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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Bearings are not only the most important element but also a common source of failures in rotary machinery. Bearing fault prognosis technology has been receiving more and more attention recently, in particular because it plays an increasingly important role in avoiding the occurrence of accidents. Therein, fault feature extraction (FFE) of bearing accelerometer sensor signals is essential to highlight representative features of bearing conditions for machinery fault diagnosis and prognosis. This paper proposes a spectral regression (SR)-based approach for fault feature extraction from original features including time, frequency and time-frequency domain features of bearing accelerometer sensor signals. SR is a novel regression framework for efficient regularized subspace learning and feature extraction technology, and it uses the least squares method to obtain the best projection direction, rather than computing the density matrix of features, so it also has the advantage in dimensionality reduction. The effectiveness of the SR-based method is validated experimentally by applying the acquired vibration signals data to bearings. The experimental results indicate that SR can reduce the computation cost and preserve more structure information about different bearing faults and severities, and it is demonstrated that the proposed feature extraction scheme has an advantage over other similar approaches.

Bearings are one of the most important components in rotating machinery [

Bearing accelerometer sensor signal analysis-based techniques, which are the most suitable and effective ones for bearing, have been extensively used since in machine prognosis it is easy to obtain sensor signals containing abundant information. These techniques mainly include three categories, namely, time domain analysis, frequency domain analysis and time-frequency domain analysis. Time domain analysis calculates characteristic features of signals statistics such as root mean squares (RMS), kurtosis value, skewness value, peak-peak value, crest factor, impulse factor, margin factor,

Feature extraction means transforming the existing features into a lower dimensional space which is useful for feature reduction to avoid the redundancy due to high-dimensional data [

Spectral methods have recently emerged as a powerful tool for dimensionality reduction and manifold learning [

The rest of the paper is organized as follows: Section 2 proposes the signal processing (including feature calculation) from accelerometer sensors according to the time domain, frequency domain and time-frequency domain. In Section 3, the graph embedding view and SR-based feature extraction approach are introduced. Section 4 gives a description of the experiments and analysis, bearing accelerometer sensor signals from bearings are employed to evaluate the effectiveness of the proposed method. Finally, concluding remarks and future work on this approach are given in Section 5.

To diagnose the abnormality, it is important to record certain physical parameters which vary according to the variation in the operation of the machine [

Time domain features often involve statistical features that are sensitive to impulse faults [

Frequency domain analysis is another description of a signal, that can reveal some information that cannot be found in the time domain [

Time-frequency domain methods are considered to be best way for analyzing non-stationary signals [

The wavelet packet coefficients of a signal

The wavelet packet node energy

In this application, we use a specific wavelet function “DB4” from the Daubechies (DB) wavelets family as the mother wavelet and decompose the vibration signals into four levels. In general, the biggest challenge in wavelet analysis is the selection and determination of the mother wavelet function as well as the decomposition level of signals for the different real-world applications [

In this section, after the graph embedding and SR method are presented, SR-based fault feature extraction approach is proposed to extract useful information from the calculated original features of vibration signals.

The SR is fundamentally based on regression theory and spectral graph analysis, so it can be incorporated into other algorithms easily [

The SR aims at finding a low-dimensional subspace _{1}, _{2}, …, _{m}_{i}^{d}_{1}, _{2}, …, _{m}_{i}^{n}_{i}_{i}_{1}, _{2}, …, _{m}^{T}_{1}, _{2}, …, _{m}^{T}_{ij}_{i}_{j}_{i}_{j}_{i}_{j}_{i}_{j}_{ii}_{j}W_{ji}

In order to remove the arbitrary scaling factor in the embedding, a constraint ^{T}

The optimal

For simply mapping for training samples and new testing samples, we choose a linear function here:
_{i}_{i}

The optimal

This maximum eigen-problem formulation in some cases can provide a more numerically stable solution. In the remainder of this paper, we will develop the SR algorithm based on

The SR has been used in various applications where it has demonstrated efficacy compared to PCA, FA, and some common manifold techniques in both feature quality and calculation efficiency [

Given a training set with _{1}, _{2}, …, _{l}_{l}_{+1}, _{l}_{+2}, …, _{m}_{i}^{n}_{k}_{k}

_{i}

If _{i}_{j}_{j}_{i}

If _{i}_{j}

Otherwise, if _{i}_{j}

_{ij}

If there is no any edge between nodes _{ij}

Otherwise, if both _{i}_{j}_{ij}_{k}_{ij}_{δ}. _{δ (}0 < _{δ} ≤ 1) is a given parameter to adjust the weight between supervised and unsupervised neighbor information. Therein, _{i}_{j}

_{0}, _{1}, …, _{c}_{−1}, which are the largest _{0} is a vector of all ones with eigenvalue 1.

