On Estimating the Cross Correlation and Least Squares Fit of One Data Set to Another With Time Shift

Abstract

Abstract Estimating the cross correlation between two data sets and the least squares fit of one data set to another shares closely related numerical procedures. Here we revisit the standard procedure of estimating the cross‐correlation function and cross‐coherence spectrum between two data sets. We attend to and run extensive Monte Carlo simulation to gain insights towards their statistical assessment with respect to the confidence level and degree of freedom which is often overlooked in published literature. We then point out the fallacy, both physically and numerically, of the estimation scheme proposed and practiced by Phillips et al. (2012, https://doi.org/10.1029/2012GL052495) that utilized Hilbert transform for least squares fitting of one time series to another with time shift (which was adopted in subsequent published work). We run Monte Carlo experiments to demonstrate that fallacy and the fact that the Phillips et al.'s scheme can only work for fitting of one trivial single‐period sinusoid while fails in any other circumstances.

Keywords

• correlation
• coherence
• least squares fit
• time shift
• Hilbert transform
• GRACE data