The center of of the Milky Way is characterized by a supermassive black hole (SMBH), which is supported by indirect but compelling observational evidence such as stellar orbits in the vicinity of Sagittarius A^* (Sgr A^*; Ghez et al., 1998; Ghez et al., 2005) and the near-infrared luminosity of Sgr A^* being consistent with the presence of an event horizon (Broderick and Narayan, 2006; Broderick et al., 2009). Similarly, we expect that active galactic nuclei (AGNs) in other galaxies host SMBHs at their centers (Kormendy, 1988; Kormendy and Richstone, 1992; Kormendy et al., 1997; Cretton and van den Bosch, 1999), which are essential to explaining the X-ray features of quasars and AGNs (see review by Mushotzky et al., 1993). Several techniques, such as the reverberation mapping (Blandford and McKee, 1982), spectral energy distribution (SED) fitting (Shields, 1978; Malkan, 1983), and broad-line region size–luminosity correlation (Vestergaard, 2002), have been developed to validate the presence of SMBHs and estimate their masses (e.g., Kormendy and Richstone, 1995; Miyoshi et al., 1995; Wandel et al., 1999; Peterson et al., 2004; Calderone et al., 2013; Capellupo et al., 2015; Bentz and Katz, 2015; Mejía-Restrepo et al., 2016). Our constraints on SMBH masses have allowed us to establish the connections between SMBHs and the evolution of their host galaxies (e.g., Magorrian et al., 1998; Ferrarese and Merritt, 2000; Häring and Rix, 2004; Heckman and Best, 2014).

Some solutions of standard general relativity simply characterize black holes using two parameters, mass and spin (Kerr, 1963), which can fully describe the properties of SMBHs. In this regard, spins of SMBHs, along with masses, could produce some of the fundamental mechanisms for powering relativistic jets (e.g., Garofalo et al., 2010; Tchekhovskoy and McKinney, 2012), as well as describing the discrepancy between radio-loud and radio-quiet AGNs (Wilson and Colbert, 1995; Moderski et al., 1998), galaxy evolution (Di Matteo et al., 2005; Volonteri et al., 2013; Sesana et al., 2014), and galaxy mergers (Hughes and Blandford, 2003; Volonteri et al., 2005; Berti and Volonteri, 2008). In particular, ultra-fast outflows (UFOs) have been detected in X-ray observations of several radio-quiet AGNs (e.g., Tombesi et al., 2010; 2011; 2012; Danehkar et al., 2018; Boissay-Malaquin et al., 2019), while extended relativistic jets have been seen in radio observations of radio-loud AGNs (see review by Blandford et al., 2019). The spins of SMBHs could have a potential role in the formation of UFOs and jets seen in AGNs and quasars (MacDonald et al., 1986; Thorne et al., 1986). These phenomena can be explained by spinning SMBHs according to the Blandford–Znajek (Blandford and Znajek, 1977) and Penrose mechanism (Penrose, 1969; 2002; Penrose and Floyd, 1971), as well as frame-dragging vortexes (e.g., Owen et al., 2011; Nichols et al., 2011; Danehkar, 2020). Alternatively, they could originate magnetically from the innermost accretion disk in the vicinity of a spinning SMBH according to the Blandford–Payne mechanism (Blandford and Payne, 1982).

