Financial Forecasting Based on Artificial Neural

2011 6th International Conference on Industrial and Information Systems, ICIIS 2011, Aug. 16-19, 2011, Sri Lanka

Financial Forecasting Based on Artificial Neural Networks: Promising Directions for Modeling W. D. S. Roshan, R. A. R. C. Gopura, Member, IEEE, A. G. B. P. Jayasekara, Member, IEEE. Abstract- Financial forecasting plays a critical role in present economic context where neural networks have become a good alternative technique over traditional methods. Vast ranges of neural models are developed to achieve better accuracy in forecasting. In addition, the ways to find out a good neural architecture is being explored by the research community. In the literature, main problems are figured out within the area of data preparing and neural network design. In this paper, the reasons that affect the performance of the models are discussed based on empirical and mathematical evidence. Finally, this paper presents the directions towards a more suitable neural model for financial forecasting by combining data preprocessing techniques, clustering techniques and support vector machine. Index Terms— Back propagation, Bias variance dilemma, Cover’s theorem, Self-organizing maps, Structural risk minimization, Support vector machine, Wavelet transform.

I.

INTRODUCTION

Forecasting exchange rates plays a major role in day-today financial markets, which becomes a difficult task due to high volatility and complexity. Even though it is easy to trade in financial markets, it is very difficult to make a profit as a result of highly unpredictability of exchange rates. During the past few years ample research has been carried out on understanding inter-relations within financial market data. Recently, there is an increasing trend on adopting artificial neural networks (ANN) and its variants, in order to explore non-linearites of the financial data [1]. Even with vast amounts of available financial data, identification of underlying patterns has become difficult due to economic, political, environmental, and even psychological factors that affect the fluctuation of exchange rates. Under these circumstances data-driven self-adaptive feature of ANN [2] and universal functional approximation capability are highly desirable in financial forecasting. Even though ANN has achieved good performance than the conventional forecasting methods there are certain limitations in designing ANN models [3], [4]. The common limitations that are highlighted in this work are within the area of data preparing and neural network design. In order to overcome those limitations, different existing data preprocessing techniques and neural network architectures are compared to highlight the strengths and weaknesses of them. Then, the directions towards a more suitable model for W. D. S. Roshan is with the Department of Mechanical Engineering, University of Moratuwa, Katubedda, Sri Lanka. (e-mail: [email protected]). R. A. R. C. Gopura is with the Department of Mechanical Engineering, University of Moratuwa, Katubedda, Sri Lanka (corresponding author:+9411-2640472; fax: +94-11-2650622; e-mail: [email protected]). A. G. B. P. Jayasekara is with the Department of Electrical Engineering, University of Moratuwa, Katubedda, Sri Lanka (e-mail: [email protected]).

forecasting are explored. Section II and section III are based on empirical evidences. Input selection and input data pre-processing are discussed in section II under data preparing. Learning algorithm selection, neural architecture and performance measure selection are discussed under neural network design in section III. Mathematical background and empirical evidence for selecting particular choices for data preparing and network design are discussed in section IV. Section V concludes the paper. II.

DATA PREPARING

A. Input selection The selection of inputs for neural networks directly affect to its performance. In the field of financial market forecasting, most of the researchers have used time delay inputs and various technical indicators (moving average, stochastic oscillator etc.) as inputs for the neural network [4]. There are large numbers of technical indicators and they got their own strengths over others. However, use of technical indicators as much as possible does not necessarily lead to better forecasting results; rather it will weaken the forecast due to curse of dimensionality [5]. Hidden relations among inputs can lead to the bias effect of neural network. Therefore, inputs should be selected carefully. Selected inputs should not be correlated with other inputs and should have maximum correlation with itself to achieve better forecast. There is no general method to select input parameters of a neural network. In [6], four technical indicators have been used. In [7] five technical indicators have been used while [8] uses 53 financial indices. Furthermore [9] used 75 technical indicators. However, most researchers focused on finding optimal input variables among the available inputs. Rescaled range analysis [10], [11], [12] and Lyapunov exponent [13] are used as a measure of time series predictability and forecastable time span is selected using those analyses. Trial and error, stepwise-regression [14], auto-regression testing [15], [16], [17] genetic algorithm based selection techniques [18], [19] are used to find most influential input variables by removing redundant ones. About 45% articles that are reviewed in this study have used technical indicators while 35% used time lagged inputs. Details of the selected inputs are presented in Table I. B. Input data preprocessing Data preprocessing is essential for better neural network performance especially for financial time series forecasting. Financial time series has highly noisy, nonlinear and nonstationary characteristics which reduces the overall

