擾動輔助經驗模態分解之邊界效應的理論與數值分析

以作者查詢圖書館館藏

、以作者查詢臺灣博碩士

、以作者查詢全國書目

、勘誤回報

、線上人數：101

、訪客IP：3.138.118.215

姓名

鄭詔合(Shao-Ho Cheng) 查詢紙本館藏

畢業系所

機械工程學系

論文名稱

擾動輔助經驗模態分解之邊界效應的理論與數值分析
(A Theoretical and Numerical Study on the Boundary Effects for Disturbance-Assisted EMD Algorithms)

相關論文

★ 二十一點遊戲之正確期望值模型：以遞迴之方式實行隨機訊號處理	★ 基於加速度計的高精度步數演算法
★ 基於加速度計低功耗精確量測步數演算法在穿戴式裝置的實現	★ 經驗模態分解局部性之近場性質的一些證明
★ 擾動輔助EMD演算法在穿戴式嵌入式裝置中MCU的即時運算	★ 總體相位經驗模態分解
★ 利用多孔介質提升甲烷熱裂解產氫之研究

檔案

[Endnote RIS 格式]

[Bibtex 格式]

[相關文章]

[文章引用]

[完整記錄]

[館藏目錄]

[檢視]

[下載]

本電子論文使用權限為同意立即開放。
已達開放權限電子全文僅授權使用者為學術研究之目的，進行個人非營利性質之檢索、閱讀、列印。
請遵守中華民國著作權法之相關規定，切勿任意重製、散佈、改作、轉貼、播送，以免觸法。

摘要(中)

經驗模態分解（empirical mode decomposition, EMD）及擾動輔助經驗模態分解（disturbanceassisted EMD, D-A EMD）為自適性分解非穩態訊號的拆解方法。它們將一個訊號拆解成若干個振盪子訊號加上一個低頻的趨勢線。過程是藉由篩選運算子（sifting operator）遞迴地將一段時間訊號拆解為若干個在時間頻率平面上分離（well separated）的本質模態函數（intrinsic mode functions, IMFs）。由演算法角度視之，它們是自適性（adaptive）和非線性（nonlinear）的方法。相較於短時傅立葉（STFT）與小波（wavelet）更能完整保留波的重要物理參數：相位以及波高，尤其可以很好地萃取後者。因此在已被應用的許多領域中，它們已經被證明在處理非穩態訊號時比傅立葉與小波優越，且其IMF 具有較強之物理或生理意義。
然而，EMD 有時會受到模態混合（mode mixing）的影響，此時可以應用D-A EMD 來解決此問題，但尚無任何一個改進方法可以解決EMD 所遭遇的所有問題，究其根本乃是因為 EMD 是一個理論遠落後於其應用的方法。
在這項研究中，研究了三項EMD 及D-A EMD 的重要應用。（a）現實世界中的訊號在訊號的某些區域有時會受到噪音的汙染，當應用EMD 或D-A EMD 對訊號進行拆解時，噪音所造成的影響會隨著疊代過程擴散到乾淨區域，導致IMF 出現誤差，意即物理意義被扭曲的IMF。（b）在即時運算的應用中，訊號往往被劃分為一系列相互重疊的時間窗口。窗口外的訊號被丟棄後將作為誤差源隨著疊代過程擴散到窗口中。這項研究中會證明上述兩項問題在數學上是相同的，它們都是由邊界效應所引起的。（c）三次木條曲線內插是EMD 及D-A EMD 演算法篩選程序中的重要步驟，它需要在曲線的兩端點代入邊界條件，自然邊界條件（natural BC）及非節點邊界條件（not-a-knot BC）是最常被採用的兩種邊界條件。這兩種邊界常與鏡像邊界、線性外插、修正線性外插等方法結合以進一步降低誤差。但是何種方法最適合EMD 目前尚無定論。
由於EMD 篩選疊代的複雜性，從數學上對EMD 邊界效應的理論分析的研究仍舊有限。先前的研究主要依賴於數值模擬，結果似乎顯示誤差只會侷限在邊界附近的區域。此研究將從 EMD 的IMF 的一次篩選疊代開始，從理論上分析邊界效應，首先證明誤差有可能隨著疊代過程影響到內部區域，有些情況下誤差甚至會隨著與邊界的距離而被放大。接著將證明分解誤差呈指數衰減的充分條件，最後透過數值實驗以分析多次篩選後的IMF 和採用不同邊界處理比較的結果。

