Investigating scaling relations in X-ray reverberating AGN using symbolic regression

Symbolic regression (SR) is a regression analysis based on genetic algorithms to search for mathematical expressions that best fit a given data set, by allowing the expressions themselves to mutate. We use the SR to analyze the parameter relations of the X-ray reverberating Active Galactic Nuclei (AGN) where the soft Fe-L lags were observed by XMM-Newton. Firstly, we revisit the lag-mass scaling relations by using the SR to derive all possible mathematical expressions and test them in terms of accuracy, simplicity and robustness. We find that the correlation between the lags, $\tau$, and the black hole mass, $M_{\rm BH}$, is certain, but the relation should be written in the form of $\log ({\tau}) = \alpha + \beta (\log{(M_{\rm BH}/M_{\odot})})^{\gamma}$, where $1 \lesssim \gamma \lesssim 2$. Moreover, incorporating more parameters such as the reflection fraction ($RF$) and the Eddington ratio ($\lambda_{\rm Edd}$) to the lag-mass scaling relation is made possible by the SR. It reveals that $\alpha$, rather than being a constant, can be $-2.15 + 0.02RF$ or $0.03(RF + \lambda_{\rm Edd})$, with the fine-tuned different $\beta$ and $\gamma$. These further support the relativistic disc-reflection framework in which such functional dependencies can be straightforwardly explained. Furthermore, we derive their host-galaxy mass, $M_{\ast}$, by fitting the spectral energy distribution (SED). We find that the SR model supports a non-linear $M_{\rm BH}$--$M_{\ast}$ relationship, while $\log (M_{\rm BH}/M_{\ast})$ varies between $-5.4$ and $-1.5$, with an average value of $\sim -3.7$. No significant correlation between $M_{\ast}$ and $\lambda_{\rm Edd}$ is confirmed in these samples.