基於整體經驗模式分解的集群分析

全文

(1)國立臺灣師範大學數學系碩士班碩士論文指導教授：蔡碧紋博士. Ensemble Empirical Mode Decomposition with Clustering Analysis. 研究生：魏韶寬中華民國 101 年 7 月.

(2) 致謝時光飛逝，研究所生活也即將結束，看似簡單的生活，若沒有順緣要過得順利並不容易。因此，對於這些順緣我很感動也很感恩。我感謝我的指導教授：蔡碧紋博士與郭志禹博士。由於他們的引導，讓我在訊號分析方面得到很大的成長，並且擴展我的視野。重要的是，兩位老師專注的研究精神與不疾不徐的處事態度讓我相當感動，讓我清楚檢視自己焦慮與慌張的行為模式並以他們為典範學習。我感謝口試委員：吳順德博士與張玉媚博士，感謝這兩位老師能來參加我的口試並給予建議，使我對於黃鍔院士所提出的 EMD 演算法有更進一步的認識，並且使我反思我製造新訊號之物理意義。在所有老師的指導之下，我的論文與學習得以延續，使我更進步。另外，我感謝家人與福智的老師，有了他們，我可以專心學習與寫論文，不受到諸多的干擾。這一段的生活將會是我無限生命中一個很重要的亮點，我很幸運在身為研究生的兩年能有一個美好記憶與經驗，體會做研究的感覺是單純且喜悅的。我希望能把這樣的喜悅與感動分享出去，期盼每個人都能得到這樣的快樂。. i.

(3) 摘要總體經驗模態分解法是一個分析訊號的方法，利用其獨特的分解方式將一個訊號分解成一組本質模態函數。然而，這個方法有多重模態函數的問題，也就是同一頻率的訊號被分解成兩個本質模態函數。將這兩個或以上函數疊加成一個單一模態的訊號來解決此問題，但是目前沒有一個一般性的準則以合併這些多重模態函數。本論文利用分群分析的方法提供一套準則以合併具有多重模態問題的本質模態函數。利用此分群方法合併所得的一組被分群的本質模態函數更為扼要且較具有實際上的物理意義。本論文所提供的方法被運用在兩個模擬的訊號、一個風力渦輪機所產生的聲音訊號以及一個由在臺灣草嶺地區一個觀測站所記錄到的地震訊號。特別的是，該地震訊號同時記錄主要的地震訊號與山崩所造成的震動訊號。這些運用的結果皆表示本論文提供的方法可以針對多重模態函數的問題提供一個決定性的改善，並且以樹狀圖的方式描述訊號的特徵。關鍵字：總體經驗模態分解法、本質模態函數、多重模態、集群分析. ii.

(4) Abstract Ensemble Empirical Mode Decomposition (EEMD) is an adaptive time-frequency data analysis method. Time series or signals can be decomposed into a collection of intrinsic mode functions (IMFs). Nevertheless, there appears a multi-mode problem where signals with a similar time scale are decomposed into different IMFs. A possible solution to this problem is to combine the multi-modes into a proper single mode, but there is no general rule on how to combine IMFs in the literature. In this paper, we propose to modify EEMD algorithm using the statistical clustering analysis and to provide a framework to combine the IMFs into a condensed set of clustered intrinsic mode functions (CIMFs). The method is applied to two artificially synthesized signals, wind turbine signal at Chunan Miaoli, and a seismic signal during the earthquake at Chi-Chi in 1999. Especially, this seismic signal contains not only the main seismic information but also the seismic motion from a landslide in Tsaoling area. The present method can separate the two signal from different sources correctly, and these applications of other examples demonstrate that, the present method offers great improvement over EEMD for extracting useful information. Keywords: Ensemble Empirical Mode Decomposition, intrinsic mode function, multi-mode, clustering analysis. iii.

(5) Contents 致謝. i. 中文摘要. ii. Abstract. iii. Contents. iv. List of Figures. vi. 1. Introduction. 1. 2. Ensemble Empirical Mode Decomposition (EEMD) 2.1 EMD algorithm . . . . . . . . . . . . . . . . . . 2.1.1 Mode mixing problem of EMD . . . . . 2.2 EEMD algorithm . . . . . . . . . . . . . . . . . 2.3 Multi-mode problem of EEMD . . . . . . . . . . 2.4 Significant test of IMFs . . . . . . . . . . . . . .. . . . . .. 3 3 4 7 8 13. 3. Method: with clustering analysis 3.1 Clustering analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The flow chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15 15 17. 4. Examples 4.1 The simulated signal: sin(t) and the intermittent signal . . . . . . . . 4.2 The simulated signal: sin(t) + 0.1 sin(10t) and the intermittent signal 4.3 The practical signal: a voice from a wind turbine at Chunan Miaoli . .. 19 19 23 26. iv. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . ..

(6) 4.4 5. The practical signal: an seismic signal from station CHY080 at Chi-Chi 32. Conclusion and discussions 5.1 Conclusions . . . . . . . . . . . . . . . . . 5.2 The white noise in EEMD . . . . . . . . . 5.3 Choice of linkage in hierarchical clustering 5.4 The number of clusters . . . . . . . . . . .. Bibliography. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 35 35 35 36 36 37. v.

(7) List of Figures 2.1 2.2 2.3 2.4 2.5 2.6 2.7 3.1 3.2 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9. The flow chart of EMD . . . . . . . . . . . . . . . . . . . . . . . . . The type of signal A . . . . . . . . . . . . . . . . . . . . . . . . . . Results of the signal A which is decomposed by EMD. The last IMF means the residue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results of that signal A is decomposed by EEMD with adding white noise with 10% of aptitude of signal A. The last IMF means the residue. Part results of signal A with adding white noise with 20% of aptitude of signal A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part results of signal A with adding white noise with 30% of aptitude of signal A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The significance test of IMFs of signal A with 10% white noise level. The energy of IMF 8 and IMF 9 are so small that they can be ignored. Dendrogram with complete linkages of IMFs of signal A with 10% white noise level . . . . . . . . . . . . . . . . . . . . . . . . . . . . The flow chart of the proposed method . . . . . . . . . . . . . . . . . The dendrogram with complete linkages of IMFs of signal A with 10% white noise level when the number of clusters is 4. . . . . . . . . . . CIMFs with 10% white noise level . . . . . . . . . . . . . . . . . . . The dendrogram with complete linkages of IMFs of signal A with 20% white noise level when the number of clusters is 5. . . . . . . . . . . CIMFs with 20% white noise level . . . . . . . . . . . . . . . . . . . The dendrogram with complete linkages of IMFs of signal A with 30% white noise level when the number of clusters is 5. . . . . . . . . . . CIMFs with 30% white noise level . . . . . . . . . . . . . . . . . . . sin(t) + 0.1 sin(10t) and the intermittent signal . . . . . . . . . . . . Results of the signal: sin(t)+0.1 sin(10t) and intermittent signal which is decomposed by EEMD. The last means the residue. . . . . . . . . . The figure is the dendrogram with complete linkages for IMFs of sin(t)+ 0.1 sin(10t) and the intermittent signal when the number of clusters is 5. vi. 5 6 7 10 11 12 14 17 18 20 20 21 21 22 22 23 24 25.

