شبکه پیوندی کاربردی ادغام شده مبتنی بر PSO و سیستم منطق فازی فاصله ای نوع 2 برای پیش بینی شاخص های بازار سهام

This paper presents an integrated functional link interval type-2 fuzzy neural system (FLIT2FNS) for predicting the stock market indices. The hybrid model uses a TSK (Takagi–Sugano–Kang) type fuzzy rule base that employs type-2 fuzzy sets in the antecedent parts and the outputs from the Functional Link Artificial Neural Network (FLANN) in the consequent parts. Two other approaches, namely the integrated FLANN and type-1 fuzzy logic system and Local Linear Wavelet Neural Network (LLWNN) are also presented for a comparative study. Backpropagation and particle swarm optimization (PSO) learning algorithms have been used independently to optimize the parameters of all the forecasting models. To test the model performance, three well known stock market indices like the Standard's & Poor's 500 (S&P 500), Bombay stock exchange (BSE), and Dow Jones industrial average (DJIA) are used. The mean absolute percentage error (MAPE) and root mean square error (RMSE) are used to find out the performance of all the three models. Finally, it is observed that out of three methods, FLIT2FNS performs the best irrespective of the time horizons spanning from 1 day to 1 month.

Financial time series is considered to be noisy, random and volatile. Stock exchange, in particular, is prone to fluctuations not only for economic factors but also for non-economic ones like political turmoil in a given country, terrorist attacks, and even individual investors’ psychology [1] and hence prediction of stock market indices is a challenging task. However, the benefits involved in accurate prediction have motivated researchers to develop newer and advanced tools and models. Out of the several techniques [2] available for stock market prediction, the statistical methods have been used extensively. The various statistical techniques include autoregressive moving average (ARMA), autoregressive integrated moving average (ARIMA), autoregressive conditional heteroscedasticity (ARCH), and generalized autoregressive conditional heteroskedasticity (GARCH) models and all these models assume the linearity of previous and current variables. Generally the financial time series data, being chaotic and noisy in nature, do not necessarily follow a fixed pattern or linearity and thus the statistical approaches do not perform very well in predicting stock market indices accurately. However, their performance improves considerably with a large volume of data, although the computational overhead becomes higher [3]. In contrast to the statistical techniques, soft and evolutionary computing methods like Artificial Neural Network (ANN), fuzzy set theory, rough set theory, support vector machine (SVM) [4], and the evolutionary learning algorithms like genetic algorithm (GA), particle swarm optimization (PSO), bacterial foraging optimization (BFO), etc. can handle the uncertain, chaotic, and nonlinear nature of the stock markets and thus have been used widely for accurate prediction of stock market indices. A survey of literature indicates that among different types of ANNs, i.e. radial basis function (RBF) Neural Network, Recurrent Neural Network (RNN), etc., multilayer perceptron (MLP) is the most popular ANN tool used for predictions of financial time series. Two artificial neural network models, ANN1 and ANN2 [5] have used to predict weekly closing price of Bombay stock exchange (BSE). However, the models suffer from computational complexity because they need as many as 800 neurons in input layer and 600 neurons in hidden layers for predicting stock price indices. To overcome these limitations, more sophisticated ANNs like Local Linear Wavelet Neural Network (LLWNN), Functional Link Artificial Neural Network (FLANN) and hybrid models [6] have been developed. A local linear wavelet neural network [7] is proposed to predict Box-Jenkins and Mackey glass time series where a hybrid training algorithm of particle swarm optimization and gradient descent method were introduced to train the models. The same LLWNN model is used [8] to predict stock market indices like NASDAQ-100 index and S&P CNX NIFTY index. They have used estimation of distribution algorithm (EDA) to optimize the parameters of the model. Results show that the LLWNN model performs marginally better than the conventional neural network models. Similarly, Functional Link Artificial Neural Network (FLANN) is used [9] and [10] to predict S&P 500 stock. Though ANN is found to be a successful forecasting tool in large number of applications, it suffers from the limitations like black box technique [11], over fitting and gets trapped in local minima. To overcome these limitations, a combination of wavelet and Takagi–Sugeno–Kang (TSK) fuzzy rule based system [12] and [13] is applied to predict financial time series data of Taiwan stock market. Fuzzy logic based models are preferred because