A Machine Learning approach for classification of accretion states of Black hole binaries
aa r X i v : . [ a s t r o - ph . H E ] J a n MNRAS , 1–10 (2020) Preprint 18 January 2021 Compiled using MNRAS L A TEX style file v3.0
A Machine Learning approach for classification of accretion states ofBlack hole binaries
H. Sreehari ★ , Anuj Nandi
1. Indian Institute of Astrophysics, Bangalore, 560034, India.2. Space Astronomy Group, ISITE Campus, U. R. Rao Satellite Centre, Outer Ring Road, Marathahalli, Bangalore, 560037, India.
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
In this paper, we employ Machine Learning algorithms on multi-mission observations for the classification of accretion statesof outbursting Black hole X-ray binaries for the first time. Archival data from
RXTE , Swift , MAXI and
AstroSat observatories areused to generate the hardness intensity diagrams (HIDs) for outbursts of the sources XTE J1859+226 (1999 outburst), GX 339-4(2002, 2004, 2007 and 2010 outbursts), IGR J17091-3624 (2016 outburst), and MAXI J1535-571 (2017 outburst). Based onvariation of X-ray flux, hardness ratios, presence of various types of Quasi-periodic Oscillations (QPOs), photon indices and disktemperature, we apply clustering algorithms like K-Means clustering and Hierarchical clustering to classify the accretion states(clusters) of each outburst. As multiple parameters are involved in the classification process, we show that clustering algorithmsclub together the observations of similar characteristics more efficiently than the ‘standard’ method of classification. We alsoinfer that K-Means clustering provides more reliable results than Hierarchical clustering. We demonstrate the importance of theclassification based on machine learning by comparing it with results from ‘standard’ classification.
Key words: methods: data analysis - accretion, accretion disc - black hole physics - X-rays: binaries.
Black hole X-ray binaries (BH-XRBs) are binary star systemsharbouring a black hole along with a secondary (donor) star.These systems undergo periods of high X-ray activity called out-bursts extending from several days to a few months in betweenthe quiescent states (Remillard & McClintock 2006; Belloni 2010;Nandi et al. 2012, 2018; Sreehari et al. 2018; Baby et al. 2020, andreferences therein). During the outburst phase, the X-ray intensityincreases gradually up to a peak value and then decays down un-til the source goes into quiescence. These sources are known toexhibit rich variability in temporal features (Belloni et al. 2000,2005; Nandi et al. 2012; Belloni & Stella 2014; Radhika et al. 2018;Sreehari et al. 2019a, and references therein). One of the salientfeatures exhibited by these sources are QPOs (van der Klis et al.1985; Casella et al. 2004; Nandi et al. 2012; Sreehari et al. 2019b;Ingram & Motta 2019). QPOs are asymmetric peaks in the powerdensity spectra of XRB sources. These features are modelled us-ing lorentzians and characterised using quality factor (Q), rmsand significance (see Casella et al. 2005; Sreehari et al. 2019b,for details). QPOs with frequency in the range 0 . −
40 Hzare considered as Low frequency QPOs (LFQPOs, Casella et al.2004; Motta et al. 2011; Yadav et al. 2016, and references therein)and those above 40 Hz are considered High frequency QPOs(HFQPOs, Morgan et al. 1997; Remillard et al. 1999; Strohmayer2001; Altamirano & Belloni 2012; Belloni & Altamirano 2013; ★ E-mail: [email protected] (HS), [email protected] (AN)
Belloni et al. 2019; Sreehari et al. 2020). LFQPOs are further clas-sified into Type-A, Type-B and Type-C (Casella et al. 2005). Type-CQPOs have higher rms (3 −
16 %) and Q (7 − − ≥ in ). Thehigh energy emission from the BH-XRB is accounted for by theinverse-Comptonization (Titarchuk 1994; Tanaka & Lewin 1995;Zdziarski et al. 1996) component represented by powerlaw basedmodels (Nandi et al. 2012; Iyer et al. 2015; Radhika et al. 2016b)having a photon-index ( Γ ) parameter.In general, a Hardness-Intensity Diagram (HID or q-diagram)provides a better idea about the accretion state transitions in aBH-XRB (Homan et al. 2001; Homan & Belloni 2005; Belloni et al.2005; Nandi et al. 2012; Radhika et al. 2016b; Sreehari et al. 2019b;Baby et al. 2020). It depicts the variation of total flux versus the hard-ness ratio during an outburst (see left panel of Figure 1). The outburstduration is classified into different spectral states based on factors likehardness ratio (ratio of high energy flux to low energy flux; hereafterHR), presence or absence of QPOs, and photon-index ( Γ ) of the en-ergy spectra along with the disk temperature (kT in ). For instance,right panel of Figure 1 shows the parameters on which the accretion © Sreehari & Anuj Nandi
Table 1.
Summary of observations used in the present work.Source Observatory/Instrument Years ReferencesXTE J1859+226
RXTE/PCA+HEXTE
RXTE/PCA+HEXTE
Swift/XRT
MAXI, AstroSat/LAXPC, Swift/XRT T o t a l F l u x Flux in units of − ergs/cm /s Type-C . . . H R . . . Γ Days ( day . MJD ) . . . T i n ( k e V ) Figure 1.