摘要(英)

Empirical mode decomposition (EMD) and disturbance-assisted EMD (D-A EMD) algorithms are adaptive and nonlinear methods that decompose a nonstationary signal into several intrinsic mode functions (IMFs) through the sifting process. From an algorithmic point of view, they are adaptive and nonlinear methods. Compared with short-time Fourier transform (STFT) and wavelet, the important physical meaning of the wave: phase and amplitude can be completely preserved, especially
the latter can be well extracted. Therefore, in many fields, EMD and D-A EMD algorithms have been applied, they have been proved to be superior to Fourier and wavelet in processing non-stationary signals, and the IMFs have strong physical or physiological significance.
However, EMD sometimes suffers mode-mixing effect. Therefore, D-A EMD algorithms can be applied to solve the problem. But there is no method that can solve all the problems encountered by EMD. Because EMD is a method that its theory is far behind its application. This study will explore three important applications of EMD. (a) When the D-A EMD algorithms are applied to decompose a signal polluted with noise in some regions, the noise will spread into the clean region and introduce errors in the IMFs. (b) During real-time computation of a D-A EMD algorithm, the signal is partitioned into a series of overlapping time windows. Points outside the window are discarded, they act as an error source and the error will spread into the current window. We will prove that the above two problems are mathematically identical. (c) The cubic spline interpolation which is an important step of EMD’s sifting procedure, it requires two boundary conditions at the two ends of the domain. The natural BC and not-a-knot BC are often applied. These two BCs are combined with mirror BC, linear extrapolation, modified linear extrapolations to further reduce errors. But the jury is still out on witch method is best for EMD.
Due to the complex nature of the sifting iteration, a mathematical theory for the error analysis of the boundary effect is still lacking. Previous studies mainly rely on simulations, and the results show that the error seems to be confined locally near the boundary. Beginning with one sifting iteration, which is the kernel of an EMD, we will theoretically analyze the boundary effects completely based on the sifting process without any approximations, prove that the error will propagate into the interior domain and may be amplified and thus obscure the meaning of an IMF. Then, prove a sufficient condition for the exponential decay of the error. When the error is amplified, we propose a method to resolve this problem. Finally, numerical experiments are conducted to analyze multiple siftings and the result of setting different boundary conditions.

關鍵字(中)

★ 經驗模態分解
★ 即時運算
★ 噪音
★ 邊界效應
★ 誤差

關鍵字(英)

★ EMD
★ real-time
★ online
★ noise
★ boundary effect
★ error

論文目次

摘要 ..................................................... i
ABSTRACT ........................................................ iii
誌謝 ..................................................... v
目錄 ......................................................... vi
圖目錄 ................................................ viii
表目錄................................................... ix
符號說明.................................................. x
一、緒論 ................................................ 1
1-1 研究動機 ............................................. 1
1-2 文獻探討 ............................................. 2
1-3 研究目的 ............................................. 4
1-4 論文章節.............................................. 8
二、經驗模態分解的基礎 ..................................... 9
2-1-1 經驗模態分解（Empirical Mode Decomposition, EMD） ....9
2-1-2 三次木條曲線內插（Cubic Spline Interpolation） ..... 10
2-2 總體經驗模態分解（Ensemble EMD, EEMD） ................11
2-3 均勻相位經驗模態分解（Uniform Phase EMD, UPEMD） ...... 12
三、 EMD 一次篩選的邊界效應 ............................... 14
3-1 ?? 的公式（Formulation of ??） ....................... 14
3-2 ??(?) 的公式（Formulation of ??(?)） ................. 17
3-3 ?? 和 ??? 的性質（Properties of ?? and ???） ......... 17
3-4 三次木條曲線邊界條件的影響 ............................ 24
四、數值實驗與討論 ...................................... 25
4-1 人造訊號以EMD 分解用於即時運算之影響 .................. 25
4-1-1 不同邊界處理的比較 ................................. 26
4-1-2 三次木條曲線邊界條件的比較 .......................... 28
4-2 腦血流速（brain blood flow velocity, BFV） ........... 29
4-2-1 不同邊界處理的比較 ................................. 29
4-2-2 噪音的影響 ........................................ 31
4-3 一天長度變化（length of day variation） .............. 31
4-3-1 以UPEMD 分析一天長度變化訊號 ....................... 32
4-3-2 不同邊界處理的比較 ................................. 33
五、結論與未來展望 ...................................... 35
5-1 結論 ................................................ 35
5-2 未來展望 ............................................ 35
參考文獻 ................................................ 36