(8) 4.10 4.11 4.12 4.13 4.14 4.15 4.16. 4.17 4.18 4.19 4.20. CIMFs of sin(t) + 0.1 sin(10t) and the intermittent signal . . . . . . . The wind turbine and the voice signal . . . . . . . . . . . . . . . . . These figures are IMF 1 ∼ 5 of the voice signal. . . . . . . . . . . . . These figures are IMF 6 ∼ 15 of the voice signal. The last IMF means the trend. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The figure is the dendrogram with complete linkages of the wind turbine when the number of clusters is 8. . . . . . . . . . . . . . . . . . CIMFs with complete linkage of the wind turbine . . . . . . . . . . . The location of CHY080 in Tsaoling area. The NE-SW profile is defined through the gravity center of the slid mass and is in parallel to the slid direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The resampled and filtered signal from the east-west direction of station CHY080 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results of the east-west earthquake signal of CHY080 which is decomposed by EEMD. The last means the trend. . . . . . . . . . . . . . . . The figure is the dendrogram with complete linkage for IMFs of the east-west signal when the number of clusters is 5. . . . . . . . . . . . CIMFs of the east-west signal . . . . . . . . . . . . . . . . . . . . .. vii. 25 26 27 28 29 31. 32 32 33 34 34.

(9) Chapter 1 Introduction Empirical Mode Decomposition (EMD) [10] is an adaptive time-frequency data analysis method which decomposes a signal into a collection of intrinsic mode functions (IMFs), based on the local characteristic time scale of the signal. It is applied in many fields such as seismic signals, voices, ozone concentration data, and the fault diagnosis of roller bearings ([8], [5], [3], [19]). One of the major drawbacks of the EMD is the frequent appearance of mode mixing, where a single IMF including oscillations of significantly different scales. In 1999, Huang [9] tried to overcome the mode mixing with a criterion based on the period length. The criterion is set as the upper limit of the period that can be included in any given IMF component. However, this criterion will only be effective when we know some characteristics of the signal, and ut is non-adaptive. To alleviate the mode mixing problem, Wu and Huang [16] proposed a noise-assisted data analysis method called ensemble empirical mode decomposition (EEMD) consisting of the original signal and adding white noise of finite amplitude. The EEMD defines components as the means of an ensemble of trials. For extracting the trend from seasonal time series, the results of the EEMD are better than of the EMD as shown in [13]. The EEMD is applied in rotor fault diagnosis of rotating machinery [12], and arrhythmia ECG noise reduction [2]. The EEMD is also extended such as post-processing of EEMD [16], noise-modulated EMD (NEMD) [14] and complementary EEMD [18]. In addition, as discussed in [7], the EEMD can separate signals of different scales without special sampling requirement. However, as discussed by Wu and Huang [16], EEMD has the multi-mode problems where a signal with similar scale appearing in different IMFs. In practice, the EEMD is run by a matlab program where. 1.

(10) the number of IMFs is determined by the length of a signal. Thus, the result may have the over-complete problem where some components are unable to provide more information after all significant meanings have been extracted from the first few IMFs. As discussed in [4], handling of multi-mode problem is still remained although these methods which Huang [16] provided have been carried out. Huang [16] provided some recommends for the multi-mode problem and the over-complete problem is to combine the multi-mode IMFs into a proper single mode. However, there is no general rule on how to combine IMFs in the literature [4]. In this manuscript, we propose using statistical clustering analysis as a tool to combine IMFs so that the multi-mode problem can be avoided. We perform hierarchical clustering analysis and use the result to describe the correlation coefficients between IMFs. When the number of clusters is given, the dendrogram provides us with a method to combine the components obtained from EEMD into a finite number of clustered intrinsic mode functions (CIMFs) having more physical meanings. We will demonstrate with our method some examples which show that the CIMFs can extract more useful information from our data. In section 4, we demonstrate that the proposed method offers great improvement over EEMD for extracting useful information from this signal. Especially, for analysing of the seismic signal which contains not only the main seismic information but also the seismic motion from a landslide in Tsaoling area, the proposed method shows the landslide information and indicates its uniqueness with a dendrogram. The structure of this manuscript is as the following. In section 2, we introduce the sifting process of EMD and EEMD algorithm and discuss the multi-mode problem and over-complete problem. In section 3, we present the background of hierarchical clustering analysis, and propose using clustering analysis to combine IMFs from into a finite set of CIMFs. In section 4, some examples of the application of the proposed method are presented. Examples are two artificially synthesized signals, wind turbine signal at Chunan Miaoli, and a seismic signal during the earthquake at Chi-Chi in 1999. We show that the CIMFs can avoid the appearance of the multi-mode problem and the result tend to have more physical meanings. Some discussions and conclusions are given in section 5.. 2.

(11) Chapter 2 Ensemble Empirical Mode Decomposition (EEMD) In this section, we firstly introduce the sifting process of EMD [10] and use a simulated signal to demonstrate the mode mixing problem of EMD. Secondly, we introduce the EEMD algorithm and use the same signal to demonstrate that EEMD algorithm improves the mode mixing problem but may cause the multi-mode problem where a signal with similar scale appearing in different IMFs. Third, the over-complete problem which occurs frequently in practice is also presented. Finally, we introduce the significant test of IMF which was suggested by [15]. The test is used to test whether the property of a IMF is close to the property of a white noise.. 2.1. EMD algorithm. The EMD decomposes a signal x(t) into intrinsic mode functions (IMFs) ci (t), as follows: n ∑ x(t) = ci (t) + rn (t), (2.1) i=1. where rn (t) is the residue of the signal after extracting the nth IMF. An IMF is a time series that represents a simple oscillatory mode and has two properties: 1. the number of extrema and the number of zero-crossings must either equal or. 3.