The left plot shows the HID of the 1999 outburst of XTE J1859+226 with LHS in magenta, HIMS in blue, SIMS in green, HSS in yellow, SIMS-decayin orange, HIMS-decay in cyan and LHS-decay in pink. The radio flares are represented by red stars. The same colour convention for accretion states is followedthrough out this paper. The evolution of physical parameters obtained from spectral fitting are presented on the right plot. The four panels are Total flux(3 −
20 keV), Hardness Ratio (6 −
20 keV/3 − Γ ) and the inner disk temperature (kT in ). In the top panel, the presence of Type-C QPOsare indicated in blue colour. state of the 1999 outburst of the source XTE J1859+226 depends on.Various branches within the q-diagram corresponding to this out-burst are classified into Low/Hard state (LHS), Hard-IntermediateState (HIMS), Soft-Intermediate State (SIMS) and High/Soft State(HSS) during the rising phase. The reverse trend is observed duringthe decay phase where the source transits to SIMS-decay, HIMS-decay and finally to LHS-decay before reaching the quiescent phase(see Figure 1 and Nandi et al. (2018)). It is observed that the LHScorresponds to maximum HR values accompanied by a low valueof flux. The flux usually reaches a high value at the end of thisstate or the beginning of the HIMS. In the HIMS, the flux valueremains the same though the hardness decreases. Generally, Radioflares (i.e. transient jets) are observed at the transition region be-tween HIMS and SIMS (Fender et al. 2004, 2009; Radhika & Nandi2014; Radhika et al. 2016a, and references therein). As the sourcetransits through the SIMS, the HR further decreases until the HSS isreached. In the decay phase, the flux level remains low and the HRgradually increases as the source transits through SIMS, HIMS andLHS. Type-C QPOs are generally associated with LHS and HIMS.Their frequency values are observed to increase during the risingphase and to decrease as the source transits through the decay phase(Rodriguez et al. 2002; Chakrabarti et al. 2008; Nandi et al. 2012,2018; Sreehari et al. 2019a). Generally, the photon index ( Γ ) ofenergy spectra increases from 1.4 to ∼ in ) is above 1 keV during the soft states(Tomsick et al. 1999; Radhika et al. 2018; Baby et al. 2020). In gen- eral, the energy spectra are dominated by the Comptonization com-ponent during the hard states (LHS and HIMS) whereas in SIMS andHSS it is dominated by thermal emission (Shakura & Sunyaev 1973;Titarchuk 1994; Tanaka & Lewin 1995; Giannios 2005; Nandi et al.2012; Sreehari et al. 2020).Though the HID comes in handy to make a good classification,the fact that the accretion states depend on more physical parametersbesides the flux, hardness ratio, presence and type of QPOs make thestandard classification tedious and time consuming. For instance, itwould be better to account for the inner disk temperature (kT in ) andthe photon index ( Γ ) from the energy spectral modelling while carry-ing out the classification. That is, if there are ‘N’ factors like HR, flux,QPOs, kT in and Γ on which the classification depends, then we canhave an N-dimensional space in which the data is distributed. Now,it is possible to classify each group of points (observations) havingsimilar characteristics in this space separately by accommodating amachine learning approach.In this paper, we attempt to make the classification of accretionstates during seven outbursts (single outbursts from XTE J1859+226,IGR J17091-3624 & MAXI J1535-571 and multiple outbursts fromGX 339-4) of BH-XRBs using Machine Learning techniques forthe first time. Specifically, we intend to use distance based methodslike the K-means clustering (MacQueen 1967) and the hierarchi-cal clustering (Ward 1963) algorithms for the classification of ac-cretion states. Prior to this, Machine Learning methods have beenemployed in astronomy for several scenarios. For instance, the K-means clustering of SDSS (Sloan Digital Sky Survey) galaxy spec-tra (Sánchez Almeida et al. 2010) were used to categorize galax- MNRAS000
20 keV/3 − Γ ) and the inner disk temperature (kT in ). In the top panel, the presence of Type-C QPOsare indicated in blue colour. state of the 1999 outburst of the source XTE J1859+226 depends on.Various branches within the q-diagram corresponding to this out-burst are classified into Low/Hard state (LHS), Hard-IntermediateState (HIMS), Soft-Intermediate State (SIMS) and High/Soft State(HSS) during the rising phase. The reverse trend is observed duringthe decay phase where the source transits to SIMS-decay, HIMS-decay and finally to LHS-decay before reaching the quiescent phase(see Figure 1 and Nandi et al. (2018)). It is observed that the LHScorresponds to maximum HR values accompanied by a low valueof flux. The flux usually reaches a high value at the end of thisstate or the beginning of the HIMS. In the HIMS, the flux valueremains the same though the hardness decreases. Generally, Radioflares (i.e. transient jets) are observed at the transition region be-tween HIMS and SIMS (Fender et al. 2004, 2009; Radhika & Nandi2014; Radhika et al. 2016a, and references therein). As the sourcetransits through the SIMS, the HR