AN APPLICATION OF FUNCTIONAL DATA ANALYSIS TO LOCAL DAMAGE DETECTION

Leśkow, Jacek; Skupień, Maria Article AN APPLICATION OF FUNCTIONAL DATA ANALYSIS TO LOCAL DAMAGE DETECTION Statistics in Transition New Series Provided in Cooperation with: Polish Statistical Association Suggested Citation: Leśkow, Jacek; Skupień, Maria (2019) : AN APPLICATION OF FUNCTIONAL DATA ANALYSIS TO LOCAL DAMAGE DETECTION, Statistics in Transition New Series, ISSN 2450-0291, Exeley, New York, NY, Vol. 20, Iss. 1, pp. 131-151, https://doi.org/10.21307/stattrans-2019-008 This Version is available at: https://hdl.handle.net/10419/207928 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ STATISTICS IN TRANSITION new series, March 2019 131 STATISTICS IN TRANSITION new series, March 2019 Vol. 20, No. 1, pp. 131–151, DOI 10.21307/stattrans-2019-008 AN APPLICATION OF FUNCTIONAL DATA ANALYSIS TO LOCAL DAMAGE DETECTION Jacek Le´ skow1,Maria Skupie´ n2 ABSTRACT Vibration signals sampled with a high frequency constitute a basic source of information about machine behaviour. Few minutes of signal observations easily translate into several millions of data points to be processed with the purpose of the damage detection. Big dimensionality of data sets creates serious difficulties with detection of frequencies specific for a particular local damage. In view of that, traditional spectral analysis tools like spectrograms should be improved to efficiently identify the frequency bands where the impulsivity is most marked (the so-called informative frequency bands or IFB). We propose the functional approach known in modern time series analysis to overcome these difficulties. We will process data sets as collections of random functions to apply techniques of the functional data analysis. As a result, we will be able to represent massive data sets through few real-valued functions and corresponding parameters, which are the eigenfunctions and eigenvalues of the covariance operator describing the signal. We will also propose a new technique based on the bootstrap resampling to choose the optimal dimension in representing big data sets that we process. Using real data generated by a gearbox and a wheel bearings we will show how these techniques work in practice. Key words: damage detection, functional data, functional principal components, informative frequency band. 1. Introduction In recent years, extensive research has been focused on big data problems related to statistical signal processing. The big data problem arises when a structural health monitoring system is supported by on-line sensors producing a signal observed with e.g. 20 kHz frequency. After several hours of observations we have millions of data points that can be used for processing. So far, many practical applications have been based on selecting some segments of data and classical analyses have then been conducted on selected segments. However, modern statistical inference can be based on the whole multi-million points sample when the functional data analysis approach is used (see, for example Horv´ ath and Kokoszka, 2012). This is especially suitable when we deal with time-varying systems and when techniques related to time-frequency analysis are used. For example, in Yang and Nagarajaiah (2014), results are presented on independent component analysis with 1Cracow University of Technology, Poland. E-mail: jlesko[email protected]. 2Pedagogical University of Cracow, Poland. E-mail: mar[email protected]. ORCID ID: https://orcid.org/0000-0003-1480-0810. 132 J. Le´ skow, M. Skupie´ n: An application of functional... wavelet transform. Interesting exploratory studies on frequency response function (FRF) can be found in Staszewski and Wallace (2014). In the case of big data and time varying systems, it is convenient to consider data as curves. For example, for a rotating element the natural data curve would be generated in the interval of the length of a cycle. For other signals, a natural interval may be a second or a minute. From this perspective, the spectrogram (see, e.g. Gryllias, et al., 2017 or Khadersab and Shivakumar, 2018) is a way of converting a big signal into segments via Fourier analysis on sliding blocks creating a time-frequency map. Such a map is exactly a collection of random curves. From this point of view, a multimillion data points segment of a signal generated by a sensor attached to some structure is seen as