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Coherent structure detection using wavelet analysis in long time-series
Journal of Wind Engineering and Industrial Aerodynamics 88 (2000) 183–195
Coherent structure detection using wavelet analysis in long time-series Xiaoning Gilliam*, James Dunyakb, Arthur Doggettc, Douglas Smitha b
a Department of Civil Engineering, Texas Tech University, Lubbock, TX 79409, USA Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA c Department of Geosciences, Texas Tech University, Lubbock, TX 79409, USA
Abstract In this paper, we introduce two statistical techniques to detect coherent structures in long time-series: an intermittency rate estimator and an event-counting technique. The intermittency rate estimator calculates the fraction of time occupied by coherent structures, in contrast to incoherent noise. The event-counting approach estimates the number of discrete structures that occurred. In both cases, the wavelet transform and extensions of the coherent structure detector (CSD) are used to develop rigorous statistical approaches. The techniques are demonstrated on a 7 h wind time-series collected at the Wind Engineering Research Field Laboratory at Texas Tech. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Wavelet transforms; Detection; Coherent structure; Exemplars
1. Introduction The wavelet transform is a popular tool for studying intermittent and localized phenomena in signals. They have been applied to windfields and pressure fields (for example, in Refs. [1–8]). First, an analyzing function, c, is chosen to capture some interesting characteristics of a signal. For example, we may use either the Haar wavelet analyzing function to detect a sustained change in signal level or the Morlet *Correspondence address: Wind Engineering Research Center, Dept. Civil, Texas Tech University, Box 41023, Lubbock, TX 79409-1023, USA. Tel.: +1-806-742-3479; fax: +1-806-742-3446. E-mail address: [email protected] (X. Gilliam). 0167-6105/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 0 0 ) 0 0 0 4 8 - 9
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wavelet to look for concentration in time-frequency of signal energy. Then the wavelet transform at time b and scale a of a given signal x is defined as Z 1 tÿb dt: xðtÞc Wx ðb; aÞ ¼ aÿ1=2 a ÿ1 This equation shows that the wavelet transform may be interpreted as a bank of matched filters between the signal and the scale a and time b version of the analyzing function. The choice of analyzing function determines the ‘‘feature’’ detected by the matched filters. A large (in magnitude) wavelet coefficient indicates the presence of this feature at that time and scale. By using a discrete implementation of the continuous wavelet transform, the appropriate wavelet analyzing function is chosen to allow the decomposition of the signal into a frame that reveals the features of the signal [9]. In simple signals, for example those with small time-bandwidth product, the wavelet transform is often easy to interpret. For complex signals, such as measurements containing turbulence, the wavelet transform replaces a complex onedimensional time-series with an even more complex two-dimensional signal representation. Interpreting the resulting pictures is problematic since even a signal with no coherent features (such as white noise) will have a complex wavelet transform with many large peaks [10]. In several earlier papers, the authors developed a rigorous statistical technique, called the coherent structures detector (CSD), for detecting coherent structures in random fluid flows. The method is based on a definition of an incoherent signal and controls the test size without assumptions about the process spectrum. The statistical technique follows from one of the oldest methods in nonparametric statistics – the development of a randomized reference distribution [11]. Detailed asymptotic proofs on the power of the test were developed and reported in [12]. The original approach focused on relatively short time-series and controlled the probability of error anywhere in the time-series. Further discussion on detection using continuous wavelets is found in Ref. [13]. The technique has been used to establish the existence of sustained (local) changes of wind-speed in hurricanes [14] and intermittent bursts of turbulence in near-ground winds [10]. A localized turbulence structure approach has also been used to define and analyze localized spatial coherency of windfields, as introduced in Ref. [15]. The statistical techniques here may easily be adapted to the discrete wavelet transform with dyadic time and scale expansions. In this case, the signal is represented in a basis instead of a frame and time-scale filtering is possible using wavelet shrinkage. (Unfortunately, the dyadic expansion can suffer from time/scale registration problems and smear important information over many coefficients. This is why we prefer the continuous wavelet transform when time-scale filtering is not an important objective.) An implementation of wavelet shrinkage using a discrete wavelet approach with the coherent structure detector has been developed by the authors [16] and applied to analysis of vertical windfields in thunderstorm genesis [17]. See also Ref. [18] for another example of time-scale filtering of turbulence using wavelets.
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For both conceptual and numerical reasons, the original CSD approach is designed for relatively short time-series of length less than approximately 30 000. (See Ref. [10] for a detailed description.) The original CSD followed the conventional statistical communications theory viewpoint: the probability of detection (correctly detecting the presence of a coherent structure anywhere in the signal) was maximized while controlling the probability of false alarm (incorrectly detecting a coherent structure anywhere in the signal when none are present). In long time-series, to avoid a single false alarm anywhere in the long time-series, CSD thresholds are pushed so high that many detections are missed. In practice, many atmospheric measurements are made over long periods of time. Instead, for long time-series, we should seek a large number of correct detections interspersed with an occasional false alarm. Thus, we develop two new approaches that are appropriate for long time-series: coherent intermittency rate estimation and coherent event counting. The coherent intermittency rate is the fraction of the flow record containing coherent structures [19]. Our new statistical technique controls the false alarm probability for each scale and time separately, thereby controlling the false alarm intermittency rate. The statistical test allows two conclusions: that a time-series contains intermittent structures at a specific scale and that the structures are localized to specific portions of the waveform. In contrast to intermittency rate, event counting identifies discrete, localized coherent structures [20]. In the CSD, the probability of a single false alarm anywhere in the time-series is controlled. For event counting, the false alarm rate, or number of false alarms per unit time, is controlled instead. Both of the new detection techniques, as well as the CSD, rely on the fundamental distinction between phase relationships in incoherent energy and coherent structures. The techniques are demonstrated on measurements from Texas Tech’s Wind Engineering Research Field Laboratory.
