Compressed-Sensing Reconstruction Based on Block Sparse ... - MDPI

sensors Article

Compressed-Sensing Reconstruction Based on Block Sparse Bayesian Learning in Bearing-Condition Monitoring Jiedi Sun 1, *, Yang Yu 2 and Jiangtao Wen 2 1 2

*

School of Information Science and Engineering, Yanshan University, 438, Hebei Avenue, Qinhuangdao 066004, China Key Laboratory of Measurement Technology and Instrumentation of Hebei Province, Yanshan University, Qinhuangdao 066004, China; [email protected] (Y.Y.); [email protected] (J.W.) Correspondence: [email protected]; Tel.: +86-335-805-7078

Received: 24 March 2017; Accepted: 15 June 2017; Published: 21 June 2017

Abstract: Remote monitoring of bearing conditions, using wireless sensor network (WSN), is a developing trend in the industrial field. In complicated industrial environments, WSN face three main constraints: low energy, less memory, and low operational capability. Conventional data-compression methods, which concentrate on data compression only, cannot overcome these limitations. Aiming at these problems, this paper proposed a compressed data acquisition and reconstruction scheme based on Compressed Sensing (CS) which is a novel signal-processing technique and applied it for bearing conditions monitoring via WSN. The compressed data acquisition is realized by projection transformation and can greatly reduce the data volume, which needs the nodes to process and transmit. The reconstruction of original signals is achieved in the host computer by complicated algorithms. The bearing vibration signals not only exhibit the sparsity property, but also have specific structures. This paper introduced the block sparse Bayesian learning (BSBL) algorithm which works by utilizing the block property and inherent structures of signals to reconstruct CS sparsity coefficients of transform domains and further recover the original signals. By using the BSBL, CS reconstruction can be improved remarkably. Experiments and analyses showed that BSBL method has good performance and is suitable for practical bearing-condition monitoring. Keywords: compressed sensing reconstruction; sparse Bayesian learning; block sparse structure; bearing condition monitoring; wireless sensor network

1. Introduction As critical components in rotating machinery, bearings that are not in a good condition can cause frequent machinery breakdowns [1], and these faults may result in equipment instability, poor efficiency, and even major production-safety accidents [2]. A stable machine-condition monitoring (MCM) system is required to guarantee the optimal states of the bearings during operation [3]. Various physical properties can be utilized to monitor and diagnose the bearing faults. Most MCM systems in the industrial field are based on vibration signals, which are easy to acquire and can provide complete information [4]. In modern industries, some problems exist in online wired MCM systems, such as installation difficulty and high cost, limited power supply and additional long cables. The wireless sensor network (WSN) offers a novel approach to improve the traditional wired MCM systems [5], and it has some advantages such as rapid deployment, removability, and low energy consumption [6]. However, the WSN manifests a number of limitations when applied in vibration-based MCM systems. According to the Nyquist–Shannon sampling theorem, an analog-to-digital converter (ADC) in WSN nodes

Sensors 2017, 17, 1454; doi:10.3390/s17061454

www.mdpi.com/journal/sensors

Sensors 2017, 17, 1454

2 of 20

needs to work at a high frequency because of the high speeds of rolling bearings [7], so does the data processing rates. For effective condition monitoring and fault diagnosis, mass memory should be used to store large packets of vibration signals. In addition, transmitting huge amounts of data via radio-frequency modules increases the time and energy consumption [8]. Thus, reducing the data size before transmission is the best way to solve the above-mentioned problems. Many compression methods have been introduced for the machinery vibration signals in WSN. However, a serious problem is that almost all these methods require collecting original vibration signals according to sample theorem, and then applying complex algorithms to compress data in node processers. Essentially, this processing does not reduce the workload. As an emerged theory, Compressed Sensing (CS) [9] shows great promise in compressing/reconstructing data with low energy consumption. CS framework utilizes a nonlinear projection to preserve the original information in a small quantity of measurement vectors, and then reconstruct the original signals through optimization algorithms [10]. When CS is introduced in WSN systems, signal-acquisition and data-compression can be processed simultaneously in a sensor node, while the reconstruction process is operated at remote host computers [11]. This huge advantage promotes the application of CS in vibration-based MCM systems using WSN. For the first few years after the introduction of CS, not much attention to its application was paid in the machinery-condition monitoring field. For the past two years, a few researchers have been studying the use of CS in machinery fault diagnosis. Zhou [12] mentioned an approach based on sparse representation for fault diagnosis in induction motors. The original feature vectors that described the different faults were extracted from the training samples and kept as the reference. Then, it compared the original feature vectors and real-time features, which are extracted from real time signals to identify the fault. The features extraction was based on orthogonal sparse basis representation, and the sparse solution problem was solved by l1-minimization. Zhang [13] proposed a bearing-fault diagnosis method, which combined the CS theory with the K-SVD dictionary learning algorithm. Several over-complete dictionaries were trained using the historical data of particular operating states. Only with the over-complete dictionary that represents the same fault type with the original signal, the reconstruction signal with matching pursuit (MP) algorithm has the smallest error. The bearing fault can be identified in accordance with the differences. Wang [14] developed a framework for remote machine-condition monitoring based on CS. The faults signals were transformed to the projection space by a measurement matrix that achieved the compressed acquisition and then the compressed data was transmitted to the host computer. Fast iterative shrinkage-thresholding algorithm and parallel proximal decomposition algorithm were combined for CS reconstruction. The reconstructed signals preserve the time-frequency representation signatures of the fault signals and the condition states were diagnosed with traditional methods. Zhu [15] introduced the CS and sparse-decomposition theory for machine and process monitoring. Reference [15] focused on several state-of-the-art applications, especially, the sparse-decomposition-based fault diagnosis. The referenced fault-diagnosis methods were all based on the fact that the fault signal could be constructed using weighted linear combinations of the fault samples in the learned dictionary. The fault types can be identified by comparing the representation coefficients and the reconstruction errors. Wang [16] studied a multiple down-sampling method combined with CS for bearing-fault diagnosis. CS was used to further reduce the amount of data which had been processed by the down-sampling method. Then, the compressive data were partially reconstructed by setting a proper sparsity degree via MP algorithm. The fault features could be identified through the incomplete reconstruction signals which included the specific harmonic components. Du [17] proposed a feature-identification scheme with CS, which did not need to recover the entire original data. The fault features could be extracted directly and quickly in the compressed domain. Reference [17] introduced an alternating direction method of multipliers to solve the CS reconstruction problem. Through the literature review, we noticed that bearings faults were diagnosed based on the features that were extracted from the entirely or partially reconstructed signals from compressed

Sensors 2017, 17, 1454

3 of 20

samples. Therefore, the accuracy level of recovered signal directly influenced the feature extraction and fault recognition. The aforementioned methods can identify faults that have demonstrated distinct and known fault features, especially they are effective when the bearings have a single point of failure. When the bearings have compound faults or unknown fault, however, the signals will present complicated waveforms. Under this circumstance, complete information recovery of the vibration signals is important. Therefore, a high-precision reconstruction algorithm plays a vital role. Almost all the above-mentioned methods use traditional CS reconstruction algorithms. Although these algorithms have good popularity, they are not quite suitable for bearing vibration signals, because these algorithms consider only the sparsity property of the vibration signals. In general, the bearing vibration signals not only have the sparsity property, but also have specific structural features. The structured sparsity model transcends the simple sparsity models. They reduce the size of measurements required to recover a signal steadily. On the other hand, during reconstruction, they enable the users to differentiate the useful signal from the interference in a better manner [18]. Thus, by utilizing the structured sparsity model, it is possible to outperform the state-of-the-art conventional reconstruction algorithms. Block sparse Bayesian learning (BSBL) [19] has the potential to solve the reconstruction problem. BSBL derives from Sparse Bayesian learning (SBL) methodology [20], which was first proposed for regression and classification in machine learning and was introduced for signals with block structures. BSBL not only recovers signals with block structures but also considers the intra-block correlation. BSBL outperforms the traditional CS algorithms and has the capacity to recover non-sparse signals with high precision. It has been successfully applied in the monitoring of fetal electrocardiogram (FECG) and electroencephalogram (EEG) via wireless body-area networks [21]. However, until now, it has not attracted much attention in the field of machinery vibration signals. Based on the properties of the bearing vibration signals and related CS theory, this paper proposed a new reconstruction method combining BSBL and sparsity in transform domain to improve the recovery accuracy. The low-dimension measurement data were acquired in the sensor nodes of WSN, transmitted to workstations via radio frequency (RF) and then recovered using proposed method in the host computer. The bearing conditions can be identified on the basis of reconstructed signals. Experimental results show the reconstruction performance is better than the conventional reconstruction algorithms. The remainder is organized as follows: Section 2 introduces the principle of Compressed Sensing and BSBL. Section 3 details the bearing vibration signal features and commonly used transform domains. The proposed method for bearing-condition monitoring via WSN is also presented in this section. Section 4 presents the experiment results and analysis for the reconstruction methods of the bearing vibration signals. We analyze the reconstruction performances and compare the results of BSBL with other methods, and also discuss the influences of the block sizes and signal-noise ratio. 2. Theoretical Background This section briefly introduces the theories: Bayesian learning.

