Predicting the Production of Total Industry in Greece with Chaos Theory and Neural Networks

Abstract

1. Introduction Non linear dynamics (Medio, 1992) in combination with neural networks had applied in a wide variety of fields, e.g. physics, engineering, ecology and economics. The economist interest is focused on the ability of forecasting an economic time series using time series analysis (Thalassinos and Pociovalisteanu, 2007). In this work we have applied non linear time series analysis in monthly values of Greece Total Industry index. We cover time period from 01.01.1962 until 01.04.2011. We have applied the method of Grassberger and Procaccia (Grassberge and Procaccia, 1983a and 1983b) to evaluate the minimum embedding dimension of each the system. In a second stage using the neural network (Hanias, Curtis, Thalassinos, 2007; Thalassinos et al., 2008 and 2009) we achieved an out of sample multi step time series prediction. 2. Time Series The data for the Production of Total Industry in Greece are collected from Organization for Economic Co-operation and Development and presented as a signal x=x(t) as it shown at Figure 1 (01.01.1962 - 01.04.2011). The sampling rate is At=1 month and the number of data are N=592. 3 [FIGURE 1 OMITTED] 3. State Space Reconstruction For a scalar time series, in our case the time series is the Production of Total Industry index, the phase space can be reconstructed using the methods of delays. The basic idea in the method of delays is that the evolution of any single variable of a system is determined by the other variables, with which it interacts. Information about the relevant variables is thus implicitly contained in the history of any single variable. On the basis of this an equivalent phase space can be reconstructed. From our data we construct a vector [[bar.X].sub.i], i=1 to N, in the m dimensional phase space given by the following relation (Kantz and Schreiber, 1997; Takens, 1981). [[bar.X].sub.i] = {xi,xi-[tau],xi-2[tau], .... xi+(m-1)x}[tau] (1) This vector represents a point to the m dimensional phase space in which the attractor is embedded each time, where [tau] is the time delay [tau]=i[DELTA]t. The element xi represents a value of the examined scalar time series in time corresponding to the ith component of the time series. Use of this method, reduces phase space reconstruction to the problem of proper determining suitable values of m and t. The choice of these values is not always simple, especially when we do not have any additional information about the original system and the only source of data is a simple sequence of scalar values, acquired from the original system. The dimension, where a time delay reconstruction of the phase space provides a necessary number of coordinates to unfold the dynamics from overlaps on itself caused by projection, is called embedding dimension m. a. Time delay [tau] Using the average mutual information we can obtain x less associated with linear point of view, and thus more suitable for dealing with nonlinear problems. The average mutual information may be expressed by the following formula (Kantz and Schreiber, 1997; Takens, 1981). [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2) where P(xi) represents probability of value xi and P(xi, xi+[tau]) is joint probability. In general, I([tau]) expresses the amount of information (in bits), which may be extracted from the value in time xi about the value in time xi+[tau]. As [tau], suitable for the phase space reconstruction, is the first minimum of I(t). [FIGURE 2 OMITTED] As shown in Figure 2, in our case the mutual information function I(t) exhibits a local minimum at 4 time steps and, thus, we shall consider [tau] = 4 to be the optimum delay time. b. Embedding dimension m One method to determine the presence of chaos is to calculate the fractal dimension, which will be non integer for chaotic systems. Even though there exists a number of definitions for the dimension of a fractal object (Box counting dimension, Information Dimension, etc. …