2. Definitions and hypotheses Many definitions of coherency are used for a variety of purposes. Here, we use a concept of coherency encountered in signal processing, based on phase relationships. We begin this section with the definition of an incoherent time-series [10–17]: Definition 2.1. Consider a discretely sampled real-valued time-series x ¼ ðx1 , ^ for the discrete Fourier transform of x, we can x2 ; :::; xm Þ. Using the notation x ^ ¼ Ai e jyi , for i ¼ 1; 2; :::; m=2. Following write for each frequency component x standard practice, we call a time-series incoherent if the yi are uniform ½0; 2p and independent of each other and the Ai . With this viewpoint, incoherent signals have no phase relationships; these phase relationships are needed to concentrate energy in time and create coherent structures. Incoherent signals can be described as linear filters driven by white noise. The
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‘‘coherency’’ in a coherent signal is between the components at different frequencies, which add up constructively and destructively to produce a localized structure. Our test hypothesis is then as follows: H0 : x is incoherent. H1 : x contains coherent structures. Our statistical tests are based on building a reference distribution through the creation of exemplars of the original time-series x. e
^el ¼ Al eyl , Definition 2.2. An exemplar of a real signal xe ðtÞ has Fourier coefficients x e where yl for l ¼ 1; 2; :::; N=2 are uniform ½0; 2p and independent of Al . We use the wavelet transform Wx ðb; aÞ as the test statistics to determine the presence of coherent structures with the characteristics of the analyzing function. A large jWx ðb; aÞj indicates a strong coherent structure. Wavelet transforms of the exemplars are used to build a reference distribution under H0 (incoherent signal), which allows the choice of an appropriate threshold thðaÞ for detection of coherent structures. The success of the test is described in the usual fashion with probability of false alarm defined as PFA ¼ Pðdetect coherency and choose H1 jH0 Þ ¼ a and the probability of detection is given by PD ¼ Pðdetect coherency and choose H1 jH1 Þ ¼ 1 ÿ b: Here a is the size of the test and b is the type II error of the test. Our goal is to develop a test that has a large probability of detection for a specific probability of false alarm. The excellent performance of CSD-style tests has been demonstrated both theoretically and numerically in Ref. [9], in which convergence to the generalized likelihood ratio test was proven in the signal-plus-colorednoise case.
3. The intermittency rate Now we are ready to introduce our first statistical technique for long time-series. Our main task is to calculate an intermittency rate while controlling the error rate for the entire window. To do so, let us first define the intermittent rate. Using the definition discussed by Hagelberg and Gamage [18,21], the intermittency rate is the fraction of the time-series containing structures. Hence, Intermittency rate=(support of structures)/(total record length). Of course, the intermittency rate is dependent on the type of structures we seek. Then Z 1 T IjWx ðb; aÞj>thðaÞ db: ð1Þ rðaÞ ¼ T 0
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Since our measurements are always discrete, we use the definition rðaÞ ¼
N 1X IjWx ðb; aÞj>thðaÞ : N b¼1
ð2Þ
Here rðaÞ is the intermittent rate; T is the time interval represented by the signal; Wx ðb; aÞ is the wavelet transform of the time-series; and thðaÞ is the threshold used to detect the intermittent coherent structures. The indicator function is 1 when jWx ðb; aÞj > thðaÞ; ð3Þ IjWx ðb; aÞj>thðaÞ ¼ 0 when jWx ðb; aÞj4thðaÞ: Note that a discrete implementation of the wavelet transform is used in practice for the discretely measured time-series. Thus, from Eqs. (1) and (2), the expected value of the intermittency rate is given by N 1X EðIjWx ðb; aÞj>thðaÞ Þ N b¼1
ð4Þ
N 1X PðjWx ðb; aÞj > thðaÞÞ: N b¼1
ð5Þ
E½rðaÞ ¼
¼
For a stationary signal, E½rðaÞ ¼ PðjWx ðb; aÞj > thðaÞÞ:
ð6Þ
Based on Eq. (5), we control the intermittency rate rðaÞ under H0 by controlling the probability PfjWx ðb; aÞj > thðaÞg. We do this through the process of building our test criteria – the threshold. Fig. 1 illustrates the process. To build our threshold thðaÞ, we first build exemplars. If the original signal is long, only one exemplar may be required. First we calculate the discrete fourier transform of the time-series, then keep the Fourier transform magnitude but randomize the phase. Hence, if x is our timeseries, each exemplar xei has exactly the same power spectrum as the given timeseries, but is known to be incoherent. Now, each exemplar is wavelet transformed to provide the wavelet-transformed exemplar Wie ðb; aÞ. (The wavelet transform may also be implemented as a convolution in the frequency domain to save some computations.) We then use order statistics from Wie ðb; aÞ to estimate the threshold thðaÞ. Thus, for a discretely sampled x and for each scale a, the threshold thðaÞ is defined as ( ! ) n X N 11X IW e ðb; aÞ>t > g : ð7Þ thðaÞ ¼ thða; g; n; NÞ ¼ sup t : n N i¼1 b¼1 i Here x is the signal with length N, g is the desired false alarm rate, n is the number of exemplars, a is the scale, and thðaÞ is the threshold. With Eq. (7), we can establish the performance of our estimation under H0 as follows: lim EðrðaÞjH0 Þ ¼ g;
n!1
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Fig. 1. Test statistics for the intermittency rate.