a collection of curves. In recent years there is a significant research dedicated to wheel bearing diagnostics. Randall and Antoni (2011) presents a review of contemporary techniques. Other publications like (Liu, et al., 2018) or (Jia, et al., 2016) present contemporary artificial intelligence technique and their applications in the wheel bearing diagnosis. However, according to our knowledge, so far no one has implemented modern statistical inference tools, based on functional data approach, to the diagnosis of wheel bearings. In this context we would like to mention research in Spiridonakos and Fassois (2014) dedicated to functional time series and their applications to nonstationary random vibrations, where a functional approach is proposed in a different context. The main line of our article is to show how to use modern statistical tools like functional data analysis or bootstrap to efficiently process big data sets and identify significant frequencies. We propose a new perspective in looking at a very popular tool in signal analysis such as the spectrogram. In Subsection 3.1. we explain the difference between classical spectrogram and new functional one proposed by us. Usually, the spectrogram is generated by a signal coming from an excavating machine or by a signal generated by a wheel bearing (see, for example Cioch, et al. 2013). Then, the proper functioning of the tested system is diagnosed by identifying the frequency band where the signal impulsivity is most marked, called in the sequel informative frequency bands (IFB) (see Randall and Antoni, 2011 or Obuchowski, et al., 2014). We propose to view the spectrogram as a collection of random curves. Using the functional data analysis approach, we are able to process such data and quickly solve the problem of identifying the IFB. The main advantage of our method is a possibility of using large data sets to generate few dimension of the diagnostic analysis. Millions of data points are represented as random curves, then in the appropriately defined infinitely dimensional Hilbert space the covariance operator is considered and its empirical counterpart is studied. Finally, only few eigenvalues and eigenfunctions of the empirical covariance operator are sufficient to efficiently represent the signal at hand. We propose a novel technique of using bootstrap resampling in deciding on dimensionality reduction. Another main advantage of our method is that no matter what frequency of signal sampling might be, our method provides a uniform result. On the diagram in Figure 1 we show the logic of our STATISTICS IN TRANSITION new series, March 2019 133 approach. Figure 1: Flowchart of the main idea of our paper. Our article is organized as follows. In Section 2, we present main elements of the functional data analysis approach as applied to signals. Basic tools such as functional principal components, Hilbert space valued random transformations, variance and covariance operators are presented there. In Section 3, we apply this approach to the problem of identifying the informative frequency bands for a spectrogram. In that context we show that without discarding any data points we are able to reduce the dimensionality of data to just a few eigenvalues and few eigenfunctions and retain more than 80% of its energy. Finally, in Section 4 we provide a short discussion of our results. 2. Functional data approach in statistical signal processing Our starting point here is the new perspective on statistical signal processing from the functional data analysis point of view. To start, assume that we observe a signal {X(s):s∈[0,T]}and the collect time Tis really huge, e.g. in the order of several millions of individual data points. We will view such a signal as a collection of random curves {Xn(s),s∈[n,n+w]}each defined on the interval [n,n+w]with the width w. These curves may be considered independent or correlated, depending on the model and a context of study. For example, the spectrogram technique Gryllias, et al., (2017), Khadersab and Shivakumar, (2018) transforms a long signal {X(s):s∈[0,T]}into a collection {x1(f),...,xN(f)}of spectral densities defined on a common frequency interval [0,Λ], where f∈[0,Λ]. In the Subsection 2.3, in the last algorithm, we explain what our functional observations are and how they were obtained from a discrete vibration signal. We assume those observations to be independent. We are