Curvature-driven smoothing: a learning algorithm for feed-forward networks

The performance of feed-forward neural networks in real applications can be often be improved significantly if use is made of a-priori information. For interpolation problems this prior knowledge frequently includes smoothness requirements on the network mapping, and can be imposed by the addition to the error function of suitable regularization terms. The new error function, however, now depends on the derivatives of the network mapping, and so the standard back-propagation algorithm cannot be applied. In this paper, we derive a computationally efficient learning algorithm, for a feed-forward network of arbitrary topology, which can be used to minimize the new error function. Networks having a single hidden layer, for which the learning algorithm simplifies, are treated as a special case.

Dynamic simulation of chemical plant

A CSSL- type modular FORTRAN package, called ACES, has been developed to assist in the simulation of the dynamic behaviour of chemical plant. ACES can be harnessed, for instance, to simulate the transients in startups or after a throughput change. ACES has benefited from two existing simulators. The structure was adapted from ICL SLAM and most plant models originate in DYFLO. The latter employs sequential modularisation which is not always applicable to chemical engineering problems. A novel device of twice- round execution enables ACES to achieve general simultaneous modularisation. During the FIRST ROUND, STATE-VARIABLES are retrieved from the integrator and local calculations performed. During the SECOND ROUND, fresh derivatives are estimated and stored for simultaneous integration. ACES further includes a version of DIFSUB, a variable-step integrator capable of handling stiff differential systems. ACES is highly formalised . It does not use pseudo steady- state approximations and excludes inconsistent and arbitrary features of DYFLO. Built- in debug traps make ACES robust. ACES shows generality, flexibility, versatility and portability, and is very convenient to use. It undertakes substantial housekeeping behind the scenes and thus minimises the detailed involvement of the user. ACES provides a working set of defaults for simulation to proceed as far as possible. Built- in interfaces allow for reactions and user supplied algorithms to be incorporated . New plant models can be easily appended. Boundary- value problems and optimisation may be tackled using the RERUN feature. ACES is file oriented; a STATE can be saved in a readable form and reactivated later. Thus piecewise simulation is possible. ACES has been illustrated and verified to a large extent using some literature-based examples. Actual plant tests are desirable however to complete the verification of the library. Interaction and graphics are recommended for future work.

Testing term structure estimation methods:evidence from the UK STRIPs market

Prices and yields of UK government zero-coupon bonds are used to test alternative yield curve estimation models. Zero-coupon bonds permit a more pure comparison, as the models are providing only the interpolation service and also not making estimation feasible. It is found that better yield curves estimates are obtained by fitting to the yield curve directly rather than fitting first to the discount function. A simple procedure to set the smoothness of the fitted curves is developed, and a positive relationship between oversmoothness and the fitting error is identified. A cubic spline function fitted directly to the yield curve provides the best overall balance of fitting error and smoothness, both along the yield curve and within local maturity regions.

Depth propagation and surface construction in 3-D vision

In stereo vision, regions with ambiguous or unspecified disparity can acquire perceived depth from unambiguous regions. This has been called stereo capture, depth interpolation or surface completion. We studied some striking induced depth effects suggesting that depth interpolation and surface completion are distinct stages of visual processing. An inducing texture (2-D Gaussian noise) had sinusoidal modulation of disparity, creating a smooth horizontal corrugation. The central region of this surface was replaced by various test patterns whose perceived corrugation was measured. When the test image was horizontal 1-D noise, shown to one eye or to both eyes without disparity, it appeared corrugated in much the same way as the disparity-modulated (DM) flanking regions. But when the test image was 2-D noise, or vertical 1-D noise, little or no depth was induced. This suggests that horizontal orientation was a key factor. For a horizontal sine-wave luminance grating, strong depth was induced, but for a square-wave grating, depth was induced only when its edges were aligned with the peaks and troughs of the DM flanking surface. These and related results suggest that disparity (or local depth) propagates along horizontal 1-D features, and then a 3-D surface is constructed from the depth samples acquired. The shape of the constructed surface can be different from the inducer, and so surface construction appears to operate on the results of a more local depth propagation process.

