Public key cryptography and error correcting codes as Ising models

We employ the methods of statistical physics to study the performance of Gallager type error-correcting codes. In this approach, the transmitted codeword comprises Boolean sums of the original message bits selected by two randomly-constructed sparse matrices. We show that a broad range of these codes potentially saturate Shannon's bound but are limited due to the decoding dynamics used. Other codes show sub-optimal performance but are not restricted by the decoding dynamics. We show how these codes may also be employed as a practical public-key cryptosystem and are of competitive performance to modern cyptographical methods.

Online approximations for wind-field models

We study online approximations to Gaussian process models for spatially distributed systems. We apply our method to the prediction of wind fields over the ocean surface from scatterometer data. Our approach combines a sequential update of a Gaussian approximation to the posterior with a sparse representation that allows to treat problems with a large number of observations.

Statistical physics of low density parity check error correcting codes

We study the performance of Low Density Parity Check (LDPC) error-correcting codes using the methods of statistical physics. LDPC codes are based on the generation of codewords using Boolean sums of the original message bits by employing two randomly-constructed sparse matrices. These codes can be mapped onto Ising spin models and studied using common methods of statistical physics. We examine various regular constructions and obtain insight into their theoretical and practical limitations. We also briefly report on results obtained for irregular code constructions, for codes with non-binary alphabet, and on how a finite system size effects the error probability.

Low density parity check codes: a statistical physics perspective

The modem digital communication systems are made transmission reliable by employing error correction technique for the redundancies. Codes in the low-density parity-check work along the principles of Hamming code, and the parity-check matrix is very sparse, and multiple errors can be corrected. The sparseness of the matrix allows for the decoding process to be carried out by probability propagation methods similar to those employed in Turbo codes. The relation between spin systems in statistical physics and digital error correcting codes is based on the existence of a simple isomorphism between the additive Boolean group and the multiplicative binary group. Shannon proved general results on the natural limits of compression and error-correction by setting up the framework known as information theory. Error-correction codes are based on mapping the original space of words onto a higher dimensional space in such a way that the typical distance between encoded words increases.

Evaluating zero error noise thresholds by replica method for Gallager code ensembles

The replica method, developed in statistical physics, is employed in conjunction with Gallager's methodology to accurately evaluate zero error noise thresholds for Gallager code ensembles. Our approach generally provides more optimistic evaluations than those reported in the information theory literature for sparse matrices; the difference vanishes as the parity check matrix becomes dense.

Supporting interoperable interpolation: the INTAMAP approach

In many Environmental Information Systems the actual observations arise from a discrete monitoring network which might be rather heterogeneous in both location and types of measurements made. In this paper we describe the architecture and infrastructure for a system, developed as part of the EU FP6 funded INTAMAP project, to provide a service oriented solution that allows the construction of an interoperable, automatic, interpolation system. This system will be based on the Open Geospatial Consortium’s Web Feature Service (WFS) standard. The essence of our approach is to extend the GML3.1 observation feature to include information about the sensor using SensorML, and to further extend this to incorporate observation error characteristics. Our extended WFS will accept observations, and will store them in a database. The observations will be passed to our R-based interpolation server, which will use a range of methods, including a novel sparse, sequential kriging method (only briefly described here) to produce an internal representation of the interpolated field resulting from the observations currently uploaded to the system. The extended WFS will then accept queries, such as ‘What is the probability distribution of the desired variable at a given point’, ‘What is the mean value over a given region’, or ‘What is the probability of exceeding a certain threshold at a given location’. To support information-rich transfer of complex and uncertain predictions we are developing schema to represent probabilistic results in a GML3.1 (object-property) style. The system will also offer more easily accessible Web Map Service and Web Coverage Service interfaces to allow users to access the system at the level of complexity they require for their specific application. Such a system will offer a very valuable contribution to the next generation of Environmental Information Systems in the context of real time mapping for monitoring and security, particularly for systems that employ a service oriented architecture.

