Khalafi, M., Reza Soltani, A., Golalipour, M., Azimmohseni, M. and Najafiamiri, F. (2021), On the spectral coherence between two periodically correlated processes. Can J Statistics. https://doi.org/10.1002/cjs.11632
Periodically correlated processes have potential applications in different fields. The univariate periodically correlated processes have been well established and developed in several theoretical and applied studies. In this article, a general class of multivariate periodically correlated processes is defined such that it enables researchers in different fields to study several periodically correlated processes simultaneously. In this class, periodically correlated processes may have different periods. As shown in this article, a multivariate periodically process corresponds to a univariate periodically correlated process. This fact leads researchers to extend the results of univariate periodically correlated to multivariate ones. The other novelty of this article is introducing spectral coherence to measure the dependence of two multivariate stationary processes with arbitrary dimensions and periodically correlated processes with possibly different periods. This measure of association can play an important role in practice. Specially, for gene expression time series of a life process which are inherently periodic, it is interesting to identify the co-expressed genes. As illustrated in this article, the interactions of the genes are effectively identified by the spectral coherence.
The statistical inferences on the spectral coherence are also provided in this article. Since the spectral coherence is a function of frequencies, to test the independence of two periodically correlated processes one should make a simultaneous hypothesis testing. To test this hypothesis a test statistic is furnished and an instruction to perform bootstrap method is also provided.