where rðaÞ is the intermittency rate and n is the number of exemplars [19]. In practice, very few exemplars are needed for long signals. To see this, consider Eq. (5) and " # N 1X IjWx ðb; aÞj>thðaÞ jH0 : lim E½rðaÞjH0 ¼ lim E n!1 n!1 N b¼1 Now, applying smoothing, we obtain " # N 1X IjWx ðb; aÞj>thðaÞ jH0 lim E n!1 N b¼1 "
# N 1X E IjWx ðb; aÞj>thðaÞ jH0 ; A1 ; :::; AN : ¼ lim E n!1 N b¼1
ð8Þ
Here Ai are defined in Definition 2.1. Under H0 , the time-series and the exemplars have the same distribution, and the phase and the amplitude of the exemplars are independent, so h i ð9Þ E IjWx ðb; aÞj>thðaÞ jH0 ; A1 ; :::; AN ¼ E IjWie ðb; aÞj>thðaÞ jH0 ; A1 ; :::; AN :
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Thus, using the dominated convergence theorem [22], from Eqs. (8) and (9), we " # obtain N h i 1X lim E IjWie ðb; aÞj>tCIED ðaÞ jH0 ; A1 ; :::; AN lim E ½rðaÞjH0 ¼ E n!1 N b ¼1 n!1 " # N 1X g ¼ g: ¼E N b ¼1 ð10Þ Now, we are ready to draw our statistical conclusions. The mean value of rðaÞ is g under H0 . A significant deviation of rðaÞ from g is statistical evidence that the signal contains coherent structures detected by the analyzing function. A theoretically precise although computationally expensive Monte Carlo algorithm may be used to develop test thresholds [20,9]. In practice, we conclude that coherent structures are present if rðaÞ4g. The coherent structure detector can be used to analyze many different phenomena, such as in the author’s papers [9,10,12–16,19,20]. The following example demonstrates both event counting and intermittency rate to analyze turbulence, using 7 h of wind-speed data collected at 50-m height on the meteorological tower at Texas Tech’s Wind Engineering Research Field Laboratory. The data were collected in the early morning hours of May 22, 1998, at 10 Hz on a uvw-anemometer. The driving mechanism was a large synoptic low moving eastward across the Rocky Mountains. Since we are interested in regions of concentrated turbulence in the signal in both techniques, we use the Morlet wavelet as our analyzing function: cðtÞ ¼ eð jctÿt
2
=2Þ
;
where c ¼ 5:4. Scales from 0:5 to 2:7 s are considered. These scales corresponded to center frequencies of 0:3 to 1:6 Hz in the scaled Morlet analyzing function. Intermittency in turbulence has been considered by many authors, with emphasis on the intermittent transfer of energy between scales [23]. These deviations from Kolmogorov’s scaling [24] become increasingly stronger at smaller scales [25,26]. Our viewpoint is different here. These analytic, numeric, and experimental studies focus on intermittency generated by the turbulence itself, such as in stationary flows in a wind tunnel. The atmospheric boundary layer also contains intermittent events driven by localized mesoscale and smaller meteorological phenomena [18], which give a much greater complexity to the turbulent wind field, even in scales much above the dissipative scales [27]. Our research using wavelets and the coherent structure detector has indicated differences in intermittent phenomena in the boundary layer due to different mesoscale phenomena such as thunderstorm outflows, deep lows, and surface heating-induced mixing. Fig. 2 shows the 7 h-long wind-speed waveform. This was measured during winds created by a low-pressure system. We now apply our algorithm to the entire 7 h timeseries. For the first example (the intermittent estimation technique), we use a false alarm intermittency rate of g ¼ 0:001. Fig. 3 is the estimated intermittency rate rðaÞ at each scale for the entire 7 h time-series, in which case rðaÞ4g. The intermittency rate ranges from four to 10 times greater than that expected by false alarms alone. We have strong statistical evidence of intermittent periods of higher turbulence. It is
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Fig. 2. Seven-hour wind time-series.
