Data analysis has wide applications in eliminating the irrelevant and redundant components in signals to reveal the important informational characteristics that are required. Conventional methods for multi-dimensional data analysis via the decomposition of time and frequency information that ignore the information in signal space include independent component analysis (ICA) and principal component analysis (PCA). We propose the processing of a signal according to the continuous wavelet transform and the construction of a three-dimensional matrix containing the time–frequency–space information of the signal. The dimensions of the three-dimensional matrix are reduced by parallel factor analysis, and the time characteristic matrix, frequency characteristic matrix, and spatial characteristic matrix are obtained with tensor decomposition. Through the comparative analysis of the simulation and the experiment, the time characteristic matrix and the frequency characteristic matrix can accurately characterize the normal and fault states of the mechanical equipment. On this basis, the authors established a probabilistic neural network classification model optimized by the improved particle swarm algorithm (IPSO). The parallel factor (PARAFAC) decomposition algorithm can extract features from the centrifugal pump experimental data for normal and multiple fault states, establish the mapping relationship of different fault features of the centrifugal pump in time, frequency, and space, and import the fault features into the model classification. The above measures can significantly improve the fault identification rate and accuracy for a centrifugal pump.