In the Boyer–Lindquist coordinates, the Kerr metric (Kerr, 1963) of a spinning black hole is expressed using the set of oblate spheroidal coordinates ( r , θ , ϕ ) as follows (Boyer and Lindquist, 1967): d s 2 = − 1 − r s r Σ c 2 d t 2 − 2 a ̄ r s r ⁡ sin 2 ⁡ θ Σ c d t   d ϕ + Σ Δ d r 2 + Σ   d θ 2 + r 2 + a ̄ 2 + a ̄ 2 r s r ⁡ sin 2 ⁡ θ Σ sin 2 ⁡ θ   d ϕ 2 , (1) where Σ = r 2 + a ̄ 2 ⁡ cos 2 ⁡ θ , Δ = r 2 − r s r + a ̄ 2 , r s = 2 G M / c 2 is the Schwarzschild radius, a ̄ = J / M c is called the Kerr spin parameter describing angular momentum per unit mass having the length dimension, J the black hole angular momentum, M the black hole mass, G the Newtonian constant of gravitation, and c the speed of light. The dimensionless spin parameter, which is frequently used in the astrophysical community, is defined as a ≡ 2 a ̄ / r s = J c / ( G M 2 ) , while − 1 ≤ a ≤ 1 ; negative values describe retrograde rotation, in which the black hole rotates in the opposite direction of the accretion disk, whereas positive values are associated with prograde rotation, and zero implies non-rotating black holes. The outer and inner event horizons are determined by the roots of Δ = 0 , which are r ± = r s ( 1 ± 1 − a 2 ) / 2 . In the case of a = 0 , Equation 1 reduces to the Schwarzschild metric (Schwarzschild, 1916) with the event horizon at r = r s . Unlike the Schwarzschild metric, which has a singularity at r = 0 , the Kerr metric has one at Σ = 0 . The innermost stable circular orbit (ISCO) of the accretion disk is located at a radius of marginal stability, r ms , which is given by (Bardeen et al., 1972): r ms a = G M c 2 3 + Z 2 − s g n a 3 − Z 1 3 + Z 1 + 2 Z 2 , (2) where s g n ( a ) is the signum function having the value − 1 , 1 or 0 according to the sign of a , while Z 1 and Z 2 are defined as, Z 1 = 1 + 1 − a 2 1 / 3 1 + a 1 / 3 + 1 − a 1 / 3 , Z 2 = 3 a 2 + Z 1 2 .

This implies that the accretion disk has a limited extent at the marginal stability radius ( r ms ) , also called the ISCO radius. The value of this radius depends upon the dimensionless spin parameter, e.g., r ms = 0.5 r s for a = 1 3 r s for a = 0 4.5 r s for a = − 1 . In prograde rotation, the ISCO radius shrinks to nearly half of the Schwarzschild radius as it approaches a near-maximal spin ( a ≈ 1 ) , while it expands in retrograde rotation ( − 1 < a < 0 ) . For a non-spinning black hole, the ISCO radius is precisely three times the Schwarzschild radius.

There are different methods available to measure the spin of a single SMBH (see review by Brenneman, 2013). All of them are based on general relativity solutions of the Kerr spacetime in the vicinity of the black hole. They use the aforementioned fact that the ISCO radius ( r ms ) of the accretion disk depends on the spin, as seen in Equation 2, and assume a geometrically thin disk that is capable of irradiating the corona radiation with light-bending in the innermost regions (see Figure 1 bottom-left). Below are the techniques that have been used to constrain the spins of SMBHs:

Various spectral models have been constructed to reproduce the general-relativistic effects of the Kerr metric on the iron K α line profile. Table 1 summarizes most of the well-known spectral models made for X-ray data analysis of relativistically blurred emission, along with their key parameters. The basic parameters in these models are the dimensionless spin parameter ( a = J c / G M 2 ) , the line-of-sight inclination angle ( i ) of the accretion disk, the spectral photon index ( Γ ) of the primary source – the corona or the base of a jet above the black hole –, as well as the boundary radii ( R in and R out ) of the innermost accretion disk. Most of these models can be loaded into tools for X-ray spectral analysis, particularly the X-ray spectral fitting package xspec ¹ (Arnaud, 1996; Arnaud et al., 1999) of HEASoft’s data analysis package for X-ray astronomy xanadu (NASA HEASARC, 2014), MIT’s Interactive Spectral Interpretation System ² (isis; Houck and Denicola, 2000), CXC ³ ’s Modeling and Fitting Package Sherpa ⁴ (Freeman et al., 2001; Doe et al., 2007) of the Chandra Interactive Analysis of Observation (ciao; Fruscione et al., 2006), and SRON ⁵ ’s X-ray high-resolution spectral modeling and fitting package spex ⁶ (Kaastra et al., 1996; de Plaa et al., 2020). These models are often applied to integrated spectra of AGNs without accounting for the variability of the X-ray sources.