(12) differ at most by one in the whole data set. 2. the mean value of the envelope defined by local maxima and is zero. In practice, the procedure for extracting an IMF is implemented throught a sifting process that works with extrema. Suppose rj−1 (t) is the remainder of x(t) after j − 1 IMFs are extracted. From the data rj−1 (t), the following procedure is used: 1. Identify all local maxima and the minima of x(t) and use cubic spline lines to form the upper and low envelops. 2. Compute the mean m(t) of the upper and lower envelops and define h(t) = rj−1 (t) − m(t) 3. Treat h(t) as the data and repeat 1 and 2 a number of time until h(t) meets the stopping criterion N ∑ ∥h(k−1) (t) − h(k) (t)∥2 < α, (2.2) h2(k−1)(t) t=0 where N is the length of the data and h(k) (t) is the kth iteration of h(t), and α is a given value. 4. The final h(t) is designated as cj (t), the IMFj . The whole sifting procedure can be repeated to all subsequent and finally stops when the residue rn become a constant, a monotonic function, or a function that contains only one extrema from which no IMF can be extracted. The procedure is also presented with a flow chart [1], as Fig. 2.1.. 2.1.1 Mode mixing problem of EMD One disadvantage of the EMD method is that a single IMF including oscillations of significantly different scales is called "mode mixing" in [16]. When the mode mixing problem occurs, an IMF can cease to have physical meaning by itself. To illustrate this problem, we consider a simulated signal which has a fundamen1 Hz and the cosine wave with 0.1 tal part as a sine wave with unit amplitude and 2π. 4.

(13) Figure 2.1: The flow chart of EMD 5.

(14) 10 amplitude and Hz riding on the six middle crests of the sine wave as Fig. 2.2. For π convenience, we call the simulated signal signal A. The function of signal A is   x(t) =. (2i − 1) π (2i − 1) π π− , π + ], i = 1, 2, ..., 6 , 0 ≤ t ≤ 6π 2 5 2 5 otherwise. (2.3). sint + 0.1cos10t t ∈ [.  sint. signal A 1.5 1 0.5 0 −0.5 −1 −1.5 0. 2. 4. 6. 8. 10. 12. 14. 16. 18. 20. Figure 2.2: The type of signal A Here, we perform EEMD/EMD algrithm by a matlab program which Haung provides (http://rcada.ncu.edu.tw/), but his program does not use the stopping criterion mentions , but it fixes the reapeat time at 10. As mentioned by Wu[17], the number of IMFs of a data set is close to log2 N where N is the number of the length of x(t). Thus, the number of IMFs is determined by rounding log2 (N ) and subtracting 1 from it in this program. The result of that signal A is decomposed by EMD in Fig. 2.3. IMF1 contains the cosine wave, but it also contains another wave which is very similar with the fundamental sine wave. The energies of IMF3 ∼9 are less than 10−4 . The residue in the EMD is sometimes regarded as the trend of x(t).. 6.

(15) IMF 1. IMF 6. −6. 0.2. 1. 0. x 10. 0. −0.2 0. 2. 4. 6. 8. 10 IMF 2. 12. 14. 16. 18. 20. 4. 6. 8. 10 IMF 3. 12. 14. 16. 18. 20. 2 0. −1 0 2 −6 x 10 0. 4. 6. 8. 10 IMF 7. 12. 14. 16. 18. 20. 4. 6. 8. 10 IMF 8. 12. 14. 16. 18. 20. 4. 6. 8. 10 IMF 9. 12. 14. 16. 18. 20. 4. 6. 8. 10 IMF 10. 12. 14. 16. 18. 20. 4. 6. 8. 10. 12. 14. 16. 18. 20. −0.5. −2 0 2 −4 x 10 2 0. −1 0 2 −6 x 10 0 −0.5. −2 0 2 −6 x 10 5. 4. 6. 8. 10 IMF 4. 12. 14. 16. 18. 20. 0. −1 0 2 −6 x 10 0 −0.5. −5 0 2 −6 x 10 0. 4. 6. 8. 10 IMF 5. 12. 14. 16. 18. 20. 4. 6. 8. 10. 12. 14. 16. 18. 20. −0.5. −1 0 2 −3 x 10 −1 −2. −1 0. 2. −3 0. 2. Figure 2.3: Results of the signal A which is decomposed by EMD. The last IMF means the residue.. 2.2. EEMD algorithm. To improve the mode mixing problem, the Ensemble Empirical Mode Decomposition (EEMD) is proposed by Wu and Huang [16]. EEMD consists of sifting an ensemble of noise-added signal and repeating several times. The procedure of the EEMD algorithm is following: 1. Add a random white noise series wi (t) with an amplitude of low proportion standard deviation of the targeted data x(t) to x∗ (t). Let x∗i (t) = x(t) + wi (t) ∑ 2. Decompose x∗ (t) into IMFs using the regular EMD. i.e. x∗i (t) = nj=1 cij (t) + rin (t) where cij (t) is the IMF j and rin (t) is the residue after extracting the nth IMF in the ith iteration 3. Repeat step 1 and 2 for a predetermined number of times, say k, but with different random white noise series generated for each time. 4. Obtain the (ensemble) means of corresponding IMFs and the residue as the final 1 ∑k 1 ∑k ∗ result. i.e c∗j = rkj i=1 ckj and rn = k k i=1 Note that the components from EEMD don't satisfy the definition of an IMF. For convenience, the resulting ensemble means are often still called IMFs. Also, the num-. 7.

(16) ber of IMFs is determined by the length of the data. Thus, it cause that the latter parts of the IMFs are unable to provide more meanings. They are presented as numerical results. We call this "over-complete problem", and this should not happen theoretically.. 2.3. Multi-mode problem of EEMD. We decompose signal A using EEMD with the added white noise in each ensemble member having a standard deviation of 0.1 and repeating 30 times. The IMFs are shown in Fig. 2.4. We find that IMF3 and IMF4 not only have the same intermittent signal but also have a half aptitude of cosine wave. This is the multi-mode problem where a signal with similar scale appearing in different IMFs. We believe this result may be due to that the dyadic filter bank [6] property of EMD which shows some overlap in scales. Signals having a scale located in the overlapping region would have a finite probability appearing in two different modes. The dyadic filter bank about EMD is discussed in [15]. Four alternative methods to improve this problem have been suggested by Wu and Huang [16]. The first is combining the two IMFs to a single component through orthogonality check. When two IMF components become grossly unorthogonal, combining the two to form a single component should be considered. However, only pairwise orthogonality checking is available. We will always combine two " most closed" IMFs to get the component. The second alternative is to tune the noise level and use more trials to reduce the root mean squared deviation in each IMF. We try to improve the multi-mode problem according to this approach. We decompose signal A with 10%, 20% and 30% levels of white-noise added and repeating 30 times. Fig. 2.4 gives the result when 10 % of a standard deviation is added, and it has been discussed as before. Fig. 2.5 and 2.6 give parts of results when 20% and 30% of a standard deviation are added respectively. In Fig. 2.5, the frequency and the shape of IMF 6 are almost identical with the frequency and the shape of IMF 7. The frequency of the intermittent signal of IMF 3 are the same with the frequency of intermittent signal of IMF 4. Fig. 2.6 also presents the same re-. 8.