further decreases until the HSS isreached. In the decay phase, the flux level remains low and the HRgradually increases as the source transits through SIMS, HIMS andLHS. Type-C QPOs are generally associated with LHS and HIMS.Their frequency values are observed to increase during the risingphase and to decrease as the source transits through the decay phase(Rodriguez et al. 2002; Chakrabarti et al. 2008; Nandi et al. 2012,2018; Sreehari et al. 2019a). Generally, the photon index ( Γ ) ofenergy spectra increases from 1.4 to ∼ in ) is above 1 keV during the soft states(Tomsick et al. 1999; Radhika et al. 2018; Baby et al. 2020). In gen- eral, the energy spectra are dominated by the Comptonization com-ponent during the hard states (LHS and HIMS) whereas in SIMS andHSS it is dominated by thermal emission (Shakura & Sunyaev 1973;Titarchuk 1994; Tanaka & Lewin 1995; Giannios 2005; Nandi et al.2012; Sreehari et al. 2020).Though the HID comes in handy to make a good classification,the fact that the accretion states depend on more physical parametersbesides the flux, hardness ratio, presence and type of QPOs make thestandard classification tedious and time consuming. For instance, itwould be better to account for the inner disk temperature (kT in ) andthe photon index ( Γ ) from the energy spectral modelling while carry-ing out the classification. That is, if there are ‘N’ factors like HR, flux,QPOs, kT in and Γ on which the classification depends, then we canhave an N-dimensional space in which the data is distributed. Now,it is possible to classify each group of points (observations) havingsimilar characteristics in this space separately by accommodating amachine learning approach.In this paper, we attempt to make the classification of accretionstates during seven outbursts (single outbursts from XTE J1859+226,IGR J17091-3624 & MAXI J1535-571 and multiple outbursts fromGX 339-4) of BH-XRBs using Machine Learning techniques forthe first time. Specifically, we intend to use distance based methodslike the K-means clustering (MacQueen 1967) and the hierarchi-cal clustering (Ward 1963) algorithms for the classification of ac-cretion states. Prior to this, Machine Learning methods have beenemployed in astronomy for several scenarios. For instance, the K-means clustering of SDSS (Sloan Digital Sky Survey) galaxy spec-tra (Sánchez Almeida et al. 2010) were used to categorize galax- MNRAS000 , 1–10 (2020)
L approach for state classification ies with similar spectral properties together. Unsupervised cluster-ing algorithm was used to categorise supernovae (Rubin & Gal-Yam2016) light curves into slow-rising, fast-rise/slow-decline and fast-rise/fast-decline groups. Huppenkothen et al. (2017) attempted thecategorisation of variability classes of GRS 1915+105 using prin-cipal component analysis (PCA) followed by logistic regression.A rejection algorithm based on machine learning is employed todetect exo-planet transits (Mislis et al. 2018) in large-scale surveydata. Teimoorinia & Keown (2018) used an Artificial Neural Net-work (ANN) method to distinguish between Active Galactic Nu-clei (AGNs) and star forming galaxies. Carruba et al. (2019) usedHierarchical clustering algorithm to group asteroids into differentasteroid-families based on their distance from the reference body.Three classes of Gamma-ray bursts (GRBs) were identified usingK-Means clustering algorithm by Chattopadhyay et al. (2007). Re-cently, George et al. (2019) applied the concept of recurrence net-works to classify binary stars into semi-detached, overcontact andellipsoidal binaries. A few other automated methods are also worthmentioning in this context. Dunn et al. (2010) used a decision treebased automated algorithm to model energy spectra and obtain thebest fitting model. Tetarenko et al. (2016) generated machine read-able tables to create a database of Galactic BH-XRBs. A novelmethod for X-ray binary variability comparison called the ‘powercolours’ method was introduced by Heil et al. (2015). The presentwork is based on two fundamentally different clustering algorithmsusing which we classify the accretion states of Black hole binaries.This paper is organised as follows. In §2, we detail the observa-tions used in this paper and the standard approach for spectral stateclassification of outbursting BH-XRBs. We introduce the machinelearning based algorithms like K-Means clustering and Hierarchicalclustering in §3. The results from the application of these two al-gorithms for multiple BH sources are presented in §4. Finally, weconclude after discussing the results in §5. We re-analyse the archival data from HEASARC and ISSDC ofmultiple outbursting BH-XRBs using various X-ray observatories.Altogether, we use data of seven outbursts (see Table 1) from fourdifferent black hole binary sources. Four of these are from the sourceGX 339-4 which was active multiple times during the RXTE era .The other sources are XTE J1859+226 ( RXTE ), IGR J17091-3624(
Swift/XRT ) and MAXI J1535-571 (
MAXI, XRT & AstroSat ). We use archival data of
RXTE for analysing the 1999 outburst of XTEJ1859+226 and multiple outbursts (2002, 2004, 2007 and 2010) ofGX 339-4.
