$\texttt{Wapiti}$: a data-driven approach to correct for systematics in RV data -- Application to SPIRou data of the planet-hosting M dwarf GJ 251

Context: Recent advances in the development of precise radial velocity (RV) instruments in the near-infrared (nIR) domain, such as SPIRou, have facilitated the study of M-type stars to more effectively characterize planetary systems. However, the nIR presents unique challenges in exoplanet detection due to various sources of planet-independent signals which can result in systematic errors in the RV data. Aims: In order to address the challenges posed by the detection of exoplanetary systems around M-type stars using nIR observations, we introduce a new data-driven approach for correcting systematic errors in RV data. The effectiveness of this method is demonstrated through its application to the star GJ 251. Methods: Our proposed method, referred to as $\texttt{Wapiti}$ (Weighted principAl comPonent analysIs reconsTructIon), uses a dataset of per-line RV time-series generated by the line-by-line (LBL) algorithm and employs a weighted principal component analysis (wPCA) to reconstruct the original RV time-series. A multi-step process is employed to determine the appropriate number of components, with the ultimate goal of subtracting the wPCA reconstruction of the per-line RV time-series from the original data in order to correct systematic errors. Results: The application of $\texttt{Wapiti}$ to GJ 251 successfully eliminates spurious signals from the RV time-series and enables the first detection in the nIR of GJ 251b, a known temperate super-Earth with an orbital period of 14.2 days. This demonstrates that, even when systematics in SPIRou data are unidentified, it is still possible to effectively address them and fully realize the instrument's capability for exoplanet detection. Additionally, in contrast to the use of optical RVs, this detection did not require to filter out stellar activity, highlighting a key advantage of nIR RV measurements.