Predicting the black hole mass and correlations in X-ray reverberating AGN using neural networks

The X-ray variability has become a powerful tool to probe the inner accretion flow around the central black holes in both active galactic nuclei (AGN) and X-ray binaries. The X-ray reverberation mapping is one of the timing analysis techniques that measures the time delays associated with the light-travel time between the X-ray photons from the direct coronal emission and back-scattered photons from the accretion disc (see Uttley et al., 2014; Cackett, Bentz, & Kara, 2021, for a review). Due to a longer distance travelled by the reflected photons, the reflection-dominated bands (the soft excess, Fe-K and Compton hump bands) lag behind the continuum-dominated bands (e.g. Fabian et al., 2009; Kara et al., 2013a, c, 2014; Zoghbi et al., 2014; Kara et al., 2019; Alston et al., 2020; Vincentelli et al., 2020). The amplitude of the lag depends on the geometry of the source, so providing us a tool to constrain the disc-corona geometry.

De Marco et al. (2013) performed a systematic look at the frequency-dependent time lags between the soft ($\sim 0.3–1$ keV) and hard ($\sim 1–4$ keV) bands in AGN and found that the amplitude of the soft lag scales with the black hole mass. Kara et al. (2016a) performed a systematic study of the X-ray reverberation lags of all Seyfert galaxies available in the XMM–Newton archive and reported a number of sources that exhibited Fe-K reverberation lags. They confirmed the lag-mass scaling relation and found the correlation between the height of the corona and the mass accretion rate of these reverberating AGN. King, Lohfink, & Kara (2017) found that the radio Eddington luminosity inversely correlates with the X-ray reflection fraction, and positively scales with the path length between the X-ray source and the accretion disc. Modelling the X-ray reverberation lags to map the extreme region near a supermassive black hole has been carried out intensively using both lamppost geometry (Wilkins & Fabian, 2013; Cackett et al., 2014; Emmanoulopoulos et al., 2014; Chainakun & Young, 2015; Chainakun, Young, & Kara, 2016; Epitropakis et al., 2016; Caballero-García et al., 2018; Ingram et al., 2019) and extended corona model (Wilkins et al., 2016; Chainakun & Young, 2017; Chainakun et al., 2019). The X-ray reverberation technique has already been applied to various scenarios such as the photon reflection off the accretion flow in the tidal disruption event (Kara et al., 2016b), the reflection from different hot-flow zones in X-ray binaries (Mahmoud, Done, & De Marco, 2019; Chainakun et al., 2021b; Kawamura et al., 2021) and the multiple scattering from the disc wind in ultraluminous X-ray sources (Luangtip et al., 2021).

Furthermore, Alston et al. (2020) suggested that the height of the X-ray corona in IRAS13224–3809 increased with the source luminosity. This is also supported by Caballero-García et al. (2020) who found significant variations in the X-ray source height from $\sim 3-5r_{\rm g}$ when the X-ray luminosity is $\sim 1.5-3$ per cent of the Eddington limit, to $\sim 10r_{\rm g}$ when the luminosity doubles. Recently, Hancock et al. (in prep) investigated the time-average and lag-frequency spectra of 20 AGN covering 121 XMM-Newton observations and separated them into 3–4 groups of similarly observed spectral states. The reflection fraction was found to be strongly correlated to the power-law photon index that suggested dynamics of the emitting region.

The X-ray reverberation features can be imprinted in the profiles of the power spectral density (PSD) that describes the variability power on different timescales. The oscillatory structures seen in the PSD of AGN can be interpreted as the reverberation signatures that relate to the geometry of the system such as the coronal height and the inclination (Papadakis et al., 2016; Emmanoulopoulos et al., 2016; Chainakun, 2019). In Chainakun et al. (2021a), we developed machine learning (ML) models, based on dictionary learning and support vector machine algorithms, to extract the X-ray reverberation signatures on the PSD profiles of AGN, and used them to predict the coronal height. The variability amplitude in light curves can be estimated using the fractional excess variance ($F_{\rm var}$), which can be constructed by integrating the PSD between two frequencies, or from the mean and the rms amplitude of the light curve (e.g. Vaughan, Fabian, & Nandra, 2003). Recently, the $F_{\rm var}$ spectra have been used to probe the intrinsic and environmental absorption origins for the X-ray variability in AGN (Parker et al., 2021).