Fig. 3. The intermittency rate during the 7 h.
impossible to effectively display the detailed images for the entire 7 h (252 000 samples) of data. Hence, from the resulting transforms, we display an interesting portion of the signal, shown in Fig. 4(a). Fig. 4(b) shows the wavelet transform of Fig. 4(a) and Fig. 4(c) shows the thresholded-wavelet transform. The information that is below the intermittent CSD threshold had been zeroed out; only the statistically relevant information remains. (No thresholded-signal or scale-integration can be developed from this transform since a continuous wavelet transform was used. See the discussion in Section 1 on applying discrete wavelet transforms to do time-scale filtering.)
4. Counting discrete events In this section, we consider the coherent event counting technique. As in the previous section, our goal is to define the false alarm rate and describe a method of
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Fig. 4. (a) Wind-speed; (b) wavalet transform; (c) threshold transform.
calculating the number of false alarms in a long time-series. Thus, we first introduce the technique of counting discrete events. Note that there is no unique definition or approach for counting discrete events, but the technique we use here is a useful interpretation for the transient phenomena (coherent events) we seek in wind signals. As we introduced in Ref. [20], we consider a time-series of length N. The entire wind flow signal x is divided into N=L ¼ M blocks of equal length L (assume N=L is integer). Hence, the wavelet transform W is split up into N=L blocks of coefficients, M W ¼ fWblockm ðb; aÞgM m¼1 ¼ fWm ðb; aÞgm¼1 :
Now, we are ready to define the event rate: Definition 4.1. Consider a fixed scale a and threshold thðaÞ. Let N be the length of time-series x with sample spacing Dt. Let Wm be the wavelet coefficients of a particular block. Then, we count an event in a block m at scale a if maxb jWm ðb; aÞj > thðaÞ. Moreover, for the entire time-series, the total number of events at scale a for threshold thðaÞ is defined as KðaÞ
M X m¼1
Imaxb jWm ðb; aÞj>thðaÞ :
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Hence, the event rate is given by gðaÞ ¼
KðaÞ ; NDt
where NDt is the time interval represented by the waveform x. The length L of the blocks plays an important role in counting false alarms. Note that there are many ways to choose the length of the block L. In this paper, we consider the following conditions as critical: *
*
The expected number of events in each block is low so that two events occurring in one block is rare; these two events would be counted as one. L is long enough that a single event rarely crosses over into adjacent blocks. This single event would be counted as two.
Note that these two conditions are opposing, so some middle ground must be found for the choice of L (see Ref. [20] for more details). We now want to compute the test criteria, the thresholds thðaÞ (see Fig. 5). The signal x is first used to generate one or more exemplars. The signal is Fourier transformed and the phase independently randomized uniformly in ½0; 2pÞ to create the exemplars xei in frequency domain. The wavelet transform is then applied in time domain or frequency domain, yielding Wie ðb; aÞ. Each transformed exemplar is then e ðb; aÞj in split into N=L blocks, and order statistics of the the maximum of jWblock m each block are accumulated. The thresholds are now chosen so that each block of length L with a false alarm rate g and sampling interval Dt will have an expected
Fig. 5. Test statistics for event counting.
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probability of false alarm gLDt. thða; L; n; gÞÞ ¼ sup t :
n X M 1 1 X Imaxb jWi;e m ðb; aÞj>t n M i¼1 m¼1
!
) > gLDt :
Here a is the scale, L is the length of a block, n is the number of exemplars used, M is the number of blocks, and g is the desired false alarm rate. At each scale a, this threshold th is used to count events in the original wavelet transform of signal x. More importantly, this threshold is used to display which portions of the original wavelet transform are statistically significant by applying a straightforward thresholding technique Wxthresholded ðb; aÞ ¼ Wx ðb; aÞIjWx ðb; aÞj>thða; L; n; gÞ : Note that lim
n!1
EðKðaÞjH0 ÞÞ ¼ g: NDt
Thus, the threshold is the correct choice to control the false alarm rate. See Ref. [9] for a detailed proof. To demonstrate event counting, we again consider the seven hours of data shown in Fig. 2. This analysis, using a false alarm event rate of one per hour, results in the total number of detections KðaÞ in 7 h at each scale shown in Fig. 6. Note that the total numbers of detections at each scale far exceed the seven expected false alarms in 7 h. See Ref. [20] for details on building a rigorous statistical test, based on the binomial distribution, from these test statistics. In Figs. 3 and 6, both intermittency rate estimation and event counting detected concentrated bursts of turbulence much more often than expected. Since both event counting and intermittency rate estimation are capturing the same phenomena, we expect similar conclusions from the two techniques.
Fig. 6. The event detections during the 7 h.
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5. Conclusions In this paper, we enhance the applicability of the coherent structure detector by developing techniques for long time-series. These techniques provide a statistically rigorous approach to extract and analyze coherent structures in incoherent noise. The first technique, intermittency rate estimation, provides a measure of the fraction of time a coherent structure is present. The second, event counting, counts coherent structures as discrete events. Both give similar overall results but provide different information about the structures. The two may be combined to develop some ideas about the magnitude, frequency, and characteristic lengths of these localized phenomena. Acknowledgements This work was performed under the Department of Commerce National Institute of Standards and Technology/Texas Tech University Cooperative Agreement Award 70NAB8H0059 and the National Science Function Grant #9980296.