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2011 6th International Conference on Industrial and Information Systems, ICIIS 2011, Aug. 16-19, 2011, Sri Lanka

TABLE I INPUT PARAMETERS USED AS INPUTS Indicator

Article

Input

[6]

Four inputs(TAIEX index futures prices and technical indicators) 53 financial indices 75 technical indicators Time delay inputs, indicators Reconstructed phase space variables, normalized daily closing price, 23 technical indicators 5 financial indices Technical indicators Daily opening, closing, highest and lowest price and daily trade volume Current fundamental asset price, strike price and time-to-maturity 27 technical indicators 4 to10 technical indicators

[8] [9] [10] [13] [19] [20] [21] [22] [23] [24] [7] [11] [25] [26] [27] [28] [29] [18] [30] [31] [14] [32] [33] [34] [15] [16] [17] [35] [36] [37] [38] [39] [40] [41] [42] [43] [12]

these models forecasts more accurately than that of back propagation (BP) according to [3], multilayer feed forward network with BP is most widely used network design and also lead to satisfactory results. However, BP algorithm has some undesirable disadvantages such as slow convergence speed, higher likeliness to get trapped in local minima, initial weight determination, and selecting transfer functions. Recurrent neural network (RNN), a variant of feed forward ANN, has effectively used in some models [13], [34]. Recently many researchers have shown that SVM is more suitable and has performed better than other algorithms in [19], [20], [29], [30], [42].

• • • 9 ~ • • 9 ’ • •

10 to 20 technical indicators • Time lagged price values 9

•

Technical Indicators

9

Time lagged inputs

’

Fundamental indicators

~

Other

network performance. Almost all recently published articles have used advanced data preprocessing techniques such as self-organizing maps (SOM) [27], [28], [33], [35], [41] clustering techniques [9], [14] genetic algorithm (GA) based systems [18], [19] principal component analysis (PCA) [43] independent component analysis (ICA) [6] and wavelet transform (WT) [34], [41]. Among those techniques SOM, PCA, ICA, WTs and combinations of these techniques have given the promising results. When selecting a specific method from above promising approaches, PCA has drawback of missing small amount of time series characteristics and better to use with nonlinear extension to deal with nonlinear characteristics of time series. When selecting ICA, nonstationary effect can affect the performance of separation of independent sources. On the other hand, WT is superior alternative for process nonstationary and noisy time series. Its multilevel decomposition and orthogonal data projection capability have been used to remove the inter relations of inputs, noise and nonstationary characteristics of the financial time series. III.

NEURAL NETWORK DESIGN

A. Learning algorithm Learning algorithm is another key factor for neural network performance. When selecting learning algorithm, although [44] has shown that scaled conjugate gradient and Bayesian regularization based models show competitive results and

B. Architecture selection Generally, neural networks are fed with past data and get the future values as outputs. The network should be able to positively identify relationships within inputs. If the network architecture become more complex it will be difficult to recognize patterns and often lead to over fit. Too many hidden neurons and higher number of training epochs can also be caused neural network to over fit. On the other hand, if the network is designed with more simple architecture it will not be able to separate correct patterns from input data and often lead to under fitting. Recently more and more researchers are realizing that using a single neural network may lead to bias effect and combining multiple neural architectures can consistently outperform the single models [3]. There is an improving attention going on for applying hybrid models and ensemble models for better performance [9], [12], [20], [40], [43]. C. Performance measure The network performance is assessed by the performance measures. Mean squared error (MSE) [17], [22], normalized mean squared error (NMSE) [10], [27], root mean square error (RMSE) [24], [34], mean absolute error (MAE), directional symmetry (DS) [6], [14], and mean absolute percentage error (MAPE) [14], [23] are widely used indicators as performance measures. In addition, [35] used profit as the performance measure. Rather than using just one performance measure, combination of performance measures like error, directional and profit [21] gives better understanding of the performance of the model. IV.