Compressed Sensing and block sparse

2.1. Compressed Sensing Framework In order to inspect the health conditions of rotating machinery thoroughly, condition monitoring systems are used to collect real-time data from machineries. The monitoring system established at the foundations of traditional Nyquist sampling will collect a large amount of data over a long period of operation. Therefore, it will require massive storage space and transmission capabilities, which prevents the successful application of WSN in condition monitoring. The CS theory breaks the conventional concepts of signal acquisition. CS incorporates data acquisition and compression into one procedure by capturing a small number of samples, which contains most of valuable information [10]. The original signal in time-frequency is mapped to the projection space by a measurement matrix according to CS and the compressed data are achieved. At host computer, the compressed signal can

Sensors 2017, 17, 1454

4 of 20

be recovered from these few samples by a particular reconstruction algorithm [22]. The fundamental noisy pattern of CS can be formed as: y = Φx + v (1) where x ∈ R N ×1 is the high-dimensional vector to be sensed, Φ ∈ R M× N is the measurement matrix where M  N, y ∈ R M×1 is the low-dimensional measurement vector, and v is a noise vector modeling errors incurred during the compression process or noise in the CS system [19]. In many practical applications, hardly any signal x is sparse in time domain, however sparse in a transform domain, thus Equation (1) can be written as: x = Ψθ (2) where Ψ ∈ R N × N is a sparse representation dictionary, which is usually an orthonormal basis matrix of a transform domain. θ is the representation coefficient vector which is sparse. If the choice of the transform basis is appropriate, most of the projection coefficients in θ will be sufficiently small. If only K coefficients are nonzero in θ, the signal is considered to be K-sparse. Equation (1) can be written as: y = ΦΨθ + v = Θθ + v

(3)

where Θ ∈ R M× N is the sensing matrix. In Equation (3), the locations of the largest coefficients in θ that contain important information are unknown. In addition, the measurement vector y is a low-dimensional signal and θ is a high-dimensional signal. Thus, solving θ using the measurement vector y and the sensing matrix Θ is a nonlinear optimization problem. The objective is to find the sparsest solution for θ. Once θ has been recovered, x can be reconstructed via Ψ. Next, we introduce three types of traditional recovery algorithms in CS [22]. The first is Convex Relaxation class which solves a convex optimization problem based on linear programming. Basis pursuit (BP), BP de-noising (BPDN) [23], least absolute shrinkage and selection operator (LASSO) [24] are most popular examples. Another class consists of Greedy Iterative algorithms, which are very dependent on the sparse degree, and the most commonly used methods are matching pursuit (MP) [25] and orthogonal matching pursuit (OMP) [26]. Some derived algorithms are regularized orthogonal matching pursuit (ROMP) [27], compressive sampling matching pursuit (CoSaMP) [28] and sparsity adaptive matching pursuit (SAMP) [29]. These algorithms have low operation costs and high speeds [22]. The third group consists of the Iterative Thresholding algorithms which utilize soft or hard thresholding to restore the signal. The thresholding function depends on the number of iterations. Message passing, expander matching pursuits, sequential sparse matching pursuits and belief propagation are some examples of this kind of algorithms [22]. The above-mentioned reconstruction algorithms are based on the fact that many signals can be described as sparse or compressible [10]. In recent years, researches have been investigating other characteristics to support the better reconstruction. Generally, the presence of large coefficients implies additional structural features [29] and this phenomenon has attracted much attention, especially in many cases where the transform coefficients of signal exhibit additional structures in the form of nonzero values occurring in clusters. Such signals are perceived as block sparse signals [30]. With these block structures, it is possible to express the signal x as a concatenation of a series of blocks. x = [ x1 ...xd ...xd+1 ...x2d ...x N −d+1 ...x N ] T | {z } | {z } | {z } x T [1]

x T [2]

(4)

x T [m]

x[i] is the ith block, and every block has the same length d. There are k nonzero blocks and k < m. The exact locations of the k blocks are uncertain. Using block-sparsity property probably improves the reconstruction performance compared with these methods which only consider the signal’s sparsity as done in the conventional methods. Several novel algorithms that achieve successes using this property have been researched, such as the model-based CoSaMP [18], joint sparsity matching pursuit (JSMP) [31], block-orthogonal matching pursuit (Block-OMP) [30], and orthogonal

Sensors 2017, 17, 1454

5 of 20

matching pursuit like algorithm combined with maximum a posteriori based on Boltzmann machine (BM-MAP-OMP) [32]. However, almost none of these existing algorithms consider the intra-block correlation, i.e., the amplitude correlation among the coefficients in each block. In practice, intra-block correlation exists extensively in various signals. Algorithms based on block sparse Bayesian Learning, which exploits the intra-block correlation, can further improve the reconstruction performance [19]. 2.2. Block Sparse Bayesian Learning BSBL is a recovery method for block-sparsity signals, which not only utilizes the block structures in the signals but also considers the intra-block correlation [19]. It needs less prior knowledge and has more excellent capabilities than conditional reconstruction algorithms. In BSBL, every block x[i] ∈ Rdi ×1 is considered as a parameterized prior model based on the multivariate Gaussian distribution: p( xi ; γi , Bi ) ∼ N (0, γi Bi ), i = 1, 2, ..., m

(5)

where γi and Bi are unknown variables. γi is a positive parameter, which controls the block-sparsity property in x. If γi = 0, the ith block will be zero. The sparsity among blocks can be ensured if most γi are close to zero. The correlation structure in a block interior is perceived via Bi ∈ Rdi ×di , which is a positive definite matrix. If we assume the blocks to be irrelevant, the prior probability distribution of x becomes p( xi ; {γi , Bi }) ∼ N (0, Σ0 ), where Σ0 = diag{γ1 B1 , γ2 B2 , . . . , γm Bm }. The noise vector also has a prior Gaussian distribution p(v; λ) ∼ N (0, λI ), where λ is a positive parameter. Then, the posterior probability of x can be obtained from p( x y; λ, {γi Bi }im=1 ) = N (µ x , Σ x )

(6)

The two critical variables in Equation (6) are unknown, but they can be obtained from µ x = Σ0 Φ T (λI + ΦΣ0 Φ T ) and Σ x = ( Σ 0 −1 +

−1

y

1 T −1 Φ Φ) λ

(7)

(8)

The two above-mentioned equations involve the parameters λ and {γi , Bi }im=1 . If they are solved, the Maximum-A-Posteriori (MAP) value of x can be expressed by Equation (7). We can estimate these parameters through a Type-II maximum likelihood procedure. This procedure can be solved to minimize the next cost function: −1 L(Ω) = log λI + ΦΣ0 Φ T + y T (λI + ΦΣ0 Φ T ) y (9) In Equation (9), Ω consists of all unknown parameters, and can be expressed as Ω , λ, {γi , Bi }im=1 . BSBL comprises three learning rules and the most important learning rule is the estimation of γi , which determines the algorithm convergence speed and recovery quality. In addition, the learning rule for λ is also important and the algorithm performance will not be good if λ is inapposite, even though γi can be perfectly estimated. If the data are noiseless, the global minimum of Equation (9) is often the true sparse solution, which is not subject to Bi . In effect, Bi just influences the local convergence. One can set different constraints for Bi to prevent overfitting [19]. We can utilize Expectation Maximization (EM) to estimate λ, γi and Bi . Consequently, a practical algorithm BSBL-EM is derived from BSBL and EM. The learning rules of BSBL-EM are detailed below. The learning rule for γi is  i 1 h T γi ← Tr Bi−1 Σix + µix (µix ) (10) di 

where Σix ∈ Rdi ×di is the ith main diagonal block in Σ x , and µix ∈ Rdi ×di is the ith block in µ x .

Sensors 2017, 17, 1454

6 of 20

The learning rule for λ is m

T

ky − Φµ x k22 + ∑ Tr (Σix (Φi ) Φi ) i =1

λ←

(11)

M

When the signal has the same length, we set Bi = B(∀i ) to prevent overfitting. The learning rule for B is B←

1 m Σix + µix (µix ) m i∑ γi =1

T

(12)

BSBL-EM assumes that the signal can be divided into a series of non-overlapping blocks. Hence, the block partitions of the original signal need to be defined in BSBL-EM. However, during reconstruction, the defined block partitions need not be the same as the true block partitions. Even though the original data have no obvious block structure, the BSBL-EM is efficient [21]. 3. Proposed Monitoring Method Based on Bearing Signal Reconstruction This paper proposes a reconstruction approach applied in a WSN-based bearing-condition monitoring based on BSBL-EM. It fully utilizes the block sparsity property of fault vibration signals in the transform domain and achieves a better reconstruction performance compared with traditional methods. This section briefly introduces the features and the block structure of bearing vibration signal and then presents the proposed bearing-vibration signal reconstruction method. 3.1. Features of Bearing Vibration Signals in Time Domain and Transformation Domains Studies show the bearing vibration signals are typical nonstationary signals and the sparsity in time-domain is not distinct [33,34]. The sparsity of the raw signal in a specific transform domain is critical in CS. The most widely used transform analysis of bearing fault is Fourier Transformation, i.e., the frequency spectra of the vibration signals. The other commonly used transform domains are discrete cosine transform (DCT) and wavelet packet transform (WPT). Taking the bearing vibration signals provided by Case Western Reserve University [35], we mainly discuss them in the transform domains and analyze the coefficients property in this paper. The data applied in this paper are the vibration signals of a deep-groove ball bearing. The specific bearing type is 6205-2RS JEM SKF. The data are collected by accelerometers, which are placed at the drive ends of the motor housing. The speed of the motor is 1750 rpm and sampling rate is 12,000 dots per second. The four states are: normal, inner-ring fault, rolling-element fault, and outer-ring fault. We used a piece of data with 2000 points as example in order to clarify the results. Figure 1 displays the original bearing vibration signals in four states and the transformation coefficients, respectively. It can be seen from Figure 1 that the fault signals are not apparently sparse, while in Figure 1b the number of large coefficients represented major information is less than the number of small coefficients. Therefore, the bearing-fault vibration signals are much sparser in the frequency domain than in time domain. Moreover, it is obvious that the coefficients in frequency domain exhibit partition characteristics, and it can be considered that the bearing signals possess block structures. A large portion of the blocks in the mid-frequency bands and a few blocks in the other positions are nonzero. Other parts can be considered zero blocks with noise. Consequently, the original signal can be modeled as a block-sparsity signal with unknown block positions and unknown noise.