Exchanging uncertainty: interoperable geostatistics?

Traditionally, geostatistical algorithms are contained within specialist GIS and spatial statistics software. Such packages are often expensive, with relatively complex user interfaces and steep learning curves, and cannot be easily integrated into more complex process chains. In contrast, Service Oriented Architectures (SOAs) promote interoperability and loose coupling within distributed systems, typically using XML (eXtensible Markup Language) and Web services. Web services provide a mechanism for a user to discover and consume a particular process, often as part of a larger process chain, with minimal knowledge of how it works. Wrapping current geostatistical algorithms with a Web service layer would thus increase their accessibility, but raises several complex issues. This paper discusses a solution to providing interoperable, automatic geostatistical processing through the use of Web services, developed in the INTAMAP project (INTeroperability and Automated MAPping). The project builds upon Open Geospatial Consortium standards for describing observations, typically used within sensor webs, and employs Geography Markup Language (GML) to describe the spatial aspect of the problem domain. Thus the interpolation service is extremely flexible, being able to support a range of observation types, and can cope with issues such as change of support and differing error characteristics of sensors (by utilising descriptions of the observation process provided by SensorML). XML is accepted as the de facto standard for describing Web services, due to its expressive capabilities which allow automatic discovery and consumption by ‘naive’ users. Any XML schema employed must therefore be capable of describing every aspect of a service and its processes. However, no schema currently exists that can define the complex uncertainties and modelling choices that are often present within geostatistical analysis. We show a solution to this problem, developing a family of XML schemata to enable the description of a full range of uncertainty types. These types will range from simple statistics, such as the kriging mean and variances, through to a range of probability distributions and non-parametric models, such as realisations from a conditional simulation. By employing these schemata within a Web Processing Service (WPS) we show a prototype moving towards a truly interoperable geostatistical software architecture.

The challenge of real-time automatic mapping for environmental monitoring network management

The automatic interpolation of environmental monitoring network data such as air quality or radiation levels in real-time setting poses a number of practical and theoretical questions. Among the problems found are (i) dealing and communicating uncertainty of predictions, (ii) automatic (hyper)parameter estimation, (iii) monitoring network heterogeneity, (iv) dealing with outlying extremes, and (v) quality control. In this paper we discuss these issues, in light of the spatial interpolation comparison exercise held in 2004.

Parallel geostatistics for sparse and dense datasets

Very large spatially-referenced datasets, for example, those derived from satellite-based sensors which sample across the globe or large monitoring networks of individual sensors, are becoming increasingly common and more widely available for use in environmental decision making. In large or dense sensor networks, huge quantities of data can be collected over small time periods. In many applications the generation of maps, or predictions at specific locations, from the data in (near) real-time is crucial. Geostatistical operations such as interpolation are vital in this map-generation process and in emergency situations, the resulting predictions need to be available almost instantly, so that decision makers can make informed decisions and define risk and evacuation zones. It is also helpful when analysing data in less time critical applications, for example when interacting directly with the data for exploratory analysis, that the algorithms are responsive within a reasonable time frame. Performing geostatistical analysis on such large spatial datasets can present a number of problems, particularly in the case where maximum likelihood. Although the storage requirements only scale linearly with the number of observations in the dataset, the computational complexity in terms of memory and speed, scale quadratically and cubically respectively. Most modern commodity hardware has at least 2 processor cores if not more. Other mechanisms for allowing parallel computation such as Grid based systems are also becoming increasingly commonly available. However, currently there seems to be little interest in exploiting this extra processing power within the context of geostatistics. In this paper we review the existing parallel approaches for geostatistics. By recognising that diffeerent natural parallelisms exist and can be exploited depending on whether the dataset is sparsely or densely sampled with respect to the range of variation, we introduce two contrasting novel implementations of parallel algorithms based on approximating the data likelihood extending the methods of Vecchia [1988] and Tresp [2000]. Using parallel maximum likelihood variogram estimation and parallel prediction algorithms we show that computational time can be significantly reduced. We demonstrate this with both sparsely sampled data and densely sampled data on a variety of architectures ranging from the common dual core processor, found in many modern desktop computers, to large multi-node super computers. To highlight the strengths and weaknesses of the diffeerent methods we employ synthetic data sets and go on to show how the methods allow maximum likelihood based inference on the exhaustive Walker Lake data set.