The early models, developed to analyze relativistic reflection, featured fiducial values for the spin parameter. Black hole spin measurement began with the diskline (Fabian et al., 1989) and laor (Laor, 1991) fixed-spin models, which were run with fiducial spin values of a = 0 and 0.998, respectively. The diskline model was based on analytic, time-consuming calculations, whereas the laor model relied on extensive pre-calculated tabulated Flexible Image Transport System (FITS) data created for different combinations of the model parameters: inclination angle, spectral photon index ( Γ ) , inner radius ( R in ) , and outer radius ( R out ) . The laor model recreates the relativistic line shape by interpolating data from the extensive FITS table. Later, Martocchia and Matt (1996) examined how the “lamp post” geometry affected the broad iron K α lines, in which the ionizing source is located on the polar axis at a height ( h ) above the black hole, as illustrated in Figure 1 (bottom-right). This investigation was followed by a comprehensive analysis for a = 0.001 and 0.9981 leading to the kerrspec model (Martocchia et al., 2000; Martocchia et al., 2002). We should note that the reflection continuum was not included in these models and needed to be handled by a separate model such as reflionx (Ross et al., 1999; Ross and Fabian, 2005; Ross and Fabian, 2007), which should be convolved with the corresponding convolution models to make a smoothed spectrum of relativistic smearing in an accretion disk. The convolution models rdblur and kdblur were prepared using diskline and laor, respectively, which can be used with a reflection model (e.g., reflionx). Nandra et al. (2007) used a modified version of the kdblur model to characterize the broad iron K α line in XMM-Newton observations of a sample of Seyfert galaxies. However, the spectral models with fixed spin rates obviously prevented us from straightforwardly measuring the black hole spin.

The next-generation of relativistic reflection models has a free parameter for the positive spin rates, which allows for the determination of the black hole spin in prograde rotation ( 0 ⩽ a < 1 ) . The kyrline (or ky) model (Dovčiak, 2004; Dovčiak et al., 2004; Dovčiak et al., 2022) was developed to incorporate extensive tables calculated for transfer functions. This model rapidly computes the shape of relativistically broadened line emission using transfer function tables without relying heavily on interpolation, yielding a significantly greater level of spectral resolution compared to the laor model. Furthermore, the black hole spin was a free parameter, ranging from a = 0 to 1. The model can therefore reproduce the relativistic blurred emission more accurately than the laor model for all positive spin rates and inclination degrees. In addition, the kyrline model features coronal emissivity law indexes ( q in as r − q in between R in and R br ; and q out as r − q out between R br and R out ) for precisely creating the emitted radiation (see Figure 1 bottom-right), describing the emissivity characteristics on both sides of the break radius ( R br ) , as well as including the limb prescriptions for limb-darkening/-brightening laws (Chandrasekhar, 1960), namely isotropic emission ( I ∝ 1 ) , Laor’s limb-darkening ( I ∝ 1 + 2.06 μ ; Laor, 1991), and Haardt’s limb-brightening ( I ∝ ln ( 1 + 1 / μ ) ; Haardt, 1993), where μ = cos ( θ e ) and θ e is the inclination angle of the emitted radiation with respect to the disk. These features were accomplished by computing them for inclusion in the comprehensive FITS table. In order to address the problems with the excessive table size and lack of smoothness in the kyrline model, Brenneman and Reynolds (2006) created an alternative model for relativistic reflection known as kerrdisk (Brenneman, 2007). Their model has a relatively smaller FITS table and a robust interpolation approach. Using a high level of smoothness in the transfer function allows for effective interpolation in kerrdisk. Furthermore, this model employs a distinct methodology, approximating the narrow line of the distant reflection from the accretion disk using a δ -function instead of a Gaussian function. Their model computes a larger portion of the integration using analytic approaches, thereby excluding the emissivity law from the calculated table. However, fitting methods can handle the modeling of the emissivity law. This model effectively reduces the table size to a fraction of the kyrline FITS table. Nevertheless, the relativistic emission produced by kerrdisk appears less smooth than those made by kyrline, with some noticeable spikes in the red wing of the relativistic line. However, data accumulated with the spectral resolutions of detectors aboard the XMM-Newton and Suzaku telescopes could not distinguish these spikes. The kyrline model has been used to conduct an XMM-Newton survey on a sample of radio-quiet Type 1 AGNs (de La Calle Pérez et al., 2010). Unlike the fixed-spin models (laor and diskline), kyrline and kerrdisk, which can relativistically be convolved with a reflection model (e.g., reflionx) using the kyconv and kerrconv models, respectively, are more accurate in producing the shape of relativistic emission for any positive spin rates. However, a black hole spinning in retrograde relative to the accretion disk ( − 1 < a < 0 ) could not have its spin constrained by the kyrline and kerrdisk models.