Sensors 2017, 17, 1454 Sensors 2017, 17, 1454

7 of 20 7 of 20

Figure 1. Bearing vibration signals in time domain and transform domains: (a) time domain; (b) Figure 1. Bearing vibration signals in time domain and transform domains: (a) time domain; frequency spectra; (c) discrete cosine transform coefficients; and (d) wavelet packet transform (b) frequency spectra; (c) discrete cosine transform coefficients; and (d) wavelet packet transform coefficients. The states in the four subpictures are: (1) normal; (2) inner ring fault; (3) rolling element coefficients. Theouter states in fault. the four subpictures are: (1) normal; (2) inner ring fault; (3) rolling element fault; and (4) ring fault; and (4) outer ring fault.

It can be seen from Figure 1 that the fault signals are not apparently sparse, while in Figure 1b theTheoretical number ofand large represented major information is less than thehave number of sparsity small realcoefficients signal analysis shows that the bearing vibration signals better coefficients.domains Therefore, the in bearing-fault vibration signals arewe much the frequency in transform than time domain. In Figure 1c,d, cansparser see theincoefficients of domain DCT and than in time domain. obvious that the coefficients in property frequencyin domain exhibit WPT showed much sparerMoreover, than thoseitofisfrequency domain and the block DCT and WPT is partition characteristics, and it can be considered that the bearing signals possess block structures. A clearly. It is noticed that the WPT coefficients only show good sparsity and block with some fault state, portion the blocksfault, in the mid-frequency bandsofand a few blocks in the other positions are forlarge example theof outer-ring while the coefficients DCT present generally spare for different nonzero. Other parts can be considered zero blocks with noise. Consequently, the original signal can fault states. Moreover DCT has the advantage of simple calculation because of the real numbers in be modeled as a block-sparsity signal with unknown block positions and unknown noise. DCT matrix and it is more beneficial to CS reconstruction algorithms. Theoretical and real signal analysis shows that the bearing vibration signals have better sparsity in transform domains than in time domain. Figure 1c,d, we can see the coefficients of DCT and 3.2. Proposed Method Based on BSBL and BearingInSignal Features WPT showed much sparer than those of frequency domain and the block property in DCT and WPT machine-condition system based WSN requires reducing thewith amount data is A clearly. It is noticed thatmonitoring the WPT coefficients only on show good sparsity and block some of fault as state, muchfor as example possiblethe before transmission. Unfortunately, most conventional data compression outer-ring fault, while the coefficients of DCT present generally spare for techniques much energy DCT and are for the WSN. calculation When CS isbecause used inof WSN-based different dissipate fault states. Moreover hasnot thesuitable advantage of simple the real conditions system, compression stage is reconstruction completed on algorithms. data acquisition module before numbersmonitoring in DCT matrix and itthe is more beneficial to CS transmission, while the reconstruction stage is completed on workstations/computers at remote 3.2. Proposed Method Based on BSBL and Bearing Features terminals. Then, the fault recognition about theSignal bearing will be made according to the reconstructed signals on the host computer. Therefore, high-precision reconstruction the original essential. A machine-condition monitoring system based on WSN requiresofreducing the signal amountis of data Theasbearing-condition method based on CS perfectly meets the above requirements. much as possible monitoring before transmission. Unfortunately, most conventional data compression Thetechniques flowchartdissipate of the method proposed this is presented in Figure 2. is used in WSN-based much energy andin are notpaper suitable for the WSN. When CS

conditions monitoring system, the compression stage is completed on data acquisition module before transmission, while the reconstruction stage is completed on workstations/computers at remote terminals. Then, the fault recognition about the bearing will be made according to the reconstructed signals on the host computer. Therefore, high-precision reconstruction of the original signal is essential. The bearing-condition monitoring method based on CS perfectly meets the above Sensors 2017, 17, 1454 8 of 20 requirements. The flowchart of the method proposed in this paper is presented in Figure 2.

Figure 2. Flowchart of a machinery condition monitoring system based on wireless sensor network. Figure 2. Flowchart of a machinery condition monitoring system based on wireless sensor network.

When the collection node is activated, a few initialization parameters such as measurement When the collection node is activated, a few initialization parameters such as measurement matrix matrix and compression ratio will be set firstly and the sensor nodes start the vibration-data and compression ratio will be set firstly and the sensor nodes start the vibration-data acquisition. acquisition. The low-dimension measurement data can be collected via the measurement matrix. The low-dimension measurement data can be collected via the measurement matrix. When a When a data-collection period is over, these temporary data are saved in memory. Next, the data-collection period is over, these temporary data are saved in memory. Next, the communication communication starts and the measured data are transmitted via RF modules. The host computer starts and the measured data are transmitted via RF modules. The host computer receives the receives the compressed data and the BSBL-EM algorithm is performed to recover the original signals. compressed data and the BSBL-EM algorithm is performed to recover the original signals. In addition, In addition, the preset sparse representation matrix is necessary when the BSBL-EM algorithm works. the preset sparse representation matrix is necessary when the BSBL-EM algorithm works. Finally, Finally, the fault diagnosis is performed on the basis of high-accuracy reconstructed signal. the fault diagnosis is performed on the basis of high-accuracy reconstructed signal. The procedure of proposed method is summarized: The procedure of proposed method is summarized: (1) Construct the measurement matrix and choose the compression ratio. (1) Construct the measurement matrix and choose the compression ratio. (2) Collect the measurement signal with measurement matrix. (2) Collect the measurement signal with measurement matrix. (3) Transmit the measurement signal to host via RF modules. (3) Transmit the the measurement signal tocombined host via RF modules. (4) Perform BSBL-EM algorithm with preset sparse representation dictionary to (4) reconstruct Perform thethe BSBL-EM combined with preset sparse representation dictionary to signal in algorithm transformation domain. reconstruct the signal in transformation (6) Apply the inverse transformation to get domain. reconstruction signal in time domain. (5) Apply the to getsignal, reconstruction in time domain. (7) Extract the inverse featurestransformation from reconstructed and applysignal diagnosis methods to judge fault types. (6) Extract the features from reconstructed signal, and apply diagnosis methods to judge fault types. The bearing-condition monitoring method proposed in this paper combines the advantages of TheBSBL. bearing-condition monitoring methodthe proposed in thisresults paperbased combines the advantages CS and In the next section, we provided experimental on practical bearingof CS andsignals BSBL.and In analyzed the next section, we of provided the experimental results based on practical vibration the impacts key factors. bearing-vibration signals and analyzed the impacts of key factors. 4. Experiments and Analysis 4. Experiments and Analysis Experiments were carried out using the bearing vibration signals provided by Case Western Experiments were using the bearing vibration signals provided by Case Western Reserve University. A carried series ofout experiments were performed to verify the performance of the Reserve University. A series of experiments were performed to verify the performance of the reconstruction methods and the BSBL-EM was compared with some typical recovery algorithms. This reconstruction methods and the BSBL-EM was compared with some typical recovery algorithms. paper also analyzed how the performance of the BSBL-EM is affected by various parameters. This paper also analyzed how the performance of the BSBL-EM is affected by various parameters. 4.1. Comparison with Traditional Reconstruction Algorithms Firstly, the BSBL-EM was verified the reconstructed performance in DCT and WPT domains. In the experiment, a bearing-vibration signal consisting of 2000 samples was chosen and in the condition of

Sensors 2017, 2017, 17, 17, 1454 1454 Sensors

of 20 20 99 of

4.1. Comparison Comparison with with Traditional Traditional Reconstruction Reconstruction Algorithms Algorithms 4.1. SensorsFirstly, 2017, 17,the 1454BSBL-EM 9 of 20 Firstly, the BSBL-EM was was verified verified the the reconstructed reconstructed performance performance in in DCT DCT and and WPT WPT domains. domains. In In

the experiment, experiment, aa bearing-vibration bearing-vibration signal signal consisting consisting of of 2000 2000 samples samples was was chosen chosen and and in in the the condition condition the of inner-ring fault. According to the features of bearing vibration signals, we chose sym6 as the the of inner-ring fault. According tofeatures the features of bearing vibration wesym6 choseassym6 as inner-ring fault. According to the of bearing vibration signals,signals, we chose the wavelet wavelet packet base function. A Gaussian random matrix of size 800 × 2000 was used as the waveletbase packet base A function. A random Gaussian random matrix sizewas 800 used × 2000 wasmeasurement used as the packet function. Gaussian matrix of size 800 ×of2000 as the measurement matrix. matrix. The The observation observation data data can can be be obtained obtained according according to to Equation Equation (3). The The block block size size measurement matrix. The observation data can be obtained according to Equation (3). The block(3). size of BSBL-EM of BSBL-EM in this experiment was set as 25. Figure 3 showed the original signal and the signal of this BSBL-EM in thiswas experiment set as 25. Figure showedsignal the original the signal in experiment set as 25.was Figure 3 showed the 3original and thesignal signaland recovered by recovered by by BSBL-EM BSBL-EM with with DCT DCT and and WPT. WPT. recovered BSBL-EM with DCT and WPT.