aware that the technic of the overlapping window may introduce some dependence into our data structure. At this point we neglect this dependence and we proceed as if the data were independent. However, in the 134 J. Le´ skow, M. Skupie´ n: An application of functional... literature there is a number of cases in which methods of functional data analysis have been adapted to series or signals by cutting them as if they were curves observed independently (see Ramsay and Silverman (2002) and (2005)). To simplify our notation and with no loss of generality, we will assume that Λ=1 so all data curves are defined on the unit interval [0,1]. The observed curves will be assumed to be square integrable. A natural choice of the realization space will therefore be the Hilbert space H=L2[0,1]. This is a consequence of the expansion methods (see Ramsay and Silverman (2002) and (2005)), where the functional form of curves is obtained by a linear span of base functions. The relevant coefficients are then estimated from discrete observation of curves at different time points by least squares methods. This point of view allows us to introduce a Hilbert space of squared integrable functions, where the theory of functional principal component analysis (and many other functional methods) can be applied. From this perspective our initial signal {X(s):s∈[0,T]}can be viewed as a collection of random curves {Xn}, each in the space H. For such random curves, we will now introduce concepts of mean, variance and covariance. Note that each random curve Xis as a random element acting from some probability space (Ω,F,P)onto L2[0,1]. If Xis integrable, then there is a unique function µ∈L2such that Ehy,Xi=hy,µifor each y∈L2. It follows that µ(t) = E[X(t)] for all t∈[0,1]. Here h·,·i is the scalar product in the Hilbert space Hdefined as hx,yi=R1 0x(s)y(s)ds with the norm defined as: kfk=qR1 0f2(t)dt for all f∈H. For more mathematical details regarding statistics on Hilbert space the reader is referred to Horv´ ath and Kokoszka (2012). We recall here the notion of spectral decomposition for matrixes and functional operator. Theorem: 1. Suppose Ais a symmetric, positive definited k×kmatrix. Then, there is an orthonormal matrix U= [u,...,uk]whose columns are the eigenvectors of A, i.e. UTU=Iand Auj=λjuj Moreover, UTAU =Diag[λ1,...,λk]The orthonormality of Uis equivalent to the assertion that the vectors u,...,ukform an orthonormal basis in the Euclidean space Rk. Theorem 1 implies that A (k×k)= k ∑ i=1 λiui (k×1) uiT (1×k) =U (k×k) Λ (k×k) UT (k×k), a representation known as a spectral decomposition of A. The above ideas can be easily extended to a separable Hilbert space. Suppose Ψ Ψ Ψis a symmetric positive–definite Hilbert–Schmidt operator in L2. Covariance operator (1) and its sample counterpart (2) are in this class, provided EkXk4<∞. The STATISTICS IN TRANSITION new series, March 2019 135 operator Ψ Ψ Ψthen admits the functional counterpart of spectral decomposition (1) hΨ Ψ Ψ(x),xi=D∞ ∑ i=j λjhx,vjivj,xE= ∞ ∑ i=j λjhx,vji2 where scalars λjare eigenvalues and vjcorresponding eigenfunctions, satisfying equation Ψ Ψ Ψ(vj) = λjvj. 2.1. Theoretical and empirical covariance operators Since we are adopting the Hilbert space approach, the usual covariance will be an operator, that is a transformation from the Hilbert space to the Hilbert space. This is analogous to the traditional concept of covariance, where a real-valued signal Xgenerates a covariance function transforming real values to real values. Let us have a closer look at the formal definition of the covariance operator. For Xintegrable and EX=0, the covariance operator of Xis defined by C(x) = E[hX,xiX],x∈L2,(1) where C(x)(t) =E[hX,xiX(t)] = EZ1 0 X(s)x(s)dsX(t) = =Z1 0 E[X(s)X(t)] | {z } =c(s,t) x(s)ds =Z1 0 c(s,t)x(s)ds. In the sequel, the covariance operator Cwill be our central point of a study as it fully describes the energy generated by the random element X, which in turn represents a signal under study. While studying the covariance operator, we will focus on characterizing its eigenvalues. They will be important in reducing the dimensionality of Cto just a few of non-negative numbers. For more theoretical properties of the covariance operators see Horv´ ath and Kokoszka (2012). The main task of statistical signal processing in the functional data analysis context will be to introduce an empirical covariance operator ˆ Cthat is fully defined by random curves x1,...,xNand for sufficiently large sample size Napproximates the theoretical covariance operator Cthat describes the signal of interest. Therefore, assume that a sample of random functions x1,...,xNcorresponds to the signal X. Recall that the spectrogram can be viewed as a collection of random curves with arguments in the frequency interval. In general, however, such random functions can represent segments of signals from different time intervals or replica of signals collected via some transformations. 