Canonical solution groups of a homomorphism : manipulator kinematics, nonparametric regression and distributed object systems

The kinematic mapping of a rigid open-link manipulator is a homomorphism between Lie groups. The homomorphisrn has solution groups that act on an inverse kinematic solution element. A canonical representation of solution group operators that act on a solution element of three and seven degree-of-freedom (do!) dextrous manipulators is determined by geometric analysis. Seven canonical solution groups are determined for the seven do! Robotics Research K-1207 and Hollerbach arms. The solution element of a dextrous manipulator is a collection of trivial fibre bundles with solution fibres homotopic to the Torus. If fibre solutions are parameterised by a scalar, a direct inverse funct.ion that maps the scalar and Cartesian base space coordinates to solution element fibre coordinates may be defined. A direct inverse pararneterisation of a solution element may be approximated by a local linear map generated by an inverse augmented Jacobian correction of a linear interpolation. The action of canonical solution group operators on a local linear approximation of the solution element of inverse kinematics of dextrous manipulators generates cyclical solutions. The solution representation is proposed as a model of inverse kinematic transformations in primate nervous systems. Simultaneous calibration of a composition of stereo-camera and manipulator kinematic models is under-determined by equi-output parameter groups in the composition of stereo-camera and Denavit Hartenberg (DH) rnodels. An error measure for simultaneous calibration of a composition of models is derived and parameter subsets with no equi-output groups are determined by numerical experiments to simultaneously calibrate the composition of homogeneous or pan-tilt stereo-camera with DH models. For acceleration of exact Newton second-order re-calibration of DH parameters after a sequential calibration of stereo-camera and DH parameters, an optimal numerical evaluation of DH matrix first order and second order error derivatives with respect to a re-calibration error function is derived, implemented and tested. A distributed object environment for point and click image-based tele-command of manipulators and stereo-cameras is specified and implemented that supports rapid prototyping of numerical experiments in distributed system control. The environment is validated by a hierarchical k-fold cross validated calibration to Cartesian space of a radial basis function regression correction of an affine stereo model. Basic design and performance requirements are defined for scalable virtual micro-kernels that broker inter-Java-virtual-machine remote method invocations between components of secure manageable fault-tolerant open distributed agile Total Quality Managed ISO 9000+ conformant Just in Time manufacturing systems.

Fast algorithms for automatic mapping with space-limited covariance functions

In this paper we discuss a fast Bayesian extension to kriging algorithms which has been used successfully for fast, automatic mapping in emergency conditions in the Spatial Interpolation Comparison 2004 (SIC2004) exercise. The application of kriging to automatic mapping raises several issues such as robustness, scalability, speed and parameter estimation. Various ad-hoc solutions have been proposed and used extensively but they lack a sound theoretical basis. In this paper we show how observations can be projected onto a representative subset of the data, without losing significant information. This allows the complexity of the algorithm to grow as O(n m 2), where n is the total number of observations and m is the size of the subset of the observations retained for prediction. The main contribution of this paper is to further extend this projective method through the application of space-limited covariance functions, which can be used as an alternative to the commonly used covariance models. In many real world applications the correlation between observations essentially vanishes beyond a certain separation distance. Thus it makes sense to use a covariance model that encompasses this belief since this leads to sparse covariance matrices for which optimised sparse matrix techniques can be used. In the presence of extreme values we show that space-limited covariance functions offer an additional benefit, they maintain the smoothness locally but at the same time lead to a more robust, and compact, global model. We show the performance of this technique coupled with the sparse extension to the kriging algorithm on synthetic data and outline a number of computational benefits such an approach brings. To test the relevance to automatic mapping we apply the method to the data used in a recent comparison of interpolation techniques (SIC2004) to map the levels of background ambient gamma radiation. © Springer-Verlag 2007.