Figure 3.3. 3.Original Original signal and sparse block Bayesian sparse Bayesian Bayesian learning with combined withmaximization expectation Figure signal and block learning combined expectation Figure Original signal and block sparse learning combined with expectation maximization (BSBL-EM) algorithm recovery signals: (a) original signal; (b) BSBL-EM recovery signal (BSBL-EM) algorithm recovery signals: (a) original signal; (b) BSBL-EM signal with discrete maximization (BSBL-EM) algorithm recovery signals: (a) original signal;recovery (b) BSBL-EM recovery signal with discrete cosine transform; and (c) BSBL-EM recovery signal with wavelet packet transform. cosine transform; andtransform; (c) BSBL-EM signal with wavelet transform. with discrete cosine andrecovery (c) BSBL-EM recovery signalpacket with wavelet packet transform.

Secondly, BSBL-EM BSBL-EM was was compared compared with with four four classical classical reconstruction reconstruction algorithms: algorithms: BP BP (Basis (Basis Secondly, Secondly, BSBL-EM was compared with four classical reconstruction algorithms: BP (Basis Pursuit) [9], OMP (Orthogonal matching pursuit) [26], IST (Iterative Soft Thresholding) [36], and Pursuit) [9], [26], ISTIST (Iterative SoftSoft Thresholding) [36], [36], and Pursuit) [9], OMP OMP (Orthogonal (Orthogonalmatching matchingpursuit) pursuit) [26], (Iterative Thresholding) LASSO (Least Absolute Shrinkage and Selection Operator) [37] in DCT and WPT domains. These LASSO (Least Absolute Shrinkage and Selection Operator) These and LASSO (Least Absolute Shrinkage and Selection Operator)[37] [37]ininDCT DCTand and WPT WPT domains. domains. These algorithms do not exploit the block structures of the signals. Another common feature in these four algorithms do do not not exploit exploit the the block block structures structures of of the the signals. signals. Another Another common common feature feature in in these these four four algorithms algorithms is that they do not need to know the sparsity degree or other prior information. Figure algorithms is is that that they they do do not not need need to to know know the the sparsity sparsity degree degree or or other other prior prior information. information. Figure Figure 444 algorithms showed the the reconstruction reconstruction results results of of the the four four algorithms. algorithms. showed showed the reconstruction results of the four algorithms.

Figure 4. 4. Reconstructed Reconstructed inner inner ring ring signals signals with with traditional traditional algorithms algorithmsin infour foursubpictures subpicturesare: are:(1) (1) Basis Basis Figure 4. Reconstructed inner ring signals with traditional algorithms in four subpictures are: (1) Basis Figure pursuit; (2) orthogonal matching pursuit; (3) Iterative Soft Thresholding; and (4) least absolute pursuit;(2) (2)orthogonal orthogonal matching pursuit; (3) Iterative Soft Thresholding; and absolute (4) leastshrinkage absolute pursuit; matching pursuit; (3) Iterative Soft Thresholding; and (4) least shrinkage and selection operator. (a) reconstruction reconstruction signals in discrete discrete cosine transform domain; (b) (b) shrinkage and selection (a) signals in cosine transform domain; and selection operator. (a)operator. reconstruction signals in discrete cosine transform domain; (b) reconstruction reconstruction signals in wavelet packet transform domain. reconstruction signals in wavelet packet transform domain. signals in wavelet packet transform domain.

Sensors 2017, 17, 1454 Sensors 2017, 17, 1454

10 of 20 10 of 20

It can be seen in Figures 3 and 4 that the reconstruction signals with traditional algorithms in DCTItdomain and WPT domain had4 larger differences. The signals using the traditional can be seen in Figures 3 and that the reconstruction signalsrecovered with traditional algorithms in algorithms in DCT domain contained large amounts of noise and the signals recovered using BSBLDCT domain and WPT domain had larger differences. The signals recovered using the traditional EM have less noise. Moreover, the reconstruction with inrecovered two domains similar algorithms in DCT domain contained large amountssignals of noise andBSBL-EM the signals usinghad BSBL-EM precision. Thus, the use of the block-sparsity property to recover the original signal is an obvious have less noise. Moreover, the reconstruction signals with BSBL-EM in two domains had similar advantageThus, of thethe BSBL precision. useframework. of the block-sparsity property to recover the original signal is an obvious In the conditioning monitoring advantage of the BSBL framework. system based on Compressed Sensing, the compression ratio is an important factor to the performance of reconstruction signals. In Equation the measurement In the conditioning monitoring system based on Compressed Sensing, the(1), compression ratio is × matrix Φ ∈ ( M ≪ N ) fulfilled the compressive sampling in the projection space. The an important factor to the performance of reconstruction signals. In Equation (1), the measurement compression ratio (CR) can be defined as: M × N matrix Φ ∈ R (M  N) fulfilled the compressive sampling in the projection space. The compression ratio (CR) can be defined as: N  M

CR 

 100%

N −NM × 100% CR = is the length of the original signal and N is the length of the compressed signal.

(13)

(13) where evaluate theoriginal performance of the BSBL, we experimented with the four traditional whereToN further is the length of the signal and N is the length of the compressed signal. algorithms andevaluate the BSBL-EM at differentofcompression ratios, i.e. 20–80%. The mean To further the performance the BSBL, we experimented with thenormalized four traditional square error ( ) was used as the evaluation criterion for reconstruction performance. The algorithms and the BSBL-EM at different compression ratios, i.e., 20–80%. The normalized mean square expression of is Equation (14). error (N MSE) was used as the evaluation criterion for reconstruction performance. The expression of N MSE is Equation (14).

2

yx 2

NMSE ky − x2k2 k xxk22

N MSE =

(14) (14)

2

where thethe original signal. Figure 5 displays the the NMSEs between the wherey isisreconstruction reconstructionsignal signaland andx is is original signal. Figure 5 displays NMSEs between original signals and the signals reconstructed via different algorithms. the original signals and the signals reconstructed via different algorithms.

Figure 5. 5. Normalized ) of of reconstructed andand original signals using five Figure Normalizedmean meansquare squareerrors errors( (N MSEs) reconstructed original signals using algorithms. (a) (a) N MSEs in discrete cosine transform domain;domain; (b) wavelet transform five algorithms. in discrete cosine transform (b) in N MSEs inpacket wavelet packet domain. domain. transform

It can be seen that the reconstructed signals with BSBL-EM had smaller errors compared with It can be seen that the reconstructed signals with BSBL-EM had smaller errors compared with the other algorithms in the given range of compression ratio. Although the NMSE can describe the the other algorithms in the given range of compression ratio. Although the NMSE can describe the reconstruction errors, NMSEs have no ability to display the similarities between original signal and reconstruction errors, NMSEs have no ability to display the similarities between original signal and the reconstructed signal. We used the Pearson correlation coefficient as Equation (14) to evaluate the the reconstructed signal. We used the Pearson correlation coefficient as Equation (14) to evaluate the similarity between the reconstructed signals and the raw signals. similarity between the reconstructed signals and the raw signals.

n  xy   x  y n∑ xy − ∑ x ∑ y rxyrxy = q 2 2q n y 2  ( y)2y ) 2 n∑x x2−( (∑ xx))2 nn ∑ y2 − (∑

(15) (15)

Sensors 2017, 17, 1454 Sensors 2017, 17, 1454

11 of 20 11 of 20

where where r xy isisthe thePearson Pearsoncorrelation correlationcoefficient, coefficient,and andx and andy are are the the reconstructed reconstructed and and raw raw signals, signals, respectively. Figure 6 displays the correlation coefficients between the original signals and the respectively. Figure 6 displays the correlation coefficients between the original signals and the signals signals reconstructed reconstructedvia viadifferent differentalgorithms algorithmsunder underthe thegiven givencompression compressionrates. rates. Sensors 2017, 17, 1454

11 of 20

where is the Pearson correlation coefficient, and and are the reconstructed and raw signals, respectively. Figure 6 displays the correlation coefficients between the original signals and the signals reconstructed via different algorithms under the given compression rates.

Figure6.6. Correlation Correlation coefficients coefficients of of reconstructed reconstructed signals signals and and original original signals signals using using five five algorithms. algorithms. Figure (a) correlation coefficients in discrete cosine transform domain; (b) correlation coefficients inwavelet wavelet (a) correlation coefficients in discrete cosine transform domain; (b) correlation coefficients in packet transform domain. packet transform domain. coefficients of reconstructed signals and original signals using five algorithms. Figure 6. Correlation (a) correlation coefficients in discrete cosine transform domain; (b) correlation coefficients in wavelet

In Figure 6,transform the experimental results indicated that the correlation coefficient between the packet domain. In Figure 6, the experimental results indicated that the correlation coefficient between the reconstructed signal using BSBL-EM and the original signal was above 90% when the compressibility reconstructed and the original wascorrelation above 90% when the compressibility In signal Figure using 6, the BSBL-EM experimental results indicatedsignal that the coefficient between the was under 70%, irrespective of whether the sparsity representation method was DCT or WPT. This reconstructed signal using BSBL-EM and the signal was above 90% when the compressibility was under 70%, irrespective of whether theoriginal sparsity representation method was DCT or WPT. impliedwas thatunder BSBL-EM can recover the bearing vibration signals with satisfactory ensuring 70%, irrespective of whether the sparsity representation was DCT orquality, WPT. This This implied that BSBL-EM can recover the bearing vibration signalsmethod with satisfactory quality, ensuring subsequent fault diagnosis with high fidelity. implied that BSBL-EMwith can recover the bearing vibration signals with satisfactory quality, ensuring subsequent fault diagnosis high fidelity. subsequent fault diagnosis with high fidelity.

