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In view of the difficulty of fault prediction for aviation hydraulic pumps and the poor real-time performance of state monitoring in practical applications, a hydraulic pump pressure signal prediction method is proposed to accomplish the monitoring and prediction of the health status of hydraulic pumps in advance. First, based on the on-line real-time acquisition of time series flight parameters and pressure signal data, the chaotic characteristics of the system are analyzed using chaos theory, so that the time series pressure signal is predictable. Second, phase space reconstruction (PSR) of the one-dimensional time series data is conducted. The embedding dimension $m$ and time delay $\tau $ are obtained by the C-C method. The reconstructed matrix is used as the training set and test set of the support vector regression (SVR) algorithm model according to a certain proportion, and the genetic algorithm (GA) is then used to optimize the parameters of the SVR model. Finally, the SVR model optimized by the genetic algorithm based on phase space reconstruction (PSR-GA-SVR) is used to test the test set data. The results show that the prediction accuracy of the proposed method is higher than that of the BP neural network based on phase space reconstruction (PSR-BPNN) and the SVR model based on phase space reconstruction (PSR-SVR). Relative to PSR-BPNN and PSR-SVR, PSR-GA-SVR produces a minimum mean square error (MSE) reduced by 73.40% and 68.0%, respectively, and a mean absolute error (MAE) decreased by 90.41% and 90.87%, respectively. The confidence level for PSR-GA-SVR was increased, and the coefficient of determination was greater than 0.98.