Article. Chaotic analysis of predictability versus knowledge discovery techniques: case study of the Polish stock market

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1 Article Chaotic analysis of predictability versus knowledge discovery techniques: case study of the Polish stock market Se-Hak Chun, 1 Kyoung-Jae Kim 2 and Steven H. Kim 3 (1) Hallym University, 1 Okchon-Dong, Chunchon, Kangwon-Do, , Korea shchun@hallym.ac.kr (2) Kyung-Hee Cyber University, 1 Hoegi-Dong, Dongdaemun-Gu, Seoul , Korea (3) Sookmyung Women s University, Chungpa-dong 2 Ka, Yongsan-Gu, Seoul, , Korea Abstract: Increasing evidence over the past decade indicates that financial markets exhibit nonlinear dynamics in the form of chaotic behavior. Traditionally, the prediction of stock markets has relied on statistical methods including multivariate statistical methods, autoregressive integrated moving average models and autoregressive conditional heteroskedasticity models. In recent years, neural networks and other knowledge techniques have been applied extensively to the task of predicting financial variables. This paper examines the relationship between chaotic models and learning techniques. In particular, chaotic analysis indicates the upper limits of predictability for a time series. The learning techniques involve neural networks and case-based reasoning. The chaotic models take the form of R/S analysis to measure persistence in a time series, the correlation dimension to encapsulate system complexity, and Lyapunov exponents to indicate predictive horizons. The concepts are illustrated in the context of a major emerging market, namely the Polish stock market. Keywords: chaotic models, knowledge discovery, backpropagation neural network, case-based reasoning 1. Introduction Increasing evidence over the past decade indicates that financial markets exhibit chaotic behavior. In general, the geometry of a chaotic process in a suitable state space exhibits a dimension of fractional rather than integer value. Consequently, such a structure is said to exhibit a fractal dimension. Traditionally, the prediction of stock markets has relied on statistical methods including multivariate statistical methods, autoregressive integrated moving average models and autoregressive conditional heteroskedasticity models. In recent years, neural networks and other knowledge techniques have been applied extensively to the task of predicting financial variables. This paper examines the relationship between chaotic models and learning techniques. In particular, chaotic analysis indicates the upper limits of predictability for a time series. The learning techniques involve neural networks and case-based reasoning. The chaotic models take the form of R/S analysis to measure persistence in a time series, the correlation dimension to encapsulate system complexity, and Lyapunov exponents to indicate predictive horizons. The concepts are illustrated in the context of a major emerging market, namely the Polish stock market. 2. Methodology The level of chaos in a data stream can be characterized by a number of methods. Two of the most popular parameters are the correlation dimension and the Lyapunov exponent Correlation dimension The correlation dimension is an estimate of the fractal dimension. More specifically, this metric considers the probability that two points chosen at random will lie within a certain distance of each other, and determines how this probability changes as the distance is increased. As an estimate of the fractal dimension, the correlation dimension is a measure of the complexity of a process (Farmer, 1982; Grassberger & Procaccia, 1983). 264 Expert Systems, November 2002, Vol. 19, No. 5

2 2. 2. Lyapunov exponent The Lyapunov exponent characterizes the dynamics of a complex process. Each dimension of the process is associated with a Lyapunov exponent. A positive exponent indi- Figure 2: Predictive procedure through case reasoning using composite neighbors. cates the sensitivity of initial conditions, i.e. how much a forecast diverges based on approximately similar starting conditions. From a slightly different perspective, a Lyapunov exponent indicates the loss of predictive ability as one looks forward in time. The largest positive exponent max determines the maximal rate of stretching. For this reason, the value of max is often used to characterize the predictability of a chaotic process. On the other hand, a negative exponent indicates the degree to which points converge toward one another. For instance, a point attractor is characterized by negative values for each exponent (Peters, 1991, 1994; Pesaran & Potter, 1993; Ott et al., 1994) Hurst exponent The Hurst exponent H is a measure of the bias in random motion. For Brownian motion, the Hurst exponent has value 0.5. For a persistent or trend-reinforcing series, 0.5 H 1.0. On the other hand, 0 H 0.5 for an antipersistent or mean reverting system. The calculation of the Hurst exponent involves a preliminary step known as rescaled (R/S) range analysis. The procedure for computing the Hurst exponent is presented in detail in Figure 1. Figure 1: Calculation of the Hurst exponent through rescaled (R/S) range analysis. In Step 2, a starting value for n of no less than 10 is recommended (Peters, 1994, p. 63) Learning techniques The learning techniques employed in this study relate to neural networks and case-based reasoning. Neural nets have been used extensively over the past decade for predicting financial markets. The application of case reasoning to forecasting, however, is an area with little prior history Neural network The most common neural network methodology employs the backpropagation (BPN) algorithm. This approach involves a layered, feedforward network structure with fully interconnected nodes from one layer to the next. The learning technique involves the back- Expert Systems, November 2002, Vol. 19, No

