Detecting temporal and spatial correlations in pseudoperiodic time series.

Recently there has been much attention devoted to exploring the complicated possibly chaotic dynamics in pseudoperiodic time series. Two methods [Zhang, Phys. Rev. E 73, 016216 (2006); Zhang and Small, Phys. Rev. Lett. 96, 238701 (2006)] have been forwarded to reveal the chaotic temporal and spatial correlations, respectively, among the cycles in the time series. Both these methods treat the cycle as the basic unit and design specific statistics that indicate the presence of chaotic dynamics. In this paper, we verify the validity of these statistics to capture the chaotic correlation among cycles by using the surrogate data method. In particular, the statistics computed for the original time series are compared with those from its surrogates. The surrogate data we generate is pseudoperiodic type (PPS), which preserves the inherent periodic components while destroying the subtle nonlinear (chaotic) structure. Since the inherent chaotic correlations among cycles, either spatial or temporal (which are suitably characterized by the proposed statistics), are eliminated through the surrogate generation process, we expect the statistics from the surrogate to take significantly different values than those from the original time series. Hence the ability of the statistics to capture the chaotic correlation in the time series can be validated. Application of this procedure to both chaotic time series and real world data clearly demonstrates the effectiveness of the statistics. We have found clear evidence of chaotic correlations among cycles in human electrocardiogram and vowel time series. Furthermore, we show that this framework is more sensitive to examine the subtle changes in the dynamics of the time series due to the match between PPS surrogate and the statistics adopted. It offers a more reliable tool to reveal the possible correlations among cycles intrinsic to the chaotic nature of the pseudoperiodic time series.