Since 2010, several spectral models have been developed for relativistic blurred emission (see Table 1) that incorporate both positive and negative spin values, enabling the measurement of the black hole’s spin in both the prograde and retrograde directions with respect to the accretion disk. To accommodate the full spin range ( − 0.998 ⩽ a ⩽ + 0.998 ) , Dauser et al. (2010) has created the relline model, together with relline_lp, featuring the “lamp post” geometry (for detail see Dauser, 2010), using FITS tables for Cunningham’s photon transfer function (Cunningham, 1975) pre-calculated with a customized version of the F77 program photon_transferfct ⁷ (also called spx; Speith, 1993; Speith et al., 1995) (a gravitationally-distorted appearance of an accretion disk made with a modified version of this program is shown in the bottom-left panel of Figure 1). The relline model employs Green’s functions to calculate the radiated radiation for arbitrary angular and radial variations, as well as robust interpolation techniques that lead to a decrease in pre-calculated tabulated data. Moreover, relativistic line profiles calculated for a hard X-ray source located on the rotational axis at a height ( h ) above the black hole, i.e., the “lamp post” geometry, are provided in the relline_lp model (Dauser et al., 2013). Both the models includes prescriptions for the limb-darkening/-brightening laws. Their corresponding convolution models, relconv and relconv_lp, are able to convolve the reflection continuum created by a reflection model such as reflionx (Ross et al., 1999; Ross and Fabian, 2005; Ross and Fabian, 2007) and xillver (García, 2010; García and Kallman, 2010; García et al., 2011; García et al., 2013). As this simple combination of the relativistic convolution model (relconv) and the reflection model (xillver) can lead to inconsistent results, García et al. (2014) made a self-consistent implementation of the spectrum reflected from the disk irradiated by an ionizing source and relativistically blurred emission in a new model called relxill, as well as an additional new model called relxill_lp for a lamp-post geometry. ⁸ These models incorporated angle-dependent reflection tables of xillver ⁹ into the relativistic blurring calculations, which exhibits like behavior to a convolution of xillver and relconv (or relconv_lp), albeit with self-consistently calculated X-ray reflection (Dauser, 2014). Subsequently, Dauser et al. (2014) and Dauser et al. (2016), further extended them to include the reflection fraction parameter ( f refl,rel ) , a flux ratio between the direct and reflected radiation that depends on the geometry and location of the radiation source. In the updated model relxillCp (Dauser et al., 2016), the primary source is made by a thermally Comptonized continuum model (nthcomp; Zdziarski et al., 1996; Życki et al., 1999) and offers a free parameter for the disk density ranging from 1 0 15 to 1 0 20 c m − 3 , whereas the previous model relxill uses a power law with a high-energy exponential cutoff (zcutoffpl) and assumes a disk density of 1 0 15 c m − 3 . Recently, the inclusion of returning radiation is implemented in the latest model of relxilllpCp by Dauser et al. (2022), featuring the “lamp post” geometry with a thermally comptonized continuum as the primary source and an unrestricted density parameter ( 1 0 15 – 1 0 20 c m − 3 ). Additionally, García et al. (2022) also present the model relxillNS featuring a black body spectrum, specifically tailored to accommodate the reflection from the disk around an accreting neutron star. The relativistic X-ray reflection models provided by the relxill package were also extended to describe the reflection spectrum in the Johannsen metric (Johannsen, 2013), referred to as relline_nk (Bambi et al., 2017; Abdikamalov et al., 2019; Abdikamalov et al., 2020; Abdikamalov et al., 2021b; Abdikamalov et al., 2021a), ^{10 , 11} whose model names contain “nk” at the end (e.g., relxill_nk and relxillCp_nk stand for non-Kerr spacetimes) to distinguish them from those in the Kerr metric (see Table 1). These models allow to validate the Kerr metric through the deformation parameters α 13 , α 22 , or ϵ 3 in the Johannsen metric, where α 13 = α 22 = ϵ 3 = 0 restores the Kerr metric (for non-Kerr metrics, see review by Bambi, 2017).