The potential use of the ML techniques in the X-ray reverberation analysis has been elaborated and discussed in Chainakun et al. (2021a). Here, we employ the key information of the X-ray reverberating AGN to develop a neural network model in order to predict the black hole mass. We consider the fundamental parameters that can be derived from the X-ray observations. These parameters include the Fe-K lag amplitude, the $F_{\rm var}$, the photon counts ($C$), the bolometric luminosity ($L_{\rm bol}$) and the redshift ($z$). The source height and the reflection fraction are not considered because their values are dependent on the assumed geometry and the choice of the reflection models.

The ultimate goal is to test how well the neural network model, trained using only fundamental X-ray observational parameters, can predict the central mass. The prediction accuracy obtained from the neural network model is reported and compared to what is obtained from the standard linear regression model. After that, we plot the mass distribution against the model features such as the $F_{\rm var}$ and the lag amplitude to investigate the global relations between them all simultaneously. From the parameter space constrained by the model, we can simulate more samples of reverberating AGN and investigate the correlations between their parameters. It provides hints of how the known-existing correlations will change if more reverberating AGN are discovered. The AGN data used for training the machine are presented in Section 2. The neural network algorithm and the development of the ML models to predict the black hole mass are explained in Section 3. We evaluate the models and present the results in Section 4. We discuss the optimization results and the obtained correlations in Section 5, while the conclusion is provided in Section 6.

We use the XMM-Newton data previously reported and analyzed by Kara et al. (2016a) where the selected samples have $\gtrsim 40$ ks exposure time and show some variability. However, we select only the data that display the Fe-K reverberation features in the lag spectra whose lower and upper bounds of the lags can be constrained. Based on these criteria, there are 22 AGN sources in total. These AGN samples and their parameter values are listed in Table 1. The Fe-K lag amplitude, $F_{\rm var}$, log($L_{\rm bol}$) and log($C$) of each AGN are average values from those of all available observations that fit the criteria. $F_{\rm var}=\sigma_{\rm rms}/\bar{x}$ is the fractional excess variance of the data in 2–10 keV band where $\sigma_{\rm rms}$ is the rms amplitude and $\bar{x}$ is the mean count rates (Vaughan, Fabian, & Nandra, 2003). $C$ is the photon counts in the 2–10 keV band. $L_{\rm bol}$ is the mean bolometric luminosities calculated from the SEDs (Wang, Watarai, & Mineshige, 2004; Vasudevan & Fabian, 2007, 2009; Vasudevan et al., 2010). The masses that are estimated by the optical reverberation techniques are from the public web data base (Bentz & Katz, 2015) using the $<f>$ = 4.3 (Grier et al., 2013), similar to Kara et al. (2016a).

There are also $\sim 21$ AGN samples left, that have $\gtrsim 40$ ks exposure time, show some variability but do not exhibit Fe-K reverberation features on the lag spectra. We spare them in a new data set (Table 4) for further analysis once the neural network model is obtained. In other words, while the reverberating AGN data in Table 1 are used during the training phase, the non-reverberating AGN data in Table 4 are kept unseen, completely new to the machine and are used only for the final evaluation of the model. Note that there are some inconsistencies in the report of the Fe-K response among previous literature. For example, Wilkins et al. (2021) recently found that there was a Fe-K reverberation lag caused by a flare in IZw1, while this source is still included in the non-reverberation sample here (Table 4). We, however, select to follow the standard analysis of Kara et al. (2016a) and discuss these inconsistencies later in the Discussion section.

We train the machine using the neural network technique so that it can make an accurate prediction of the black hole mass. The flowchart illustrating the training and testing process is presented in Fig. 1. The following subsections outline step-by-step the methodology used to explore the nature of the data as well as to develop and evaluate our neural network models.

According to our results, the developed ML models can make higher accurate predictions of the black hole mass than the linear regression model. The combination of the solver and activation function is dependent on the characteristics of the data. All neural network models here require L-BFGS-B solver. Generally, L-BFGS-B solver is a suitable second-order optimization method and is fastest for small convex problems or small data size, appropriate for our data set. It employs the full training set to obtain the later update to parameters at every iteration (Le et al., 2011). If the size of the dataset is large, L-BFGS-B computing time can be very long on a single machine, so it is hard to incorporate new data in an online environment. On the other hand, tanh is found to be the best activation function that provides high accuracy for generalized MLP architectures of neural networks (Karlik & Olgac, 2011; Montavon, Orr, & Müller, 2012). If the $F_{\rm var}$ is only a single feature, the correlation between $F_{\rm var}$ and log($M/M_{\odot}$) is linearly dependent. This is why the ReLu which is rectified linear unit activation functions is better than tanh (full non-linearity unit). All combined-feature models ($R^{2}\gtrsim 0.9$) developed here require the L-BFGS-B solver and tanh activation function, which agree with other works that use multi-feature/variable regression in MLP neural networks (e.g. Gheorghe & Badea, 2014; Artrith, Urban, & Ceder, 2017).

