Description and training of neural network dynamics

Attractor properties of a popular discrete-time neural network model are illustrated through numerical simulations. The most complex dynamics is found to occur within particular ranges of parameters controlling the symmetry and magnitude of the weight matrix. A small network model is observed to produce fixed points, limit cycles, mode-locking, the Ruelle-Takens route to chaos, and the period-doubling route to chaos. Training algorithms for tuning this dynamical behaviour are discussed. Training can be an easy or difficult task, depending whether the problem requires the use of temporal information distributed over long time intervals. Such problems require training algorithms which can handle hidden nodes. The most prominent of these algorithms, back propagation through time, solves the temporal credit assignment problem in a way which can work only if the relevant information is distributed locally in time. The Moving Targets algorithm works for the more general case, but is computationally intensive, and prone to local minima.

A novel neural network technique for exploratory data analysis

This paper surveys the context of feature extraction by neural network approaches, and compares and contrasts their behaviour as prospective data visualisation tools in a real world problem. We also introduce and discuss a hybrid approach which allows us to control the degree of discriminatory and topographic information in the extracted feature space.

Theoretical foundations of neural networks

Neural networks have often been motivated by superficial analogy with biological nervous systems. Recently, however, it has become widely recognised that the effective application of neural networks requires instead a deeper understanding of the theoretical foundations of these models. Insight into neural networks comes from a number of fields including statistical pattern recognition, computational learning theory, statistics, information geometry and statistical mechanics. As an illustration of the importance of understanding the theoretical basis for neural network models, we consider their application to the solution of multi-valued inverse problems. We show how a naive application of the standard least-squares approach can lead to very poor results, and how an appreciation of the underlying statistical goals of the modelling process allows the development of a more general and more powerful formalism which can tackle the problem of multi-modality.

Topographic mappings and feed-forward neural networks

This thesis is a study of the generation of topographic mappings - dimension reducing transformations of data that preserve some element of geometric structure - with feed-forward neural networks. As an alternative to established methods, a transformational variant of Sammon's method is proposed, where the projection is effected by a radial basis function neural network. This approach is related to the statistical field of multidimensional scaling, and from that the concept of a 'subjective metric' is defined, which permits the exploitation of additional prior knowledge concerning the data in the mapping process. This then enables the generation of more appropriate feature spaces for the purposes of enhanced visualisation or subsequent classification. A comparison with established methods for feature extraction is given for data taken from the 1992 Research Assessment Exercise for higher educational institutions in the United Kingdom. This is a difficult high-dimensional dataset, and illustrates well the benefit of the new topographic technique. A generalisation of the proposed model is considered for implementation of the classical multidimensional scaling (¸mds}) routine. This is related to Oja's principal subspace neural network, whose learning rule is shown to descend the error surface of the proposed ¸mds model. Some of the technical issues concerning the design and training of topographic neural networks are investigated. It is shown that neural network models can be less sensitive to entrapment in the sub-optimal global minima that badly affect the standard Sammon algorithm, and tend to exhibit good generalisation as a result of implicit weight decay in the training process. It is further argued that for ideal structure retention, the network transformation should be perfectly smooth for all inter-data directions in input space. Finally, there is a critique of optimisation techniques for topographic mappings, and a new training algorithm is proposed. A convergence proof is given, and the method is shown to produce lower-error mappings more rapidly than previous algorithms.

Estimating conditional volatility with neural networks

It is well known that one of the obstacles to effective forecasting of exchange rates is heteroscedasticity (non-stationary conditional variance). The autoregressive conditional heteroscedastic (ARCH) model and its variants have been used to estimate a time dependent variance for many financial time series. However, such models are essentially linear in form and we can ask whether a non-linear model for variance can improve results just as non-linear models (such as neural networks) for the mean have done. In this paper we consider two neural network models for variance estimation. Mixture Density Networks (Bishop 1994, Nix and Weigend 1994) combine a Multi-Layer Perceptron (MLP) and a mixture model to estimate the conditional data density. They are trained using a maximum likelihood approach. However, it is known that maximum likelihood estimates are biased and lead to a systematic under-estimate of variance. More recently, a Bayesian approach to parameter estimation has been developed (Bishop and Qazaz 1996) that shows promise in removing the maximum likelihood bias. However, up to now, this model has not been used for time series prediction. Here we compare these algorithms with two other models to provide benchmark results: a linear model (from the ARIMA family), and a conventional neural network trained with a sum-of-squares error function (which estimates the conditional mean of the time series with a constant variance noise model). This comparison is carried out on daily exchange rate data for five currencies.

Practical assessment of neural network applications

This paper reports the initial results of a joint research project carried out by Aston University and Lloyd's Register to develop a practical method of assessing neural network applications. A set of assessment guidelines for neural network applications were developed and tested on two applications. These case studies showed that it is practical to assess neural networks in a statistical pattern recognition framework. However there is need for more standardisation in neural network technology and a wider takeup of good development practice amongst the neural network community.

Structured neural network modelling of multi-valued functions for wind retrieval from scatterometer measurements

A conventional neural network approach to regression problems approximates the conditional mean of the output vector. For mappings which are multi-valued this approach breaks down, since the average of two solutions is not necessarily a valid solution. In this article mixture density networks, a principled method to model conditional probability density functions, are applied to retrieving Cartesian wind vector components from satellite scatterometer data. A hybrid mixture density network is implemented to incorporate prior knowledge of the predominantly bimodal function branches. An advantage of a fully probabilistic model is that more sophisticated and principled methods can be used to resolve ambiguities.

Neural network-based wind vector retrieval from satellite scatterometer data

Obtaining wind vectors over the ocean is important for weather forecasting and ocean modelling. Several satellite systems used operationally by meteorological agencies utilise scatterometers to infer wind vectors over the oceans. In this paper we present the results of using novel neural network based techniques to estimate wind vectors from such data. The problem is partitioned into estimating wind speed and wind direction. Wind speed is modelled using a multi-layer perceptron (MLP) and a sum of squares error function. Wind direction is a periodic variable and a multi-valued function for a given set of inputs; a conventional MLP fails at this task, and so we model the full periodic probability density of direction conditioned on the satellite derived inputs using a Mixture Density Network (MDN) with periodic kernel functions. A committee of the resulting MDNs is shown to improve the results.

Neural network regression with input uncertainty

It is generally assumed when using Bayesian inference methods for neural networks that the input data contains no noise or corruption. For real-world (errors in variable) problems this is clearly an unsafe assumption. This paper presents a Bayesian neural network framework which allows for input noise given that some model of the noise process exists. In the limit where this noise process is small and symmetric it is shown, using the Laplace approximation, that there is an additional term to the usual Bayesian error bar which depends on the variance of the input noise process. Further, by treating the true (noiseless) input as a hidden variable and sampling this jointly with the network's weights, using Markov Chain Monte Carlo methods, it is demonstrated that it is possible to infer the unbiassed regression over the noiseless input.