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This paper presents an integration prediction method which is called a hybrid forecasting system based on multiple scales. In this method, the original data are decomposed into multiple layers by the wavelet transform and the multiple layers are divided into low-frequency, intermediate-frequency and high-frequency signal layers. Then autoregressive moving average models, Kalman filters and Back Propagation neural network models are employed respectively for predicting the future value of low-frequency, intermediate-frequency and high-frequency signal layers. An effective algorithm for predicting the stock prices is developed. The price data with the Shandong Gold Group of Shanghai stock exchange market from 28<sup>th</sup> June 2011 to 24<sup>th</sup> June 2012 are used to illustrate the application of the hybrid forecasting system based on multiple scales in predicting stock price. The result shows that time series forecasting can be produced by forecasting on low-frequency, intermediate-frequency and high-frequency signal layers separately. The actual value and the forecasting results are matching exactly. Therefore, the forecasting result of simulation experiments is excellent.

Forecasting is the process of making projections about future performance based on existing historical data. Stock market prediction is regarded as a challenging task in financial time-series forecasting, primarily due to uncertainties involved in the movement of the market. Many factors influence the behavior of the stock market, including both economic and noneconomic. So, stock price time-series data are characterized by nonlinearities, discontinuities, and high-frequency multi-polynomial components and predicting market price movements is quite difficult [

Methods of forecasting stock prices can be classified into two categories: statistical and artificial intelligence (AI) models. The statistical methods include the autoregressive (AR) model [

Different forecasting models can complement each other in capturing different patterns appearing in one-time series, as a combination of forecast outperforms individual forecasting models. In this paper, we construct a hybrid forecasting system on multiple scales. In this system the original data are first decomposed into multiple layers by the wavelet transform, and those layers are divided into low-frequency, intermediate- frequency and high-frequency signal layers. Then autoregressive moving average (ARMA) models are employed for predicting the future value of low-frequency layers; Kalman filters are designed to predict the future value of intermediate-frequency layers; Back Propagation (BP) neural network models are established by the high-frequency signal of each layer for predicting the future value. Finally, those predictions of the future values are restructured and corrected. Furthermore, the empirical data set of Shandong Gold Group of Shanghai Stock Exchange (SSE) closings prices from 28^{th} June 2011 to 24^{th} June 2012 is used to illustrate the application of the forecasting system.

The stock market is made up of short-term, middle-term and long-term dealers etc., Short-term dealers only pay close attention to the short-term price changes in the market, the price fluctuations caused by this behavior has only a short-term memory; by contrast, the price that long-term dealers pay close attention to is the market price changing over a long-term range, the price fluctuations caused by this behavior has a long-term memory. As the dealers’ investment behaviors are under the influence of the outer environment and their chosen investment tactics, in turn generating completely different characteristics in stock price fluctuations, they are dispersed and reflected correspondingly in different time scales [

The multiple scales forecasting system is mainly made up of five parts: scale decomposition, high-frequency data forecast, intermediate frequency data forecast, low frequency data forecast, and data composition. The input data is the real stock price, the output data is the predict stock price, its flow diagram is as shown in

Wavelet analysis is based on wavelet, which is a wave form that tends to be irregular and asymmetric it is capable of separating a signal into shifted and scaled versions of the original (or mother) wavelet. Wavelet function

where

where a and b are real numbers;

(a < 1) factor of the wavelet function

For the time series

where

The wavelet transformation seeks out the level of similarity between the time series data and wavelet function at different scales and translation and generates wavelet coefficient

where m and n are integers that control the wavelet scale/dilation and translation, respectively; a_{0} is a specified fined scale step greater than 1; and b_{0} is the location parameter and must be greater than zero. The most common and simplest choice for parameters are a_{0} = 2 and b_{0} = 1.

This power-of-two logarithmic scaling of the dilations and translations is known as dyadic grid arrangement and is the simplest and most efficient method for practical purposes. For a discrete time series, f(t) when occurs at a different time t (i.e. here integer time steps are used), the discrete wavelet transform becomes:

where ^{m}n. f(t) is a finite time series (t = 0, 1, 2, ∙∙∙, N ? 1), and N is an integer power of 2 (N = 2^{m}); n is the time translation parameter, which changes in the range, 0 < n < 2^{M}m ? 1, where 1 < m < M.

