Time Series Data Analysis by Ultra-High Frequency Trigonometric Higher Order Neural Networks

Introduction

Conventional ANN models are incapable of handling discontinuities in the input training data. ANNs do not always perform well because of the complexity of the patterns. Artificial neural networks function as “black boxes”, and thus are unable to provide explanations for their behavior. The “black boxes” characteristic is seen as a disadvantage by users, who prefer to be presented with a rationale for the recognition being generated. To overcome these limitations, interest has recently been expressed in using Higher Order Neural Network (HONN) models for data simulation and pattern recognition. Such models can provide information concerning the basis of the data they are recognition, and hence can be considered as ‘open box’ rather than ‘black box’ models. Furthermore, HONN models are also capable of simulating higher frequency and higher order nonlinear data, thus producing superior data recognition, compared with those derived from ANN-based models. This is the motivation therefore for developing the Polynomial Higher Order Neural Network (PHONN) model for data simulation and recognition (Zhang, Murugesan, and Sadeghi, 1995). Zhang & Fulcher (1996b) extend this idea to group PHONN models for data simulation. Zhang, Zhang and Fulcher (2000) develop higher order neural network group models for data approximation. The problem we need to address is to devise a neural network structure that will not only act as an open box to simulate modeling and recognition functions, but which will also facilitate learning algorithm convergence. We would like todemonstrate how it is possible to simulate discontinuous functions, to any degree accuracy, using higher orderneural network group theory, even at points of discontinuity.

Time series models are the most studied models in macroeconomics as well as in financial economics. Nobel Prize in Economics in 2003 rewards two contributions: nonstationarity and time-varying volatility. These contributions have greatly deepened our understanding of two central properties of many economic time series (Vetenskapsakademien, 2003). Nonstationarity is a property common to many macroeconomic and financial time series models. It means that a variable has no clear tendency to return to a constant value or a linear trend. Examples include the value of the US dollar expressed in Japanese yen and consumer price indices of the US and Japan. Granger (1981) changes the way of empirical models in macroeconomic relationships by introducing the concept of cointegrated variables. Granger and Bates (1969) research the combination of forecasts. Granger and Weiss (1983) show the importance of cointegration in the modeling of nonstationary economic series. Granger and Lee (1990) studied multicointegration. Granger and Swanson (1996) further develop multicointegration in studying of cointegrated variables. The first motivation of this chapter is to develop a new nonstationary data analysis system by using new generation computer techniques that will improve the accuracy of the analysis.