(17) sults. Because of that EEMD is constrained by the dyadic filter bank property of EMD. Although the noise level is changed, the multi mode problem can not be improved. The third alternative is sifting a low but fixed number of times for obtaining each IMFs. This is used in the matlab program, and the fixed number is 10. The four method is to use rigorous check of each component against the definition,and divide the outcome into different groups according to the total number of IMFs generated. However, this method does not provide a criterion to divide them into groups.. 9.

(18) IMF 1 0.05 0 −0.05 0. 2. 4. 6. 8. 10 IMF 2. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10 IMF 3. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10 IMF 4. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10 IMF 5. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10. 12. 14. 16. 18. 20. 0.05 0 −0.05 0 0.05 0 −0.05 0 0.1 0 −0.1 0 0.5 0 −0.5 0. IMF 6 1 0 −1 0. 2. 4. 6. 8. 10 IMF 7. 12. 14. 16. 18. 20. 4. 6. 8. 10 IMF 8. 12. 14. 16. 18. 20. 4. 6. 8. 10 IMF 9. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10 IMF 10. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10. 12. 14. 16. 18. 20. 0.01 0 −0.01 0 2 −4 x 10 1 0.5 0 0 2 −5 x 10 5 0 0 0 −0.05 −0.1 0. Figure 2.4: Results of that signal A is decomposed by EEMD with adding white noise with 10% of aptitude of signal A. The last IMF means the residue.. 10.

(19) IMF 3 0.1. 0. −0.1 0. 2. 4. 6. 8. 10. 12. 14. 16. 18. 20. 12. 14. 16. 18. 20. 12. 14. 16. 18. 20. 12. 14. 16. 18. 20. 12. 14. 16. 18. 20. 12. 14. 16. 18. 20. IMF 4 0.2. 0. −0.2 0. 2. 4. 6. 8. 10 IMF 5. 0.05. 0. −0.05 0. 2. 4. 6. 8. 10 IMF 6. 1. 0. −1 0. 2. 4. 6. 8. 10 IMF 7. 0.2. 0. −0.2 0. 2. 4. 6. 8. 10 IMF 8. 0.05. 0. −0.05 0. 2. 4. 6. 8. 10. Figure 2.5: Part results of signal A with adding white noise with 20% of aptitude of signal A. 11.

(20) IMF 3 0.1. 0. −0.1 0. 2. 4. 6. 8. 10. 12. 14. 16. 18. 20. 12. 14. 16. 18. 20. 12. 14. 16. 18. 20. 12. 14. 16. 18. 20. 12. 14. 16. 18. 20. 12. 14. 16. 18. 20. IMF 4 0.1. 0. −0.1 0. 2. 4. 6. 8. 10 IMF 5. 0.05. 0. −0.05 0. 2. 4. 6. 8. 10 IMF 6. 1. 0. −1 0. 2. 4. 6. 8. 10 IMF 7. 0.5. 0. −0.5 0. 2. 4. 6. 8. 10 IMF 8. 0.1. 0. −0.1 0. 2. 4. 6. 8. 10. Figure 2.6: Part results of signal A with adding white noise with 30% of aptitude of signal A. 12.

(21) 2.4. Significant test of IMFs. Significant test of IMFs which Wu and Huang suggested by [15] and [3]. This is a method to distinguish whether an IMF has physical meanings or it is just a white noise. This test is based on deriving the energy density spread function of IMFs and assigning statistical significance of information content for IMFs from a white noise. ∑N 2 j=1 [ci (j)] Let Ei = be the energy density of the IMF i, and T¯i is the average N period of it. Based on numerical experiments on white noise with EMD, we can get lnE¯i + lnT¯i = 0. (2.4). where E¯i is the mean of Ei when N approaches infinity. As discussed in [3], for a time series that has a normal distribution, its energy should have a χ2 distribution with the degrees of freedom equal to the mean of energy. Thus, the probability density function of its energy is (2.5) ρ(N Ei ) ∝ (N Ei )N E i /2−1 e−N Ei /2 and ρ(Ei ) ∝ N (N Ei )N E i /2−1 e−N Ei /2 .. (2.6). ¯. Also, using the Taylor quadratic approximation for elnEi −lnEi , the following equation is derived. ρ(lnEi ) ∝ exp(− where σ 2 = Thus,. 2 N Ei. =. N Ei (lnEi − lnEi )2 ) ∼ N (lnEi , σ 2 ) 4. (2.7). 2T i N. √. 2 lnT¯i /2 e (2.8) N where zα is the α percentile of the standard normal distribution. The method is based on the rejection of a null hypothesis, which states that all IMFs are the components of a pure white noise. If the null hypothesis is rejected, we conclude that the corresponding IMFs contain information against white noise. Fig. 2.7 gives the significant test of the IMFs from signal A with 10% white noise level. We use signal A to illustrate that how to test IMFs of signal A. Fig. 2.7 plots lnEi = −lnT¯i + ±zα. 13.

(22) the spread lines for 95th and 99th percentiles based on the equation, and the star points present the energy of IMFs corresponding their periods. when the star point exceeds the 99th percentile line, we can know that the IMF correponding to the star point is not a white noise. Significance test of IMFs 15. LOG2 ( Mean Normalized Energy ). 10. 5. 0. −5. −10. 0. 2. 4. 6 LOG2 ( Mean Period ). 8. 10. 12. Figure 2.7: The significance test of IMFs of signal A with 10% white noise level. The energy of IMF 8 and IMF 9 are so small that they can be ignored.. 14.