Kara et al. (2016a) found that the correlation coefficient of the frequency–mass relation ($r_{s}=-0.68$) is stronger than that of the lag–mass relation ($r_{s}=0.60$). They suggested using the frequency–mass relation for determining the black hole mass instead of the lag–mass relation. This argument, however, may be specific to the averaging scheme applied to the observational data (Hancock et al., in prep.). Here, we find the correlation between the Fe-K lag amplitude and the mass for AGN sources to be $r_{s}=0.589$, confirming what was reported by Kara et al. (2016a). Even though both $F_{\rm var}$ and log($C$) are produced from the lightcurves extracted in the same energy band of 2–10 keV, it is clear that the black hole mass shows a significantly stronger correlation with $F_{\rm var}$ than log($C$). Also, both log($L_{\rm bol}$) and log($C$) show a relatively weak correlation with the black hole mass. This suggests that the X-ray timing data may be more useful for predicting the black hole mass than the time-average X-ray data.

Hancock et al. (in prep.) studied the correlations between the central mass and the lags, but using the lags measured between the soft excess 0.3–0.8 keV and continuum-dominated 1–4 keV bands. The correlation coefficient was found to be $r_{s}=0.72$. This may suggest that the lags in the soft band are stronger correlated to the mass than the lags in Fe-K band. Although the Fe-K band is sometimes contaminated by the neutral distant reflection, the distant reflection varies on different timescales from the timescales of the inner-disc reflection. Therefore, unlike the soft excess lags whose origin is ambiguous, the Fe-K lags are likely to be a clean signature of X-ray reverberation. The Fe-K band then may provide a more accurate reflection of how much the signals produced via X-ray reverberation correlate with the black hole mass.

We also analyze the correlation between the black hole mass and other features of the ML model. Interestingly, we find that the correlation between the mass and the $F_{\rm var}$ is $r_{s}=-0.718$, which is stronger than the lag–mass relation and even stronger than the frequency–mass relation reported by Kara et al. (2016a). However, the prediction accuracy is higher when using the lags alone than when using the $F_{\rm var}$ alone. Nevertheless, for a specific source, different individual or combined observations into, e.g., high flux and low flux states result in different time lag estimates for a single mass. Therefore, only one parameter cannot be a good predictor for the mass since its value changes with how the data of each AGN are selected and combined while the mass remains constant. It is then better to use both $F_{\rm var}$ and lag amplitude as the mass predictors that, evidently, can also improve the model accuracy ($R^{2}=0.9124$).

Based on the $F_{\rm var}$–lag–mass relation predicted by the ML model, the reverberating AGN with a specific black hole mass can exhibit $F_{\rm var}$ and lags within a limited regime. The model can rule out the possibility for the majority of the non-reverberating AGN to display the lags (Table 4 and Fig. 7), revealing its potential to make accurate predictions for new data. Since the model can be successfully applied to both reverberating and non-reverberating AGN, it suggests the common origins of main variability that contributes to $F_{\rm var}$ for both AGN groups. This suits the framework of the X-ray variability driven by, e.g, the disc propagating-fluctuations that can operate in both reverberating and non-reverberating AGN.

Note that the model still predicts the presence of the Fe-K lags for some sources included in the non-reverberation group (Table 4). For example, the Fe-K lags of IZw1 are predicted to be $\lesssim 600$ s. In fact, this is in agreement with Wilkins et al. (2021) who reported observations of X-ray flares around the central black hole in IZw1 and detected the Fe-K reverberation lags of $\sim 746\pm 157$ s by using a different method to the standard Fourier analysis. Moreover, based on the XMM-Newton observations, Zoghbi et al. (2013) reported the Fe-K reverberation lags in MCG–5–23–16 and SWIFT J2127.4+5654, but new analysis using NuSTAR data suggested no strong evidence for relativistic reverberation in both AGN (Zoghbi, Miller, & Cackett, 2021). It is not straightforward to determine if these AGN exhibit no intrinsic reverberation lags or if the inconsistencies arise due to different methods and model assumptions used to quantify the significance of the lag. Nevertheless, we try moving these samples to the non-reverberating AGN group, and find only a slight change in the lag–mass correlations inferred (e.g., lag-mass correlation coefficient $r_{s}$ changes less than $\sim 4$ per cent). This is because these inconsistencies are found only in the minority of the samples. Therefore, moving them across different groups (reverberation and non-reverberation) does not change the trend of the key results here.