Different families of wavelets whose equalities vary according to several criteria can be used for analyzing sequences of data points. The main criteria are: 1) the speed of convergence to 0 of these functions when the time t or the frequency ω reaches infinity, which quantifies time and frequency localizations, 2) the symmetry, 3) the number of vanishing moments of C and 4) the regularity, which is useful for obtaining nice features, such as smoothness of the reconstructed signal. The most commonly used wavelets are the orthogonal ones. Because the Daubechies wavelets, which are shown in

The low frequency data which is gain by wavelet decomposing change slowly, so that it can be regard as steady time array, and forecasted with ARMA model. The low frequency data which was received using the wavelet decomposing methods were changing slowly, so it was regarded as a steady time array, so ARMA model was adopted to predict the share price.

The ARMA model is usually applied to auto correlated time series data. This model is a great tool for understanding and predicting the future value of a specified time series. ARMA is based on two parts: autoregressive (AR) part and moving average (MA) part [_{t}; t = 0, ±1, ±2, ∙∙∙} is ARMA (p, q) if it is stationary and:

The parameters p and q are called the autoregressive and the moving average orders, respectively. {e_{t}; t = 0, ±1, ±2, ∙∙∙} is a Gaussian white noise sequence.

The Akaike information criterion (AIC) can also be applied to decide the order of ARMA model. AIC is a measure of the goodness of fitting an estimated model. It is based on the concept of entropy. Entropy is a measure of the information lost when a mathematical model is used to describe the actual data. AIC is a powerful tool for model selection. The model with the lowest AIC has the best performance. The AIC is defined by the following equation:

where

The intermediate frequency data which is gain by wavelet decomposing is non-steady, so that it can be regard as steady time array, and forecasted with Kalman filter. The intermediate frequency data that received by using wavelet decomposing method was non-steady, so adopting the Kalman Filter was used to predict future forecasting.

Kalman Filter is introduced and developed by Kalman [

where x(k) is system state vector, A and B are state transition matrix, z(k) is measurement vector, H is output matrix, W(k) is system error and V(k) is measurement error.

The recursive equations of Kalman filter are as following

The high-frequency data that was received by using wavelet decomposing methods was changing more violent with obvious randomness and non-linear characters, thus adopting the BP nerve network was used to predict future forecasts. In general, artificial neural networks (ANNs) possess attributes of learning, generalizing, parallel processing and error endurance. These attributes make the ANNs powerful in solving complex problems. Our study employs a BP neural network which is widely used in business situations.

A back-propagation network consists of at least three layers of units: an input layer, intermediate hidden layer, and an output layer (see

output layer, the forward pattern is then compared with the correct (or observed) output pattern to calculate an error signal. The error signal for each such target output pattern is then back-propagated from the output layer to the input layer in order to appropriately amend or tune the weights in each layer of the network. After a B-P network has learned the correct classification for a set of inputs, it can be tested on a second set of inputs to see how well it classifies untrained patterns. Thus, an important consideration in applying B-P learning is how well the network generalizes. The detailed algorithm can be found elsewhere [

The data for our experiments are Shandong gold group closing prices, collected on the Shanghai Stock Exchange (SSE) market. The total number of values for the Shandong gold group closing prices is 230 trading prices, from 28^{th} June 2011 to 24^{th} June 2012. The first 200 data was used for testing, the last 30 data was the predicting results, and then made a comparison.

There are two criteria for the selection of the mother wavelet. Firstly, the shape and the mathematical expression of the wavelet must be selected correctly so that the physical interpretation of the wavelet coefficients is easy. Secondly, the chosen wavelet must allow a fast computation of the required wavelet coefficients. In this paper, the discrete approximation of Meyer wavelet (D-Meyer) is hence selected as it is a fast algorithm which also supports discreet transformation [

The ARMAS are employed to forecast low-frequency layers, and the forecasting results of A5 and D5 are shown in

forecasting system the predicting results matching the reality data exactly. Therefore, multi-scale forecasting system was very effective.

The stock market data are highly random and non-stationary, and thus contain much noise. The lack of a good forecasting model motivates us to find an improved method of making forecasts called a hybrid forecasting system based on multiple scales. In this method, the original data are decomposed into multiple layers by the wavelet transform. And the multiple layers were divided into low-frequency, intermediate-frequency and high-frequency signal layers. Then autoregressive moving average (ARMA) models are employed for predicting the future value of low-frequency layers; Kalman filters are designed to predicting the future value of intermediate-frequency layers; and Back Propagation (BP) neural network models are established by the high-frequency signal of each layer for predicting the future value. Real data are used to illustrate the application of the hybrid forecasting system based on multiple scales, and the result of simulation experiments is excellent.

Li, Y.Q., Li, X.B. and Wang, H.F. (2016) Based on Multiple Scales Forecasting Stock Price with a Hybrid Forecasting System. American Journal of In- dustrial and Business Management, 6, 1102- 1112. http://dx.doi.org/10.4236/ajibm.2016.611103