(23) Chapter 3 Method: with clustering analysis Cluster analysis is a set of statistical methods that identifies and classifies objects or variables into classes (called clusters) so that objects in the same clustering are similar to each other where the dissimilarities or distances between objects are used when forming the clusters. The two most common clustering procedures are hierarchical and non-hierarchical. In this paper, we use the hierarchical clustering procedure. Hierarchical clustering produce a set of several level of nested clusters, in which cluster is the union of its subclusters. At the lowest level, each individual object is a cluster, and at the highest level the cluster containing all the objects. A hierarchical clustering is often displayed graphically using a tree-like diagram, called a dendrogram, which displays both the cluster-subcluster relationships and the order in which the clusters are merged or split.. 3.1. Clustering analysis. The clustering rocess[11] is following: 1. Start with all objects in separate clusters. Denote these clusters C1 ,C2 ,...,Cn . 2. Define the distance di,j between Ci and Cj . Here, di,j = 1 − corr(Ci , Cj ). 15. (3.1).

(24) where corr(Ci , Cj ) is the correlation coefficient between Ci and Cj . let t = 1 be an index of the iterative process. 3. Find the smallest distance between two clusters and denote these two clusters Ci and Cj . 4. Amalgamate clusters Ci and Cj to form a new cluster Cn+t . 5. Define the distance between the new cluster Cn+t and all remaining clusters Ck as follows: d(n+t),k = f (di,k , dj,k ) 6. Add cluster Cn+t as a new cluster and remove clusters Ci and Cj . Let t = t + 1. 7. Return to step 3 and continue until one cluster remains. Remarks: 1. Decision of the distance di,j : For any two time series, if they have high negative correlations, the period of one is very close to the period of another. Based on EMD algorithm, the existence of that the two different IMFs have high negative correlations is impossible. Thus, the distance which we choose is available. 2. Choice of f in step 5 is Complete Linkage: f (di,k , dj,k ) = max{di,k , dj,k }. (3.2). 3. Since the residue presents a trend of the data, it is not treated as a component in clustering process Fig. 3.1 describes the relationships between IMFs of signal A with 10% white noise level to describe the and that shows the dendrogram for hierarchical clustering. We describe and provide the orthogonalities between all IMFs with correlation coefficients of all IMFs. Also, as the number of clusters is given, all IMFs in the same cluster are combined into a single component. This single component is called Clustered IMF. 16.

(25) (CIMF) in this paper. Due to this method, the multi-mode problem can be improved, and the method provides a general criterion of combining IMFs until all IMFs are combined into a single component. In addition, if the number of clusters is appropriate, over-complete problem can be improved simultaneously. The Fig. 3.2 presents that the flow chart of the method in this manuscript.. Figure 3.1: Dendrogram with complete linkages of IMFs of signal A with 10% white noise level. 3.2. The flow chart. The flow chart of this method is shown in Fig. 3.2. Step 1 is loading the targeted original data. Step 2 is to pre-process for the data . For example, resample or filter. This step is not necessary, but it can remove some information which we don't want. Step 3 is to decompose the data by EEMD algorithm. The outputs of step 3 are IMFs and a residue. The step 4 is to cluster outputs of step 3 with complete linkage and the output of it is a dendrogram for outputs of step 3. When the number of clusters is given, step 5 is to combine IMFs in the same cluster to a single data according to the dendrogram from step 4. Finally, the outputs of step 5 are CIMFs.. 17.

(26) Figure 3.2: The flow chart of the proposed method. 18.

(27) Chapter 4 Examples In this section, the proposed method is applied to two simulated signals, wind turbine and an earthquake signals. First, signal A is decomposed by EEMD with three different white noise levels. The results indicates that the proposed method can improve the multi-mode problem when signal A is decomposed by EEMD with differnt white noise levels. Second, there are two events of multi-mode problem in different frequencies of the IMFs of the other simulated simultaneously, and the proposed method also can recover the sources of this simulated. Third, a practical signal from a wind turbine located at Chunan Miaoli is considered. The dendrogram which the proposed method provides indicates the multi-mode problem happens one time and two particular signals in IMFs of the wind turbine signal. Fourth, we consider the earthquake signal from station CHY080 at Chi-Chi. It recorded the landslide signature when the earthquake is happening, and IMF 2 of it indicates the landslide. The dendrogram provides shows that IMF 2 is particular with each other. It is consistent with the reality in the earthquake signal.. 4.1. The simulated signal: sin(t) and the intermittent signal. We start with our demonstration for the simulated signal A. The results of the EEMD is given in Fig. 2.4 and the dendrogram after clustering IMFs shows in Fig. 4.1. When the number of clusters is specified to be 4, the IMFs are combined into CIMFs as Fig.. 19.

(28) 4.2. The CIMF 1 is the white noise from EEMD algorithm. The CIMF 2 is the intermittent signal and CIMF 3 is a sine wave which is the main signal of this simulation. The CIMF 4 presents the numeric errors. Each CIMFs presents the different source from signal A. The proposed method improves the multi-mode problem and the overcomplete problem for signal A.. Figure 4.1: The dendrogram with complete linkages of IMFs of signal A with 10% white noise level when the number of clusters is 4. CIMF1 0.1 0 −0.1 0. 2. 4. 6. 8. 10 CIMF2. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10 CIMF3. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10 CIMF4. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10. 12. 14. 16. 18. 20. 0.2 0 −0.2 0 2 0 −2 0 −4. 1. x 10. 0.5 0 0. Figure 4.2: CIMFs with 10% white noise level. 20.

(29) As discussed in section 2.3, we have shown that the change of the noise level can't not improve the multi-mode problem for signal A. Also, this problem may happen in other IMFs, when the different white noise level changes. We get the dendrogram from clustering IMFs of signal A with 20 % white noise level as shown in Fig. 4.3. When the number of clusters is specified to be 5, we combine them into CIMFs as Fig.4.4. According to the significant test, CIMF 1 (IMF 1) and CIMF 2 (IMF 2) are close to white noise. The other characteristics of CIMFs in Fig.4.4 are consistent with the CIMFs of signal A with 10% white noise level.. Figure 4.3: The dendrogram with complete linkages of IMFs of signal A with 20% white noise level when the number of clusters is 5. CIMF1 0.1 0 −0.1 0. 2. 4. 6. 8. 10 CIMF2. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10 CIMF3. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10 CIMF4. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10 CIMF5. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10. 12. 14. 16. 18. 20. 0.05 0 −0.05 0 0.2 0 −0.2 0 2 0 −2 0 0.05 0 −0.05 0. Figure 4.4: CIMFs with 20% white noise level. 21.