136 J. Le´ skow, M. Skupie´ n: An application of functional... For x1,...,xNwe define the sample covariance operator as: ˆ C(x) = 1 N N ∑ i=1 hxi,xixi,x∈H.(2) It is important to note that in the formula (2) the symbol xcorresponds to any function xfrom the Hilbert space Hwhile xiis the observed random function xigenerated by the signal of interest. In such a way the estimator ˆ Cgiven in (2) approximates the theoretical covariance operator Cdefined in (1). For more mathematical theory related to this approximation the reader is refereed to (Bosq, 2000). For covariance operators which are symmetric, positive defined and are defined on a Hilbert space and are Hilbert-Schmidt operators we have a very interesting property. Suppose Ψ Ψ Ψis a symmetric, positive definite Hilbert–Schmidt operator with eigenfunctions vjand eigenvalues λj, satisfying λ1>λ2>···. Then, sup kxk=1 {hΨ Ψ Ψ(x),xi:hx,vji=0,1≤j≤i−1,i<p}=λi and the supremum is reached if x=vi. The maximizing function xis unique up to a sign. The upper bound pfor index iis defined in Subsection 2.2. Of course, the empirical covariance operator ˆ Cgiven in (2) satisfies the above property. This in turn gives us the following facts fundamentally important in the subsequent statistical considerations: •the empirical covariance operator ˆ Cdefined in (2) is fully defined by its eigenfunctions and eigenvalues, •the covariance and the total variance of the sample (thus the signal) will be described by the estimated eigenvalues. In what follows, we will show how to apply these facts. To start, recall that the random functions x1,...,xNcorrespond to the signal of interest. Now, fix the integer number pN. Next, choose the basis u1,u2,... in Hsuch that: ˆ S2=N ∑ i=1  xi− p ∑ k=1 hxi,ukiuk   2←min Then, each curve xican be approximated by p ∑ k=1 hxi,ukiuk= p ∑ k=1 ckuk. Hence, an infinite dimensional curve xiis represented by a pvariate vector (hxi,u1i,...,hxi,upi). Now we will use the fundamental fact that the basis elements u1,...,upcan be chosen to correspond to the eigenfunctions of the sample covariance ˆ C. More precisely, functions ˆu1,ˆu2,..., ˆupminimizing ˆ S2are equal (up to a sign) to normalized eigenfunctions of the sample covariance operator ˆ C. Note that scores hx,ˆuki=R1 0x(t)ˆuk(t)dt measure the importance of the kth function ˆukin the representation STATISTICS IN TRANSITION new series, March 2019 137 x≈ p ∑ k=1 hx,ˆukiˆuk.(3) In the sequel, we will call ˆukthe k−th functional principal component. One of its important properties is orthonormality. The eigenvalues are extremely important in describing the total energy of the signal. For a random element Xwith values in the Hilbert space Hwe have: EkXk2= ∞ ∑ j=1 EhX,vji2= ∞ ∑ j=1 hC(vj),vji= ∞ ∑ j=1 λj. The quantity EkXk2can be called theoretical total variance. Its empirical equivalent, sample total variance, based on a sample of random functions x1,...,xNis defined as: 1 N N ∑ i=1 kxik2=1 N N ∑ i=1 hxi,xii=1 N N ∑ i=1DN ∑ j=1 hxi,ˆujiˆuj, N ∑ j=1 hxi,ˆujiˆujE= = N ∑ j=1 1 N N ∑ i=1 hxi,ˆuji2= N ∑ j=1 hˆ C(ˆuj),ˆuji= N ∑ j=1 ∞ ∑ k=1 ˆ λkhˆuj,ˆuki2 |{z } =δjk = N ∑ j=1 ˆ λj, where δjk is the Kronecker delta - a function of two variables, defined as follows: δjk =δ(j,k) = (1,if k=j 0,if k6=jand ˆ λjis interpreted as variance in the direction ˆuj. In other words, the empirical functional principal component ˆujgenerated by the empirical covariance operator ˆ Cexplains the fraction of the total sample variance equal to ˆ λj/N ∑ k=1 ˆ λk. The above approach will be referred to as the Functional Principal Component Analysis or FPCA for short. 