參考文獻

［1］ Huang, Norden E., et al. "The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis." Proceedings of the Royal Society of London. Series A: mathematical, physical and engineering sciences 454.1971 (1998): 903-995.
［2］ Li, Ruoyu, and David He. "Rotational machine health monitoring and fault detection using EMD-based acoustic emission feature quantification." IEEE Transactions on Instrumentation and Measurement 61.4 (2012): 990-1001.
［3］ Guo, Runxia, et al. "Remaining useful life prediction for rolling bearings using EMD-RISI-LSTM." IEEE Transactions on Instrumentation and Measurement 70 (2021): 1-12.
［4］ Bashan, Amir, et al. "Comparison of detrending methods for fluctuation analysis." Physica A: Statistical Mechanics and its Applications 387.21 (2008): 5080-5090.
［5］ Hu, Kun, et al. "Altered phase interactions between spontaneous blood pressure and flow fluctuations in type 2 diabetes mellitus: nonlinear assessment of cerebral autoregulation." Physica A: Statistical Mechanics and its Applications 387.10 (2008): 2279-2292.
［6］ Wu, Zhaohua, Norden E. Huang, and Xianyao Chen. "The multi-dimensional ensemble empirical mode decomposition method." Advances in Adaptive Data Analysis 1.03 (2009): 339-372.
［7］ Wu, Zhaohua, and Norden E. Huang. "Ensemble empirical mode decomposition: a noise-assisted data analysis method." Advances in adaptive data analysis 1.01 (2009): 1-41.
［8］ Hu, Xiyuan, Silong Peng, and Wen-Liang Hwang. "EMD revisited: A new understanding of the envelope and resolving the mode-mixing problem in AM-FM signals." IEEE Transactions on Signal Processing 60.3 (2011): 1075-1086.
［9］ Motin, Mohammod Abdul, Chandan Kumar Karmakar, and Marimuthu Palaniswami. "Selection of empirical mode decomposition techniques for extracting breathing rate from PPG." IEEE Signal Processing Letters 26.4 (2019): 592-596.
［10］ Huang, Norden E., Zheng Shen, and Steven R. Long. "A new view of nonlinear water waves: the Hilbert spectrum." Annual review of fluid mechanics 31.1 (1999): 417-457.
［11］ Wang, Yung-Hung, Kun Hu, and Men-Tzung Lo. "Uniform phase empirical mode decomposition: An optimal hybridization of masking signal and ensemble approaches." IEEE Access 6 (2018): 34819-34833.
［12］ Yeh, Jia-Rong, Jiann-Shing Shieh, and Norden E. Huang. "Complementary ensemble empirical mode decomposition: A novel noise enhanced data analysis method." Advances in adaptive data analysis 2.02 (2010): 135-156.
［13］ Torres, María E., et al. "A complete ensemble empirical mode decomposition with adaptive noise." 2011 IEEE international conference on acoustics, speech and signal processing (ICASSP). IEEE, 2011.
［14］ Colominas, Marcelo A., Gastón Schlotthauer, and María E. Torres. "Improved complete ensemble EMD: A suitable tool for biomedical signal processing." Biomedical Signal Processing and Control 14 (2014): 19-29.
［15］ Wang, Yung-Hung, et al. "On the computational complexity of the empirical mode decomposition algorithm." Physica A: Statistical Mechanics and its Applications 400 (2014): 159-167.
［16］ Oberlin, Thomas, Sylvain Meignen, and Valérie Perrier. "An alternative formulation for the empirical mode decomposition." IEEE Transactions on Signal Processing 60.5 (2012): 2236-2246.
［17］ Yang, Hui-Wen, et al. "A minimum arclength method for removing spikes in empirical mode decomposition." IEEE Access 7 (2019): 13284-13294.