(30) Similarly, in the case of IMFs of signal A with 30% white noise level, when the number of clusters is specified to be 5, the CIMFs of signal A with 30% white noise level are also consistent with the CIMFs of signal A with 20% white noise level as Fig. 4.5 and 4.6. In the view of these results, the proposed method can improve the multimode problem when signal A is decomposed by EEMD with differnt white noise levels.. Figure 4.5: The dendrogram with complete linkages of IMFs of signal A with 30% white noise level when the number of clusters is 5. CIMF1 0.2 0 −0.2 0. 2. 4. 6. 8. 10 CIMF2. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10 CIMF3. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10 CIMF4. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10 CIMF5. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10. 12. 14. 16. 18. 20. 0.1 0 −0.1 0 0.2 0 −0.2 0 2 0 −2 0 0.1 0 −0.1 0. Figure 4.6: CIMFs with 30% white noise level. 22.

(31) 4.2. The simulated signal: sin(t) + 0.1 sin(10t) and the intermittent signal. Another simulated signal is discussed. It is the combination of a sine wave with 1 5 unit amplitude and Hz and a sine wave with 0.1 unit amplitude and Hz. At six 2π π middle crests of the sine wave with unit amplitude, the cosine wave with a 0.1 unit 20 amplitude and Hz are riding on it as Fig.4.7. The function of the simulated signal π is  (2i − 1) π (2i − 1) π  sint + 0.1sin10t + 0.1cos40t t ∈ [ π− , π + ], i = 1, 2, ...6 x(t) = , 0 ≤ t ≤ 6π 2 5 2 5  sint + 0.1sin10t otherwise. (4.1) The simulated signal is decomposed by EEMD with a white noise level at 10% of the standard deviation of the signals and ensembling 30 times. In Fig. 4.8, the frequency of intermittent signal in IMF 2 is the same to the frequency in IMF 3. In addition IMF 4 and IMF 5 have the same period. There are two events of multi-mode problem. That is the multi-mode problem could happen not only one time. We cluster the IMFs to get the dendrogram (Fig. 4.9) and give 5 which is the number of clusters. All IMFs are combined into CIMFs according to the dendrogram. CIMF 1 is a white noise from EEMD algorithm. CIMF 2 is the intermittent signal. CIMF 3 is the sine wave with 5 0.1 unit amplitude and Hz. CIMF 4 is the main sine wave of the simulated signal, π and CIMF 5 is the numeric error signal. CIMFs are shown in Fig. 4.10. The results demonstrate that the proposed method is better than only EEMD algorithm for extracting useful information. 1.5 1 0.5 0 −0.5 −1 −1.5 0. 2. 4. 6. 8. 10. 12. 14. 16. 18. 20. Figure 4.7: sin(t) + 0.1 sin(10t) and the intermittent signal. 23.

(32) IMF 1 0.05 0 −0.05 0. 2. 4. 6. 8. 10 IMF 2. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10 IMF 3. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10 IMF 4. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10 IMF 5. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10. 12. 14. 16. 18. 20. 0.1 0 −0.1 0 0.1 0 −0.1 0 0.1 0 −0.1 0 0.1 0 −0.1 0. IMF 6 2 0 −2 0. 2. 4. 6. 8. 10 IMF 7. 12. 14. 16. 18. 20. 4. 6. 8. 10 IMF 8. 12. 14. 16. 18. 20. 4. 6. 8. 10 IMF 9. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10 IMF 10. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10. 12. 14. 16. 18. 20. 0.02 0 −0.02 0 2 −3 x 10 5 0 −5 0 2 −5 x 10 2 1 0 0 0.05 0 −0.05 0. Figure 4.8: Results of the signal: sin(t) + 0.1 sin(10t) and intermittent signal which is decomposed by EEMD. The last means the residue.. 24.

(33) Figure 4.9: The figure is the dendrogram with complete linkages for IMFs of sin(t) + 0.1 sin(10t) and the intermittent signal when the number of clusters is 5 CIMF1 0.05 0 −0.05 0. 2. 4. 6. 8. 10 CIMF2. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10 CIMF3. 12. 14. 16. 18. 20. 2. 4. 6. 8. 10 CIMF4. 12. 14. 16. 18. 20. 4. 6. 8. 10 CIMF5. 12. 14. 16. 18. 20. 4. 6. 8. 10. 12. 14. 16. 18. 20. 0.2 0 −0.2 0 0.2 0 −0.2 0 2 0 −2 0 2 −5 x 10 2 1 0 0. 2. Figure 4.10: CIMFs of sin(t) + 0.1 sin(10t) and the intermittent signal. 25.

(34) 4.3. The practical signal: a voice from a wind turbine at Chunan Miaoli. This subsection presents an application for a voice signal of a wind turbine at Chunan Miaoli. The wind turbine has 67 meters high, with three rotors diameter of 70 meters, as shown in Fig.4.11a(The figure is from http://www.bhes.ntpc.edu.tw/ ~bhesnet9/windpower3.html). The voice signal was recorded with sampling frequency 1087 Hz in 5 minutes, and we take the 30 second ∼ 90 second part of it as Fig. 4.11b. We decomposed it into 14 IMFs and a trend through EEMD algorithm with a white noise level at 10% of the standard deviation of the signals and ensembling 30 times, as Fig. 4.12 and 4.13. The wind turbine 6. 4. 2. 0. −2. −4. −6. −8 30. (a) The wind turbine at Chunan Miaoli. 40. 50. 60. 70. 80. 90. (b) The voice signal from The wind turbine at Chunan Miaoli. Figure 4.11: The wind turbine and the voice signal In Fig.4.11b, we find that the recorded signal has two patterns. One happens about 1 second,and is from a rotor pass through the voice recorder since the period of this turbine is 3 second, and it is also shown in IMF 6. The other happens about 20 second, it likely the main wave of the original signal, and is also recorded in IMF 12. According to the significant test of IMFs, IMF 1 and IMF 2 are close to the white noise.. 26.

(35) IMF 1 2 0 −2 30. 40. 50. 60 IMF 2. 70. 80. 90. 40. 50. 60 IMF 3. 70. 80. 90. 40. 50. 60 IMF 4. 70. 80. 90. 40. 50. 60 IMF 5. 70. 80. 90. 40. 50. 60. 70. 80. 90. 1 0 −1 30 2 0 −2 30 5 0 −5 30 5 0 −5 30. Figure 4.12: These figures are IMF 1 ∼ 5 of the voice signal.. 27.

