A Comparative Study on Methods to Handle Wavelet Edge Distortion

AWSY Wickramasinghe¹ and WMND Basnayake^1,

¹Department of Statistics, University of Colombo, Sri Lanka

Cite this paper

AWSY Wickramasinghe and WMND Basnayake. A Comparative Study on Methods to Handle Wavelet Edge Distortion. American Journal of Applied Mathematics and Statistics. 2022; 10(3):95-102. doi: 10.12691/AJAMS-10-3-4

Abstract

The discrete wavelet transformation (DWT) is of considerable use in the domain of time series analysis. A fundamental problem in DWT is the distortions occurring at the edges while utilizing finite-length series. Insufficient information at boundary regions can lead to questionable accuracies of the transformation at the edges which will thus make a profound effect on further applications. Since the generally used methods to handle edge distortion such as zero padding has their own drawbacks, there are studies that have been done to alleviate the problem using mathematical techniques. With the main objective of finding an evidence-based strategy to reduce the edge effect inherent to DWT using statistical terminologies, this research compares the effect of statistical and machine learning-based denoising and extrapolating techniques in reducing-edge distortion using daily catchment flow series. The most suitable mother wavelet function and the decomposition level for the given series were considered as “biorthogonal 3.1” and level 2 and the edge effect was quantified using MAPE metric. The extrapolating techniques outperformed the denoising methods resulting Vanilla LSTM model with the lowest MAPE values and according to the averaged results taken considering 10 different points of the series, the Vanilla-LSTM and the SARIMA-LSTM hybrid model convincingly alleviate the edge distortion of all coefficients in a more generalized manner.

Keywords

discrete wavelet transformation, neural networks, LSTM, time series

Copyright

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References

[1]	Boashash, B. (1987). Theory, Implementation and Application of Time-Frequency Signal-Analysis using the Wigner-Ville Distribution. Retrieved from https://www.researchgate.net/profile/Boualem_Boashash/publication/265752441 _Theory_Implementation_and_Application_of_Time-Frequency_Signal_-Analysis_using_the_Wigner-Ville_Distribution/links/573f481508ae9f741b321c4b/Theory-Implementation-and-Application-o.
[2]	Gabor, D. (1946). Theory of communication. Part 1: The analysis of information. Journal of the Institution of Electrical Engineers-Part III: Radio and Communication Engineering, pp. 429-441.
[3]	Daubechies, I. (1992). Ten Lectures on Wavelets. Ten Lectures on Wavelets
[4]	Daubechies, I. (1996). Where do wavelets Come From? 84(4).
[5]	Valens, C. (1999). A really friendly guide to wavelets. Ed. Clemens Valens.
[6]	Kaiser, G. (2010). A Friendly Guide to Wavelets. In Springer Science & Business Media. Retrieved from https://books.google.lk/books.
[7]	Varma, T., Chitre, V., & Patil, D. (2012). The haar wavelet and the biorthogonal wavelet trransforms of an image. International Journal of Engineering Research and Applications, 268-291.
[8]	Cohen, A., Daubechies, I., & Vial, P. (1993). Wavelets on the interval and fast wavelet transforms. Applied and Computational Harmonic Analysis, 1(1), 54-81.
[9]	De Queiroz, R. L. (1992). Subband processing of finite length signals without border distortions. ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings, 4, 613-616.
[10]	Su, H., Liu, Q., & Li, J. (2012). Boundary effects reduction in wavelet transform for time-frequency analysis. WSEAS Transactions on Signal Processing, 8(4), 169-179.
[11]	Addison, P. S. (2002). The illustrated Wavelet Transform Handbook. In Biomedical Instrumentation and Technology.
[12]	Su, H., Liu, Q., & Li, J. (2013). Adaptive approach for boundary effects reduction in rotating machine signals analysis. Advances in Mechanical Engineering, 2013.
[13]	Strang, G., & Nguyen, T. (1996). Wavelets and Filter Banks. In SIAM. Retrieved from. (1996).
[14]	Basnayake, W., Attygalle, M. D. T., Liyanage-hansen, L., & Nandalal, K. D. W. (2019). Modified 1D Multilevel DWT Segmented ANN Algorithm to Reduce Edge Distortion. 7(1), 25-31.
[15]	Mallat, S. (1999). Fourier kingdom. A Wavelet Tour of Signal Processing, 20-41.
[16]	Williams, J. R., & Amaratunga, K. (1997). A Discrete Wavelet Transform Without Edge Effects Using Wavelet Extrapolation. Journal of Fourier Analysis and Applications, 3(4), 1-19.
[17]	Polikar, R. (2019). the Wavelet Tutorial Second Edition Part I. 1-67.
[18]	Oliveira, M. O., & Bretas, A. S. (2009). Application of discrete wavelet transform for differential protection of power transformers. 2009 IEEE Bucharest PowerTech: Innovative Ideas Toward the Electrical Grid of the Future.
[19]	Chaovalit, P., Gangopadhyay, A., Karabatis, G., & Chen, Z. (2011). Discrete wavelet transform-based time series analysis and mining. ACM Computing Surveys, 43(2).
[20]	Armstrong, B. J. S., & Collopy, F. (1992). Error Measures For Generalizing About Forecasting Methods: Empirical Comparisons By J. Scott Armstrong and Fred Collopy Reprinted with permission form. International Journal of Forecasting, 8(1), 69-80.
[21]	Chatfield, C. (2003). The Analysis of Time Series: An Introduction. In The Analysis of Time Series: An Introduction.
[22]	Kalekar, P. (2004). Time series forecasting using Holt-Winters exponential smoothing. Kanwal Rekhi School of Information Technology, (04329008), 1-13.
[23]	Chatfield, Chris. (2000). TIME-SERIES FORECASTING.
[24]	Längkvist, M., Karlsson, L., & Loutfi, A. (2014). A review of unsupervised feature learning and deep learning for time-series modeling. Pattern Recognition Letters, 42(1), 11-24.
[25]	Gers, F. A., Schmidhuber, J., & Cummins, F. (1999). Learning to forget: Continual prediction with LSTM. IEE Conference Publication, 2(470), 850-855.
[26]	Casella, G., Fienberg, S., & Olkin, I. (2006). Springer Texts in Statistics. In Design (Vol. 102).
[27]	Wang, L., Zou, H., Su, J., Li, L., & Chaudhry, S. (2013). An ARIMA-ANN Hybrid Model for Time Series Forecasting Li. Systems Research and Behavioral Science, 30(27), 244-259.