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Coherent structure detection using wavelet analysis in long time-series Xiaoning Gilliam*, James Dunyakb, Arthur Doggettc, Douglas Smitha b
a Department of Civil Engineering, Texas Tech University, Lubbock, TX 79409, USA Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA c Department of Geosciences, Texas Tech University, Lubbock, TX 79409, USA
Abstract In this paper, we introduce two statistical techniques to detect coherent structures in long time-series: an intermittency rate estimator and an event-counting technique. The intermittency rate estimator calculates the fraction of time occupied by coherent structures, in contrast to incoherent noise. The event-counting approach estimates the number of discrete structures that occurred. In both cases, the wavelet transform and extensions of the coherent structure detector (CSD) are used to develop rigorous statistical approaches. The techniques are demonstrated on a 7 h wind time-series collected at the Wind Engineering Research Field Laboratory at Texas Tech. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Wavelet transforms; Detection; Coherent structure; Exemplars
1. Introduction The wavelet transform is a popular tool for studying intermittent and localized phenomena in signals. They have been applied to windfields and pressure fields (for example, in Refs. [1–8]). First, an analyzing function, c, is chosen to capture some interesting characteristics of a signal. For example, we may use either the Haar wavelet analyzing function to detect a sustained change in signal level or the Morlet *Correspondence address: Wind Engineering Research Center, Dept. Civil, Texas Tech University, Box 41023, Lubbock, TX 79409-1023, USA. Tel.: +1-806-742-3479; fax: +1-806-742-3446. E-mail address: [email protected] (X. Gilliam). 0167-6105/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 0 0 ) 0 0 0 4 8 - 9
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wavelet to look for concentration in time-frequency of signal energy. Then the wavelet transform at time b and scale a of a given signal x is defined as Z 1 tÿb dt: xðtÞc Wx ðb; aÞ ¼ aÿ1=2 a ÿ1 This equation shows that the wavelet transform may be interpreted as a bank of matched filters between the signal and the scale a and time b version of the analyzing function. The choice of analyzing function determines the ‘‘feature’’ detected by the matched filters. A large (in magnitude) wavelet coefficient indicates the presence of this feature at that time and scale. By using a discrete implementation of the continuous wavelet transform, the appropriate wavelet analyzing function is chosen to allow the decomposition of the signal into a frame that reveals the features of the signal [9]. In simple signals, for example those with small time-bandwidth product, the wavelet transform is often easy to interpret. For complex signals, such as measurements containing turbulence, the wavelet transform replaces a complex onedimensional time-series with an even more complex two-dimensional signal representation. Interpreting the resulting pictures is problematic since even a signal with no coherent features (such as white noise) will have a complex wavelet transform with many large peaks [10]. In several earlier papers, the authors developed a rigorous statistical technique, called the coherent structures detector (CSD), for detecting coherent structures in random fluid flows. The method is based on a definition of an incoherent signal and controls the test size without assumptions about the process spectrum. The statistical technique follows from one of the oldest methods in nonparametric statistics – the development of a randomized reference distribution [11]. Detailed asymptotic proofs on the power of the test were developed and reported in [12]. The original approach focused on relatively short time-series and controlled the probability of error anywhere in the time-series. Further discussion on detection using continuous wavelets is found in Ref. [13]. The technique has been used to establish the existence of sustained (local) changes of wind-speed in hurricanes [14] and intermittent bursts of turbulence in near-ground winds [10]. A localized turbulence structure approach has also been used to define and analyze localized spatial coherency of windfields, as introduced in Ref. [15]. The statistical techniques here may easily be adapted to the discrete wavelet transform with dyadic time and scale expansions. In this case, the signal is represented in a basis instead of a frame and time-scale filtering is possible using wavelet shrinkage. (Unfortunately, the dyadic expansion can suffer from time/scale registration problems and smear important information over many coefficients. This is why we prefer the continuous wavelet transform when time-scale filtering is not an important objective.) An implementation of wavelet shrinkage using a discrete wavelet approach with the coherent structure detector has been developed by the authors [16] and applied to analysis of vertical windfields in thunderstorm genesis [17]. See also Ref. [18] for another example of time-scale filtering of turbulence using wavelets.
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For both conceptual and numerical reasons, the original CSD approach is designed for relatively short time-series of length less than approximately 30 000. (See Ref. [10] for a detailed description.) The original CSD followed the conventional statistical communications theory viewpoint: the probability of detection (correctly detecting the presence of a coherent structure anywhere in the signal) was maximized while controlling the probability of false alarm (incorrectly detecting a coherent structure anywhere in the signal when none are present). In long time-series, to avoid a single false alarm anywhere in the long time-series, CSD thresholds are pushed so high that many detections are missed. In practice, many atmospheric measurements are made over long periods of time. Instead, for long time-series, we should seek a large number of correct detections interspersed with an occasional false alarm. Thus, we develop two new approaches that are appropriate for long time-series: coherent intermittency rate estimation and coherent event counting. The coherent intermittency rate is the fraction of the flow record containing coherent structures [19]. Our new statistical technique controls the false alarm probability for each scale and time separately, thereby controlling the false alarm intermittency rate. The statistical test allows two conclusions: that a time-series contains intermittent structures at a specific scale and that the structures are localized to specific portions of the waveform. In contrast to intermittency rate, event counting identifies discrete, localized coherent structures [20]. In the CSD, the probability of a single false alarm anywhere in the time-series is controlled. For event counting, the false alarm rate, or number of false alarms per unit time, is controlled instead. Both of the new detection techniques, as well as the CSD, rely on the fundamental distinction between phase relationships in incoherent energy and coherent structures. The techniques are demonstrated on measurements from Texas Tech’s Wind Engineering Research Field Laboratory.
