Forecasting Chaotic and Non-Linear Time Series with Artificial Intelligence and Statistical Measures

Abstract

In this chapter was presented a summary of how use the intelligent computational modelling for time series forecasting and the importance of the correct choice of the fitness function. Three methodology were employed for adjust the parameters of an ANN, a Modified Genetic Algorithm (MGA) (Section 4.2), a Particle Swarm Optimization (PSO) (Section 4.3) and the GRASPES Method (Section 5). The success of evolution control is highly dependent of the optimization algorithm and the fitness function complexity (Buche et al., 2005). As affirmed in the Section 7, the fitness function assigns a fitness value to each individual in the population of Evolutionary Algorithm, at any time, measuring how good is a solution. This solution is represented by a point in the landscape. When an algorithm has a possibility to be guided by different fitness functions, each one will walk on the space, pointing in a different way. The experiments presented here show that the choice of the fitness function is also very important as the choice of the intelligent method employed for time series modelling. The results reached with the MGA and PSO methods were developed in independent way and can be found at (de Mattos Neto et al., 2009; Rodrigues et al., 2009), respectively. In the Rodrigues et al. (2009) was employed eight different fitness functions, where the fitness function of Equation 32 achieved the best performance for the S&P500 index series and the fitness function 3 obtained the best result for the Dow Jones Industrial Average index series. However, in the (de Mattos Neto et al., 2009) the main goal was to develop the combination between the intelligent techniques ANN and PSO. For this reason, only one fitness function was applied (fitness function given by Equation 33) Analyzing the S&P500 index results obtained here is possible to observe that the statistical error measures have a strong accomplished. These statistical measures present many times a competitive behavior. For example, observing the Table 1 for the GRASPES method, the fitness function f 2 (Equation 31) is directly based on MSE error, but f 2 has a inferior performance for the MSE error than the fitness function f 1 (Equation 30) which is based on ARV error. This observation shows that the choice of the fitness function is not a trivial decision