Anger and perceived legitimacy of aggression in male Hong Kong Chinese athletes: Effects of type of sport and level of competition

Problem: The vast majority of research examining the interplay between aggressive emotions, beliefs, behaviors, cognitions, and situational contingencies in competitive athletes has focused on Western populations and only select sports (e.g., ice hockey). Research involving Eastern, particularly Chinese, athletes is surprisingly sparse given the sheer size of these populations. Thus, this study examines the aggressive emotions, beliefs, behaviors, and cognitions, of competitive Chinese athletes. Method: Several measures related to aggression were distributed to a large sample (N ¼ 471) of male athletes, representing four sports (basketball, rugby union, association football/soccer, and squash). Results: Higher levels of anger and aggression tended to be associated with higher levels of play for rugby and low levels of play for contact (e.g., football, basketball) and individual sports (e.g., squash). Conclusions: The results suggest that the experience of angry emotions and aggressive behaviors of Chinese athletes are similar to Western populations, but that sport psychology practitioners should be aware of some potentially important differences, such as the general tendency of Chinese athletes to disapprove of aggressive behavior.

Capacity improvements for a direct-sequence code division multiple access network

The rapidly increasing demand for cellular telephony is placing greater demand on the limited bandwidth resources available. This research is concerned with techniques which enhance the capacity of a Direct-Sequence Code-Division-Multiple-Access (DS-CDMA) mobile telephone network. The capacity of both Private Mobile Radio (PMR) and cellular networks are derived and the many techniques which are currently available are reviewed. Areas which may be further investigated are identified. One technique which is developed is the sectorisation of a cell into toroidal rings. This is shown to provide an increased system capacity when the cell is split into these concentric rings and this is compared with cell clustering and other sectorisation schemes. Another technique for increasing the capacity is achieved by adding to the amount of inherent randomness within the transmitted signal so that the system is better able to extract the wanted signal. A system model has been produced for a cellular DS-CDMA network and the results are presented for two possible strategies. One of these strategies is the variation of the chip duration over a signal bit period. Several different variation functions are tried and a sinusoidal function is shown to provide the greatest increase in the maximum number of system users for any given signal-to-noise ratio. The other strategy considered is the use of additive amplitude modulation together with data/chip phase-shift-keying. The amplitude variations are determined by a sparse code so that the average system power is held near its nominal level. This strategy is shown to provide no further capacity since the system is sensitive to amplitude variations. When both strategies are employed, however, the sensitivity to amplitude variations is shown to reduce, thus indicating that the first strategy both increases the capacity and the ability to handle fluctuations in the received signal power.

Equilibrium properties of disordered spin models with two-scale interactions

Methods for understanding classical disordered spin systems with interactions conforming to some idealized graphical structure are well developed. The equilibrium properties of the Sherrington-Kirkpatrick model, which has a densely connected structure, have become well understood. Many features generalize to sparse Erdös- Rényi graph structures above the percolation threshold and to Bethe lattices when appropriate boundary conditions apply. In this paper, we consider spin states subject to a combination of sparse strong interactions with weak dense interactions, which we term a composite model. The equilibrium properties are examined through the replica method, with exact analysis of the high-temperature paramagnetic, spin-glass, and ferromagnetic phases by perturbative schemes. We present results of replica symmetric variational approximations, where perturbative approaches fail at lower temperature. Results demonstrate re-entrant behaviors from spin glass to ferromagnetic phases as temperature is lowered, including transitions from replica symmetry broken to replica symmetric phases. The nature of high-temperature transitions is found to be sensitive to the connectivity profile in the sparse subgraph, with regular connectivity a discontinuous transition from the paramagnetic to ferromagnetic phases is apparent.

Minimizing unsatisfaction in colourful neighbourhoods

Colouring sparse graphs under various restrictions is a theoretical problem of significant practical relevance. Here we consider the problem of maximizing the number of different colours available at the nodes and their neighbourhoods, given a predetermined number of colours. In the analytical framework of a tree approximation, carried out at both zero and finite temperatures, solutions obtained by population dynamics give rise to estimates of the threshold connectivity for the incomplete to complete transition, which are consistent with those of existing algorithms. The nature of the transition as well as the validity of the tree approximation are investigated.

Sparsely spread CDMA - A statistical mechanics-based analysis

Sparse code division multiple access (CDMA), a variation on the standard CDMA method in which the spreading (signature) matrix contains only a relatively small number of nonzero elements, is presented and analysed using methods of statistical physics. The analysis provides results on the performance of maximum likelihood decoding for sparse spreading codes in the large system limit. We present results for both cases of regular and irregular spreading matrices for the binary additive white Gaussian noise channel (BIAWGN) with a comparison to the canonical (dense) random spreading code. © 2007 IOP Publishing Ltd.

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%.