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.

Sequential, Bayesian geostatistics:a principled method for large data sets

The principled statistical application of Gaussian random field models used in geostatistics has historically been limited to data sets of a small size. This limitation is imposed by the requirement to store and invert the covariance matrix of all the samples to obtain a predictive distribution at unsampled locations, or to use likelihood-based covariance estimation. Various ad hoc approaches to solve this problem have been adopted, such as selecting a neighborhood region and/or a small number of observations to use in the kriging process, but these have no sound theoretical basis and it is unclear what information is being lost. In this article, we present a Bayesian method for estimating the posterior mean and covariance structures of a Gaussian random field using a sequential estimation algorithm. By imposing sparsity in a well-defined framework, the algorithm retains a subset of “basis vectors” that best represent the “true” posterior Gaussian random field model in the relative entropy sense. This allows a principled treatment of Gaussian random field models on very large data sets. The method is particularly appropriate when the Gaussian random field model is regarded as a latent variable model, which may be nonlinearly related to the observations. We show the application of the sequential, sparse Bayesian estimation in Gaussian random field models and discuss its merits and drawbacks.

Bayesian analysis of the scatterometer wind retrieval inverse problem:Some new approaches

The retrieval of wind vectors from satellite scatterometer observations is a non-linear inverse problem. A common approach to solving inverse problems is to adopt a Bayesian framework and to infer the posterior distribution of the parameters of interest given the observations by using a likelihood model relating the observations to the parameters, and a prior distribution over the parameters. We show how Gaussian process priors can be used efficiently with a variety of likelihood models, using local forward (observation) models and direct inverse models for the scatterometer. We present an enhanced Markov chain Monte Carlo method to sample from the resulting multimodal posterior distribution. We go on to show how the computational complexity of the inference can be controlled by using a sparse, sequential Bayes algorithm for estimation with Gaussian processes. This helps to overcome the most serious barrier to the use of probabilistic, Gaussian process methods in remote sensing inverse problems, which is the prohibitively large size of the data sets. We contrast the sampling results with the approximations that are found by using the sparse, sequential Bayes algorithm.

Data visualisation and exploration with prior knowledge

Visualising data for exploratory analysis is a major challenge in many applications. Visualisation allows scientists to gain insight into the structure and distribution of the data, for example finding common patterns and relationships between samples as well as variables. Typically, visualisation methods like principal component analysis and multi-dimensional scaling are employed. These methods are favoured because of their simplicity, but they cannot cope with missing data and it is difficult to incorporate prior knowledge about properties of the variable space into the analysis; this is particularly important in the high-dimensional, sparse datasets typical in geochemistry. In this paper we show how to utilise a block-structured correlation matrix using a modification of a well known non-linear probabilistic visualisation model, the Generative Topographic Mapping (GTM), which can cope with missing data. The block structure supports direct modelling of strongly correlated variables. We show that including prior structural information it is possible to improve both the data visualisation and the model fit. These benefits are demonstrated on artificial data as well as a real geochemical dataset used for oil exploration, where the proposed modifications improved the missing data imputation results by 3 to 13%.

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.

Statistical physics of error-correcting codes

In this thesis we use statistical physics techniques to study the typical performance of four families of error-correcting codes based on very sparse linear transformations: Sourlas codes, Gallager codes, MacKay-Neal codes and Kanter-Saad codes. We map the decoding problem onto an Ising spin system with many-spins interactions. We then employ the replica method to calculate averages over the quenched disorder represented by the code constructions, the arbitrary messages and the random noise vectors. We find, as the noise level increases, a phase transition between successful decoding and failure phases. This phase transition coincides with upper bounds derived in the information theory literature in most of the cases. We connect the practical decoding algorithm known as probability propagation with the task of finding local minima of the related Bethe free-energy. We show that the practical decoding thresholds correspond to noise levels where suboptimal minima of the free-energy emerge. Simulations of practical decoding scenarios using probability propagation agree with theoretical predictions of the replica symmetric theory. The typical performance predicted by the thermodynamic phase transitions is shown to be attainable in computation times that grow exponentially with the system size. We use the insights obtained to design a method to calculate the performance and optimise parameters of the high performance codes proposed by Kanter and Saad.

Various techniques and procedures for refining ERS-1 orbits

The satellite ERS-1 was launched in July 1991 in a period of high solar activity. Sparse laser tracking and the failure of the experimental microwave system (PRARE) compounded the orbital errors which resulted from mismodelling of atmospheric density and hence surface forces. Three attempts are presented here to try and refine the coarse laser orbits of ERS-1, made prior to the availability of the full altimetric dataset. The results of the first attempt indicate that by geometrically modelling the satellite shape some improvement in orbital precision may be made for any satellite; especially one where no area tables already exist. The second and third refinement attempts are based on the introduction of data from some second satellite; in these examples SPOT-2 and TOPEX/Poseidon are employed. With SPOT-2 the method makes use of the orbital similarities to produce along-track corrections for the more fully tracked SPOT-2. Transferring these corrections to ERS-1 produces improvements in the precise orbits thus determined. With TOPEX/Poseidon the greater altitude results in a more precise orbit (gravity field and atmospheric errors are of less importance). Thus, by computing height differences at crossover points of the TOPEX/Poseidon and ERS-1 ground tracks the poorer orbit of ERS-1 may be improved by the addition of derived radial corrections. In the positive light of all three results several potential modification are suggested and some further avenues of investigation indicated.