Step4: Regularized least squares: Calculate _{1}, …, _{c}_{−1} with _{k}^{n}_{k}_{k}_{k}

_{1}, …, _{c}_{−1}]. The testing samples or new sample can be embedded into

Feature extraction, which is a mapping process from the measured signal space to the feature space, can be regarded as the most important step for intelligent fault diagnosis systems [

Firstly, we calculate 10 features of the time domain directly from bearing vibration signals and three features of the frequency domain based on FFT. Subsequently, we decompose vibration signals into four scales using WPT with ‘DB4’, and then calculate wavelet packet nodes energy in the fourth level as 16 features of the time-frequency domain. So far, we have obtained 29 initial features from vibration signals (see

Secondly, we extract the most representative features from 29 initial features via the SR-based method. Obviously, very large initial features' dimension will result in decreasing performance of bearing prognosis and therefore also increasing computational costs. How to extract the really effective information of bearing fault is a challenging problem. In this paper, if we choose the first _{1}, …, _{c}_{−1}] in

Based on the new projection data set z using SR-based method, the high-dimensional data space is reduced to a low-dimensional data space, however, retaining the majority of local variation information in the projected data set. With the reduced dimensions and local variance information preservation, the extracted features z will be used as the new input features of pattern recognizers for bearing faults.

Finally, we validate the SR-based method using

Data acquisition is a process of collecting and storing useful data from targeted physical assets for the purpose of Condition-based Maintenance (CBM). This process is an essential step in implementing a CBM program for machinery fault diagnosis and prognosis. To evaluate the effectiveness of the signal processing and feature extraction methods for bearings, the vibration data related to the bearing and the system investigation in this paper were provided by the Bearing Data Center of the Case Western Reserve University (CWRU), and acquired by bearing accelerometer sensors under different operating loads and bearing conditions [

The test-rig shown in

The vibration signals were collected through accelerometers using a 16 channel digital audio tape (DAT) recorder at the sampling frequency 12 kHz. In order to evaluate the performance of the SR-based feature extraction approach proposed in this paper, we separate the experimental vibration data into four datasets, named as D_IRF, D_ORF, D_BF and D_MIX. Specifically, similar to the ORF and BF datasets, the IRF dataset includes five severity conditions,

For the obtained vibration signal data, we calculate original features following the time domain, frequency domain and time-frequency domain for the next feature extraction. Time domain features could be calculated directly from vibration signals using

As mentioned earlier, furthermore, some statistical features of time domain are sensitive to inchoate faults, for instance, RMS and kurtosis values should be able to capture the mutual difference in the time domain signal for the fault and healthy bearings.

The advantage of frequency domain analysis over time domain analysis is its ability to easily identify and isolate certain frequency components of interest. The most widely used conventional analysis is the spectrum analysis by means of fast Fourier transform (FFT), which is a well-established method because of its simplicity.

The Fourier spectrum analysis provides a general method for examining the global energy-frequency distribution. The main idea of spectrum analysis is to either look at the whole spectrum or look closely at certain frequency components of interest and thus extract features from the obtained vibration signal data. On this basis, we calculate frequency domain features, such as frequency center, RMS frequency and root variance frequency using

Time-frequency analysis, which investigates waveform signals in both time and frequency domain, has been developed for non-stationary waveform signals. Traditional time-frequency analysis uses time-frequency distributions, which represent the energy or power of waveform signals in two-dimensional functions of both time and frequency to better reveal fault patterns for more accurate diagnosis. In this study, we decompose vibration signals obtained from the test-rig into four scales using WPT with mother wavelet ‘DB4’,

From

In the technique presented in this paper, the total 29 features were calculated from 10 time domain features, three frequency domain features and 16 time-frequency domain features. In general, it is difficult to estimate which features are more sensitive to fault development and propagation in a machine system, furthermore, the effectiveness of these original features could change under different working conditions. In addition, this amount of original features is too many, thus it could be a burden and decrease the performance of the classifier or recognizer. Therefore, feature extraction and dimension reduction using some related technique are proposed in this study, so that more salient and low dimensional features are extracted for performing bearings fault diagnosis or prognosis.

At first, we take two experiments, each select randomly three features from the total 29 features in the D_MIX dataset, which are illustrated in

In order to validate the performance of SR-based method for feature extraction, SR is originally implemented in the D_MIX dataset, the first

We generally keep the first several eigenvectors corresponding to the large eigenvalues which can keep most variance information of the given data. However, high input data dimensions could decrease the recognition performance of the classifiers and result in more training time cost. Thus, the selection of the number of the eigenvectors should be based on the requirement of the real-world applications [

Similarly, we also perform these four feature extraction algorithms in the D_IRF dataset, the extracted first two and three features are compared in

From the corresponding compared results, we can observe that SR has better projection performance over other three methods, as it can obtain a more clear separation of the clustering on the map for the corresponding severity recognition. This is due to the fact that SR is capable of discovering local structured information of the data manifold. However, PCA aims to discover the global structure of the Euclidean space. For the D_IRF dataset, each of fault severity classes is a local structure, SR preserves the intrinsic geometry structure of the dataset in a low-dimensional space. This illustrates that the local information could be more meaningful than the global information from given dataset in some industrial situations. In addition, LPP shows better performance than PCA and FA, since LPP is also graph embedding method based on the local structure of the manifold. This result indicates that features extracted via spectral graph embedding analysis could be more effective than which extracted via global structure by PCA and FA, which illustrates that SR-based feature extraction is very effective to extract most sensitive features for fault classification and severity recognition tasks. As we know, the clearer the separation, the more robust a classifier is. Consequently, the extracted features by SR are able to improve the performance of the classifiers more effectively, which further proves that SR is capable of extracting the most effective features from original features without too much calculation cost.