2.2. Reduction of dimensionality While working with big data sets generated by signals, the crucial point is to select the number pof eigenvalues that give a reasonable approximation of the sample total variance. One of the methods of selecting p, for which function xihas the best approximation given by the formula p ∑ j=1 hxi,ˆujiˆujis the CPV method. This method is based on calculating the cumulative percentage of the total variance 138 J. Le´ skow, M. Skupie´ n: An application of functional... (CPV) explained by the first pempirical functional principal components CPV (p) = p ∑ i=1 ˆ λi N ∑ i=1 ˆ λi .(4) We choose pfor which CPV(p)exceeds a desired level. Ideally, one would like to recover 100% of the total variance, however in practical situations we usually settle with 80% or higher. Such approach has a dramatic effect on our ability to process big data generated by the signals observed with high frequency over long periods of time. First, we split the signal into a sequence of random functions and then we follow the approach above to identify the first peigenvalues. In the following section we will show that for vibration data coming from the gearbox of the excavating machine choosing pas small as 8 retrieves a large percentage of the total variance. Below, we present our original method, based on bootstrap technique, which allows us to precisely evaluate the percentage of variance explained with a confidence interval. CPV bootstrap algorithm. Step 1. We start from the initial sample of random functions x1,...,xN. We sample with replacement the first bootstrap sample x∗1 1,...,x∗1 Nfrom the initial set x1,...,xN. It is important that the bootstrap sample is of the same size as the original one. For such bootstrap sample we calculate the first bootstrap value CPV (p)∗1 of CPV (p)(see formula (4)). Step 2. We repeat Step 1 B times. Usually, we take B = 1000. As a result, we get B bootstrap replications {CPV(p)∗1,...,CPV (p)∗B}. Step 3. We produce a 95% confidence interval for CPV(p)using 2.5% and 97.5% empirical quantiles from the replications {CPV (p)∗1,...,CPV (p)∗B}. We illustrate the logic of our bootstrap procedure on the diagram in Figure 2. The below procedure is admissible from the statistical point of view as it is reconstructing the true unknown distribution of the CPV(p), which in turn is based on the unknown distribution of eigenvalues. For a more detailed discussion related to eigenvalues distribution in the functional approach see e.g. (Mas, 2002). We would like to emphasize that our method allows us to analyse big data sets generated by signals observed over a long period of time using just 8 eigenvalues and 8 associated eigenfunctions. In general, one can start with even 2 eigenvalues, calculate the CPV (3)/CPV (2)and bootstrap it to get its confidence intervals and then see whether adding third eigenvalue significantly improves CPV(3) as compared to STATISTICS IN TRANSITION new series, March 2019 145 Figure 6: First four FPC with scatterplots of scores. 146 J. Le´ skow, M. Skupie´ n: An application of functional... 3.2. Cumulative percentage of variance (CPV) study The previous subsection was devoted to illustrating how our FPCA method works in practice. For the first eight empirical functional eigenvalues ˆ λ1,...,ˆ λ8we have obtained a quite reassuring result: they represent as much as 81.88% of variance. From the statistical perspective, however, we would like to get more information on the variability of the CPV (p)coefficient. In other words, we would like to be able to measure the variability of our estimate with the point value of 81.88%. To answer this question, we will apply the CPV bootstrap algorithm introduced in the previous Section 2. The statistical features of the bootstap distribution of CPV(8) are shown in Figure 7. Descriptive statistics of CPV(8) Values Minimum 80.43000 2.5% quantile 81.02000 Median 81.78000 Mean 81.79348 97.5% quantile 82.54025 Maximum 82.98000 Variance 0.14648 Skewness −0.00285 Kurtosis 3.07514 Figure 7: Descriptive statistics for CPV (8)based on bootstrap samples. The most important message from the above calculations is that the 95% confidence interval for CPV(8) is from 81.02% (the 2.5% quantile) to 82.54% (the 97.5% quantile). This means that CPV as a random variable is quite concentrated around its point value 81.88% and that our results are quite reliable and have only a small spread. To analyse the speed of convergence CPV (p+1)/CPV (p)p→∞ −→ 1we have performed the bootstrap distribution study of this ratio and have obtained the results presented in Figure 8. STATISTICS IN TRANSITION new series, March 2019 147 Descriptive statistics of CPV(9)/CPV(8) Values Minimum 1.01612 2.5% quantile 1.01719 Median 1.01859 Mean 1.01864 97.5% quantile 1.02023 Maximum 1.02113 Variance 6.56e−07 Skewness 0.19956 Kurtosis 2.95591 Figure 8: Normality of CPV (9)/CPV (8)based on bootstrap samples. Again, we see the usefulness of the bootstrap method. From the bootstrap method we get the 95% confidence interval for the proportion CPV (9)/CPV (8)is [1.017;1.020]which means that increasing p from 8 to 9 we will get only 2% more of the variance explained. This is sufficient argument to stop at p=8. Adding more FPC’s dose not improve significantly total variance explained. 