg grating reflectors.

Tunable random fiber laser

An optical fiber is treated as a natural one-dimensional random system where lasing is possible due to a combination of Rayleigh scattering by refractive index inhomogeneities and distributed amplification through the Raman effect. We present such a random fiber laser that is tunable over a broad wavelength range with uniquely flat output power and high efficiency, which outperforms traditional lasers of the same category. Outstanding characteristics defined by deep underlying physics and the simplicity of the scheme make the demonstrated laser a very attractive light source both for fundamental science and practical applications.

Ultra-long raman laser with a feedback based on the Rayleigh scattering

This paper reports the Rayleigh scattering effects in ultra-long Raman fibre laser. It has been found that in a long fibre cavity (-100 km) the distributed feedback due to Rayleigh back scattering at propagation of light between fibre Bragg grating reflectors may be comparable with the lumped feedback provided by the FBG itself. As a result, Raman lasing in the fibre span limited by lumped (FBG) reflector at one side only appears possible due to significant reflection from the RS-based "random" distributed mirror at the other side. Thus, it concludes that a distributed Rayleigh scattering "random" mirror can form a cavity together with a single FBG spliced to the opposite cavity end.

Random distributed feedback fibre lasers

The concept of random lasers exploiting multiple scattering of photons in an amplifying disordered medium in order to generate coherent light without a traditional laser resonator has attracted a great deal of attention in recent years. This research area lies at the interface of the fundamental theory of disordered systems and laser science. The idea was originally proposed in the context of astrophysics in the 1960s by V.S. Letokhov, who studied scattering with "negative absorption" of the interstellar molecular clouds. Research on random lasers has since developed into a mature experimental and theoretical field. A simple design of such lasers would be promising for potential applications. However, in traditional random lasers the properties of the output radiation are typically characterized by complex features in the spatial, spectral and time domains, making them less attractive than standard laser systems in terms of practical applications. Recently, an interesting and novel type of one-dimensional random laser that operates in a conventional telecommunication fibre without any pre-designed resonator mirrors-random distributed feedback fibre laser-was demonstrated. The positive feedback required for laser generation in random fibre lasers is provided by the Rayleigh scattering from the inhomogeneities of the refractive index that are naturally present in silica glass. In the proposed laser concept, the randomly backscattered light is amplified through the Raman effect, providing distributed gain over distances up to 100km. Although an effective reflection due to the Rayleigh scattering is extremely small (~0.1%), the lasing threshold may be exceeded when a sufficiently large distributed Raman gain is provided. Such a random distributed feedback fibre laser has a number of interesting and attractive features. The fibre waveguide geometry provides transverse confinement, and effectively one-dimensional random distributed feedback leads to the generation of a stationary near-Gaussian beam with a narrow spectrum. A random distributed feedback fibre laser has efficiency and performance that are comparable to and even exceed those of similar conventional fibre lasers. The key features of the generated radiation of random distributed feedback fibre lasers include: a stationary narrow-band continuous modeless spectrum that is free of mode competition, nonlinear power broadening, and an output beam with a Gaussian profile in the fundamental transverse mode (generated both in single mode and multi-mode fibres).This review presents the current status of research in the field of random fibre lasers and shows their potential and perspectives. We start with an introductory overview of conventional distributed feedback lasers and traditional random lasers to set the stage for discussion of random fibre lasers. We then present a theoretical analysis and experimental studies of various random fibre laser configurations, including widely tunable, multi-wavelength, narrow-band generation, and random fibre lasers operating in different spectral bands in the 1-1.6μm range. Then we discuss existing and future applications of random fibre lasers, including telecommunication and distributed long reach sensor systems. A theoretical description of random lasers is very challenging and is strongly linked with the theory of disordered systems and kinetic theory. We outline two key models governing the generation of random fibre lasers: the average power balance model and the nonlinear Schrödinger equation based model. Recently invented random distributed feedback fibre lasers represent a new and exciting field of research that brings together such diverse areas of science as laser physics, the theory of disordered systems, fibre optics and nonlinear science. Stable random generation in optical fibre opens up new possibilities for research on wave transport and localization in disordered media. We hope that this review will provide background information for research in various fields and will stimulate cross-disciplinary collaborations on random fibre lasers. © 2014 Elsevier B.V.