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Time-series forecasting is the core of the prognostics and health management (PHM) of turbomachinery. However, missing data often occurs due to several reasons, such as the failure of sensors. These partially missing and irregular data greatly affect the quality of time-series modeling and prediction as most time-series models assume that the data are sampled uniformly over time. Meanwhile, the training process of models requires a large number of samples and time. Due to various reasons, it is difficult to obtain a significant amount of monitoring data, and the PHM of turbomachinery has high timeliness and accuracy requirements. To fix these problems, we propose a multi-task Gaussian process (MTGP)-based data imputation method that leverages knowledge transfer across multiple sensors and even equipment. Thereafter, we adopt long short-term memory (LSTM) neural networks to build time-series forecasting models based on the imputed data. In addition, the model integrates the methods of denoising and dimensionality reduction. The superiority of this integrated time-series forecasting framework, termed MT-LSTM, has been verified in various data imputation scenarios of a synthetic dataset and a real turbomachinery case.