2. Definitions and hypotheses Many definitions of coherency are used for a variety of purposes. Here, we use a concept of coherency encountered in signal processing, based on phase relationships. We begin this section with the definition of an incoherent time-series [10–17]: Definition 2.1. Consider a discretely sampled real-valued time-series x ¼ ðx1 , ^ for the discrete Fourier transform of x, we can x2 ; :::; xm Þ. Using the notation x ^ ¼ Ai e jyi , for i ¼ 1; 2; :::; m=2. Following write for each frequency component x standard practice, we call a time-series incoherent if the yi are uniform ½0; 2p and independent of each other and the Ai . With this viewpoint, incoherent signals have no phase relationships; these phase relationships are needed to concentrate energy in time and create coherent structures. Incoherent signals can be described as linear filters driven by white noise. The
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‘‘coherency’’ in a coherent signal is between the components at different frequencies, which add up constructively and destructively to produce a localized structure. Our test hypothesis is then as follows: H0 : x is incoherent. H1 : x contains coherent structures. Our statistical tests are based on building a reference distribution through the creation of exemplars of the original time-series x. e
^el ¼ Al eyl , Definition 2.2. An exemplar of a real signal xe ðtÞ has Fourier coefficients x e where yl for l ¼ 1; 2; :::; N=2 are uniform ½0; 2p and independent of Al . We use the wavelet transform Wx ðb; aÞ as the test statistics to determine the presence of coherent structures with the characteristics of the analyzing function. A large jWx ðb; aÞj indicates a strong coherent structure. Wavelet transforms of the exemplars are used to build a reference distribution under H0 (incoherent signal), which allows the choice of an appropriate threshold thðaÞ for detection of coherent structures. The success of the test is described in the usual fashion with probability of false alarm defined as PFA ¼ Pðdetect coherency and choose H1 jH0 Þ ¼ a and the probability of detection is given by PD ¼ Pðdetect coherency and choose H1 jH1 Þ ¼ 1 ÿ b: Here a is the size of the test and b is the type II error of the test. Our goal is to develop a test that has a large probability of detection for a specific probability of false alarm. The excellent performance of CSD-style tests has been demonstrated both theoretically and numerically in Ref. [9], in which convergence to the generalized likelihood ratio test was proven in the signal-plus-colorednoise case.
3. The intermittency rate Now we are ready to introduce our first statistical technique for long time-series. Our main task is to calculate an intermittency rate while controlling the error rate for the entire window. To do so, let us first define the intermittent rate. Using the definition discussed by Hagelberg and Gamage [18,21], the intermittency rate is the fraction of the time-series containing structures. Hence, Intermittency rate=(support of structures)/(total record length). Of course, the intermittency rate is dependent on the type of structures we seek. Then Z 1 T IjWx ðb; aÞj>thðaÞ db: ð1Þ rðaÞ ¼ T 0
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Since our measurements are always discrete, we use the definition rðaÞ ¼
N 1X IjWx ðb; aÞj>thðaÞ : N b¼1
ð2Þ
Here rðaÞ is the intermittent rate; T is the time interval represented by the signal; Wx ðb; aÞ is the wavelet transform of the time-series; and thðaÞ is the threshold used to detect the intermittent coherent structures. The indicator function is 1 when jWx ðb; aÞj > thðaÞ; ð3Þ IjWx ðb; aÞj>thðaÞ ¼ 0 when jWx ðb; aÞj4thðaÞ: Note that a discrete implementation of the wavelet transform is used in practice for the discretely measured time-series. Thus, from Eqs. (1) and (2), the expected value of the intermittency rate is given by N 1X EðIjWx ðb; aÞj>thðaÞ Þ N b¼1
ð4Þ
N 1X PðjWx ðb; aÞj > thðaÞÞ: N b¼1
ð5Þ
E½rðaÞ ¼
¼
For a stationary signal, E½rðaÞ ¼ PðjWx ðb; aÞj > thðaÞÞ:
ð6Þ
Based on Eq. (5), we control the intermittency rate rðaÞ under H0 by controlling the probability PfjWx ðb; aÞj > thðaÞg. We do this through the process of building our test criteria – the threshold. Fig. 1 illustrates the process. To build our threshold thðaÞ, we first build exemplars. If the original signal is long, only one exemplar may be required. First we calculate the discrete fourier transform of the time-series, then keep the Fourier transform magnitude but randomize the phase. Hence, if x is our timeseries, each exemplar xei has exactly the same power spectrum as the given timeseries, but is known to be incoherent. Now, each exemplar is wavelet transformed to provide the wavelet-transformed exemplar Wie ðb; aÞ. (The wavelet transform may also be implemented as a convolution in the frequency domain to save some computations.) We then use order statistics from Wie ðb; aÞ to estimate the threshold thðaÞ. Thus, for a discretely sampled x and for each scale a, the threshold thðaÞ is defined as ( ! ) n X N 11X IW e ðb; aÞ>t > g : ð7Þ thðaÞ ¼ thða; g; n; NÞ ¼ sup t : n N i¼1 b¼1 i Here x is the signal with length N, g is the desired false alarm rate, n is the number of exemplars, a is the scale, and thðaÞ is the threshold. With Eq. (7), we can establish the performance of our estimation under H0 as follows: lim EðrðaÞjH0 Þ ¼ g;
n!1
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Fig. 1. Test statistics for the intermittency rate.
