Generalized Optimal Wavelet Decomposing Algorithm for Big Financial Market Data

1. Generalized Optimal Wavelet Decomposing Algorithm for Big Financial Market Data , and National Chiao Tung University, Taiwan KEDGE Business School, France

2. Overview • High Frequency Financial Data • Wavelet Based Denoising Approach • Smoothness Oriented Wavelet DenoisingAlgorithm • Experimental Result • Conclusion

3. High Frequency Financial Data • Stylized Facts of High-Frequency Financial Data • Distributional properties • Fatter tails in the unconditional return distributions. (Bollerslev et al. (1992), Marinelli et al. (2000)) • Stock returns are not independently and identically distributed. (Burnecki and Weron (2004), Sun et al. (2007)) • Long-range dependence. (Robinson (2003), Teyssiére and Kirman (2006), Sun et al. (2007))

4. Data Denoising • A classic assumption for data mining is that the data is generated bycertain systematic patterns plus random noise.

5. Trinity of Wavelet Denoising • Wavelet, : • Level of decomposition, : • Threshold,: • i.e., soft or hard thresholding • Donoho et al. (1994, 1995) address details about implementation of the operators and selection of thresholding rules.

6. Wavelet Based Denoising Approach • The observed data X can be decomposed as follows: where is the true trend and is the additive noise sampled at time t. • The general orthogonal wavelet denosing procedure is as follows: , , • We intend to wavelet denoise in order to recover as an estimate of .

7. Smoothness Oriented Wavelet Denoising Algorithm (SOWDA) • In order to evaluate the denoising performance, i.e., to see how close toward , we define denoising properties as follows: • Let be a random variable showing the difference between and and there exist constants and .

8. Wavelet Decomposition • Choosing or designing the right wavelet, thresholding and determining level of decomposition is crucial for a successful wavelet transform of a specific application.

9. Evaluation of Smoothness • Global extrema, : • based on Grubbs test for outliers • Local extrema, : • Let be a function detect local maxima: • Let be a function detect local maxima: • = • Denoising performance: = )

10. Summary of denoising factors of SOWDA • Wavelet, • Level of decomposition, • Thresholding rules • Denoising performance: = )

11. Simulated Data Pattern (1/2) • QQ Plot of Simulated Data versus Standard Normal: Pattern 1 Pattern 2

12. Simulated Data Pattern (2/2) • Original signal:

13. Comparison of denoisingperformances of – The simulated pattern 1 Mean ofRMSEunder Variance of RMSEunder

14. Comparison of denoisingperformances of – The simulated pattern 1 Mean ofRMSEunder Variance of RMSEunder

15. Comparison of denoising performances of – The simulated pattern 2 Mean ofRMSEunder Variance of RMSEunder

16. Comparison of denoising performances of – The simulated pattern 2 Mean ofRMSEunder Variance of RMSEunder

17. Moving Window Design for the Numerical Studies • E is the length of the data used for training (approximation). • V is the length for one-step ahead forecasting (validation). • F is the length for the two-step ahead forecasting.

18. Empirical Experimental Result – goodness of fit of In-Sample (DJIA30 stocks 2010 60-minute data)

19. Empirical Experimental Result – goodness of fit of Validation (DJIA30 stocks 2010 60-minute data)

20. Empirical Experimental Result – goodness of fit of Forecasting (DJIA30 stocks 2010 60-minute data)