RXTE/PCA light curves in the energy range 3 −
20 keVare extracted for these outbursts following Radhika & Nandi (2014)and Sreehari et al. (2019a). The power spectra for these light curvesare generated using the powspec tool of the
XRONOS package.The power spectral features (QPOs and continuum) are mod-elled with multiple lorentzians. QPOs are identified based on thevalue of quality factor (Q = 𝜈 / 𝐹𝑊 𝐻𝑀 ), rms and significance( 𝜎 = 𝑛𝑜𝑟𝑚 /( 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 _ 𝑒𝑟𝑟𝑜𝑟 ) ). Lorentzian features with Q and https://heasarc.gsfc.nasa.gov/cgi-bin/W3Browse/w3browse.pl https://astrobrowse.issdc.gov.in/astro_archive/archive/Home.jsp https://heasarc.gsfc.nasa.gov/docs/xte/XTE.html https://heasarc.gsfc.nasa.gov/xanadu/xronos/xronos.html 𝜎 above 3 are generally considered as QPOs. Similarly, QPO pa-rameters for the 2016 outburst of the source IGR J17091-3624 areobtained from Radhika et al. (2018). For the source MAXI J1535-571, we have used QPO information from AstroSat and
Swift/XRT following Sreehari et al. (2019b). We use presence of different typesof QPOs along with flux and HR values as inputs to our MachineLearning algorithm in order to classify the spectral states of eachsource.
We follow the standard data extraction procedure (Sreehari et al.2019a) of
RXTE/PCA+HEXTE to extract spectra in the range3 −
150 keV for GX 339-4 and XTE J1859+226. Similarly,the
Swift/XRT spectra are extracted following Radhika et al.(2018). The energy spectra thus extracted are modelled using phabs(diskbb+smedge × powerlaw) following Radhika & Nandi(2014) and Aneesha et al. (2019). The total flux in 3 −
20 keV corre-sponding to the best fitting model is used to generate the HID. In thecase of Swift/XRT data, we use flux in the energy range 0 . −
10 keV.Additionally, we use inner-disc temperature (kT in ) from diskbb andphoton index ( Γ ) from powerlaw as inputs for our analysis. For thesource MAXI J1535-571, we have used the flux and hardness ratiosobtained from MAXI webpage following Sreehari et al. (2019b).Following the spectral data modelling, we compute the HR as theratio of flux in higher energy band (6 −
20 keV) to the flux in lowerenergy band (3 − Swift/XRT , this ratio is calcu-lated considering 4 −
10 keV and 0 . − Γ ) increases as thesource becomes softer.As we have to simultaneously take into account several parametersmentioned above in order to classify the accretion states, it would bebetter if we develop a standard algorithm to carry out this classifica-tion. In this paper, we attempt to incorporate all these factors (flux,HR, presence of QPOs, photon index, inner disk temperature) alongwith the observation time (MJD), in order to get a faster, easier andmore reliable classification scheme based on clustering techniques. In this section, we introduce two of the commonly used clusteringtechniques in machine learning that we will employ to carry out theclassification of accretion states of BH-XRBs. The algorithms thatwe use are K-Means clustering (MacQueen 1967) and Hierarchicalclustering (Ward 1963). We classify accretion states for seven out-bursts with both these methods and compare it with the ‘standard’results. Our intention is to identify the clustering method that is moresuitable to address the accretion state classification problem. http://maxi.riken.jp/mxondem/ MNRAS , 1–10 (2020)
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Figure 2.
The flowchart of K-Means clustering algorithm is shown. WCSS(Within Cluster Sum of Squared) distances is the sum of squared distancesbetween each point in the clusters and the corresponding cluster centroid.Once the number of clusters (K) are chosen, the algorithm starts with Krandomly chosen centroids. Each point is associated to the nearest centroidand then the centroid is updated as the mean of coordinates of the points withinthe cluster. This process is repeated until consistent results are obtained.
In order to pre-process the data to be clustered, we have normal-ized all numerical parameters like time (MJD), flux, hardness ratio(HR), photon index ( Γ ) and temperature (kT in ). After this, we con-vert the categorical (non-numerical) entries like presence of QPOsand Instrument used for observations (eg: RXTE/PCA , Swift/XRT )into dummy variables using the pandas (McKinney 2010) function get_dummies . This function performs a ‘one-hot’ encoding on thecategorical data. In ‘one-hot’ encoding, we represent a categoricalvariable which can take four values (say) as a vector of length four.Suppose we have QPO’s column in our data which can take values‘A’, ‘B’, ‘C’ and ‘No’ corresponding to Type-A, Type-B, Type-C andNo_QPOs respectively. If only a Type-C QPO is present in an ob-servation, the ‘one-hot’ encoded vector representing QPOs for thatobservation is [0, 0, 1, 0], where all entries are zero except for theposition indicating a Type-C QPO. The input parameters for the clus-tering algorithm used in each case are chosen based on the ExplainedVariance Ratio (EVR, Achen 1982). The EVR is the ratio of varianceof a parameter to the net variance of all the parameters. The varia-tions in the output variable are influenced mostly by the parameterswhich have the largest EVR. The same pre-processed data are used
Figure 3.