Automatic processing, quality assurance and serving of real-time weather data over lightweight protocols

Recent advances in technology have produced a significant increase in the availability of free sensor data over the Internet. With affordable weather monitoring stations now available to individual meteorology enthusiasts a reservoir of real time data such as temperature, rainfall and wind speed can now be obtained for most of the United States and Europe. Despite the abundance of available data, obtaining useable information about the weather in your local neighbourhood requires complex processing that poses several challenges. This paper discusses a collection of technologies and applications that harvest, refine and process this data, culminating in information that has been tailored toward the user. In this case we are particularly interested in allowing a user to make direct queries about the weather at any location, even when this is not directly instrumented, using interpolation methods. We also consider how the uncertainty that the interpolation introduces can then be communicated to the user of the system, using UncertML, a developing standard for uncertainty representation.

Automated mapping of environmental variables from a SEIS or SISE perspective

The INTAMAP FP6 project has developed an interoperable framework for real-time automatic mapping of critical environmental variables by extending spatial statistical methods and employing open, web-based, data exchange protocols and visualisation tools. This paper will give an overview of the underlying problem, of the project, and discuss which problems it has solved and which open problems seem to be most relevant to deal with next. The interpolation problem that INTAMAP solves is the generic problem of spatial interpolation of environmental variables without user interaction, based on measurements of e.g. PM10, rainfall or gamma dose rate, at arbitrary locations or over a regular grid covering the area of interest. It deals with problems of varying spatial resolution of measurements, the interpolation of averages over larger areas, and with providing information on the interpolation error to the end-user. In addition, monitoring network optimisation is addressed in a non-automatic context.

The n-tuple network: coping with sub-optimality

The subject of this thesis is the n-tuple net.work (RAMnet). The major advantage of RAMnets is their speed and the simplicity with which they can be implemented in parallel hardware. On the other hand, this method is not a universal approximator and the training procedure does not involve the minimisation of a cost function. Hence RAMnets are potentially sub-optimal. It is important to understand the source of this sub-optimality and to develop the analytical tools that allow us to quantify the generalisation cost of using this model for any given data. We view RAMnets as classifiers and function approximators and try to determine how critical their lack of' universality and optimality is. In order to understand better the inherent. restrictions of the model, we review RAMnets showing their relationship to a number of well established general models such as: Associative Memories, Kamerva's Sparse Distributed Memory, Radial Basis Functions, General Regression Networks and Bayesian Classifiers. We then benchmark binary RAMnet. model against 23 other algorithms using real-world data from the StatLog Project. This large scale experimental study indicates that RAMnets are often capable of delivering results which are competitive with those obtained by more sophisticated, computationally expensive rnodels. The Frequency Weighted version is also benchmarked and shown to perform worse than the binary RAMnet for large values of the tuple size n. We demonstrate that the main issues in the Frequency Weighted RAMnets is adequate probability estimation and propose Good-Turing estimates in place of the more commonly used :Maximum Likelihood estimates. Having established the viability of the method numerically, we focus on providillg an analytical framework that allows us to quantify the generalisation cost of RAMnets for a given datasetL. For the classification network we provide a semi-quantitative argument which is based on the notion of Tuple distance. It gives a good indication of whether the network will fail for the given data. A rigorous Bayesian framework with Gaussian process prior assumptions is given for the regression n-tuple net. We show how to calculate the generalisation cost of this net and verify the results numerically for one dimensional noisy interpolation problems. We conclude that the n-tuple method of classification based on memorisation of random features can be a powerful alternative to slower cost driven models. The speed of the method is at the expense of its optimality. RAMnets will fail for certain datasets but the cases when they do so are relatively easy to determine with the analytical tools we provide.

