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The Laser Interferometer Space Antenna (LISA) will provide us with a uniqueopportunity to observe the early inspiral phase of supermassive binary blackholes (SMBBHs) in the mass range of $10^5-10^6\,M_{\odot}$, that lasts forseveral years. It will also detect the merger and ringdown phases of thesesources. Therefore, such sources are extremely useful for multiparameter testsof general relativity (GR), where parametrized deviations from GR at multiplepost-Newtonian orders are simultaneously measured, thus allowing for a rigoroustest of GR. However, the correlations of the deviation parameters with theintrinsic parameters of the system make multiparameter tests extremelychallenging to perform. We demonstrate the use of principal component analysis(PCA) to obtain a new set of deviation parameters, which are best-measuredorthogonal linear combinations of the original deviation parameters. With theobservation of an SMBBH of total redshifted mass,$\sim\mathrm{7\times10^5\,M_{\odot}}$ at a luminosity distance of 3 Gpc, we canestimate the five most dominant PCA parameters, with 1-$\sigma$ statisticaluncertainty of $\lesssim 0.2$. The two most dominant PCA parameters can bebounded to $\sim \mathcal{O}(10^{-4})$, while the third and fourth-dominantones to $\sim \mathcal{O}(10^{-3})$. Measurement of the PCA parameters withsuch unprecedented precision with LISA makes them an excellent probe to testthe overall PN structure of the GW phase evolution.

Enhancing the performance of multiparameter tests of general relativity with LISA using Principal Component Analysis