where rðaÞ is the intermittency rate and n is the number of exemplars [19]. In practice, very few exemplars are needed for long signals. To see this, consider Eq. (5) and " # N 1X IjWx ðb; aÞj>thðaÞ jH0 : lim E½rðaÞjH0 ¼ lim E n!1 n!1 N b¼1 Now, applying smoothing, we obtain " # N 1X IjWx ðb; aÞj>thðaÞ jH0 lim E n!1 N b¼1 "
# N 1X E IjWx ðb; aÞj>thðaÞ jH0 ; A1 ; :::; AN : ¼ lim E n!1 N b¼1
ð8Þ
Here Ai are defined in Definition 2.1. Under H0 , the time-series and the exemplars have the same distribution, and the phase and the amplitude of the exemplars are independent, so h i ð9Þ E IjWx ðb; aÞj>thðaÞ jH0 ; A1 ; :::; AN ¼ E IjWie ðb; aÞj>thðaÞ jH0 ; A1 ; :::; AN :
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Thus, using the dominated convergence theorem [22], from Eqs. (8) and (9), we " # obtain N h i 1X lim E IjWie ðb; aÞj>tCIED ðaÞ jH0 ; A1 ; :::; AN lim E ½rðaÞjH0 ¼ E n!1 N b ¼1 n!1 " # N 1X g ¼ g: ¼E N b ¼1 ð10Þ Now, we are ready to draw our statistical conclusions. The mean value of rðaÞ is g under H0 . A significant deviation of rðaÞ from g is statistical evidence that the signal contains coherent structures detected by the analyzing function. A theoretically precise although computationally expensive Monte Carlo algorithm may be used to develop test thresholds [20,9]. In practice, we conclude that coherent structures are present if rðaÞ4g. The coherent structure detector can be used to analyze many different phenomena, such as in the author’s papers [9,10,12–16,19,20]. The following example demonstrates both event counting and intermittency rate to analyze turbulence, using 7 h of wind-speed data collected at 50-m height on the meteorological tower at Texas Tech’s Wind Engineering Research Field Laboratory. The data were collected in the early morning hours of May 22, 1998, at 10 Hz on a uvw-anemometer. The driving mechanism was a large synoptic low moving eastward across the Rocky Mountains. Since we are interested in regions of concentrated turbulence in the signal in both techniques, we use the Morlet wavelet as our analyzing function: cðtÞ ¼ eð jctÿt
2
=2Þ
;
where c ¼ 5:4. Scales from 0:5 to 2:7 s are considered. These scales corresponded to center frequencies of 0:3 to 1:6 Hz in the scaled Morlet analyzing function. Intermittency in turbulence has been considered by many authors, with emphasis on the intermittent transfer of energy between scales [23]. These deviations from Kolmogorov’s scaling [24] become increasingly stronger at smaller scales [25,26]. Our viewpoint is different here. These analytic, numeric, and experimental studies focus on intermittency generated by the turbulence itself, such as in stationary flows in a wind tunnel. The atmospheric boundary layer also contains intermittent events driven by localized mesoscale and smaller meteorological phenomena [18], which give a much greater complexity to the turbulent wind field, even in scales much above the dissipative scales [27]. Our research using wavelets and the coherent structure detector has indicated differences in intermittent phenomena in the boundary layer due to different mesoscale phenomena such as thunderstorm outflows, deep lows, and surface heating-induced mixing. Fig. 2 shows the 7 h-long wind-speed waveform. This was measured during winds created by a low-pressure system. We now apply our algorithm to the entire 7 h timeseries. For the first example (the intermittent estimation technique), we use a false alarm intermittency rate of g ¼ 0:001. Fig. 3 is the estimated intermittency rate rðaÞ at each scale for the entire 7 h time-series, in which case rðaÞ4g. The intermittency rate ranges from four to 10 times greater than that expected by false alarms alone. We have strong statistical evidence of intermittent periods of higher turbulence. It is
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Fig. 2. Seven-hour wind time-series.
