## 6 Conclusions  
In this work, a hybrid bearing fault diagnosis approach that combines the improved fast spectral correlation and PSO-RF was proposed and investigated. The main innovations of this work are as follows: (1) an improved fast spectral correlation was developed, which is more immune to noise compared to fast spectral correlation and can accurately extract the fault features of bearings; (2) an  
Figure 19: The classification results of (**a**) extreme learning machine (ELM); (**b**) support vector machine (SVM). Figure 18: The recognition results of: (**a**) empirical mode decomposition-singular value decomposition (EMD-SVD); (**b**) multi-scale permutation entropy (MPE). effective classifier named PSO-RF was developed, which improves the precision of RF for identifying and classifying bearing faults. Two experimental datasets of bearing faults were studied. The analysis results demonstrate the presented approach can accurately identify the health states of rolling bearings. Moreover, the presented approach was compared to some state-of-art works to validate its effectiveness from two main aspects. One focuses fault feature extraction, in which the improved fast spectral correlation used as the fault feature method in this paper was compared with the EMD-SVD and MPE methods. The results reflected that the improved fast spectral correlation method has higher classification accuracies than the EMD-SVD and MPE methods. The other focuses on bearing faults classification, in which the classifier of PSO-RF was compared with RF, ELM and SVM. The results show that the PSO-RF method has the highest classification accuracy out of these. Therefore, this paper provides a good method for rolling bearing fault identification, which can be applied to the actual fault identification. **Author Contributions:** [PERSON] and [PERSON] conceived and performed the experiment; [PERSON] proposed the method, analyzed the data and wrote the paper; [PERSON] gave some valuable suggestions; [PERSON] contributed the material of the experiment. **Funding:** This work was supported by the Fundamental Research Funds for the Central Universities (No. 2017 XS134) and the National Natural Science Foundation of China (Grant No. 51777074, 51475164). **Conflicts of Interest:** The authors declare no conflicts of interest. **Nomenclature**  
\(F_{s}\)  
sampling frequency  
\(t_{n}\)  
time instants  
\(R_{\pi}(t_{n},\tau)\)  
cyclic autocorrelation function of \(x(t_{n})\)  
\(\tau\)  
time delay  
\(\alpha\)  
cyclic frequency  
\(f\)  
frequency  
STFT  
Short time Fourier transform  
\(N_{w}\)  
window length of STFT  
\(N_{0}\)  
central time index of window  
\(R\)  
block shift in STFT  
\(w[n]\)  
function of time index \(n\)  
\(x[n]\)  
abbreviated form of \(x(t_{n})\)  
\(f_{k}\)  
the \(k\)-th discrete frequency  
\(\Delta f\)  
frequency resolution  
\(X_{w}\)  
Gabor coefficient at time index \(i\) and frequency \(f_{k}\)  
\(L\)  
length of signal \(x(t_{n})\)  
\(T\)  
period of \(x(t_{n})\)  
\(p\)  
index of STFT frequency closest to a given cyclic frequency \(\alpha\)  
\(\delta\)  
residue  
\(R_{w}(\alpha)\)  
kernel function  
KA(\(\alpha\))  
kurtosis value  
\(\mathrm{KA^{\prime}}(\alpha)\)  
weighting factor  
\(S_{x}^{Fast}(\alpha,f)\)  
fast spectral correlation  
\(S_{x}^{Fast}(\alpha,f)\)  
improved fast spectral correlation  
\(f_{d}\)  
natural frequency  
\(M\)  
repeated times of impact  
\(D\)  
single pulse strength  
\(A\)  
amplitude  
\(\xi\)  
system damping ratio  
\(f_{n}\)  
natural frequency  
\(f_{o}\)  
outer race fault frequency  
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