There are lots of non-equidistant sequences in actual applications due to random sampling, imperfect sensors, event-triggered phenomena, and so on. A new grey prediction method for non-equidistant sequences (r
-NGM(1,1)) is proposed based on the basic grey model and the developed fractional-order non-equidistant accumulated generating operation (r
-NAGO), and the accumulated order is extended from the positive to the negative. The whole r
-NAGO deletes the randomness of original sequences in the form of weighted accumulation and improves the exponential law of accumulated sequences. Furthermore, the Levenberg–Marquardt algorithm is used to optimize the fractional order. The optimal r
-NGM(1,1) can enhance the predicting performance of the non-equidistant sequences. Results of three practical cases in engineering applications demonstrate that the proposed r
-NGM(1,1) provides the significant predicting performance compared with the traditional grey model.
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