MATHEMATICAL BACKGROUND AND EMPIRICAL EVIDENCE

A. Cover’s theorem Determining number of hidden neurons is critical for good accuracy. According to Cover’s theorem [45] “A complex pattern-classification problem cast in a high-dimensional space nonlinearly is more likely to be linearly separable than in low dimensional space”. If the input space is transformed into higher dimensional space according to Cover’s theorem, it is easy to linearly separate those patterns and advisable to choose higher number of hidden neurons than number of input nodes. Higher number of hidden nodes than the input

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2011 6th International Conference on Industrial and Information Systems, ICIIS 2011, Aug. 16-19, 2011, Sri Lanka

nodes has been found in [15], [16], [17], [21], [22], [25] while [26] claimed best neural model as 9-3-1 architecture. Radial basis functions (RBF) [31] and SVM have more principal basis for construction and often create their architecture according to Cover’s theorem. B. Structural risk minimization Generalization error should be kept in minimum possible value for best performance. Therefore, reduction of upper bound of the generalization error is essential for every forecasting model. Vapnik–Chervonenkis (VC) dimension, first proposed by [46], is the representation of maximum number of examples that can be learned by a learning network without an error. VC dimension plays major role in finding upper bound of generalization error. They have defined the guaranteed risk (upper bound of the , , is as in (2), where generalization error) confidence interval and it depends on training sample size N, VC-dimension , and probability . , ,

1 2

, ,

(1)

1

1

, ,

(2)

Here, training error is denoted by and represents weight vector for the fixed training set. There is a value of h, which reduces the upper bound of generalization error to its minimum possible value. According to [47], VC dimension, h of purely linear network is proportional to number of free parameters , purely threshold neural network is proportional to , while Sigmoidal activation function based feed forward neural architectures has VC dimension proportional to W . Going through above-mentioned proofs, upper bound of generalization error can be varied by varying number of hidden neurons. Due to the fact most articles related to feed forward neural networks have used trial and error methods to find the architecture with best generalizing capability. Results have shown significant improvement of the generalization capability [26], [40]. Unlike feed forward neural networks and RBF networks, SVM has interesting property of structural risk minimization. SVM’s VC dimension is bounded from above as shown in (3). can be minimized by properly selecting the optimal margin of separation and it is independent of the dimensionality of the and thus immune to the curse of input space dimensionality. Let denote the diameter of smallest ball containing all the input vectors can be written as, ,

1

(3)

Empirical evidences [19], [29], [30], [38], [42] along with the structural risk minimization property have shown that SVM achieved good generalization performance than feed forward ANN and RBF networks in their benchmark performance comparisons. Significant comparison was not performed

within SVM and recurrent models. However [9] claimed that ensemble recurrent model outperform single SVM. Mathematical evidence for ensemble performance is discussed in next section. C. Neural network generalization and Bias variance dilemma Generalization of neural network is still a challenging problem. For a good generalization, the model should give low training error as well as low testing error. Cross validation is the most widely used generalization method for feed forward neural networks. Input data set is divided into three sets called training, validation and testing. Training set is used to train the neural network. Validation set contains the data that has never seen by the neural network. Validation set is used in training along with training set to calculate out of sample error. If the output of the sample error increases for consecutive iterations, it will be the stopping criteria for network training. However, there is a risk of early stopping the training. Even using optimized VC dimension to reduce the upper bound of generalization error as presented in section IV-B, once it reaches its minimum point, generalization error cannot be reduced further more for a single network. This is due to well-known bias variance dilemma. Average value of the estimation error between the | regression function and the approximating function , evaluated over entire training set can be written as follows. , And

,

(4)

are defined by , ,

|

(5) ,

(6)

