The adaptive and sparsest time-frequency analysis（ASTFA） can realize signal spare decomposition adaptively by using Gauss-Newton iteration,but Gauss-Newton iteration is very sensitive to initial value which is determined by experience. In order to keep Gauss-Newton iteration converge,an improved method based on Firefly Algorithm（FA） is introduced to ASTFA,the validity of improvement is proved by simulation and experiment. Compared to traditional time-domain features,the minimum convex hull is capable to extract spatial information of the signal. One the basis,the forecasting model of gear crack fault based on FA-ASTFA and minimum convex hull is presented. The experimental analysis shows that the proposed model is more reliable and accurate than the traditional forecasting model for predicting the degree of gear incipient crack fault.