X-ray time-resolved observations of AGNs have shown variability in the relativistically blurred reflection (e.g., MCG–6-30-15; Fabian and Vaughan, 2003; Vaughan and Fabian, 2004; Larsson et al., 2007; Miller et al., 2008) that could be caused by general relativistic effects, particularly light bending near the black hole event horizon. Niedźwiecki and Życki (2008) and Niedźwiecki and Miyakawa (2010) investigated variability patterns of the red wing in X-ray reflection of the AGN in the Seyfert 1 galaxy MCG–6-30-15 using a detailed light-bending model (Miniutti and Fabian, 2004), which led to the development of the spectral model reflkerr and its corresponding lamp-post model reflkerr_lp (Niedźwiecki et al., 2016; Niedźwiecki et al., 2019). ¹² In particular, Niedźwiecki et al. (2016) identified some inconsistencies between reflkerr and relxilllp owing to the neglect of the general-relativistic redshift of the direct coronal radiation in relxilllp, though they found that the relxilllp model still produces acceptable results in weak-gravity in the energies below 80 keV. Moreover, the lamp-post model reflkerr_lp developed by Niedźwiecki et al. (2019) demonstrated a departure from relxilllp in the energies above 30 keV. Another model-family for spectral and timing variability in accreting black holes has been developed (Ingram et al., 2019; Mastroserio et al., 2021; 2022), named reltrans and reltransCp, ¹³ which calculated the emergent reflection spectrum using xillver (or xillverCp in the case of reltransCp). The reltrans model considers all the general-relativistic effects to calculate the time delays and energy changes that occur when X-ray photons from the corona reflect from the accretion disk and scatter towards the observer. The calculations of reltrans incorporate both continuum lags and reverberation lags in a self-consistent manner to produce most of the practical X-ray variability time scales.

Principal component analysis (PCA; Hotelling, 1933), ¹⁴ also referred to as the “Hotelling transform,” is a well-known method in multivariate statistics relying on eigenvalues and eigenvectors (see review by Jolliffe and Cadima, 2016) that has been extensively discussed in detail in the literature (e.g., Mardia et al., 1979; Jolliffe, 2002; Izenman, 2008; Rencher and Christensen, 2012). It bears a close relation to the “Kosambi–Karhunen–Loève transform” (Kosambi, 1943; Karhunen, 1947; Loève, 1948) in probability theory, and is among three classical techniques in multivariate analysis to determine the principal dimensions of large data, along with independent component analysis (ICA; Hérault and Ans, 1984; Hérault et al., 1985; Hérault and Jutten, 1986) and non-negative matrix factorization (NMF; Lee and Seung, 1999; Lee and Seung, 2000). PCA can be employed to separate various characteristics that are mostly responsible for complex variations in large data in astronomy (e.g., Wall and Jenkins, 2012; Ivezić et al., 2020) as well as to simplify complex data for machine learning approaches (e.g., Bishop, 2006; Müller and Guido, 2016; Witten et al., 2017; Géron, 2019). This is implemented by reducing the number of available data into a group of independent PCA components, which then provide information about the different levels of their contributions to the complexity of the entire data. Astronomers have extensively employed it as a practical multivariate method. The early application of this technique in astronomy (see review by Francis and Wills, 1999) can be traced back to some studies on spectral analyses of stars (Deeming, 1964; Whitney, 1983), galaxies (Faber, 1973; Bujarrabal et al., 1981; Efstathiou and Fall, 1984), and quasars (Mittaz et al., 1990; Francis et al., 1992; Boroson and Green, 1992). This approach was also employed for imaging analysis of the interstellar medium (Heyer and Schloerb, 1997; Brunt et al., 2009). It was later used for X-ray binaries (e.g., Malzac et al., 2006; Koljonen et al., 2013; Koljonen, 2015) and blazars (Gallant et al., 2018), and more recently for X-ray variability in symbiotic stars (Danehkar et al., 2024a) and starburst regions (Danehkar et al., 2024b). Especially, it has extensively been leveraged for X-ray data analysis of AGNs in Seyfert 1 galaxies (e.g., Vaughan and Fabian, 2004; Miller et al., 2008; Parker et al., 2014b; Gallo et al., 2015).