3 ward propagation of errors to aid in updating internode weights. As a model of biological systems, artificial neural networks have learning capabilities which can be applied to the task of prediction. Unfortunately, BPN models suffer from protracted training periods. Thousands of trials are usually required for satisfactory performance in various tasks. The time and effort required for training have hindered their widespread application to practical domains. Fortunately, certain other learning techniques such as probabilistic neural networks and case-based reasoning offer much swifter response Case-based reasoning and composite neighbors Conventional methods of prediction based on discrete logicusuallyseekthesinglebestinstance,oraweighted combination of a small number of neighbors in the observational space. For instance, the rule of thumb in casebased reasoning is to seek the nearest neighbor to a target case. In an analogous way, certain algorithms in neural networks seek a fixed number of the closest neighbors; this approach is illustrated by the use of self-organizing maps for pattern recognition tasks (Kohonen, 1984). An intelligent learning algorithm should therefore take account of a virtual or composite neighbor whose parameters are defined by some weighted combination of actual neighbors in the case base. In this way, the algorithm can utilize the knowledge reflected in a larger subset of the case base than the immediate collection of proximal neighbors. The procedure for case reasoning using composite neighbors is presented in Figure 2. The key to the composite approach lies in the determination of the most effective set of weights to use in order to construct the virtual neighbor. Learning the optimal set of weights is the primary challenge, and the particular values of the weights may well evolve over time as the experience base expands. A promising way to address this task lies in simulated annealing: the weights for constructing the composite neighbor may be perturbed randomly and the advantageous trends pursued, as in the quest for effective parameters in a neural network algorithm. 3. Application to stock market data The data The case study examined the relationship between chaotic models and learning techniques. The application involved the prediction of the Polish stock price index, for which the input variables were as follows. Stock price index (SI): Polish stock price index with a baseline of 100 for 1994 Total return index (RI): cumulative stock index return, including dividends Dividend yield (DY): dividend yield of the Polish stock market Turnover volume (VO): trading volume of the Polish stock market Price/earnings ratio (PE): price/earnings ratio for the Polish stock price index Interest rate (INT): interest rate for Poland The learning phase consisted of 569 observations from 1 March 1994 to 30 April 1996, while the testing phase consisted of 100 observations from 1 May 1996 to 15 September Model construction An exploratory plot of the Polish stock price index is presented in Figure 3. During the period of the study, the stock market first declined over the course of a year, and then began to recover slowly after March Other exploratory plots for the raw data series are shown in Figures 4 8. Figure 4 depicts the trajectory of the total return index for the stock market. Figure 5 plots the interest rate for Poland, while Figure 6 displays the turnover by volume for the exchange. Figure 7 depicts the price/ earnings ratio for the Polish stock price index. Figure 8 plots the time series of the dividend yield for the bourse. For the BPN architectures, the transfer function took the form of a sigmoidal curve. The learning rate and momentum were both set to 0.1. The training proceeded until there was no perceptible change in the error (at a threshold of 10 8 ). The network architecture was determined in part by the domain variables. Since six variables were selected as input data streams to provide a multivariate forecast for the stock index, the architecture would be 6*h*1 where h denotes the number of neurons in the hidden layer. 1 In this paper, the value of 4 was chosen for h. In constructing the predictive model for Poland, the input variables were first transformed. The modifications involved a logarithmic transformation (L), a differencing operation (D) and a standardization operation (Z) as appropriate. The correlation coefficients between the transformed variables are shown in Table 1 along with their significance levels. The input variables for predicting the Polish stock market were as follows: 1 A common difficulty in applying neural networks lies in overfitting the data. A rule of thumb in the field of statistical modeling specifies that, for a case base of N observations, the degrees of freedom in the model should not exceed N 0.5. For the Polish model using neural networks, the training set contained 569 observations; consequently an upper bound would be about 24 weights, each corresponding to a degree of freedom in the neural network to be trained. The neural network model with a 6*4*1 configuration (corresponding to = 28 weights) was selected and evaluated. 266 Expert Systems, November 2002, Vol. 19, No. 5

4 Figure 3: Polish stock price index (SI). Figure 4: Total return index (RI) for the Polish stock price index. Figure 5: Polish interest rate (INT). Expert Systems, November 2002, Vol. 19, No