The flowchart of Hierarchical clustering algorithm is shown. Tobegin with, each data point has been considered as a cluster. Then distancebetween each pair of points (clusters) is found and stored as a matrix (D).The points separated by the shortest distance are then merged to form a newcluster and D is updated. Again, the clusters separated by the shortest distanceare identified and merged together based on the complete linkage criterion.This process is repeated until we get the desired number of clusters. See textfor details. as inputs for both K-Means and Hierarchical clustering, which aredescribed below.
We have used the scikit-learn (Pedregosa et al. 2011) implementationof K-Means clustering (MacQueen 1967) algorithm to classify thespectral states of BH-XRBs. The flowchart for this algorithm is pre-sented in Figure 2. Given a set of points in an N-dimensional space,the K-Means algorithm starts by randomly (based on a seed) guessingK (must be pre-defined) centroids. The algorithm finds the distanceto each point from each of the centroids and groups each point to thenearest centroid. At this instance, we get K groups or clusters. Noweach centroid is updated to be the mean value coordinate within eachgroup. Then the distances to each point from the new centroid arere-computed and the cluster is updated. This process continues untilwe get same centroids and clusters in consecutive iterations. Thevalue of K can be an informed guess or it can be obtained from theelbow method (see Figure 4). In the elbow method, the Within Clus-ter Sum of Squared (WCSS) distances for several number of clustersis calculated and the perpendicular from the ‘elbow’ of the plot isdrawn on to the X-axis. The elbow is the region of this plot wherethe rate of change of WCSS with respect to the number of clustersreduces considerably. The point where the perpendicular from theelbow meets the X-axis is chosen as K as shown in Figure 4.
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L approach for state classification W C SS Number of Clusters (K) − − − − − − R a t eo f c hangeo f W C SS Figure 4.
Within cluster sum of squared (WCSS) distances versus number ofclusters. The elbow of this plot is to be identified and a perpendicular is tobe drawn from the elbow on to the X-axis to obtain the required number ofclusters. The plot indicates the number of clusters to be around 8. The bottompanel shows the derivative of WCSS w.r.t. number of clusters. It is evidentthat the curve flattens around 8 clusters indicating the optimal number ofclusters required. See text for details.
Unlike K-Means clustering, Hierarchical clustering (Ward 1963;Duda et al. 2000) is based on a bottom-up algorithm. The algorithmis initiated with each point representing a cluster. At every furtheriteration the algorithm merges the nearest clusters together. The dis-tance calculation between clusters can be based on different linkagesnamely single, complete, ward and average. We use complete linkagewhere the distance between two clusters is the largest point-wise sep-aration in between clusters. The process stops once we get the desirednumber of clusters. This method is computationally more expensivebecause of the requirement of a linkage function besides the distancemeasure. Flowchart for this method is presented in Figure 3.In the following section, we present the results of clustering meth-ods when applied to the scenario of classification of accretion statesin BH-XRB sources.
In this section, we present the outcomes of the application of cluster-ing algorithms to accretion state classification of black hole binaries.Unless otherwise mentioned, we present only one result based onwhich algorithm produced the better classification corresponding toeach outburst.
We start with the comparison of accretion state clustering using K-Means with the ‘standard’ classification carried out for the sameoutburst (1999) of the source XTE J1859+226. In order to determinethe number of clusters required, we make use of the WCSS plot (theelbow method, see Figure 4). On visual inspection of this diagram,the elbow corresponds to an X-value (Number of clusters) in between6 to 9. The bottom panel of Figure 4 shows the derivative of WCSSwhich flattens considerably as the number of clusters approaches
Hardness Ratio T o t a l f l u x ( − e r g s / c m / s ) XTE J1859+226: HID K-Means Clustering
1: SIMS2: HIMS3: LHS decay4: HIMS decay5: LHS6: SIMS decay7: HSS
Figure 5.
Outcome of K-Means clustering of the q-diagram for the 1999outburst of XTE J1859+226. Coloured stars within white circles indicate theposition of centroid of each cluster. The legend is provided in the order inwhich each cluster is identified. The colour convention is the same as thatused in Figure 1. H I M S H I M S de c a y H S S L H S L H S de c a y S I M S S I M S de c a y Standard State H I M S H I M S de c a y H S S L H S L H S de c a y S I M SS I M S de c a y C l u s t e r ed S t a t e Figure 6.
The standard results are shown along the X-axis and K-Means clus-tering results are shown along the Y-axis of the confusion matrix. Diagonalentries denote correct or matching classification whereas off-diagonal entriesdenote a mismatch in classification. The ratio of diagonal counts to the totalcounts is the percentage match between the two classifications. The colourbar varies from yellow to blue as the counts increases.