PCA can decompose time-resolved spectroscopic data into groups of PCA components and eigenvectors, yielding eigenvalues in the process. Normalized eigenvalues can yield the contribution of each eigenvector to the temporal evolution of the whole data over time. Each decomposed PCA component and eigenvector can be referred to as a principal spectrum with its corresponding light curve. The process of conducting PCA requires performing the decomposition of a matrix into its eigenvectors and eigenvalues. To analyze variability of a source in astronomy, this data matrix for PCA contains a set of spectroscopic data collected at n time intervals, each binned into m spectral channels. Let consider a rectangular ( m × n ) matrix X consisting of n rows by m columns, one can determine the principal components of X through the following three methods:

SVD and PCA decomposition closely relate to the optimal solution for neural networks in auto-association mode (Bourlard and Kamp, 1988; Baldi and Hornik, 1989). As discussed by Hertz et al. (1991) in the context of unsupervised Hebbian learning, PCA can be used for dimensionality reduction of large data before proceeding with machine learning algorithms, such as artificial neural networks (ANNs). PCA can indeed alleviate the “curse of dimensionality” (coined by Bellman, 1957; Bellman, 1961), also known as the “Hughes phenomenon” (Hughes, 1968) or “peaking phenomenon” (Trunk, 1979), which often arises when searching for patterns in unknown large data. It has been extensively demonstrated in the literature that PCA can be utilized as a pre-processing step to simplify complex data prior to machine learning (e.g., Bishop, 2006), data mining (Witten et al., 2017), and deep learning (Goodfellow et al., 2017). Recently, Ivezić et al. (2020) also discussed in detail the applications of PCA, ICA, and NMF in dimensionality reduction for data mining and machine learning in astronomy.

Using PCA for the pre-processing of astronomical data enables a significant reduction in dimensionality and complexity of data, leading to an improvement in machine learning performance. The use of PCA to reduce the dimensionality of the data for training ANNs can be traced back to earlier efforts on the classification of galaxy spectra (Folkes et al., 1996; Lahav et al., 1996) and stellar spectra (Bailer-Jones et al., 1998; Singh et al., 1998). Later, Zhang and Zhao (2003) applied PCA to the multiwavelength data of AGNs, stars, and normal galaxies in order to reduce the dimensionality of the parameter space for support vector machines (SVM) and learning vector quantization (LVQ), two supervised classification algorithms in machine learning, resulting in the classification of stars, AGNs, and normal galaxies. PCA also reduced the complexity of image data for the morphological classification of galaxies with an ANN (de la Calleja and Fuentes, 2004). Moreover, Bu and Pan (2015) deployed PCA to pre-assemble stellar atmospheric parameters from spectra for Gaussian process regression (GPR) and then compared the results of GPR with those from ANNs, kernel regression (KR), and support-vector regression (SVR). Kuntzer et al. (2016) also conducted stellar classification from single-band images using pre-processed data from PCA to train ANNs to determine the spectral type. More recently, we see the application of PCA to construct input data for ANNs in stellar population synthesis modeling (Alsing et al., 2020), finding thermal components in X-ray spectra of the Perseus cluster (Rhea et al., 2020), and finally X-ray spectral analysis of AGN (Parker et al., 2022).

The avenue of automated spectral analysis with machine learning algorithms has not yet been fully explored for constraining the relativistically broadened iron emission in AGN, mostly because of the complicated variability seen in the X-rays over the course of observations. X-ray observations of AGNs have shown some X-ray changes in power-law continua, which were ascribed to so-called transient obscuration events caused by eclipsing material near the primary source, such as NGC 3783 (Mehdipour et al., 2017), NGC 3227 (Turner et al., 2018), and Mrk 335 (Longinotti et al., 2019; Parker et al., 2019), or flaring variations in the corona in the innermost central regions, e.g., PDS 456 (Matzeu et al., 2017; Reeves et al., 2021) and NGC 3516 (Mehdipour et al., 2022). This kind of change in X-rays over time, along with a relatively large number of parameters in relativistic reflection models (see Table 1), makes it much more complicated for machine learning algorithms to automatically determine the spins of SMBHs from the archival X-ray data. Nevertheless, as seen in Figure 2, the dimensionality reduction offered by PCA can avoid the curse of dimensionality in the X-ray data of AGNs. In the future, we will be able to use machine learning to automatically conduct the spin analysis of SMBHs in AGNs thanks to the principal spectra of relativistic reflection disentangled by PCA from X-ray observations.