Here, bias is the representation of inability to approximate the regression function by given network and variance is the representation of lack of information contained in the training sample. A single neural network achieves low bias result for higher variance and vice versa, unless training set is infinitely large. However, infinitely large training set cause slow convergence due to its size [48]. In a neural architecture design, bias/variance dilemma should be considered seriously. Therefore, both bias and variance should be kept in small values in order to achieve good generalization. According to principal of divide and concur, dividing a computationally complex task into computationally simple tasks and combining its output to take the solution is a better method for solving such task [49] .This method allows reduction in the variance by combining multiple over-trained models, which are trained to zero bias. It will reduce both bias and variance at once for a finite training set, thus remove the bias variance dilemma. Empirical evidence for reducing the bias variance dilemma be found in [14], [27], [28], [33], [35], [41]. Summary of all reviewed articles are presented in Table II.

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2011 6th International Conference on Industrial and Information Systems, ICIIS 2011, Aug. 16-19, 2011, Sri Lanka

TABLE II SUMMARY OF MODELING TECHNIQUES Article [6]

Data preprocessing Normalize, ICA

Network type SVM

Training method Support vector regression(SVR)

[7]

Normalized

Dynamic Ridge Polynomial ANN

Constructive learning algorithm

[8] [9]

Grey Relational Analysis Euclidean distance based partitioning

BP Genetic algorithm (GA)

[10]

Rescaled range analysis

FFANN Recurrent ensemble model FFANN

BP

[11]

FFANN

BP

Ensemble

vary

Probabilistic neural network TDNN,RNN RBF

[15]

Rescaled range analysis+ Normalizing+ sensitive analysis Normalizing+ Hurst exponent Phase Diagrams, Correlation Dimension, Lyapunov exponent Step wise regression analysis +Normalizing+ Dynamic learning algorithm based clustering Auto-regression testing

FFANN

Temporal BP, Extended Kalman filter, Particle-swarm optimization (PSO)+ adaptive recursive least-squares Adaptive BP

[16]

Auto-regression testing

FFANN

[17]

Auto-regression testing

FFANN

[18] [19]

GA based instant selection GA

FFANN SVM

[20]

-

[21]

-

Stochastic Volatility (SV) model with jumps +SVM ANN

[22]

Normalize

FFANN

[23] [24]

Grey-GJR–GARCH Particle swarm optimization + normalize -

FFANN FFANN

[12] [13]

[14]

[26]

[27] [28] [29] [30] [31]

Normalize, Self-organizing feature maps Kohonen self-organizing map Z-score normalization method Normalize Extract indicators with better performance

FFANN

Benchmark models Random walk (RW), SVR

Multilayer perceptron (MLP), Functional Link Neural Network (FLNN), Pi-Sigma Neural Network (PSNN), Ridge Polynomial Neural Network (RPNN) ANN, ANN / Grey model (1, N ) Genetic programming prediction, SVM

Performance measures RMSE, NMSE MAD, DS ,correct down trend (CD) , correct up trend (CP) NMSE, Annualized Return

RMSE Frequency of correct prediction

Auto regression integrated moving average(ARIMA) ARIMA 3-Benchmark models for profit calculation

NMSE, Gradient

FFANN, K-nearest neighbor, Decision tree, Native Bayesian classifier Fisher linear classifier

Average error rate

Type 2 fuzzy time series model, Fuzzy time series model, Fuzzy dual-factor timeseries

MAD,MAPE,DS,CP, CD

Standard BP, LM-based learning, Extended Kalman filter (EKF), BP with optimal learning rates Batch learning, EKF-based learning, Levenberg-Marquardt(LM)based learning, Standard BPNN

NMSE, Directional change statistics (DS) NMSE, DS

Four different networks with modified back propagation for each

MSE Hit ratio Hit ratio

SVR

GA-ANN RW ,ARIMA , linear discriminant analysis, BPANN, SVM, GA-SVM ANN-SV, Garman–Kohlhagen model

GMDH algorithm

FFANN

BP+ Stochastic time effective function BP BP

-

RME, MAPE,DS, Profitability MSE

BP Learning Algorithm with Adaptive Forgetting Factors BP + Adaptive Smoothing+ Momentum Terms GA SVR