8, indicating that the rate of change of WCSS reduces significantlyaround 8 clusters. It suggests an optimum number of clusters for theclassification to be equal to 8. However, as we are aware of the sevencanonical states of BH-XRBs we choose the number of clusters (K)to be seven. The normalised parameters along with the calculatedEVRs are MJD (0.139), Total flux (0.096), Hardness Ratio (0.099),Photon index (0.060), Inner Disk Temperature (0.106) and presenceof Type-C QPOs (0.248). The calculated EVRs for this outburst arementioned in Table 2.Figure 5 shows the output of K-Means clustering with 7 clusters. Itindicates that the LHS of rising phase is an isolated observation unlike
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Table 2.
The Explained Variance Ratios (EVRs) of parameters for each outburst considered in this paper are presented here. For the source MAXI J1535-571,we have used
MAXI flux to generate the complete q-diagram and hence spectral parameters were not taken into account.Source Outburst MJD Flux HR Γ kT in Type-C/Type-BXTE J1859+226 1999 0.139 0.096 0.099 0.060 0.106 0.248GX 339-4 2002 0.158 0.107 0.128 0.067 - 0.268GX 339-4 2004 0.175 0.083 0.200 0.109 0.139 0.145GX 339-4 2007 0.099 0.119 0.170 0.130 - 0.240GX 339-4 2010 0.170 0.087 0.169 0.045 0.116 0.204IGR J17091-3624 2016 0.090 0.059 0.097 0.121 0.196 0.133MAXI J1535-571 2017 0.195 0.120 0.051 - - 0.116/0.211
Hardness Ratio T o t a l f l u x ( − e r g s / c m / s ) GX 339-4: 2002: HID K-Means Clustering
1: HIMS decay2: HSS3: HIMS4: SIMS5: LHS decay6: LHS
Hardness Ratio T o t a l f l u x ( − e r g s / c m / s ) GX 339-4:2004: HID K-Means Clustering
1: HSS2: LHS3: SIMS4: HIMS5: LHS decay6: SIMS decay7: SIMS8: HIMS decay
Hardness Ratio T o t a l f l u x ( − e r g s / c m / s ) GX 339-4: 2007: HID K-Means Clustering
1: LHS decay2: HSS3: HIMS4: LHS5: SIMS decay6: HIMS decay7: LHS8: SIMS
Hardness Ratio T o t a l f l u x ( − e r g s / c m / s ) GX 339-4: 2010: HID K-Means Clustering
1: HSS2: LHS3: HIMS4: LHS decay5: SIMS6: HIMS decay7: SIMS decay
Figure 7.
Results of K-Means clustering of the q-diagram for multiple outbursts of GX 339-4 are presented with sequence as Top-left: 2002 outburst, Top-right:2004 outburst, Bottom-left: 2007 outburst and Bottom-right: 2010 outburst. Colour convention is same as that used in Figure 5. See text for details. the standard method (left panel of Figure 1). The HIMS of the risingphase matches with the standard classification except for the initialtwo points. The green coloured points in Figure 5 specify the SIMS,which extends only for a small duration. This is because several pointsthat were earlier classified as SIMS were more closely related to theproperties of the yellow points indicating HSS. Following this, wehave the decay phase with SIMS-decay (orange), HIMS-decay (cyan)and LHS-decay (pink). We bring to the reader’s notice that the legendentries in each figure are presented in the order in which each clusteris detected. We use the same colour convention for marking each state in all the clustering results. The accuracy in classification iscalculated as the ratio of diagonal entries to the total entries in theconfusion matrix (Fawcett 2006). The confusion matrix (see Figure6) portrays the counts of predicted labels (Clustered States) versus thetrue labels (Standard States). It is evident that there are 8 off-diagonalentries and a total of 44 observations in the confusion matrix whichcorresponds to an accuracy of 82%. This means 82% of K-Meansclustered states are correct when compared with the standard results.It must be noted that maximum discrepancy (6) between clusteringresults and standard results occurred with the classification of SIMS
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L approach for state classification and HSS which generally happens to be the region where HR valuesfluctuate the most. Similar observations are also noted in the case ofGX 339-4 and IGR J17091-3624 (see Figures 7 and 8). The precision(ratio of true positive detections to the total positive detections) forthe K-Means classification is 91% and the f1-score which is theharmonic mean of precision and sensitivity (ratio of true positivedetections to actual positives) is 0.82. However, with hierarchicalclustering only 45% accuracy is obtained. The BH-XRB source GX 339-4 was discovered by the OrbitingSolar Observatory-7 (Markert et al. 1973). During the
RXTE era,it has gone into multiple outbursts in 1999, 2002, 2004, 2007 and2010 (Sreehari et al. 2019a). We use complete q-diagrams during2002, 2004, 2007 and 2010 (Belloni et al. 2005; Nandi et al. 2012;Aneesha et al. 2019) to classify the spectral states with the aid of K-Means clustering algorithm. The EVRs for the parameters consideredfor all outbursts are tabulated in Table 2. The outcomes of K-Meansclustering for this source are shown in Figure 7. It is evident thatGX 339-4 has completed the q-diagram in all the four outburstsconsidered.Top-left panel of Figure 7 corresponds to the HID of 2002 outburstof GX 339-4. The clustering results have 90% matching instanceswith the standard classification (Belloni et al. 2005). We have fol-lowed Belloni et al. (2005) by considering only six clusters for thisoutburst, as in the standard classification the source has not enteredthe SIMS decay state. The top-right panel corresponds to the 2004outburst of GX 339-4. Though the source completes a q-profile,unlike other cases the flux of the source in HIMS is lower duringthis outburst. Moreover, it is the SIMS that has the peak flux duringthis outburst. The accuracy of classification using K-Means methodfor this outburst when compared to the standard results is 95%. Itshould be noted that we have considered two regions as SIMS in thiscase. This was required to address the non-canonical behaviour ofthe source during the rising phase. Generally, we need to use onlyseven clusters. However, in the case of 2007 outburst, due to thedata gap in the LHS of rising phase, the algorithm represents theLHS as two separate clusters. In order to overcome this problem weuse an additional cluster which is also labelled as LHS. As standardclassification details of this outburst are unavailable, we present theK-Means results without comparison. The q-diagram of the 2010outburst has a canonical profile and is presented in the bottom-rightpanel of Figure 7. The percentage of matching instances between K-Means clustering and standard classification (Nandi et al. 2012) forthis outburst is 87%. Precision, accuracy and f1-score correspondingto these results are summarised in Table 3. The accuracy score for the2002, 2004, 2007 and 2010 outbursts of GX 339-4 with Hierarchicalclustering are only 48%, 47%, 57% and 54% respectively.
