The Disc Miner I: A statistical framework to detect and quantify kinematical perturbations driven by young planets in discs

[Abridged] The study of disc kinematics has recently opened up as a promising method to detect unseen planets. However, a systematic, statistically meaningful analysis of such an approach remains missing. The aim of this work is to devise an automated, statistically robust technique to identify kinematical perturbations induced by the presence of planets in a gas disc, and to accurately infer their location. For this purpose, we produce hydro simulations of planet-disc interactions with different planet masses, 0.3, 1.0 and 3.0 $M_{Jup}$, at a radius of $R=100$ au in the disc, and perform radiative transfer calculations of CO to simulate observables for 13 planet azimuths. Using the DISCMINER package, we fit the synthetic data cubes with a Keplerian model of the channel-by-channel emission to study line profile differences, including deviations from Keplerian rotation. The detection technique, based on line centroid differences, captures localised planet-driven perturbations, and can distinguish them from axisymmetric velocity perturbations. The method can detect all three simulated planets, at all azimuths, with an average accuracy of $\pm3^\circ$ in azimuth and $\pm8$ au in radius. Owing to disc structure and line-of-sight projection effects, planets at azimuths close to $\pm45^\circ$ yield the highest velocity fluctuations, whereas those at limiting cases, $0^\circ$ and $\pm90^\circ$, drive the lowest. The observed peak velocities range within 40$-$70, 70$-$170 and 130$-$450 m s$^{-1}$ for 0.3, 1.0 and 3.0 $M_{Jup}$ planets. Our analysis indicates that the variance of peak velocities is boosted near planets due to organised gas motions prompted by their localised gravitational well. We propose an approach that exploits this velocity coherence to provide, for the first time, statistically significant detections of localised planet-driven perturbations in the gas disc kinematics.