In this study,

In order to further evaluate the proposed SR-based method, we adopt other experimental data, in particular, the bearing fault data acquired from an accelerated bearing life tester (ABLT-1) at the Hangzhou Bearing Test and Research Center in China (detailed information is described in [

In this case, we not only compare with PCA, FA and LPP, but also compare with some other Graph Embedding based approaches, such as Laplacian Eigenmap (LE) and Linear Discriminant Analysis (LDA). The experimental results of

It is noted that we tested the performance of the SR processing using the whole training and testing data for feature extraction in this experiment, which is not related to new test samples. In fact, handling data out samples (

This paper has proposed a novel fault feature extraction scheme by adopting SR for bearing accelerometer sensor signals, and is the first time SR was applied to feature extraction of bearing faults. SR combines the spectral graph analysis and regression to provide an efficient and effective approach for regularized subspace learning problems, so that it can extract the most representative features from original calculated features. We adopt

This work was supported by National Natural Science Foundation of China (50674086) and National Students' Innovative Entrepreneurship Training Program of China University of Mining and Technology (CUMT).

An example of three-level wavelet packet decomposition.

The flow chart of the proposed scheme.

The test-rig.

The vibration signal waveforms from the different fault types: (

The normalized time domain features in the D_MIX dataset: (

The normalized time domain features in the D_IRF dataset: (

The single-sided amplitude spectrum based on FFT in the D_MIX dataset: (

The single-sided amplitude spectrum based on FFT in the D_ORF dataset: (

The signals of the decomposed by WPT from: (

The normalized feature analysis of wavelet packet node energy in

The randomly selected three features from the total 29 features in the D_MIX dataset: (

The randomly selected three features from the total 29 features in the D_IRF dataset: (

The data projection result with the first two eigenvectors in the D_MIX dataset using: (

The data projection result with the first three eigenvectors in the D_MIX dataset using: (

The data projection result with the first two eigenvectors in the D_IRF dataset using: (

The data projection result with the first three eigenvectors in the D_IRF dataset using: (

The comparison of the accuracy rate in four datasets.

The 29 initial features from vibration signals.

RMS, SRA, KV, SV, PPV, CF, IF, MF, SF, KF | |

FC, RMSF, RVF | |

WPNE(4,1), WPNE(4,2), …, WPNE(4,16) |

The experimental datasets.

500 | Normal, IRF07, IRF14, IRF21, IRF28 | inner race fault severity | |

400 | Normal, ORF07, ORF14, ORF21 | outer race fault severity | |

500 | Normal, BF07, BF14, BF21, BF28 | ball fault severity | |

400 | Normal, IRF14, ORF14, BF14 | mixed fault classification |

The average value of the time domain features in the D_MIX dataset.

0.073 ± 0.003 | 0.194 ± 0.017 | 0.100 ± 0.004 | 0.141 ± 0.054 | |

0.050 ± 0.003 | 0.0878 ± 0.007 | 0.068 ± 0.003 | 0.080 ± 0.023 | |

2.760 ± 0.192 | 22.252 ± 5.486 | 3.003 ± 0.237 | 6.509 ± 4.491 | |

−0.032 ± 0.098 | −0.050 ± 0.187 | −0.001 ± 0.066 | 0.052 ± 0.196 | |

0.419 ± 0.031 | 3.028 ± 0.390 | 0.645 ± 0.069 | 1.408 ± 0.844 | |

3.038 ± 0.289 | 8.079 ± 0.937 | 3.411 ± 0.344 | 5.061 ± 1.291 | |

3.770 ± 0.389 | 13.603 ± 1.797 | 4.277 ± 0.448 | 7.134 ± 2.533 | |

4.429 ± 0.480 | 17.870 ± 2.389 | 5.051 ± 0.536 | 8.907 ± 3.571 | |

1.240 ± 0.015 | 1.682 ± 0.060 | 1.254 ± 0.012 | 1.382 ± 0.130 | |

2.755 ± 0.198 | 20.998 ± 5.237 | 3.000 ± 0.220 | 6.508 ± 4.492 |

The accuracy rate of classification by

0.9882 | 0.9881 | 0.9457 | 0.9893 | ||

0.9678 | 0.9676 | 0.9331 | 0.9793 | ||

0.9716 | 0.9715 | 0.9536 | 0.9782 | ||

0.9194 | 0.9191 | 0.9085 | 0.9324 |

The accuracy rate of classification by

4.296 | 0.8163 | 0.7231 | |

6.348 | 0.7615 | 0.6912 | |

4.973 | 0.8651 | 0.7984 | |

4.397 | 0.8425 | 0.7661 | |

7.242 | 0.8987 | 0.8214 |