The BH-XRB source IGR J17091-3624 was discovered in 2003 bythe
INTEGRAL observatory (Kuulkers et al. 2003). It has gone intooutburst multiple times after its discovery. The source completes aq-profile during the 2016 outburst (Radhika et al. 2018). After cal-culating the EVRs, we use the K-Means clustering algorithm on theHID (q-diagram) of its 2016 outburst. Left panel of Figure 8 showsthe result of clustering using the K-Means algorithm. There is only57% accuracy in this classification. On the right panel of Figure 8,we present the Hierarchical clustering results for the same outburst. Hierarchical clustering outcome shows 71% match (accuracy) withstandard classification. This is the only instance where hierarchicalclustering gave a quantitatively better outcome than the K-Meansmethod in the scenario of classification of accretion states. The pre-cision for Hierarchical clustering outcome is 0.92 and f1-score is0.77 for this outburst.
MAXI J1535-571 was discovered in 2017 simultaneously by
MAXI (Negoro et al. 2017) and
Swift (Kennea et al. 2017) observatories.We performed both K-Means clustering and hierarchical clusteringon the HID of this source generated using
MAXI data. Figure 9 showsthe different spectral states and the legend shows the correspondingcolours. It should be noted that in the rising phase, the source goesthrough LHS -> HIMS –> SIMS –> HIMS2 –> SIMS2 –> HSS as isreported in Tao et al. (2018) and Sreehari et al. (2019b). In the decayphase, the source transits through SIMS-decay, HIMS-decay andfinally to LHS-decay. The splits in HIMS were automatically detectedby the algorithm. This is because, we have used QPO informationfrom
AstroSat/LAXPC and
Swift/XRT as well (Sreehari et al. 2019b).The advantage here is that the decay phase data could be easilyclassified by clustering as opposed to the standard method whereinthe large error bars on decay phase data made it difficult to classifythe states. Clustering outcomes indicate 81% match with standardresults (Sreehari et al. 2019b) for the case of K-Means algorithm,while Hierarchical clustering results show only 58% accuracy. TheK-Means method has a precision of 89% and f1-score equal to 0.82when applied to this outburst, as summarised in Table 3. It is to benoted that although unlike the other cases the HID is plotted in units ofPhotons/cm /s, the available data from MAXI and QPO informationfrom
AstroSat and
XRT are enough to classify the states. So we havenot looked into the archival data from
NICER (Miller et al. 2018) forthis source.
In this paper, we have applied Machine Learning based clustering al-gorithms in the scenario of accretion state classification of BH-XRBsfor the first time. We have considered 7 outbursts from four differentBH-XRB sources (XTE J1859+226, GX 339-4, IGR J17091-3624and MAXI J1535-571) in order to check the feasibility of classify-ing accretion states using clustering algorithms. For this, we haveconsidered two algorithms namely K-Means clustering and Hierar-chical clustering. The algorithms require the number of clusters asan input parameter. The elbow method (see Figure 4) indicates 8clusters to be a reasonable choice. However, the optimal number ofclusters required is known to be seven from standard classifications(Belloni et al. 2005; Nandi et al. 2012). For some cases, we have usedeight clusters as was required due to the presence of data gaps in ob-servations (see bottom-left panel of Figure 7) or due to non-canonicalevolution of accretion state (see Figure 9). Further, we pre-processthe data by normalising all numerical parameters and applying ‘one-hot’ encoding to categorical or non-numerical parameters like typeof QPOs and instrument used for observation.The application of K-Means clustering for classification of spec-tral states for several outbursts as shown in this paper implies thatit is a dependable method, especially when there are multiple cri-teria to be considered. The method has also proved its robustnesswhen applied to the
MAXI data which has large uncertainties for thecase of the 2017 outburst of the source MAXI J1535-571. The only
MNRAS , 1–10 (2020)
Sreehari & Anuj Nandi
Hardness Ratio T o t a l f l u x ( − e r g s / c m / s ) IGR J17091-3624: HID K-Means Clustering
1: SIMS decay2: LHS decay3: HIMS4: HSS5: SIMS6: LHS7: HIMS decay
Hardness Ratio T o t a l f l u x ( − e r g s / c m / s ) IGR J17091-3624: HID Hierarchical Clustering
LHS decayHIMS decayHSSHIMSSIMSSIMS decayLHS
Figure 8.