［18］ Rilling, Gabriel, Patrick Flandrin, and Paulo Goncalves. "On empirical mode decomposition and its algorithms." IEEE-EURASIP workshop on nonlinear signal and image processing. Vol. 3. No. 3. Grado: IEEER, 2003.
［19］ Fontugne, Romain, Pierre Borgnat, and Patrick Flandrin. "Online empirical mode decomposition." 2017 IEEE international conference on acoustics, speech and signal processing (ICASSP). IEEE, 2017.
［20］ Liang, Sheng-Fu, et al. "A hierarchical approach for online temporal lobe seizure detection in long-term intracranial EEG recordings." Journal of neural engineering 10.4 (2013): 045004.
［21］ Chang, Nai-Fu, et al. "On-line empirical mode decomposition biomedical microprocessor for Hilbert Huang transform." 2011 IEEE biomedical circuits and systems conference (BioCAS). IEEE, 2011.
［22］ Liu, Zhenyu, et al. "Vital Sign Extraction in the Presence of Radar Mutual Interference." IEEE Signal Processing Letters 27 (2020): 1745-1749.
［23］ Faltermeier, Rupert, et al. "Weighted sliding empirical mode decomposition." Advances in Adaptive Data Analysis 3.04 (2011): 509-526.
［24］ Faltermeier, Rupert, et al. "Sliding empirical mode decomposition." The 2010 international joint conference on neural networks (IJCNN). IEEE, 2010.
［25］ Stallone, Angela, Antonio Cicone, and Massimo Materassi. "New insights and best practices for the successful use of Empirical Mode Decomposition, Iterative Filtering and derived algorithms." Scientific reports 10.1 (2020): 1-15.
［26］ Wang, Yung-Hung, Hsu-Wen Vincent Young, and Men-Tzung Lo. "The inner structure of empirical mode decomposition." Physica A: Statistical Mechanics and its Applications 462 (2016): 1003-1017.
［27］ Hu, Kun, et al. "Nonlinear pressure-flow relationship is able to detect asymmetry of brain blood circulation associated with midline shift." Journal of Neurotrauma 26.2 (2009): 227-233.
［28］ Lo, Men-Tzung, et al. "Nonlinear phase interaction between nonstationary signals: a comparison study of methods based on Hilbert-Huang and Fourier transforms." Physical Review E 79.6 (2009): 061924.
［29］ Lambeck, Kurt. The Earth′s variable rotation: geophysical causes and consequences. Cambridge University Press, 2005.
［30］ https://www.iers.org/IERS/EN/DataProducts/EarthOrientationData/eop.html
［31］ Baldwin, M. P., et al. "The quasi‐biennial oscillation." Reviews of Geophysics 39.2 (2001): 179-229.
［32］ Currie, Robert G. "Detection of the 11-yr sunspot cycle signal in Earth rotation." Geophysical Journal International 61.1 (1980): 131-140.3
［33］ Le Mouël, Jean‐Louis, Fernand Lopes, and Vincent Courtillot. "A solar signature in many climate indices." Journal of Geophysical Research: Atmospheres 124.5 (2019): 2600-2619.
［34］ Le Mouël, J. L., et al. "On forcings of length of day changes: From 9-day to 18.6-year oscillations." Physics of the Earth and Planetary Interiors 292 (2019): 1-11.
［35］ Huang, Norden E., et al. "On Holo-Hilbert spectral analysis: a full informational spectral representation for nonlinear and non-stationary data." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 374.2065 (2016): 20150206.
［36］ Shen, Wenbin, and Cunchao Peng. "Detection of different-time-scale signals in the length of day variation based on EEMD analysis technique." Geodesy and Geodynamics 7.3 (2016): 180-186.

指導教授

王淵弘(Yung-Hung Wang)

審核日期

2022-8-30

推文