(36) IMF 6 5 0 −5 30. 40. 50. 60 IMF 7. 70. 80. 90. 40. 50. 60 IMF 8. 70. 80. 90. 40. 50. 60 IMF 9. 70. 80. 90. 40. 50. 60 IMF 10. 70. 80. 90. 40. 50. 60. 70. 80. 90. 5 0 −5 30 2 0 −2 30 2 0 −2 30 1 0 −1 30. IMF 11 1 0 −1 30. 40. 50. 60 IMF 12. 70. 80. 90. 40. 50. 60 IMF 13. 70. 80. 90. 40. 50. 60 IMF 14. 70. 80. 90. 40. 50. 60 IMF 15. 70. 80. 90. 40. 50. 60. 70. 80. 90. 2 0 −2 30 0.5 0 −0.5 30 0.1 0 −0.1 30 0 −0.5 −1 30. Figure 4.13: These figures are IMF 6 ∼ 15 of the voice signal. The last IMF means the trend.. 28.

(37) We produce the dendrogram as Fig.4.14 through clustering analysis with complete linkage. According to Fig. 4.14, we find that there is a multi-mode problem between IMF 13 and IMF 14. In addition, we also find that IMF 7 and 10 relative to the others are self-reliant, and this indicates that these play an important role in the signal. This dendrogram presents some characteristics of IMFs from EEMD in the view of correlation coefficients.. Figure 4.14: The figure is the dendrogram with complete linkages of the wind turbine when the number of clusters is 8.. 29.

(38) According to the dendrogram (Fig.4.14), when the number of clusters 8 is given, IMF 1 and 2 are combined into CIMF 1. As we discuss before, CIMF 1 is like white noise. IMF 3 and 4 are combined into CIMF 2; 5 and 6 into CIMF 3; 8 and 9 into CIMF 5; 11 and 12 into CIMF 7 ;13 and 14 into CIMF 8. Especially, IMF 7 is CIMF 4 and IMF 10 is CIMF 6. For CIMFs, the peak which happens about 1 second is presented in CIMF 3, and the other which happens about 20 second is also shown in CIMF 8. All CIMFs are shown in Fig. 4.15. The results of this subsection indicate that the method which we suggest improves the multi-mode problem of the signal and also provides more information from the results of EEMD through presenting the correlations between IMFs.. 30.

(39) CIMF1 2 0 −2 30. 40. 50. 60 CIMF2. 70. 80. 90. 40. 50. 60 CIMF3. 70. 80. 90. 40. 50. 60 CIMF4. 70. 80. 90. 40. 50. 60. 70. 80. 90. 5 0 −5 30 5 0 −5 30 5 0 −5 30. CIMF5 5 0 −5 30. 40. 50. 60 CIMF6. 70. 80. 90. 40. 50. 60 CIMF7. 70. 80. 90. 40. 50. 60 CIMF8. 70. 80. 90. 40. 50. 60. 70. 80. 90. 1 0 −1 30 2 0 −2 30 1 0 −1 30. Figure 4.15: CIMFs with complete linkage of the wind turbine. 31.

(40) 4.4. The practical signal: an seismic signal from station CHY080 at Chi-Chi. In 1999, the Chi-Chi earthquake triggered a major landslide in Tsaoling area, with a source volume 125 × 106 m3 . Near the scar area, there is a station, CHY080, recording the seismic signals during the earthquake and the recorded data exhibit some distinctive signatures. The location of CHY080 is shown in Fig.4.16. The record has three directions: east-west, north-South, and vertical. This section presents the analysis of its east-west direction. Before analysing the east-west direction signal, we change the sample frequency of it from 200 points per second to 100 with applying an anti-aliasing FIR filter and take frequencies it contains below 20 with a quadric low-pass filter. The resampled and filtered signal is shown in Fig. 4.17.. Figure 4.16: The location of CHY080 in Tsaoling area. The NE-SW profile is defined through the gravity center of the slid mass and is in parallel to the slid direction.. resampled and filtered chy080e 40. 30. 20. 10. 0. −10. −20. −30. −40 70. 75. 80. 85. Figure 4.17: The resampled and filtered signal from the east-west direction of station CHY080. 32.

(41) Applying EEMD to this signal with a white noise level at 10% of the standard deviation of the signals and ensembling 30 times, we find that the signal is decomposed into 9 IMFs, see Fig. 4.18. The main earthquake contents are likely to be classified into the third to the four IMF because they total contain more than 81% of the spectral energy of the acceleration. Another distinctive characteristic is that outstanding wave packets are found between 76 and 78 second in IMF 2. IMF 2 is likely from the landslide. The significance of IMF white noise is tested. IMF 1 is close to the margin of white noise and hence is likely a white noise. We get the dendrogram (Fig. 4.19) through clustering analysis with complete linkage. When the number of clusters, 5, is given, we combine IMFs into CIMFs, as shown in Fig. 4.20. CIMF 1 is the IMF 1, and CIMF 2 is the IMF 2 which contains landslide messages, and other CIMFs are from IMFs which contain the main earthquake signal. The results, as described above, show the method which we suggest decomposes this earthquake signal into some components appropriately. In addition, we can find characteristics of this earthquake signal easily according to the dendrogram. IMF 1. IMF 6. 1. 5. 0. 0. −1 70. −5 70. 75. 80. 85. 75. IMF 2 0. 0 −5 70. 75. 80. 85. 75. IMF 3. 80. 85. 80. 85. 80. 85. 80. 85. IMF 8. 50. 1. 0. 0 75. 80. 85. −1 70. 75. IMF 4. IMF 9. 20. 0.5. 0 −20 70. 85. 5. −10 70. −50 70. 80 IMF 7. 10. 0 75. −0.5 85 70. 80. 75. IMF 5. IMF 10. 10. 2. 0. 0. −10 70. −2 70. 75. 80. 85. 75. Figure 4.18: Results of the east-west earthquake signal of CHY080 which is decomposed by EEMD. The last means the trend.. 33.

(42) Height 1.1 1.0 0.9 0.8 IMF1 0.7. &,0). IMF2. &,0) &,0). &,0). 0.6 0.5. &,0) IMF9. IMF5. IMF6 IMF3. IMF4. 0.4. IMF7. IMF8. Figure 4.19: The figure is the dendrogram with complete linkage for IMFs of the eastwest signal when the number of clusters is 5. CIMF1 1 0 −1 70. 75. 80. 85. 80. 85. 80. 85. 80. 85. 80. 85. CIMF2 10 0 −10 70. 75 CIMF3. 50 0 −50 70. 75 CIMF4. 10 0 −10 70. 75 CIMF5. 5 0 −5 70. 75. Figure 4.20: CIMFs of the east-west signal. 34.

