Spectrum sensing has been identified as a key enabling functionality to ensure cognitive radios would not interfere with primary users; however, it always faces the hard challenge in a low signal-to-noise (SNR) radio region. In this paper, we optimize spectrum sensing algorithm which combines backward linear prediction and QR decomposition of the over-sampled received signals. We propose an adaptive spectrum sensing strategy to adapt the algorithm to the wireless environment and get a proper trade off between the probability of detection and the algorithm complexity. We claim some key parameters of the algorithm, including over-sampling factor, length of statistical sampling sequence and linear prediction order, and evaluate how these parameters affect the performance of the algorithm. Simulation results show that the proposed method can archive a desired detection performance at very low SNR and reduce the algorithm complexity when the SNR is in a high level.