Satellite borne radar altimetry : empirical orbit refinement and ocean signal recovery techniques

Measurements of the sea surface obtained by satellite borne radar altimetry are irregularly spaced and contaminated with various modelling and correction errors. The largest source of uncertainty for low Earth orbiting satellites such as ERS-1 and Geosat may be attributed to orbital modelling errors. The empirical correction of such errors is investigated by examination of single and dual satellite crossovers, with a view to identifying the extent of any signal aliasing: either by removal of long wavelength ocean signals or introduction of additional error signals. From these studies, it was concluded that sinusoidal approximation of the dominant one cycle per revolution orbit error over arc lengths of 11,500 km did not remove a significant mesoscale ocean signal. The use of TOPEX/Poseidon dual crossovers with ERS-1 was shown to substantially improve the radial accuracy of ERS-1, except for some absorption of small TOPEX/Poseidon errors. The extraction of marine geoid information is of great interest to the oceanographic community and was the subject of the second half of this thesis. Firstly through determination of regional mean sea surfaces using Geosat data, it was demonstrated that a dataset with 70cm orbit error contamination could produce a marine geoid map which compares to better than 12cm with an accurate regional high resolution gravimetric geoid. This study was then developed into Optimal Fourier Transform Interpolation, a technique capable of analysing complete altimeter datasets for the determination of consistent global high resolution geoid maps. This method exploits the regular nature of ascending and descending data subsets thus making possible the application of fast Fourier transform algorithms. Quantitative assessment of this method was limited by the lack of global ground truth gravity data, but qualitative results indicate good signal recovery from a single 35-day cycle.

Mean sea surface modelling from multi-satellite altimetry utilising frequency domain techniques

A technique is presented for the development of a high precision and resolution Mean Sea Surface (MSS) model. The model utilises Radar altimetric sea surface heights extracted from the geodetic phase of the ESA ERS-1 mission. The methodology uses a modified Le Traon et al. (1995) cubic-spline fit of dual ERS-1 and TOPEX/Poseidon crossovers for the minimisation of radial orbit error. The procedure then uses Fourier domain processing techniques for spectral optimal interpolation of the mean sea surface in order to reduce residual errors within the model. Additionally, a multi-satellite mean sea surface integration technique is investigated to supplement the first model with additional enhanced data from the GEOSAT geodetic mission.The methodology employs a novel technique that combines the Stokes' and Vening-Meinsz' transformations, again in the spectral domain. This allows the presentation of a new enhanced GEOSAT gravity anomaly field.

Neuro-electromagnetic imaging of the human visual cortex

Methods of solving the neuro-electromagnetic inverse problem are examined and developed, with specific reference to the human visual cortex. The anatomy, physiology and function of the human visual system are first reviewed. Mechanisms by which the visual cortex gives rise to external electric and magnetic fields are then discussed, and the forward problem is described mathematically for the case of an isotropic, piecewise homogeneous volume conductor, and then for an anisotropic, concentric, spherical volume conductor. Methods of solving the inverse problem are reviewed, before a new technique is presented. This technique combines prior anatomical information gained from stereotaxic studies, with a probabilistic distributed-source algorithm to yield accurate, realistic inverse solutions. The solution accuracy is enhanced by using both visual evoked electric and magnetic responses simultaneously. The numerical algorithm is then modified to perform equivalent current dipole fitting and minimum norm estimation, and these three techniques are implemented on a transputer array for fast computation. Due to the linear nature of the techniques, they can be executed on up to 22 transputers with close to linear speedup. The latter part of the thesis describes the application of the inverse methods to the analysis of visual evoked electric and magnetic responses. The CIIm peak of the pattern onset evoked magnetic response is deduced to be a product of current flowing away from the surface areas 17, 18 and 19, while the pattern reversal P100m response originates in the same areas, but from oppositely directed current. Cortical retinotopy is examined using sectorial stimuli, the CI and CIm ;peaks of the pattern onset electric and magnetic responses are found to originate from areas V1 and V2 simultaneously, and they therefore do not conform to a simple cruciform model of primary visual cortex.