The X-ray reverberation produces the dip in the PSD profiles so it dilutes the $F_{\rm var}$ in a particular timescale where reverberation dominates (Papadakis et al., 2016; Chainakun, 2019). The amplitude of the dip increases with increasing the reflection fraction, resulting in a decrease in $F_{\rm var}$ on a particular reverberation timescales. The mass, on the other hand, scales up the intrinsic lags. However, $F_{\rm var}$ here represents the fractional excess variance contributed by all mechanisms that affect the X-ray variability in 2–10 keV band, not only by the reverberation. For example, the $F_{\rm var}$ can be affected by the constant or less-variable emission component, e.g. from outflowing gas, that is varied among different AGN (Parker et al., 2021). The constant emission component raises the mean but does not affect the standard deviation of the data, so the $F_{\rm var}$ decreases. On the other hand, the constant absorption lowers both mean and standard deviation of the data proportionally, hence its presence has small effects on the $F_{\rm var}$. The deviation of the $F_{\rm var}$–lag relation then may be induced by, e.g., the presence of the constant or less-variable emission component, resulting in the change of the lag–mass correlation coefficient.

Our results suggest that to maintain the lag–mass scaling relation, the anti-correlation between the lag and $F_{\rm var}$ must preserve. Contrarily, the $F_{\rm var}$–mass anti-correlation seems to be stronger even if new reverberating AGN show more independence of their lags on $F_{\rm var}$. Moreover, the model suggests the regime of $F_{\rm var}\gtrsim 0.5$, lags $\gtrsim 1000$ s and $M\lesssim 10^{6.5}M_{\odot}$ where the reverberating AGN are not yet observed (top-right portion of Fig. 6). It possibly represents a forbidden regime that is unphysical, which is why there is still no reverberating AGN sample fitting into. Although the AGN with constant emission component can introduce the deviation of the $F_{\rm var}$–lag correlation coefficient, they may not fit into this ambiguous regime. This regime corresponds to the samples with high $F_{\rm var}$, while constant emission produces negative effects on $F_{\rm var}$.

Furthermore, the observed flux in a particular energy band always contains both continuum and reflection flux. This introduces dilution effects to the lags so that the observed lags are smaller than the intrinsic lags (e.g. Wilkins & Fabian, 2013; Chainakun & Young, 2015). Chainakun et al. (2019) studied the lags produced by an extended corona under the inverse-Compton scattering scenario and found that, with a fixed coronal size and geometry, the coronal temperature and optical depth can affect the lags. The more the complex model is, the less the measured reverberation lags can straightforwardly relate to the true light-travel distance. In fact, Hinkle & Mushotzky (2021) investigated the fundamental X-ray corona properties of 33 AGN observed under the 105-month Swift/BAT campaign together with the archival XMM-Newton and NuSTAR data. They found no strong correlations between the black hole mass, coronal compactness and coronal temperature. Therefore, if the lags significantly depend on the coronal compactness or temperature, the lags may not strongly correlate with the mass. The less correlation between the lags and the mass, if observed, then may suggest the extended corona framework where the lag amplitudes of the majority of the sources are more strongly affected by the properties of the coronal extent rather than the geometry.

According to our results, the $F_{\rm var}$ and lag amplitude are the key parameters of the X-ray variability data to predict the black hole mass. The clear trend of how the mass varies with log($C$) and $z$ is not clearly seen. Based on the current XMM-Newton observations, we find $z$ is correlated with $L_{\rm bol}$ ($r_{s}=0.796$), but there is no clear pattern of associations between z and other parameters (time lags, mass, and $F_{\rm var}$). Perhaps, this is because the samples probed by XMM-Newton are only those relatively nearby and are not distributed widely enough across a broad range of redshift. Fig. 12 represents the distribution of our reverberating AGN samples into the parameter space when the redshift $z$ is discretized into 2 groups: $z<0.015$ (low redshift) and $z\geq 0.015$ (high redshift). The results show that we tend to observe the high-redshift sources when they exhibit relatively large $L_{\rm bol}$. The pattern of associations between $z$ and other parameters cannot be easily revealed, however, even though the samples are just grouped into low and high redshifts. This is why we still cannot obtain a meaningful interpretation for the redshift associated with the lag and the mass. Note that these results do not imply that there is no effect of redshift at all, they instead suggest that the amount of current samples is not enough for the model to gain useful information relating to $z$. In the Athena era, the model should be able to provide more insights into not only the trend of scaling relations, but also the redshift dependence, when we can probe more reverberating sources at higher redshift.

