The spatial information of the signal is neglected by the conventional frequency/time decompositions such as the fast Fourier transformation, principal component analysis, and independent component analysis. Framing of the data being as a three-way array indexed by channel, frequency, and time allows the application of parallel factor analysis, which is known as a unique multi-way decomposition. The parallel factor analysis was used to decompose the wavelet transformed ongoing diagnostic channel–frequency–time signal and each atom is trilinearly decomposed into spatial, spectral, and temporal signature. The time–frequency–space characteristics of the single-source fault signal was extracted from the multi-source dynamic feature recognition of mechanical nonlinear multi-failure mode and the corresponding relationship between the nonlinear variable, multi-fault mode, and multi-source fault features in time, frequency, and space was obtained. In this article, a new method for the multi-fault condition monitoring of slurry pump based on parallel factor analysis and continuous wavelet transform was developed to meet the requirements of automatic monitoring and fault diagnosis of industrial process production lines. The multi-scale parallel factorization theory was studied and a three-dimensional time–frequency–space model reconstruction algorithm for multi-source feature factors that improves the accuracy of mechanical fault detection and intelligent levels was proposed.

The monitoring of mechanical equipment systems contains an increasing number of complex content, expanding from traditional time, and frequency information to three-dimensional data of the time, space, and frequency information, and even higher-dimensional data containing subjects, experimental conditions. For high-dimensional data analysis, traditional decomposition methods such as Hilbert transform, fast Fourier transformation, and Gabor transformation not only lose the integrity of the data, but also increase the amount of calculation and introduce a lot of redundant information. The phenomenon of feature coupling, aliasing, and redundancy between the mechanical multi-source data signals will cause the inaccuracy of the evaluation, diagnosis, and prediction of industrial production operation status. The analysis of the three-way tensor composed of channel, frequency, and time is called parallel factor analysis (PARAFAC). The properties between the parallel factor analysis results and the input signals are studied through simulation experiments. Parallel factor analysis is used to decompose the third-order tensor composed of channel-time-frequency after continuous wavelet transformation of vibration signal into channel, time, and frequency characteristics. Multi-scale parallel factor analysis successfully extracted non-linear multi-dimensional dynamic fault characteristics by generating the spatial, spectral, time-domain signal loading value and three-dimensional fault characteristic expression. In order to verify the effectiveness of the space, frequency, and time domain signal loading values of the fault characteristic factors generated by the centrifugal pump system after parallel factor analysis, the characteristic factors obtained after parallel factor analysis are used as the SPRT test sequence for identification and verification. The results indicate that the method proposed in this article improves the measurement accuracy and intelligence of mechanical fault detection.