Adaptation of the Phase Distance Correlation Periodogram to Account for Measurement Uncertainties

We present an improvement of the phase distance correlation (PDC) periodogram to account for uncertainties in the time-series data. The PDC periodogram introduced in our previous papers is based on the statistical concept of distance correlation. By viewing each measurement and its accompanying error estimate as a probability distribution, we are able to use the concept of energy distance to design a distance function (metric) between measurement-uncertainty pairs. We used this metric as the basis for the PDC periodogram, instead of the simple absolute difference. We demonstrate the periodogram's performance using both simulated and real-life data. This adaptation makes the PDC periodogram much more useful, demonstrating it can be helpful in the exploration of large time-resolved astronomical databases, ranging from Gaia radial velocity and photometry data releases to those of smaller surveys, such as APOGEE and LAMOST. We have made a public GitHub repository available, with a Python implementation of the new tools available to the community.