Galactic cosmic ray (GCR) is usually assumed as a stable "background," with solar influence considered as a modulation. The violent solar energetic particle (SEP) events associated with solar activities change particle fluxes by several orders of magnitude in a few minutes. Thus, the flux observation of GCR provided by satellites may be heavily contaminated by spurious spikes due to SEPs, and that provided by ground-based neutron monitors (NMs) may be contaminated by the system error spikes and the ground level enhancement effect. To obtain the "pure" background GCR flux for modulation research, the removal of multifarious spikes is necessary. In this article, we use a robust automatic despiking algorithm based on the Poincare map thresholding method provided by Goring and Nikora for "purification" of the time-series GCR flux observations. We can show that the algorithm is good at cleaning up the heavily contaminated GCR intensity rates measured by both spacecraft and NMs without artificial parameters. In addition, using the algorithm to despike the spacecraft observations of relatively lower energetic proton flux, we get both 11 year and 27 day period cycles comparable to the much higher energy GCR flux data measured by the ground-based NMs.