ANN models with different indicators as inputs BPANN

SVM

Improved bacterial chemotaxis optimization (IBCO)+BP SVR

FFANN

BP

BSE-30 SENSEX

SVM

Adaptive SVR

SVM RBF

SVR Artificial fish swarm algorithm +K means clustering

Three-layer BP neural network, Regularized RBF neural network BPANN, Case based reasoning RBF optimized by GA ,PSO, ARIMA,ANN,SVM

325

SVR

NMSE, Sign statistics

False alarms

MAE, MAPE

RMSE , MAE, MAPE MSE, RMSE, MAE MSE

NMSE,MAE, DS,WDS Not specified NMSE, DS, MAE Hit ratio Error ratio

2011 6th International Conference on Industrial and Information Systems, ICIIS 2011, Aug. 16-19, 2011, Sri Lanka

[32] [33]

Mean removal Normalize + Selforganizing feature maps Wavelet transform

SVM SVM

SVR SVR + grid search

5 SVM based feature selection methods Single SVR

DS MSE, MAE, MAPE

RNN FFANN FFANN

BP

BP-ANN, conventional ANN optimized by the ABC algorithm, two conventional fuzzy time-series models FFANN, Highly Granular Unsupervised time Lagged Network Adaptive exponential smoothing

RMSE, MAE, MAPE, Theil’s inequality coefficient Profit

[36]

Normalization + Kohonen SOM -

artificial bee colony algorithm(ABC) temporal BP

[37]

-

BP

BPNN, Exponential Smoothing Forecast model

[38]

Normalizing + Interval variation

Exponential Smoothing +FFANN Ensemble SVM

[39]

GARCH(1,1) Volatility

FFANN

Proposed multistage SVM-based nonlinear ensemble model Conjugate gradient based method

Ensemble models trained using simple averaging , Simple MSE, Stacked regression, Variance-based weighting , Artificial neural network BS model with historical volatility ,BS model with GARCH(1,1), SV, SVJ

[40]

Proposed interval sampling method

FFANN

PCA+ Meta-modeling

ARIMA,FNN,SVM, 4 meta-models

[41]

Recurrent Self-Organizing Map + Wavelet transform

-

ANN, SVMs, Generalized autoregressive conditional heteroskedasticity (GARCH)

RMSE

[42]

Chaos-based delay coordinate embedding Normalize

kernel partial least square regressions SVM

SVR

Pure SVR, Chaos-BPNN, BPNN

MSE, RMSE, MAE

GLAR+PCA + FFANNensemble

BP

Generalized auto regression (GLAR),ANN,Hybrid, Equal weights, Minimum error

NMSE, DS

[34] [35]

[43]

V.

CONCLUSION AND SUGGESTIONS

RMSE, Correct tendencies number RMSE, DS RMSE

Average absolute error, average squared errors NRMSE, DS

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Financial forecasting is a challenging problem and still an open research area. This paper presents the empirical and mathematical evidences for working towards most suitable forecasting model. Data preparing and neural architecture design are identified as two main problems. ICA, PCA, WT are superior to available various data preprocessing techniques. However, any comparison between ICA, PCA and WT has not performed in previous approaches. The nonstationary effect can be removed by using WT and clustering input space into multiple clusters. Then the bias variance dilemma can be removed by combining individual networks which are trained for each cluster. SVM is proposed to be the most suitable model based on empirical and mathematical evidence. Structural risk minimization property raises SVM performs better than other algorithms. By going through above mentioned mathematical and empirical evidence, it can be concluded that wavelet decomposition or ICA and PCA with nonlinear and nonstationary extensions can be utilized to preprocess input data. Preprocessed input space clustered with clustering techniques like SOM or its variants, the use of SVM model for each cluster, and combining these models will lead to superior performance and will be a good area for future researches. REFERENCES [1] L. Yu, S. Y. Wang, and K. K. Lai, "Forecasting Foreign Exchange Rates via Artificial Neural Networks," Springer, New York, forthcoming., 2006.

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