4.2. Comparison Comparison with with Other Other Reconstruction Reconstruction Algorithms Algorithms Utilizing Utilizing Block-Sparsity Block-SparsityProperty Property 4.2. 4.2. Comparison with Other Reconstruction Algorithms Utilizing Block-Sparsity Property

In Section Section 4.1, 4.1, BSBL-EM BSBL-EM was was compared comparedwith with four traditional traditional CS algorithms algorithmswhich whichdid did not not utilize utilize In In Section 4.1, BSBL-EM was compared withfour four traditional CSCS algorithms which did not utilize the block-sparsity block-sparsity property the bearing vibration signal.InIn this section, we compared the BSBLthe property ofofthe vibration signal. this wecompared compared BSBL-EM the block-sparsity property ofbearing the bearing vibration signal.In this section, section, we thethe BSBLEM with some algorithms that are based on the block structure of the original signal. EMalgorithms with some algorithms that areon based the block structure of the originalsignal. signal. with some that are based theon block structure of the original Sparsity representations dictionaries were DCT matrix and WPT matrix inexperiment this experiment as Sparsity representations dictionaries were DCT matrix and WPT matrix as as well. Sparsity representations dictionaries were DCT matrix and WPT matrix in in this this experiment well. A segment of bearing the bearing vibrationsignal, signal, consisting of of 2000 samples, was chosen for the for the well. A segment of the vibration consisting 2000 samples, was chosen A segment of the bearing vibration signal, consisting of 2000 samples, was chosen for the experiment, experiment, andsignal this signal in the condition of of the outer-ring fault. Other parameters were set to set to experiment, and this waswas in of the condition the outer-ring fault. Other parameters were and this be signal was inthose the condition the outer-ring fault. Other parameters set torecovered be the same as the same as used in Section 4.1. Figure 7 shows the original signal andwere the signal be the same as those used in Section 4.1. Figure 7 shows the original signal and the signal recovered those used inBSBL-EM. Section 4.1. Figure 7 shows the original signal and the signal recovered using BSBL-EM. using using BSBL-EM.

Figure 7. Outer fault signal and BSBL-EM recovery signals: (a) original signal; (b) BSBL-EM recovery

Figure 7. Outer fault signal and BSBL-EM recovery signals: (a) original signal; (b) BSBL-EM recovery signal with discrete cosine transform; and (c) BSBL-BO recovery signal with wavelet packet transform. signal with discrete cosine transform; and (c) BSBL-BO recovery signal with wavelet packet transform.

Figure 7. Outer fault signal and BSBL-EM recovery signals: (a) original signal; (b) BSBL-EM recovery signal with discrete cosine transform; and (c) BSBL-BO recovery signal with wavelet packet transform.

Sensors 2017, 17, 1454 Sensors 2017, 17, 1454

12 of 20 12 of 20

The compared algorithms were Block-OMP [30], BM-MAP-OMP [32], JSMP [31], Group Lasso [38],The Group BP [39], and StructOMP [40]. All[30], these algorithms exploit the block structures the compared algorithms were Block-OMP BM-MAP-OMP [32], JSMP [31], Group Lassoof[38], signals. A common feature of all these compared algorithms is that they all need to know the prior Group BP [39], and StructOMP [40]. All these algorithms exploit the block structures of the signals. Figure 8 shows the reconstruction theall compared algorithms with DCT and Ainformation. common feature of all these compared algorithmsresults is thatof they need to know the prior information. WPT. Figure 8 shows the reconstruction results of the compared algorithms with DCT and WPT.

Figure 8. Reconstructed signals with block structure algorithms and BSBL-EM. Compared algorithms Figure 8. Reconstructed signals with block structure algorithms and BSBL-EM. Compared algorithms are: (1) Block orthogonal matching pursuit; (2) orthogonal matching pursuit like algorithm combined are: (1) Block orthogonal matching pursuit; (2) orthogonal matching pursuit like algorithm combined with maximum a posteriori based on Boltzmann machine; (3) joint sparsity matching pursuit; (4) with maximum a posteriori based on Boltzmann machine; (3) joint sparsity matching pursuit; (4) Group Group least absolute shrinkage and selection operator; (5) Groupbasis pursuit; and (6) Struct least absolute shrinkage and selection operator; (5) Groupbasis pursuit; and (6) Struct orthogonal orthogonal matching pursuit. (a) reconstruction signals in discrete cosine transform domain; (b) matching pursuit. (a) reconstruction signals in discrete cosine transform domain; (b) reconstruction reconstruction signals in wavelet packet transform domain. signals in wavelet packet transform domain.

It can be seen in Figure 8 that the additional noise was serious when using the DCT. Although It can is beless seen in Figure that the additional noise was in serious using the be DCT. Although the noise when using8WPT, most detail coefficients WPT when domain cannot reconstructed the noise is less using WPT, most detail coefficients WPTobvious. domain The cannot be reconstructed successfully andwhen the distortion in the reconstructed signalsinwere signals recovered via successfully and the distortion in the reconstructed signals were obvious. The signals recovered via the the Block OMP algorithm lost all the original information. In order to evaluate the quality of Block OMP algorithm lost all the original information. In order to evaluate the quality of reconstruction reconstruction via these block-sparsity algorithms, the outer-ring fault signal was also processed via these algorithms,mentioned the outer-ring fault signal was also processed using theblock-sparsity traditional algorithms in Section 4.1. Figure 9 shows the using resultsthe of traditional traditional algorithms mentioned in Section 4.1. Figure 9 shows the results of traditional algorithms in DCT and algorithms in DCT and WPT domains. WPT domains.

Sensors 2017, 17, 1454 Sensors 2017, 17, 1454

13 of 20 13 of 20

Figure9.9.Reconstructed Reconstructedouter outerring ringfault faultsignals signalswith withtraditional traditionalalgorithms. algorithms.Compared Comparedalgorithms algorithmsin in Figure four subpictures are: (1) Basis pursuit; (2) orthogonal matching pursuit; (3) Iterative Soft four subpictures are: (1) Basis pursuit; (2) orthogonal matching pursuit; (3) Iterative Soft Thresholding; Thresholding; and (4)shrinkage least absolute shrinkageoperator. and selection operator. (a)signals reconstruction signals in and (4) least absolute and selection (a) reconstruction in discrete cosine discrete cosine transform domain; (b) reconstruction signals in wavelet packet transform domain. transform domain; (b) reconstruction signals in wavelet packet transform domain.

Figures 8 and 9 display an unexpected phenomenon. In general, the algorithms that utilize the Figures 8 and 9 display an unexpected phenomenon. In general, the algorithms that utilize the structural properties should demonstrate better performances. However, except for the BSBL-EM, structural properties should demonstrate better performances. However, except for the BSBL-EM, none of the algorithms using the block-sparsity property showed a precision higher than that of the none of the algorithms using the block-sparsity property showed a precision higher than that of the traditional algorithms, in particular, the signals recovered in the wavelet packet domain. This traditional algorithms, in particular, the signals recovered in the wavelet packet domain. This problem problem can be analyzed from three angles. Firstly, it has been mentioned that the block-sparsity can be analyzed from three angles. Firstly, it has been mentioned that the block-sparsity reconstruction reconstruction methods need to know some prior information, such as the block position and size, methods need to know some prior information, such as the block position and size, which will directly which will directly affect the recovery results. However, the block sparse structures are actually affect the recovery results. However, the block sparse structures are actually obscure even though obscure even though the energy clusters at some positions. There are also vast numbers of small the energy clusters at some positions. There are also vast numbers of small coefficients at other coefficients at other positions, regardless of the DCT or WPT domain. Most algorithms that use the positions, regardless of the DCT or WPT domain. Most algorithms that use the block sparse structure block sparse structure have been experimented under ideal conditions that have less noise, in the past have been experimented under ideal conditions that have less noise, in the past researches. However, researches. However, practical applications are different from numerical simulations. The structural practical applications are different from numerical simulations. The structural properties of the original properties of the original signal are often implicit and noise is inevitable. Secondly, these blocksignal are often implicit and noise is inevitable. Secondly, these block-sparsity algorithms require sparsity algorithms require corresponding structural features matched with the specific signals. corresponding structural features matched with the specific signals. However, the characteristics of the However, the characteristics of the bearing vibration signals in DCT and WPT domain have great bearing vibration signals in DCT and WPT domain have great differences, as displayed in Figure 1c,d. differences, as displayed in Figure 1c,d. Finally, the bearing vibration signals are typical nonFinally, the bearing vibration signals are typical non-stationary signals; therefore, the sparsity appears stationary signals; therefore, the sparsity appears divergent in different transform domains, for divergent in different transform domains, for example in this paper the coefficients in WPT domain example in this paper the coefficients in WPT domain showed better sparsity and block partitions showed better sparsity and block partitions than DCT domain, therefore the reconstructed results than DCT domain, therefore the reconstructed results under most circumstances had better under most circumstances had better performances. Most algorithms that utilize block structures performances. Most algorithms that utilize block structures cannot achieve good reconstructed cannot achieve good reconstructed performances for the complex real bearing signals if it lacks the prior performances for the complex real bearing signals if it lacks the prior information about the block information about the block structures. In other words, excessive dependence on prior information structures. In other words, excessive dependence on prior information results in the algorithms using results in the algorithms using the block-sparsity property demonstrating worse adaptability than the the block-sparsity property demonstrating worse adaptability than the traditional algorithms in traditional algorithms in practical applications practical applications BSBL framework is different from most of the algorithms that utilize the block-sparsity property. BSBL framework is different from most of the algorithms that utilize the block-sparsity property. As we can see in Figure 7, the signals recovered by the BSBL-EM were almost the same whatever As we can see in Figure 7, the signals recovered by the BSBL-EM were almost the same whatever in in DCT domain or WPT domain. BSBL-EM can well reconstruct the signal when the coefficients DCT domain or WPT domain. BSBL-EM can well reconstruct the signal when the coefficients have have inconspicuous blocks in transformation domain, even if the coefficients are non-stationary. inconspicuous blocks in transformation domain, even if the coefficients are non-stationary. Moreover, Moreover, this algorithm only needs a minimum number of prior knowledge, namely the signals this algorithm only needs a minimum number of prior knowledge, namely the signals characterizing sparsity and block partitions, so the application of BSBL is as simple as that of the traditional CS algorithms.

