Innovative Methods in the Prediction and Analysis of Solar-Terrestrial Time Series

The aim of this thesis is to explore the application of feed forward neural networks, and other numerical methods, to the prediction and analysis of solar terrestrial time series. The three time series under scrutiny are the sunspot number, the 10.7cm solar flux and the geomagnetic Kp index. Each time series will be predicted and examined on time scales of days, months and years. As the work of the thesis unfolds, new perspectives on the time series of interest will be afforded, fueling the prediction intiatives of the later Chapters. New techniques for analysing time series are proposed and applied, as well as some new methods of using neural networks to make predictions. Chapter 1 reviews the three main fields of interest. The first field is that of the statistical theory of time series modelling. The basic concepts and terminology are introduced, followed by a review of various time series models and prediction schemes. The more recent topic of neural networks is the second reviewed field. Again the basic ideas are introduced, and the defining equations of feed forward neural networks are stated along with a complete description of the training algorithm known as back propagation. To link these first two fields I suggest how the neural network can be viewed as a statistical time series model. Next, the current understanding of the solar terrestrial environment is reviewed, starting with an overview of solar activity, with particular attention paid to the phenomena associated with the solar cycle. The terrestrial environment is then discussed, focussing on how the Sun and its activity affects the Earth's magnetic field. Finally, a selection of past attempts at predicting solar terrestrial time series are described and discussed. Chapter 2 is where the analysis of the three time series is documented. The work of this chapter is concerned with providing an impression of matters such as: the accumulation and formatting of the data; the search for periodicities; the nature of any periodicities; the non-stationarity of sunspot number; the stationary aspects of sunspot number; the auto-correlation of the time series; the cross-correlation of the time series, especially in relation to the Sun's influence on the Earth; and the use of wavelet transform in analysing time series. Apart from being of intrinsic interest in itself this work provides a familiarity with the data that will directly and indirectly fuel the prediction initiatives of the following chapters. Chapter 3 is an exploration of feed forward neural networks and back propagation. In this chapter some simple FFNNs performing some simple problems are investigated, as well as the (not simple) training algorithm, back propagation. For several cases it is shown what networks can, and cannot, be expected to do because of limitations of numbers of neurons or the activation function used. It is also shown that back-propagation is not always reliable as a training algorithm, as it sometimes completely fails in training networks to perform tasks that they should be able to perform in theory. An important new method, that of analytic training, is also introduced. This method shows how to "train" a neural network to perform any analytic function, by way of constructing and solving a set of linear equations. Chapter 4 further bridges the gap between neural network methods and the statistical models of time series. In this chapter, various artificial time series are predicted using neural networks, and in a few cases, analytic training is used to prescribe what the minimum requirements are for a network to be able to predict a given class and order of statistical time series model. The ability to compare theory and practice makes the results of this chapter very interesting. At the end of the Chapter, the problem of delayed prediction is highlighted. Delayed predictions are predictions in which events in the time series (such as peaks or troughs) are predicted late. (Abstract shortened by ProQuest.)