5 Figure 6: Turnover by volume (VO) for the Polish exchange. Figure 7: Price/earnings ratio (PE) for the Polish stock price index. Figure 8: Dividend yield (DY) for the Polish stock price index. 268 Expert Systems, November 2002, Vol. 19, No. 5

6 Table 1: Kendall correlation coefficients with their significance levels in parentheses for the transformed input variables for the Polish stock market ZPE (0.000) ZDLRI (0.000) (0.000) ZDLSI (0.000) (0.000) (0.000) ZDLVO (0.000) (0.290) (0.000) (0.000) ZINT (0.000) (0.000) (0.000) (0.000) (0.892) ZDY ZPE ZDLRI ZDLSI ZDLVO The notation LA denotes the logarithm of variable A, while ZB denotes the standardized version of variable B. Moreover the notation DC denotes the difference of variable C over the previous period. For instance, ZDLSI is the standardized form of the difference of the logarithm of variable SI while ZDY is the standardized form of variable DY. ZDLSI: the standardized form of the difference of the logarithm of the stock price index ZDLVO: the standardized form of the difference of the logarithm of the trading volume ZDLRI: the standardized form of the difference of the logarithm of the cumulative stock index return ZDY: the standardized form of the dividend yield ZPE: the standardized form of the price/earnings ratio ZINT: the standardized form of the interest rate 4. Results of study A rescaled (R/S) range analysis for the raw Polish index is shown in Figure 9. An estimate of the Hurst exponent is given by the slope of the dotted line in the figure. The slope for Poland tends to be higher than 0.5; hence the data series is persistent. However, the estimate of the slope varies as a function of the predictive horizon T. For the observations in Figure 9, a series of incremental estimates was obtained for the Hurst exponent. Table 2 presents the estimates of H as a function of the predictive horizon T i. The estimates begin with H(T 1 ) = and then tend to decline as the horizon T i increases. A Hurst exponent larger than 0.5 indicates persistence, while the converse indicates reversion. Hence the deviation from 0.5 is a measure of predictability: G(T i ) H(T i ) 0.5 The predictability G(T i ) is also shown in Table 2. Although the pattern is not clear-cut, the predictability tends to fall as a function of the horizon T i. Table 3 presents the rates of change in the value of R/S as a function of the predictive horizon, as well as corresponding changes in accuracy due to forecasts from the knowledge systems. The change in accuracy was traced by the mean absolute percentage error (MAPE) for each predictive technique, namely BPN and case-based reasoning. As expected, the estimated values of the slope of R/S tended to fall with T i, while the errors in both predictive techniques rose. The correlation coefficients between the estimated Hurst exponent and the forecast error are presented in Table 4. The correlation with BPN is weakly significant, while that with case-based reasoning is indeterminate. The next table lists the correlations between the predictability G and the forecast errors. As anticipated, a rise Table 2: The error MAPE i and incremental Hurst exponent H(T i ) asafunctionofthehorizont i for the ith observation i T i log T i log(r/s) H(T i ) G(T i ) Figure 9: R/S analysis to compute the Hurst exponent versus random walk (H random = 0.5) as a function of predictive horizon T The incremental Hurst exponent is calculated as H(T i ) [log(r/s) i log(r/s) i1 ]/(log T i log T i1 ). Expert Systems, November 2002, Vol. 19, No

7 Table 3: The slopes of MAPE i and of log(r/s) i1 as a function of the horizon T i i T i log(r/s) i slog(r/s) i MAPEFBPN i smapefbpn i MAPEFCBR i smapefcbr i N/A N/A N/A The slope of MAPE i was calculated as smape(t i ) (MAPE i MAPE i1 )/(T i T i1 ), while that of log(r/s) i was slog(r/s) i [log(r/s) i log(r/s) i1 ]/(T i T i1 ). CBR, case-based reasoning. Table 4: Correlation coefficients between the estimated Hurst exponent and forecast error (MAPE) for each technique Method p value BPN CBR Table 5: Correlation coefficients between the predictability G and forecast error (MAPE) for each technique Method p value Figure 10: Error as a function of the horizon T for each learning method. BPN CBR Here, G H 0.5. Table 6: Correlation coefficients between the slope of log(r/s) and the slope of the forecast error (MAPE) for each technique Method p value BPN CBR in G was correlated with a fall in error for both BPN and case-based reasoning. However, the pattern is not statistically significant. Table 6 exhibits the correlation coefficients between the incremental slope of log(r/s) and the corresponding fore- cast errors at various horizons T i. For BPN, the fall in error with a rise in log(r/s) is statistically significant, but the pattern for case-based resasoning is inconclusive. Figure 10 compares the performance of the two learning techniques as a function of the horizon T. As expected, the errors rose with an increase in T. The neural and case reasoning methods performed approximately equally up to a period of T = 16, after which the errors due to BPN rose faster than those in case-based reasoning. The shortest horizon for which the differential performance appears to be substantive was T = 20 days. A pairwise t test shown in Table 7 indicates that the difference was significant at level p Table 8 presents a comparative set of nonlinear characteristics for the stock indexes of several emerging markets. The Hurst exponent is highest for the Philippines, but its correlation dimension and Lyapunov exponent are only moderate. With the highest correlation dimension of 5.358, Singapore s stock market appears to be the most complex. The market exhibits weakly reverting behavior, as indicated 270 Expert Systems, November 2002, Vol. 19, No. 5