Sensors 2017, 17, 1454

14 of 20

characterizing sparsity and block partitions, so the application of BSBL is as simple as that of the Sensors 2017, CS 17, 1454 14 of 20 traditional algorithms.

4.3. Effect Effect of of Block Block Sizes Sizes in in BSBL-EM BSBL-EM 4.3. Theimproved improvedperformance performanceof ofBSBL-EM BSBL-EMover overmost mostof ofthe thetypical typicalalgorithms algorithmshad hadbeen beendisplayed displayed The inthe thepast past two comparison experiments. Except the sparsity, block structure is prior the only prior in two comparison experiments. Except the sparsity, block structure is the only condition condition needed when reconstructing the signal with BSBL-EM. In this we investigated needed when reconstructing the signal with BSBL-EM. In this section, wesection, investigated the effectsthe of effects ofblock different sizes in and BSBL-EM and showed different sizesblock in BSBL-EM showed the results.the results. In the the previous experiments, 25.25. ForFor thethe block-sparsity algorithm, the In experiments, the thesize sizeofofeach eachblock blockwas was block-sparsity algorithm, block sizesize is aiskey factor, so how does it affect the the performance of BSBL-EM? To examine this,this, we the block a key factor, so how does it affect performance of BSBL-EM? To examine used various block sizes toto complete satisfactory we used various block sizes completereconstruction, reconstruction,and andthe thereconstructed reconstructed results results were satisfactory forall allconditions. conditions. The The raw raw data data in in this this experiment experiment are are the the inner-ring inner-ring fault fault state state and and the the measurement measurement for matrix is is aa Gaussian random matrix 800 × × 2000 2000and andthe thesparse sparserepresentation representationdictionary dictionaryisis matrix matrix of of size size 800 DCTmatrix. matrix. The The criteria criteria for for evaluating evaluating the the reconstruction reconstruction performance performance were were NMSE NMSE and and correlation correlation aaDCT coefficient.The Theblock blocksizes sizesranged rangedfrom from5 5toto100. 100.Figure Figure displays reconstructed performances coefficient. 1010 displays thethe reconstructed performances of of BSBL-EM various block partitions. BSBL-EM for for various block partitions.

Figure10. 10.NMSE NMSE and correlation coefficients of reconstructed and original signal with Figure and correlation coefficients of reconstructed signalssignals and original signal with different different block sizes. block sizes.

Accordingtotothe the two evaluation criteria, the differences between the various reconstructed According two evaluation criteria, the differences between the various reconstructed signals signals were very small. Therefore, we can consider that the performance of BSBL-EM is not sensitive were very small. Therefore, we can consider that the performance of BSBL-EM is not sensitive to the to thepartitions block partitions the effects block partitions can be ignored. block and theand effects of blockofpartitions can be ignored. 4.4. Effect Effect of of SNR SNR 4.4. Inmost mostindustrial industrialfields, fields, bearing vibration signals contain different of noise. Most In thethe bearing vibration signals contain different levelslevels of noise. Most often, often, denoising processes for acquiring data arefor necessary for thefault subsequent diagnosis. denoising processes for acquiring data are necessary the subsequent diagnosis.fault Therefore, it is Therefore, it is natural to question whether the BSBL-EM can reconstruct the original signal with natural to question whether the BSBL-EM can reconstruct the original signal with less noise, whileless at noise, while atpreserve the sameimportant time, preserve important information. the same time, information. Westudied studiedthe theeffects effectsof ofsignal-to-noise signal-to-noise ratio ratio (SNR) (SNR) and andBSBL-EM BSBL-EM will will estimate estimate noise noise model model We parameter λ according to the learning rule in Equation (11). This section explores the reconstruction parameter λ according to the learning rule in Equation (11). This section explores the reconstruction effectsunder undervarious variousnoise noiselevels. levels. The The original original data data were were bearing bearing outer-ring outer-ringfault faultsignals signalsadded addedwhite white effects Gaussiannoise noisewith withdifferent differentSNR, SNR, measurement matrix was same Gaussian random matrix as Gaussian measurement matrix was thethe same Gaussian random matrix as the the previous experiment and the sparse representation dictionary was DCT matrix. Figure 11 previous experiment and the sparse representation dictionary was DCT matrix. Figure 11 compares compares original and DCT coefficients with the reconstructed under different the originalthe signal and signal DCT coefficients with the reconstructed ones under ones different noise levels.noise levels.

Sensors 2017, 17, 1454 Sensors 2017, 17, 1454 Sensors 2017, 17, 1454

15 of 20 15 of 20 15 of 20

Figure thethe reconstructed signals andand discrete cosine transform coefficients under under three Figure11. 11.Original Originaland and reconstructed signals discrete cosine transform coefficients Figure 11. Original and the reconstructed signals and discrete cosine transform coefficients under signal-to-noise ratios (SNRs): (a) original signal; signal; (b) SNR(b) = 1SNR dB; =(c)1 SNR = 10 dB;= and (d)and SNR(d) = 20 dB. three signal-to-noise ratios (SNRs): (a) original dB; (c) SNR 10 dB; SNR three signal-to-noise ratios (SNRs): (a) original signal; (b) SNR = 1 dB; (c) SNR = 10 dB; and (d) SNR == (a) sample points; (b) discrete cosine transform coefficients. 20 dB. (a) sample points; (b) discrete cosine transform coefficients. 20 dB. (a) sample points; (b) discrete cosine transform coefficients.

As shown shown in in Figure Figure 11, the reconstruction effects were good using BSBL-EM with lowlow SNR. The As in Figure11, 11,the thereconstruction reconstruction effects were good using BSBL-EM with SNR. As effects were good using BSBL-EM with low SNR. The reconstructed DCT coefficients and signals were similar to the original signals when the SNR is The reconstructed coefficients signals were similar theoriginal originalsignals signalswhen when the the SNR is reconstructed DCTDCT coefficients andand signals were similar totothe is higher than 10 dB. Figure 12 shows the correlation coefficient between the original signals without higher higher than than 10 10 dB. dB. Figure Figure 12 12 shows shows the the correlation correlation coefficient coefficient between between the the original original signals signals without without noise and the reconstruction signals at different noise levels with SNR less than 20 dB. noise and the reconstruction signals at different noise levels with SNR less than 20 dB. noise and the reconstruction signals at different noise levels with SNR less than 20 dB.

Figure 12. 12. Correlation Correlation coefficients coefficients between between original original signal signal and and reconstructed reconstructed signal signal with with different different Figure Figure 12. Correlation coefficients between original signal and reconstructed signal with different SNRs. SNRs. SNRs.

There Therewas wasaaagreat greatdifference differencebetween betweenthe thereconstructed reconstructedsignal signaland andoriginal originalone oneunder underthe thelow low There was great difference between the reconstructed signal and original one under the low SNR. However, the reconstructed DCT coefficients still had similarity. The reconstruction performance SNR. However, the reconstructed DCT coefficients still had similarity. The reconstruction SNR. However, the reconstructed DCT coefficients still had similarity. The reconstruction was significantly withimproved the increase ofthe SNR. When of theSNR. SNRWhen exceeds dB,exceeds the reconstructed performance wasimproved significantly improved with the increase of SNR. When the10 SNR exceeds 10 dB, dB, the the performance was significantly with increase the SNR 10 signal had small difference with original ones and the correlation coefficient can reach more than 0.9. reconstructed signal had small difference with original ones and the correlation coefficient can reach reconstructed signal had small difference with original ones and the correlation coefficient can reach more than than 0.9. more 4.5. Effect of0.9. Wavelet Packet Paremeters

4.5. Effect Effect of Wavelet Wavelet Packet Paremeters Paremeters In the past experiments, we used wavelet packet transformation matrix as the sparse 4.5. of Packet representation matrix. Actually, the packet transformation matrixesmatrix have different In the the past past experiments, we wavelet used wavelet wavelet packet transformation transformation as the the forms, sparse In experiments, we used packet matrix as sparse which depend on the wavelet kernels and decomposition levels. In the section, we discuss the CS representation matrix. Actually, the wavelet packet transformation matrixes have different forms, representation matrix. Actually, the wavelet packet transformation matrixes have different forms, reconstruction influenced various wavelet packet which depend dependperformance on the the wavelet wavelet kernels by and decomposition levels.parameters. In the the section, section, we we discuss discuss the the CS CS which on kernels and decomposition levels. In The experiment data were the same as in Section 4.1. The measurement matrix was Gaussian reconstruction performance performance influenced influenced by by various various wavelet wavelet packet packet parameters. parameters. reconstruction random matrix and thedata reconstruction algorithm was BSBL-EM. Firstly, we set thematrix basic wavelet types The experiment were the same as in Section 4.1. The measurement was Gaussian Gaussian The experiment data were the same as in Section 4.1. The measurement matrix was random matrix matrix and and the the reconstruction reconstruction algorithm algorithm was was BSBL-EM. BSBL-EM. Firstly, Firstly, we we set set the the basic basic wavelet wavelet random types as as variable. variable. The The contrastive contrastive wavelet wavelet kernels kernels were were db4, db4, db5, db5, db6, db6, sym4, sym4, sym5 sym5 and and sym6. sym6. The The types

Sensors 2017, 17, 1454 Sensors 2017, 17, 1454 Sensors 2017, 17, 1454

16 of 20 16 of 20 16 of 20

wavelet kernels were similarwavelet to the waveforms of bearing vibration signals. The correlation coefficient as variable. The contrastive kernels were db4, db5, db6, sym4, sym5 and sym6. The wavelet wavelet kernels were similar to the waveforms of bearing vibration signals. The correlation coefficient of reconstruction signals and original signal with different compression ratios is displayed in Figure kernels were similar to the waveforms of bearing vibration signals. The correlation coefficient of of reconstruction signals and original signal with different compression ratios is displayed in Figure 13. reconstruction signals and original signal with different compression ratios is displayed in Figure 13. 13.