(43) Chapter 5 Conclusion and discussions 5.1. Conclusions. Signals can be decomposed into a collection of IMFs with EEMD. The proposed method provides an another process for EEMD to improve these problems through statistical clustering analysis. According to a dendrogram with the distance which we define in this paper, this method provides more information about the relationships between IMFs. As discussed for the an seismic signal from station CHY080, the dendrogram of IMFs of the an seismic signal indicates the uniqueness of the landslide message. In addition, the CIMFs with different white noise levels are consistent. In fact, performing this method well depends on an appropriate number of clusters, and this number is equivalent to the definition of multi-mode. As a result, this method could need more information to decide the number of clusters. Nevertheless, the proposed method still offers a deterministic interpolation for the multi-mode IMFs and significantly benefit many in-situ applications.. 5.2. The white noise in EEMD. As discussed in section 4.1, because of the proportion of white noise, we find that the more IMFs are the white noises. In addition, since the white noises are random, these IMFs have poor correlation to each other. As a result, the white noise IMFs are hard to be combined together. In the view of this, the white noise test of IMFs can be. 35.

(44) performed and containing IMFs which are not white noises before clustering them.. 5.3. Choice of linkage in hierarchical clustering. In this paper, we use clustering analysis with complete linkage. The goal of using this linkage is to distinguish the difference between IMFs and CIMFs. In fact, we can choose other linkages to achieve specific goals. For example, single linkage, average linkage, etc. However, as [11] have discussed, some characteristics of other linkages should be considered. The single linkage: f (di,k , dj,k ) = min{di,k , dj,k } is closely related to the minimal spanning dendrogram. The average linkage is taking average distance from individuals in one cluster to individuals in another. This linkage tends to combine clusters with small variances. We may choose linkage according to the characteristics of objects, but there is no best linkage in clustering analysis.. 5.4. The number of clusters. The method which we suggest can provide an objective criterion to merge IMFs into CIMFs. However, the number of clusters effects the result in this method obviously. If the number is too small, the CIMFs have mode mixing problem again. If the number is too large, the over-complete problem can not be improved. This method is based on that the signals from the same source are similar. i.e. the signals have more high correlation coefficient than others which is not similar to them. Through characteristics of a signal, we can appoint a correlation coefficient as criterion to cluster IMFs. However, there is no best objective criterion to cluster them.. 36.

(45) Bibliography [1] A. AYENU-PRAH and N. ATTOH-OKINE. A criterion for selecting relevant intrinsic mode functions in empirical mode decomposition. Advances in Adaptive Data Analysis, 2(1):1--24, 2010. [2] K.-M. Chang. Arrhythmia ecg noise reduction by ensemble empirical mode decomposition. Sensors, 10(6):6063--6080, 2010. [3] Y.-M. Chang, Z. Wu, , J. Chang, and N. E. Huang. Model validation based on ensemble empirical mode decomposition. Advances in Adaptive Data Analysis, 2(4):415--428, 2010. [4] W. Y. L. Dennis E. B. Tan. Sleep disorder detection and identification. Journal of Biomedical Science and Engineering, 5:330--340, 2010. [5] Z. Feng, X. Ding, and Y. Jiang. Pitch period estimation of voice signal based on eemd and hilbert transform. In Informatics in Control, Automation and Robotics (CAR), 2010 2nd International Asia Conference on, volume 1, pages 365 --368, march 2010. [6] P. Flandrin, G. Rilling, and P. Goncalves. Empirical mode decomposition as a filter bank. World Scientific, 11(2):112--114, 2004. [7] D. HUANG. and Y. XU. A new application of ensemble emd ameliorating the error from insufficient sampling rate. Advances in Adaptive Data Analysis, 3(4): 493--508, 2011. [8] N. E. Huang, C. C. Chern, K. Huang, L. W. Salvino, S. R. Long, and K. L. Fan. A new spectral representation of earthquake data: Hilbert spectral analysis of station. 37.

(46) tcu129, chi-chi, taiwan, 21 september 1999. Bulletin of the Seismological Society of America, 91(5):1310--1338, 2001. [9] N. E. Huang, Z. Shen, and S. R. Long. A new view of nonlinear water waves: the hilbert spectrum. Annual Review of Fluid Mechanics, 31:417 – 457, 1999. [10] N. E. Huang, Z. Shen, S. R. Long, M. L. Wu, H. H. Shih, Q. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu. The empirical mode decomposition and hilbert spectrum for nonlinear and nonstationary time series analysis. Proc. R. Soc. London, A454:903--995, 1998. [11] J. Joseph F. Hair, R. E. Anderson, R. L. Tatham, and W. C. Black. MUTIVARIATE DATA ANALYSIS with Reading. Macmillan Publishing Company, 1992. [12] Y. Lei, Z. He, and Y. Zi. Application of the eemd method to rotor fault diagnosis of rotating machinery. Mechanical Systems and Signal Processing, 23:1327--1338, 2011. [13] F. MHAMDI, J.-M. POGG, and M. J. IDANE. Trend extraction for seasonal time series using ensemble empirical mode decomposition. Advances in Adaptive Data Analysis, 3(3):363--383, 2011. [14] P.-H. TSUI and C.-C. CHANG. Noise-modulated empirical mode decomposition. Advances in Adaptive Data Analysis, 2(1):25--37, 2010. [15] Z. Wu and N. E. Huang. A study of the characteristics of white noise using the empirical mode decomposition method. Proc. R. Soc. London, A460:1597 – 1611, 2004. [16] Z. Wu and N. E. Huang. Ensemble empirical mode decomposition:a noiseassisted data analysis method. Advances in Adaptive Data Analysis, 1(1):1--41, 2009. [17] Z. Wu, N. E. Huang, S. R. Long, and C.-K. Peng. On the trend, detrending, and variability of nonlinear and nonstationary time series. PNAS, 104(38):14889 – 14894, 2007.. 38.

(47) [18] J.-R. YEH, J.-S. SHIEH, and N. E. Huang. Complementary ensemble empirical mode decomposition: A novel noise enhanced data analysis method. Advances in Adaptive Data Analysis, 2(2):135--156, 2010. [19] D. Yu, J. Cheng, and Y. Yang. Application of emdmethod and hilbert spectrum to the fault diagnosis of roller bearings. Mechanical Systems and Signal Processing, 19:259--270, 2005.. 39.

(48)