as they offer an efficient tool to handle uncertainties. A fuzzy neural network is used [14] to forecast financial time series where genetic algorithm and gradient descent learning algorithm are used alternatively in an iterative manner to adjust the parameters until the error is less than a certain threshold value. A hybrid neuro fuzzy architecture based on Kalman filter [15] has been applied to predict financial time series taking Mackey glass time series as experimental data. Fu-Yuan has adopted an improved PSO algorithm and fuzzy neural network [11] to predict Shanghai stock market indices and genetic fuzzy neural network [16] to forecast Shenzhen stock indices. A hybrid forecasting model [17] based on fuzzy time series and particle swarm optimization technique has been used to forecast Taiwan stock exchange. A survey of 100 published articles [18] related to prediction of stock market indices has concluded that neural networks and fuzzy models are most suitable for stock market prediction. In this paper, a hybrid functional link and interval type-2 fuzzy neural system (FLIT2FNS) [19], [20], [21], [22] and [23] is used to forecast stock market indices. Interval Type-2FLS is a simplified version of Type-2FLS. A type-2 fuzzy set [24] is more capable to incorporate uncertainties compared to type-1 fuzzy logic system. Due to computational complexity of type-2 fuzzy logic system, Liang and Mendel proposed interval Type-2FLS, i.e. a simplified version of Type-2FLS which possesses all the advantages of Type-2FLS sans its computational complexity. This has been dealt with in Section 3. However, a detailed discussion can be seen in ref. [25] Further, the interval Type-2FLS used in this paper belongs to TSK (Takagi–Sugano–Kang) type [26], [27] and [28] yielding an easier defuzzification procedure for forecasting stock market indices. The parameters of all three models (FLIT2FNS, FLANN and Type-1FLS, and LLWNN) are optimized by the commonly used backpropagation (BP) learning algorithm. Further, to overcome two major limitations of backpropagation learning algorithm, i.e. slowness in error convergence speed and its inability to escape local optima, particle swarm optimization (PSO) [29] and [30], a population based, self-adaptive search optimization technique is used to optimize the parameters of all the models. PSO will be discussed in details in Section 5.2. Three stock market indices Standard's & Poor's 500 (S&P 500), Dow Jones industrial average (DJIA) and the Bombay stock exchange (BSE) are used as the experimental data. The S&P 500 is an index of the prices of 500 large publicly held companies publishing its stock market fluctuations since 1957. The DJIA known as “the barometer of stock market” began publishing the composite list of stocks of major companies in the year 1984. In 1997, it appeared in Wall Street Journal. In the beginning, it averaged the stocks of 12 companies with the idea of giving a picture of the trend in the stock market which can help in the forecasting. The BSE is the oldest stock exchange in Asia and on August 2007 as many as 4700 companies were listed in the Exchange which is the largest stock exchange in the world [31]. This paper is organized as follows: In Section 2, a comparative analysis is made between Type-1FLS and interval Type-2FLS. Section 3 deals with the proposed FLIT2FNS model. A brief note on LLWNN model is given in Section 4. Section 5 deals with both backpropagation (BP) and particle swarm optimization (PSO) learning algorithms. Section 6 provides the original datasets and an overview of input selection for all the three models. In Section 7, empirical results are given and they are discussed and analyzed in Section 8. Finally, Section 9 draws the conclusion.

An integrated model comprising Functional Link Artificial Neural Network and interval Type-2fuzzy logic system has been used to predict three stock market indices, S&P 500, BSE, and DJIA for 1 day, 1 week, and 1 month in advance. For comparison the paper also presents the performance of the integrated FLANN and Type-1FLS, and Local Linear Wavelet Network (LLWNN) models. From the numerical experimentation results, it is clearly established that the interval Type-2FLS helps to deal with the uncertainties present in the stock market in an efficient manner. Backpropagation learning algorithm is used to train all the models. However, to improve the accuracy in prediction and reduce MAPE, Particle swarm optimization technique (PSO) is used to train the neural models separately. The simulation results clearly show that FLIT2FNS model performs better than that of FLANN and Type-1FLS followed by LLWNN model irrespective of the learning algorithms used or irrespective of the periodicity of the prediction. The average MAPE obtained from FLIT2FNS-PSO model is 0.32% for 1 day ahead prediction for all three datasets whereas, it is 1.1% and 1.4% for FLANN and Type-1FLS-PSO, and LLWNN-PSO models, respectively.