3.3. Application to informative frequency bands In our experiment, we set the threshold Ldefined in (5) as 80%. Therefore, for our vibration data coming from the gearbox we will be looking for two sets: Aspectr and AFPCA such that IE E(Aspectr)≥80% and IE E(AFPCA)≥80%.We will see how close those two sets are on real applications using vibration signals. Searching for Aspectr that satisfies (5) may be quite time consuming as we have to consider all possible subsets of frequencies from the set [0,Λ]. For example, for the vibration gearbox data one would have to deal with 2513 combinations! As described in the previous Section, we start the search by identifying the frequency f1 that maximizes ∑T t=0|ST FT(t,f)|2, then select the second in the order of the energy contribution and so on. The analysis based on CPV (p)presented in the previous subsection has shown that p=8first functional principal components reproduce as much as 81.88% of the total variability in the data. Therefore, the FPCA method induced by the spectrogram of gearbox signal creates the eight-dimensional vector of scores, taken from the functional expansion. This means reducing the dimensionality of our problem 148 J. Le´ skow, M. Skupie´ n: An application of functional... from 1265 windows with 513 frequencies in each to 8×513 considering spectrogram data as a functions of frequencies. This means, that finding AFPCA is much faster than Aspectr, especially for large data sets. Below, we provide a listing of all frequencies pertaining to Aspectr and AFPCA. Aspectr ={447.13,455.11,439.14,463.1,431.16,471.08,423.17,479.06,415.19, 542.94,550.92,534.96,487.05,526.97,558.91,518.99,566.89,407.2, 495.03,511,574.88,503.02,335.35,343.33,327.36,399.22,351.31, 319.38,359.3,391.24}. Aspectr has 30 elements. AFPCA ={447.13,455.11,439.14,463.1,431.16,471.08,423.17,479.06,415.19, 542.94,534.96,550.92,487.05,526.97,558.91,407.2,518.99,566.89}. AFPCA has 18 elements. Note that the first 10 frequencies coincide, up to the second decimal point (in Hz). If we consider frequencies from the intersection B=Aspectr ∩AFPCA , then the percentage of signal energy describing IFB is already at the level of 74,03%, not very far from the threshold of 80%. We have applied the above analysis to the data set generated by a wheel bearing and described in Cioch et al. (2013). The sets Aspectr and AFPCA are shown below for these data Aspectr ={1272.51,1291.23,1253.8,1309.94,3106.43,3125.15,3087.72,1235.09,1328.65 3143.86,3069.01,1347.37,3162.57,3050.29,1216.37,1366.08,3031.58,3181.29 1197.66,1384.8,} AFPCA ={1272.51,1291.23,1253.8,1309.94,3106.43,1235.09,3125.15,1328.65,3087.72 3143.86,3069.01,1347.373162.57,1216.37,3050.29,1366.08,3181.29,3031.58 1384.8,1197.66}. A careful examination of the above listings for both data sets shows that both sets contain the same frequencies. They are shown in the order of their importance in energy explained in Section 2. Therefore, the only change we have using the STATISTICS IN TRANSITION new series, March 2019 149 FPCA is the change of the order of the energy importance of the frequency. Observe, however, that such a change is not dramatic. FPCA preserves the order of the first five frequencies and then makes only small changes, never bigger than two places in the order of energy. 4. Conclusions Our article is devoted to introducing the functional data approach to analyse big data generated by signals available for structural health monitoring. We show that applying the FPCA - the functional principal component approach - we can reduce the dimensionality of the data from several millions to several thousands. Using such an approach we show the importance of the eigenfunctions and eigenvalues calculated for functions generated by observing the signal. 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