Figure 13. 13. Correlation coefficients coefficients between original original signal and and reconstructed signal signal via different different Figure Figure 13. Correlation Correlation coefficients between between original signal signal and reconstructed reconstructed signal via via different wavelet kernels. wavelet waveletkernels. kernels.

It can be seen in Figure 13 that various wavelets kernels have little influence to reconstruction ItItcan canbe beseen seenin inFigure Figure13 13that thatvarious variouswavelets waveletskernels kernelshave havelittle littleinfluence influenceto toreconstruction reconstruction performance. Figure 14 displays the CS reconstruction performance with different levels. In this performance. performance. Figure 14 14 displays displays the the CS CS reconstruction reconstruction performance performance with withdifferent differentlevels. levels. In In this this experiment, the chosen wavelet kernel was sym6. experiment,the thechosen chosenwavelet waveletkernel kernelwas wassym6. sym6. experiment,

Figure 14. Correlation coefficients between original signal and reconstructed signal via different Figure 14. Correlation coefficients between original signal and reconstructed signal via different wavelet14. decomposition levels. Figure Correlation coefficients between original signal and reconstructed signal via different wavelet decomposition levels. wavelet decomposition levels.

In Figure 14, the correlation coefficients showed slight improvements with the increase of In Figure 14, the correlation coefficients showed slight improvements with the increase of wavelet decomposition levels; however, thisshowed change was not obvious. Therefore, is no necessary to In Figure 14, the correlation coefficients improvements with the it increase of wavelet wavelet decomposition levels; however, this changeslight was not obvious. Therefore, it is no necessary to increase the decomposition levels to improve the performance. Actually, the reconstruction decomposition levels; however, this change was not obvious. Therefore, it is no necessary to increase increase the decomposition levels to improve the performance. Actually, the reconstruction algorithms have thelevels greatest influence onperformance. CS reconstruction performance. the decomposition to improve the Actually, the reconstruction algorithms have algorithms have the greatest influence on CS reconstruction performance. the greatest influence on CS reconstruction performance. 4.6. Faults Classification 4.6. Faults Classification 4.6. Faults Classification In the above sections, we only evaluated the CS performance with NMSE and correlation In the above sections, we only evaluated the CS performance with NMSE and correlation coefficients, whichsections, only quantitatively presentedthe the CS reconstruction The ultimate purpose In the above we only evaluated performanceaccuracy. with NMSE and correlation coefficients, which only quantitatively presented the reconstruction accuracy. The ultimate purpose of machinery condition monitoringpresented system is the accuracy. bearing The faults types.purpose Will this coefficients, which only quantitatively theidentifying reconstruction ultimate of of machinery condition monitoring system is identifying the bearing faults types. Will this compressed sampling and reconstruction affect the identification accuracy? In this section, we machinery condition monitoring system is identifying the bearing faults types. Will compressed compressed sampling and reconstruction affect the identification accuracy? In this section, we perform the classification using test signals, which were reconstructed BSBL-EM. sampling andfaults reconstruction affect the the identification accuracy? In this section, weby perform the faults perform the faults classification using the test signals, which were reconstructed by BSBL-EM. The faults classification method waswere based on feature extraction and pattern recognition. We classification using the test signals, which reconstructed by BSBL-EM. The faults classification method was based on feature extraction and pattern recognition. We decomposed the bearing vibration signals with sym6 wavelet packet by five levels and extracted the decomposed the bearing vibration signals with sym6 wavelet packet by five levels and extracted the wavelet packet energy spectrum in 32 decomposition nodes. This wavelet packet energy spectrum wavelet packet energy spectrum in 32 decomposition nodes. This wavelet packet energy spectrum was used as the feature vector. The pattern recognition method was support vector machine (SVM). was used as the feature vector. The pattern recognition method was support vector machine (SVM).

Sensors 2017, 17, 1454

17 of 20

The faults classification method was based on feature extraction and pattern recognition. We decomposed the bearing vibration signals with sym6 wavelet packet by five levels and extracted the wavelet packet energy spectrum in 32 decomposition nodes. This wavelet packet energy spectrum was used as the feature vector. The pattern recognition method was support vector machine (SVM). The training data were bearing vibration signals which include seven states. The states were normal, and six faults. The fault types consisted of inner-ring fault, rolling-element fault, and outer-ring fault. Each fault type was divided into two fault diameters, 7 mils and 21 mils. The number of training data totally was 1400, which consists of 200 sets of data for each bearing state. The number of testing data was totally 280, which consists of 40 sets of data for each bearing states. In each set of data, the sample points were 2000. We tested the faults classification result at different compression ratios. The measurement matrix was a Gaussian random matrix and the reconstruction algorithm was BSBL-EM. The training samples and testing sample are shown in Tables 1 and 2, respectively. Table 1. Training samples. State of the Signal

Motor Load (hp)

Motor Speed (r/min)

Number of Signals

Normal state

0 1 2 3

1797 1772 1750 1730

50 50 50 50

Inner ring fault (2 diameters)

0 1 2 3

1797 1772 1750 1730

100 100 100 100

Rolling element fault (2 diameters)

0 1 2 3

1797 1772 1750 1730

100 100 100 100

Outer ring fault (2 diameters)

0 1 2 3

1797 1772 1750 1730

100 100 100 100

Table 2. Testing samples. State of the Signal

Motor Load (hp)

Motor Speed (r/min)

Number of Signals

Normal state

0 1 2 3

1797 1772 1750 1730

10 10 10 10

Inner ring fault (2 diameters)

0 1 2 3

1797 1772 1750 1730

20 20 20 20

Rolling element fault (2 diameters)

0 1 2 3

1797 1772 1750 1730

20 20 20 20

Outer ring fault (2 diameters)

0 1 2 3

1797 1772 1750 1730

20 20 20 20

In this experiment, we test the faults classification accuracy with different CS compression ratios. Figure 15 displays the fault classification result.

Sensors 2017, 17, 1454 Sensors 2017, 17, 1454

18 of 20 18 of 20

Figure 15. 15. Faults Faults classification classification accuracy accuracy with with various various CS CScompression compressionratio. ratio. Figure

It can be seen that reconstruction signals were able to preserve most of the fault information. It can be seen that reconstruction signals were able to preserve most of the fault information. When the compression ratio was very high, for example 80%, some information was lost and there When the compression ratio was very high, for example 80%, some information was lost and there was the lowest accuracy value; however, when the CR was lower than 70%, all types of bearing faults was the lowest accuracy value; however, when the CR was lower than 70%, all types of bearing faults can be identified accurately. The successful rate of faults classification can be close to 100% when the can be identified accurately. The successful rate of faults classification can be close to 100% when the CR is 40%. CR is 40%. 5. Conclusions 5. Conclusions This paper proposed a bearing-condition monitoring method for WSN using BSBL based on This paper proposed a bearing-condition monitoring method for WSN using BSBL based on Compressed Sensing, which could reduce the burden on the WSN nodes. The compressed data Compressed Sensing, which could reduce the burden on the WSN nodes. The compressed data acquisition in the projection space could acquire fewer measurement data with all the fault acquisition in the projection space could acquire fewer measurement data with all the fault information. information. To accurately reconstruct the original and identify the bearing conditions, we To accurately reconstruct the original and identify the bearing conditions, we investigated BSBL-EM investigated BSBL-EM as the reconstruction algorithm, which considers both the sparsity and the as the reconstruction algorithm, which considers both the sparsity and the block property of the block property of the compressed data. The reconstruction performances of the proposed method compressed data. The reconstruction performances of the proposed method were compared with were compared with that of several traditional CS reconstruction algorithms and some algorithms that of several traditional CS reconstruction algorithms and some algorithms based on block sparsity. based on block sparsity. BSBL-EM demonstrated the best properties in the interpretation of the BSBL-EM demonstrated the best properties in the interpretation of the results. This paper also results. This paper also discussed the performance of BSBL-EM influenced by main factors. The discussed the performance of BSBL-EM influenced by main factors. The experimental results showed experimental results showed the algorithm based on BSBL framework could achieve good the algorithm based on BSBL framework could achieve good reconstruction results under a broad reconstruction results under a broad range of compression ratios and various SNRs; especially, it was range of compression ratios and various SNRs; especially, it was insensitive to block partitions. In the insensitive to block partitions. In the last section, we used SVM to identify the bearing faults. Most last section, we used SVM to identify the bearing faults. Most faults can be classified. When the CR is faults can be classified. When the CR is low, the identification accuracy is close to 100%. low, the identification accuracy is close to 100%. Acknowledgments: The The Project Project Supported Supported by by National National Natural Natural Science Science Foundation Foundation of of China China (No. (No. 51204145), 51204145), Acknowledgments: and Natural Science Foundation of Hebei Province of China (Nos. E2016203223 and E2013203300). We deeply and Natural Science Foundation of Hebei Province of China (Nos. E2016203223 and E2013203300). We deeply appreciate appreciatethe theCase CaseWestern WesternReserve ReserveUniversity Universityfor forproviding providingthe thebearing bearingvibration vibrationdata dataon ontheir theirwebsite. website. Author Contributions: J.S. conceived and designed the experiments. Y.Y. performed the experiments and wrote Author Contributions: and designed the experiments. Y.Y. performed the experiments and wrote the paper. J.W. analyzedJ.S. theconceived data and contributed analysis tools. the paper. J.W. analyzed the data and contributed analysis tools. Conflicts of Interest: The authors declare no conflicts of interest.

