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A computationally quick and conceptually simple method to recover time delayof the chaotic system from scalar time series is developed in this paper. Weshow that the orbits in the incomplete two-dimensional reconstructedphase-space will show local clustering phenomenon after the componentpermutation procedure proposed in this work. We find that information capturedby the incomplete two-dimensional reconstructed phase-space, is related to thetime delay ${\tau _0}$ present in the system, and will be transferred to thepermutation component by the procedure of component permutation. We thenpropose the segmented mean-variance (SMV) from the permutation component toidentify the time delay ${\tau _0}$ of the system. The proposed SMV shows clearmaximum when the embedding delay $\tau $ of the incomplete reconstructionmatches the time delay ${\tau _0}$ of the chaotic system. Numerical datagenerated by a time-delay system based on the Mackey-Glass equation operatingin the chaotic regime are used to illustrate the effectiveness of the proposedSMV. Experimental results show that the proposed SMV is robust to additiveobservational noise and is able to recover the time delay of the chaotic systemeven though the amount of data is relatively small and the feedback strength isweak. Moreover, the time complexity of the proposed method is quite low.

Incomplete Phase-Space Method to Reveal Time Delay From Scalar Time-series