Fig. 3. The intermittency rate during the 7 h.
impossible to effectively display the detailed images for the entire 7 h (252 000 samples) of data. Hence, from the resulting transforms, we display an interesting portion of the signal, shown in Fig. 4(a). Fig. 4(b) shows the wavelet transform of Fig. 4(a) and Fig. 4(c) shows the thresholded-wavelet transform. The information that is below the intermittent CSD threshold had been zeroed out; only the statistically relevant information remains. (No thresholded-signal or scale-integration can be developed from this transform since a continuous wavelet transform was used. See the discussion in Section 1 on applying discrete wavelet transforms to do time-scale filtering.)
4. Counting discrete events In this section, we consider the coherent event counting technique. As in the previous section, our goal is to define the false alarm rate and describe a method of
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Fig. 4. (a) Wind-speed; (b) wavalet transform; (c) threshold transform.
calculating the number of false alarms in a long time-series. Thus, we first introduce the technique of counting discrete events. Note that there is no unique definition or approach for counting discrete events, but the technique we use here is a useful interpretation for the transient phenomena (coherent events) we seek in wind signals. As we introduced in Ref. [20], we consider a time-series of length N. The entire wind flow signal x is divided into N=L ¼ M blocks of equal length L (assume N=L is integer). Hence, the wavelet transform W is split up into N=L blocks of coefficients, M W ¼ fWblockm ðb; aÞgM m¼1 ¼ fWm ðb; aÞgm¼1 :
Now, we are ready to define the event rate: Definition 4.1. Consider a fixed scale a and threshold thðaÞ. Let N be the length of time-series x with sample spacing Dt. Let Wm be the wavelet coefficients of a particular block. Then, we count an event in a block m at scale a if maxb jWm ðb; aÞj > thðaÞ. Moreover, for the entire time-series, the total number of events at scale a for threshold thðaÞ is defined as KðaÞ
M X m¼1
Imaxb jWm ðb; aÞj>thðaÞ :
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Hence, the event rate is given by gðaÞ ¼
KðaÞ ; NDt
where NDt is the time interval represented by the waveform x. The length L of the blocks plays an important role in counting false alarms. Note that there are many ways to choose the length of the block L. In this paper, we consider the following conditions as critical: *
*
The expected number of events in each block is low so that two events occurring in one block is rare; these two events would be counted as one. L is long enough that a single event rarely crosses over into adjacent blocks. This single event would be counted as two.
Note that these two conditions are opposing, so some middle ground must be found for the choice of L (see Ref. [20] for more details). We now want to compute the test criteria, the thresholds thðaÞ (see Fig. 5). The signal x is first used to generate one or more exemplars. The signal is Fourier transformed and the phase independently randomized uniformly in ½0; 2pÞ to create the exemplars xei in frequency domain. The wavelet transform is then applied in time domain or frequency domain, yielding Wie ðb; aÞ. Each transformed exemplar is then e ðb; aÞj in split into N=L blocks, and order statistics of the the maximum of jWblock m each block are accumulated. The thresholds are now chosen so that each block of length L with a false alarm rate g and sampling interval Dt will have an expected
Fig. 5. Test statistics for event counting.
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probability of false alarm gLDt. thða; L; n; gÞÞ ¼ sup t :
n X M 1 1 X Imaxb jWi;e m ðb; aÞj>t n M i¼1 m¼1
!
) > gLDt :
Here a is the scale, L is the length of a block, n is the number of exemplars used, M is the number of blocks, and g is the desired false alarm rate. At each scale a, this threshold th is used to count events in the original wavelet transform of signal x. More importantly, this threshold is used to display which portions of the original wavelet transform are statistically significant by applying a straightforward thresholding technique Wxthresholded ðb; aÞ ¼ Wx ðb; aÞIjWx ðb; aÞj>thða; L; n; gÞ : Note that lim
n!1
EðKðaÞjH0 ÞÞ ¼ g: NDt
Thus, the threshold is the correct choice to control the false alarm rate. See Ref. [9] for a detailed proof. To demonstrate event counting, we again consider the seven hours of data shown in Fig. 2. This analysis, using a false alarm event rate of one per hour, results in the total number of detections KðaÞ in 7 h at each scale shown in Fig. 6. Note that the total numbers of detections at each scale far exceed the seven expected false alarms in 7 h. See Ref. [20] for details on building a rigorous statistical test, based on the binomial distribution, from these test statistics. In Figs. 3 and 6, both intermittency rate estimation and event counting detected concentrated bursts of turbulence much more often than expected. Since both event counting and intermittency rate estimation are capturing the same phenomena, we expect similar conclusions from the two techniques.
Fig. 6. The event detections during the 7 h.
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5. Conclusions In this paper, we enhance the applicability of the coherent structure detector by developing techniques for long time-series. These techniques provide a statistically rigorous approach to extract and analyze coherent structures in incoherent noise. The first technique, intermittency rate estimation, provides a measure of the fraction of time a coherent structure is present. The second, event counting, counts coherent structures as discrete events. Both give similar overall results but provide different information about the structures. The two may be combined to develop some ideas about the magnitude, frequency, and characteristic lengths of these localized phenomena. Acknowledgements This work was performed under the Department of Commerce National Institute of Standards and Technology/Texas Tech University Cooperative Agreement Award 70NAB8H0059 and the National Science Function Grant #9980296.
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