Conflicts of Interest: The authors declare no conflicts of interest.

References

References 1. Randall, R.B.; Antoni, J. Rolling element bearing diagnostics-A tutorial. Mech. Syst. Signal Process. 2011, 25, 485–520. [CrossRef] 1. Randall, R.B.; Antoni, J. Rolling element bearing diagnostics-A tutorial. Mech. Syst. Signal Process. 2011, 25, 2. Rai, A.; Upadhyay, S.H. A review on signal processing techniques utilized in the fault diagnosis of rolling 485–520. element bearings. Tribol. Int. 2016, 96, 289–306. [CrossRef] 2. Rai, A.; Upadhyay, S.H. A review on signal processing techniques utilized in the fault diagnosis of rolling 3. El-Thalji, I.; Jantunen, E. A summary of fault modelling and predictive health monitoring of rolling element element bearings. Tribol. Int. 2016, 96, 289–306. bearings. Mech. Syst. Signal Process. 2015, 60–61, 252–272. [CrossRef] 3. El-Thalji, I.; Jantunen, E. A summary of fault modelling and predictive health monitoring of rolling element 4. Patil, M.S.; Mathew, J.; RajendraKumar, P.K. Bearing signature analysis as a medium for fault detection: bearings. Mech. Syst. Signal Process. 2015, 60–61, 252–272. A review. J. Tribol.-Trans. Asme 2008, 130, 7. [CrossRef] 4. Patil, M.S.; Mathew, J.; RajendraKumar, P.K. Bearing signature analysis as a medium for fault detection: A review. J. Tribol.-Trans. Asme 2008, 130, 7.

Sensors 2017, 17, 1454

5. 6. 7. 8. 9. 10. 11.

12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

22. 23. 24. 25. 26. 27. 28. 29.

30. 31.

19 of 20

Hou, L.Q.; Bergmann, N.W. Novel Industrial Wireless Sensor Networks for Machine Condition Monitoring and Fault Diagnosis. IEEE Trans. Instrum. Meas. 2012, 61, 2787–2798. [CrossRef] Gungor, V.C.; Hancke, G.P. Industrial Wireless Sensor Networks: Challenges, Design Principles, and Technical Approaches. IEEE Trans. Ind. Electron. 2009, 56, 4258–4265. [CrossRef] Huang, Q.Q.; Tang, B.P.; Deng, L. Development of high synchronous acquisition accuracy wireless sensor network for machine vibration monitoring. Measurement 2015, 66, 35–44. [CrossRef] Buratti, C.; Conti, A.; Dardari, D.; Verdone, R. An Overview on Wireless Sensor Networks Technology and Evolution. Sensors 2009, 9, 6869–6896. [CrossRef] [PubMed] Donoho, D.L. Compressed sensing. IEEE Trans. Inf. Theory 2006, 52, 1289–1306. [CrossRef] Baraniuk, R.G. Compressive sensing. IEEE Signal Proc. Mag. 2007, 24, 118–121. [CrossRef] Zhang, Z.L.; Jung, T.P.; Makeig, S.; Rao, B.D. Compressed Sensing for Energy-Efficient Wireless Telemonitoring of Noninvasive Fetal ECG Via Block Sparse Bayesian Learning. IEEE Trans. Biomed. Eng. 2013, 60, 300–309. [CrossRef] [PubMed] Zhou, Z.H.; Zhao, J.W.; Cao, F.L. A novel approach for fault diagnosis of induction motor with invariant character vectors. Inf. Sci. 2014, 281, 496–506. [CrossRef] Zhang, X.P.; Hu, N.Q.; Hu, L.; Chen, L.; Cheng, Z. A bearing fault diagnosis method based on the low-dimensional compressed vibration signal. Adv. Mech. Eng. 2015, 7, 12. [CrossRef] Wang, Y.X.; Xiang, J.W.; Mo, Q.Y.; He, S.L. Compressed sparse time-frequency feature representation via compressive sensing and its applications in fault diagnosis. Measurement 2015, 68, 70–81. [CrossRef] Zhu, K.P.; Lin, X.; Li, K.X.; Jiang, L.L. Compressive sensing and sparse decomposition in precision machining process monitoring: From theory to applications. Mechatronics 2015, 31, 3–15. [CrossRef] Tang, G.; Hou, W.; Wang, H.; Luo, G.; Ma, J. Compressive Sensing of Roller Bearing Faults via Harmonic Detection from Under-Sampled Vibration Signals. Sensors 2015, 15, 25648–25662. [CrossRef] [PubMed] Du, Z.H.; Chen, X.F.; Zhang, H.; Miao, H.H.; Guo, Y.J.; Yang, B.Y. Feature Identification with Compressive Measurements for Machine Fault Diagnosis. IEEE Trans. Instrum. Meas. 2016, 65, 977–987. [CrossRef] Baraniuk, R.G.; Cevher, V.; Duarte, M.F.; Hegde, C. Model-Based Compressive Sensing. IEEE Trans. Inf. Theory 2010, 56, 1982–2001. [CrossRef] Zhang, Z.L.; Rao, B.D. Extension of SBL Algorithms for the Recovery of Block Sparse Signals with Intra-Block Correlation. IEEE Trans. Signal Process. 2013, 61, 2009–2015. [CrossRef] Wipf, D.P.; Rao, B.D. Sparse Bayesian learning for basis selection. IEEE Trans. Signal Process. 2004, 52, 2153–2164. [CrossRef] Zhang, Z.L.; Jung, T.P.; Makeig, S.; Rao, B.D. Compressed Sensing of EEG for Wireless Telemonitoring with Low Energy Consumption and Inexpensive Hardware. IEEE Trans. Biomed. Eng. 2013, 60, 221–224. [CrossRef] [PubMed] Qaisar, S.; Bilal, R.M.; Iqbal, W.; Naureen, M.; Lee, S. Compressive Sensing: From Theory to Applications, a Survey. J. Commun. Netw. 2013, 15, 443–456. [CrossRef] Chen, S.; Ma, S.; Dl, D. Atomic decomposition by basis pursuit. SIAM Rev. 2006, 20, 33–61. Tibshirani, R. Regression Shrinkage and Selection via the Lasso. J. R. Stat. Soc. 1996, 58, 267–288. Mallat, S.G.; Zhifeng, Z. Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 1993, 41, 3397–3415. [CrossRef] Tropp, J.A.; Gilbert, A.C. Signal recovery from random measurements via orthogonal matching pursuit. IEEE Trans. Inf. Theory 2007, 53, 4655–4666. [CrossRef] Needell, D.; Vershynin, R. Uniform Uncertainty Principle and Signal Recovery via Regularized Orthogonal Matching Pursuit. Found. Comput. Math. 2009, 9, 317–334. [CrossRef] Needell, D.; Tropp, J.A. CoSaMP: Iterative Signal Recovery from Incomplete and Inaccurate Samples. Commun. ACM 2010, 53, 93–100. [CrossRef] Do, T.T.; Gan, L.; Nguyen, N.; Tran, T.D. Sparsity Adaptive Matching Pursuit Algorithm for Practical Compressed Sensing. In Proceedings of the Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, USA, 26–29 October 2008; IEEE: Pacific Grove, CA, USA, 2008; pp. 581–587. Eldar, Y.C.; Kuppinger, P.; Bolcskei, H. Block-Sparse Signals: Uncertainty Relations and Efficient Recovery. IEEE Trans. Signal Process. 2010, 58, 3042–3054. [CrossRef] Baron, D.; Duarte, M.F.; Wakin, M.B.; Sarvotham, S.; Baraniuk, R.G. Distributed Compressive Sensing. arXiv, 2009, arXiv:0901.3403.

Sensors 2017, 17, 1454

32. 33.

34.

35. 36. 37. 38. 39. 40.

20 of 20

Peleg, T.; Eldar, Y.C.; Elad, M. Exploiting Statistical Dependencies in Sparse Representations for Signal Recovery. IEEE Trans. Signal Process. 2012, 60, 2286–2303. [CrossRef] Jiang, Y.; Li, Z.X.; Zhang, C.; Hu, C.; Peng, Z. On the bi-dimensional variational decomposition applied to nonstationary vibration signals for rolling bearing crack detection in coal cutters. Meas. Sci. Technol. 2016, 27, 16. [CrossRef] Liu, T.; Yan, S.Z.; Zhang, W. Time-frequency analysis of nonstationary vibration signals for deployable structures by using the constant-Q nonstationary gabor transform. Mech. Syst. Signal Process. 2016, 75, 228–244. [CrossRef] Case Western Reserve University Bearing Data Center Website. Available online: http://csegroups.case. edu/bearingdatacenter (accessed on 10 January 2017). Bredies, K.; Lorenz, D.A. Linear Convergence of Iterative Soft-Thresholding. J. Mach. Learn. Res. 2008, 14, 813–837. [CrossRef] Efron, B.; Hastie, T.; Johnstone, I.; Tibshirani, R. Least angle regression. Ann. Stat. 2004, 32, 407–451. Yuan, M.; Lin, Y. Model selection and estimation in regression with grouped variables. J. R. Stat. Soc. B 2006, 68, 49–67. [CrossRef] Van den Berg, E.; Friedlander, M.P. Probing the Pareto Frontier for Basis Pursuit Solutions. SIAM J. Sci. Comput. 2008, 31, 890–912. [CrossRef] Huang, J.Z.; Zhang, T.; Metaxas, D. Learning with Structured Sparsity. J. Mach. Learn. Res. 2011, 12, 3371–3412. © 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).