Circular pearson correlation using cosine series expansion

In resting-state fMRI, there is no external anchor that will lock brain activation across voxels. Thus, correlation of fMRI time series between voxels is often done by computing coherence in the frequency domain. However, such approach ignores the time lag of fMRI time series across voxels. To address the problem, we propose to use the concept of circular Pearson correlation in determining the time lag, which locks the time series, and the maximum correlation at locking. We further express the circular Pearson correlation analytically in terms of cosine series expansion. The proposed method is applied to 208 twin pairs to determine if the time lag and the maximum correlation are heritable genetic features.