Support vector regression with chaos-based firefly algorithm for stock market price forecasting

Due to the inherent non-linearity and non-stationary characteristics of financial stock market price time series, conventional modeling techniques such as the Box–Jenkins autoregressive integrated moving average (ARIMA) are not adequate for stock market price forecasting. In this paper, a forecasting model based on chaotic mapping, firefly algorithm, and support vector regression (SVR) is proposed to predict stock market price. The forecasting model has three stages. In the first stage, a delay coordinate embedding method is used to reconstruct unseen phase space dynamics. In the second stage, a chaotic firefly algorithm is employed to optimize SVR hyperparameters. Finally in the third stage, the optimized SVR is used to forecast stock market price. The significance of the proposed algorithm is 3-fold. First, it integrates both chaos theory and the firefly algorithm to optimize SVR hyperparameters, whereas previous studies employ a genetic algorithm (GA) to optimize these parameters. Second, it uses a delay coordinate embedding method to reconstruct phase space dynamics. Third, it has high prediction accuracy due to its implementation of structural risk minimization (SRM). To show the applicability and superiority of the proposed algorithm, we selected the three most challenging stock market time series data from NASDAQ historical quotes, namely Intel, National Bank shares and Microsoft daily closed (last) stock price, and applied the proposed algorithm to these data. Compared with genetic algorithm-based SVR (SVR-GA), chaotic genetic algorithm-based SVR (SVR-CGA), firefly-based SVR (SVR-FA), artificial neural networks (ANNs) and adaptive neuro-fuzzy inference systems (ANFIS), the proposed model performs best based amel on two error measures, n .  Stock market price prediction is regarded as one of the most hallenging tasks of financial time series prediction. The diffiulty of forecasting arises from the inherent non-linearity and on-stationarity of the stock market and financial time series. In he past, Box–Jenkins models , such as the autoregressive (AR) odel and the autoregressive integrated moving average (ARIMA) odel, were proposed to tackle this problem. However, these modls were developed based on the assumption that the time series eing forecasted are linear and stationary. In recent years, nonlinear pproaches have been proposed, such as autoregressive conditional eteroscedasticity (ARCH) , generalized autoregressive condiional heteroscedasticity (GARCH) , artificial neural networks ∗ Corresponding author. E-mail address: Farookh.Hussain@uts.edu.au (F.K. Hussain). 568-4946/$ – see front matter.  . Previous researchers have indicated that ANN, which implements the empirical risk minimization principle in its learning process, outperforms traditional statistical models . However, ANN suffers from local minimum traps and the difficulty of determining the hidden layer size and learning rate [24,25]. By contrast, support vector regression, originally introduced by Vapnik [24,26], has a global optimum and exhibits better prediction accuracy due to its implementation of the structural risk minimization principle which considers both the training error and the capacity of the regression model [25,27]. The main problem with SVR is the determination of its hyperparameters, which requires practitioner experience. Unsuitably chosen kernel functions or hyperparameter settings may lead to significantly poor performance [27–30].