The research described in this chapter is concerned with the development of a novel artificial higherorder neural networks architecture called the recurrent Pi-sigma neural network. The proposed artificial neural network combines the advantages of both higher-order architectures in terms of the multi-linear interactions between inputs, as well as the temporal dynamics of recurrent neural networks, and produces highly accurate one-step ahead predictions of the foreign currency exchange rates, as compared to other feedforward and recurrent structures.
Time Series Analysis
A time series is a set of observations xt, each one being recorded at a specific time t (Anderson, 1976). A discrete time series is one where the set of times at which observations are made is a discrete set. Continuous time series are obtained by recording observations continuously over some time interval.
Analysing time series data leads to the decomposition of time series into components (Box & Jenkins, 1976). Each component is defined to be a major factor or force that can affect any time series. Three major components in time series may be identified. Trend refers to the long-term tendency of a time series to rise or fall. Seasonality refers to the periodic behaviour of a time series within a specified period or time. The fluctuation in a time series after the trend and seasonal components have been removed is termed as the irregular component.
If a time series can be exactly predicted from past knowledge, it is termed as deterministic. Otherwise, it is termed as statistical, where past knowledge can only indicate the probabilistic structure of future behaviour. A statistical series can be considered as a single realisation of some stochastic process. A stochastic process is a family of random variables defined on a probability space. A realisation of a stochastic process is a sample path of this process.
The prediction of time series signals is based on their past values. Therefore, it is necessary to obtain a data record. When obtaining a data record, the objective is to have data that are maximally informative and an adequate number of records for prediction purposes. Hence, future values of a time series x(t) can be predicted as a function of past values (Brockwell & Davis, 1991): (1)
where τ refers to the number of prediction steps ahead, and φ is the number of past observations taken into consideration (also known as the order of the predictor).