Higher-Order Spectra of Cluster Point Processes Generating 1/f Fluctuations

1/f fluctuations have been extensively observed in various fields including both biology and physics. Recently, a cluster point process has been successfully applied to modeling the physically and the physiologically observed series of events which exhibit 1/f fluctuations. In this paper, the higher-order spectra up to the 3rd order are derived for the general cluster point processes whose primary and secondary processes are renewal according to the heuristic shot noise approach. As far as the cluster point process whose primary process is Poissonian, the results coincide with those obtaind by the probability generating functional. In the context of 1/f fluctuations, the power spectral densities (PSD) and the bispectra are given for varied parameters, especially for the power law distribution of cluster size. In addition to the rising landscape in the lower frequency range, these 2nd- and 3rd-order spectra are shown to have undulating structures that reflect the periodicity of event occurrence within the clusters, which indicates that there are quadratic correlations between the dominant frequency components that construct the 1/f profile in the PSD. Although several point process models that can produce the 1/f PSD have been proposed, the bispectra could be a possible measure for selecting an appropriate model from these candidates and may contribute to identifying the generation mechanism of the 1/f fluctuation.