It is also possible to calculate $F_{\rm var}$ in the frequency domain by integrating the corresponding power spectrum between two frequencies, so that the obtained $F_{\rm var}$ is specific to a particular range of timescales, which will be investigated in the future work. Finally, there were also reports on the correlation between the central black hole mass and the host galaxy total stellar mass (e.g. Reines & Volonteri, 2015). Investigating the relationship between the AGN and host-galaxy parameters using machine learning techniques is also planned for the future.

We investigate several scenarios of applying the neural network to predict the black hole mass and correlations in AGN. The model does not require the assumed source geometry in advance, but this does not mean that the inferred correlations are independent of source geometry. The predictions by the neural network models are much more accurate than those from the standard linear regression in all cases, providing an independent way to predict the AGN mass. The best model that uses either $F_{\rm var}$ or the lag alone contains $\sim 90$–130 neurons ($R^{2}\sim 0.6$–0.7 and MAE $\sim 0.2$–0.3). It is, however, better to use both $F_{\rm var}$ and the lag amplitude to predict the central mass ($R^{2}=0.9124$ and MAE = 0.1077), which requires 168 neurons. The mass distribution predicted by the model satisfies the lag–mass scaling relation regarding to the observed reverberating AGN samples. Despite of the limited number of available samples for training the machine, we illustrate that the model can potentially be used to exclude and identify the AGN samples that do not exhibit the lags. This, in turn, suggests the common origins of the variability, e.g. disc propagating-fluctuations, that drives $F_{\rm var}$ in both reverberating and non-reverberating AGN.

The $F_{\rm var}$–mass anti-correlation is always true and probably stronger with an increasing number of sources. The model reveals the regime in the parameter space where there is still no reverberating AGN samples lying into ($F_{\rm var}\gtrsim 0.5$ with the lags $\gtrsim 1000$ s and $M\lesssim 10^{6.5}M_{\odot}$). In fact, there is no straightforward mechanism for the AGN to operate high $F_{\rm var}$ and large lags with such a small mass. Perhaps, this is the reason why we do not observe reverberating AGN in this regime. The model shows that the less correlation between the lag and the mass is possible with increasing number of the newly-discovered reverberating AGN. The AGN inducing this may require an extended corona whose property (e.g. temperature and compactness) rather than geometry is more dominant to the measured time lags. The hints of their presence then may be a noticeable decrease in the lag–mass correlation coefficient.

The more accurate mass can be retrieved using all features investigated here ($R^{2}=0.9993$ and MAE = 0.0100). The neural network in this case contains 249 neurons. The model with all features coupled together is probably too complex, so the clear pattern of association to the relation between $z$ and log($C$) is not observed. The obtained neural network models are observational driven since they are trained and tested using the real observational data. The current Fe-K reverberation samples observed so far are representative enough in the way that the model can draw some meaningful correlations between their observable variables. It is possible that the source spectral state may have an effect on the detection probability. For example, super soft sources occupying at the high end of the $L_{\rm bol}$ parameter space may have poor count rates in the hard band so detection of Fe-K lags become more difficult. Regarding the obtained correlation between $L_{\rm bol}$ and the mass which is very weak (Table 2), it will require a very large number of these sources in order to drive the significant and meaningful relationship between $L_{\rm bol}$ and the mass in the samples. We suspect that these extreme sources are likely not the majority of populations, hence do not significantly alter the inferred correlations. This work illustrates an application of the ML technique to construct the multi-feature parameter space where the samples of reverberating AGN can be simulated. More number of newly-discovered reverberating AGN will help place robust constraint on the extrapolating results predicted by the models.

This research has received funding support from the NSRF via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation (grant number B16F640076). The ML models in this work are adopted from the sklearn software package. The calculations were carried out using the high performance computing resources in the Institute of Science and the Centre for Scientific and Technological Equipment, Suranaree University of Technology.

The data underlying this article can be accessed from XMM-Newton Observatory (http://nxsa.esac.esa.int). The neural network algorithm adopted here is available in scikit-learn at https://scikit-learn.org/. The derived data and developed model underlying this article will be shared on reasonable request to the corresponding author.