8 Table 7: Pairwise t tests of the predictive models for predictive horizon T = 20 Variable Number of pairs Correlation Two-tailed Mean SD SE of mean significance NN CBR Paired differences Mean SD SE of mean t value Degrees of freedom Two-tailed significance % CI (2.831, 3.808) The comparison is based on the MAPE of the residuals. Table 8: Hurst exponents, correlation dimension and Lyapunov exponents for each stock market a larger collection of learning techniques represents a rich area for future investigation. H Correlation Lyapunov dimension exponent Singapore Malaysia Philippines China Hong Kong Poland Korea by a Hurst exponent of Moreover, it has the highest Lyapunov exponent, indicating a rapid loss of information over time. Poland has a relatively complex market, with a correlation dimension of However, it has the second largest Hurst exponent of , thereby indicating a moderate level of predictability. Moreover, Poland has the smallest Lyapunov exponent: the inverse value of 1/0.053 = indicates that the predictive horizon for Poland is about 19 days. 5. Concluding remarks and future work In this paper we investigated the predictability of a complex process based on chaotic analysis and knowledge based tools. Case-based reasoning outperformed neural nets, but the performance of both techniques declined with an increase in predictive horizon, as expected from informal considerations as well as chaotic analyses. The literature contains numerous varieties of neural networks as well as other learning techniques such as induction and genetic algorithms. The systematic evaluation of Acknowledgements This research was supported by the Hallym Academy of Science at Hallym University, Korea, References Famer, J.D. (1982) Chaotic attractors of an infinite-dimensional dynamical system, Physica, 4D (3), Grassberger, P. and I. Procaccia (1983) Measuring the strangeness of strange attractors, Physica, 9D, Kohonen, T. (1984) Self-Organization and Associative Memory, New York: Springer. Ott, E., T. Sauer and J.A. Yorke (1994) Coping with Chaos, New York: Wiley. Pesaran, M.H. and S.M. Potter (1993). Nonlinear Dynamics, Chaos and Econometrics, New York: Wiley. Peters, E.E. (1991) Chaos and Order in the Capital Markets, New York: Wiley. Peters, E.E. (1994) Fractal Market Analysis, New York: Wiley. The authors Se-Hak Chun Se-Hak Chun is a Professor at Hallym University. His research interests include financial forecasting, complex theory, data mining and electronic commerce. He received a BS degree in economics at Korea University (1994), an ME degree in management engineering at the Korea Advanced Institute of Science and Technology (1997) and a PhD degree in management engineering at the Korea Advanced Institute of Science and Technology (2002). Expert Systems, November 2002, Vol. 19, No

9 Kyoung-Jae Kim Kyoung-Jae Kim is a full-time lecturer in the Department of Management Information Systems in Kyung Hee Cyber University. He received his MS and PhD degrees in management information systems from the Graduate School of Management of the Korea Advanced Institute of Science and Technology and his BA degree from Chung-Ang University. He has published in Applied Intelligence, Expert Systems, Expert Systems with Applications, Intelligent Data Analysis, etc. His research interests include data mining, knowledge management and intelligent agents. Steven Kim Steven Kim is Distinguished Professor at Sookmyung Women s University. His research interests include knowl- edge discovery, online multimedia and technological innovation. From 1995 to February 2001, he taught at the Graduate School of Management at the Korea Advanced Institute of Science and Technology. Between 1993 and 1995, he was President of Lightwell Inc., a company dedicated to high performance devices for intelligent automation and photonic computing. Prior to a one-year position as a Visiting Fellow at Cornell University, he served as an Assistant Professor of Mechanical Engineering at Massachusetts Institute of Technology from 1985 to In 1989 he received the Presidential Young Investigator award from the US National Science Foundation. He earned a BS degree in mechanical engineering at Columbia University (1977) and a PhD in management and mechanical engineering at MIT (1985). In between, he received master s degrees in mechanical engineering (1978), operations research (1980), and management (1983) at the two universities. Dr Kim has written over 100 technical articles. 272 Expert Systems, November 2002, Vol. 19, No. 5