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In the field of signal processing such as system identification, the affine projection algorithm (APA) is extensively implemented. However, running such algorithms in a non-Gaussian scenario may degrade its performance, since the second-order moment cannot extract all information from the signal. To prevent performance degradation of the algorithm in system identification tasks, we propose a novel APA based on least mean fourth (LMF) algorithm. The new algorithm, namely affine projection least mean fourth algorithm (APLMFA) is based on the high-order error power (HOEP) criterion and as such, can achieve improved performance. We also provide a convergence analysis for APLMFA. Numerical simulation results verify the presented APLMFA achieves smaller steady-state error as compared with the state-of-the-art algorithms.