Detecting critical transitions in the case of moderate or strong noise by binomial moments.

Detecting critical transitions in the case of moderate or strong noise (collectively referred to as big noise) is challenging, since such noise can make a critical transition point far from the bifurcation point, leading to the failure of traditional small-noise methods. To handle this tough issue, we first transform a generic noisy system into a linear set of binomial moment equations (BMEs). Then, we can solve a closed set of BMEs obtained by truncation and use the resulting binomial moments to reconstruct a joint probability distribution of the state variables of the original system. Third, we derive a leading indicator from the closed set of BMEs. Importantly, the reconstructed distribution determines the way of critical transition (i.e., critical transition is distribution transition rather than state transition in the strong-noise case) as the system comes close to the critical transition point, whereas the derived indicator anticipates when the distribution transition occurs. Our theory has broad applications, and artificial and data examples exhibit its power.