Results of K-Means clustering of the q-diagram for the 2016 outburst of IGR J17091-3624 on the left and results of Hierarchical clustering of thesame outburst on the right. Among the seven outbursts considered in this paper, this is the only case where Hierarchical clustering results have better accuracythan K-Means clustering when compared with the standard classification. See text for details.
Hardness Ratio P h o t o n s / c m / s MAXI J1535-571: K-Means Clustering
1: SIMS decay2: SIMS3: HIMS4: HSS5: LHS decay6: HIMS decay7: SIMS28: LHS
Figure 9.
Results of K-Means clustering of the q-diagram for the 2017 out-burst of MAXI J1535-571. See text and Table 3 for details. disadvantage seems to be its inability to match clusters with similarcharacteristics when they are separated by data gaps as in the case ofthe 2007 outburst of GX 339-4 (see §4.2). This is because, the prox-imity in time (MJD) is one major factor for classification. When wetry excluding the time (MJD) as a parameter, the algorithm classifiesboth LHS-rise (LHS) and LHS-decay into the same cluster. As wehave to distinguish between these two states, we consider MJD asa necessary parameter. The inclusion of categorical parameters likepresence and type of QPOs and the instrument used which in turnspecifies the energy range helps in carrying out a better classification.However, we have not made use of Type-B QPO information exceptfor MAXI J1535-571. This is because though Type-B QPOs are usu-ally observed in SIMS, they are not present in all SIMS observations(Casella et al. 2005). It implies that the presence of Type-B QPOs areonly sufficient and not a necessary parameter for state classification.We have also applied Hierarchical clustering for all seven outburstdata. The percentage of matching classifications from Hierarchicalclustering with ‘standard’ results are lower than that obtained from
Table 3.
The weighted average precision, accuracy and f1-score for K-Meansclustering based classification with respect to the standard classification arepresented in this table. The cases where Hierarchical clustering is used aredenoted with † symbol. We point out that these values are not a measureof accuracy in classification, as we claim that the K-Means clustering basedmethod gives more reliable results as is discussed in §5.Source Outburst Precision Accuracy f1-scoreXTE J1859+226 1999 0.91 0.82 0.82GX 339-4 2002 0.93 0.90 0.90GX 339-4 2004 0.96 0.95 0.95GX 339-4 2010 0.90 0.87 0.87IGR J17091-3624 † K-Means clustering. Also, from visual inspection besides low preci-sion and accuracy, it was clear that Hierarchical clustering results aresub-optimal in majority of the cases and hence we have not includedthose results in this paper. Reasonable outcomes with Hierarchicalclustering are obtained only for 2016 outburst of IGR J17091-3624(see §4.3 and Figure 8). Besides this, unlike K-Means clustering,Hierarchical clustering cannot correct itself as it proceeds in an ag-glomerative way (Chattopadhyay et al. 2007). In Table 3, we provideonly the results from the quantitatively better (i.e. more accurate) ofthe two methods for each outburst.Results obtained from K-means clustering method are consistentwith the standard results (see Table 3) and so it can be considered asan alternative to the ‘standard’ approach. However, pure observableslike flux and hardness ratios are not enough for a good classification.As mentioned before, the presence of QPOs and spectral parameterslike Γ and kT in are also to be considered. For instance, Type-C QPOsare usually found only in the LHS and HIMS states. Similarly, Γ haslow values in the harder states, while it is higher in the soft states asmentioned in the §1. The weightage of these parameters are evidentfrom the EVRs mentioned in Table 2. In most of the cases the Type-CQPOs are found to have higher weightage than the other parameters.Anyway the values indicate that all the tabulated parameters aresignificant. The minimum weightage is for photon index with a value MNRAS000
The weighted average precision, accuracy and f1-score for K-Meansclustering based classification with respect to the standard classification arepresented in this table. The cases where Hierarchical clustering is used aredenoted with † symbol. We point out that these values are not a measureof accuracy in classification, as we claim that the K-Means clustering basedmethod gives more reliable results as is discussed in §5.Source Outburst Precision Accuracy f1-scoreXTE J1859+226 1999 0.91 0.82 0.82GX 339-4 2002 0.93 0.90 0.90GX 339-4 2004 0.96 0.95 0.95GX 339-4 2010 0.90 0.87 0.87IGR J17091-3624 † K-Means clustering. Also, from visual inspection besides low preci-sion and accuracy, it was clear that Hierarchical clustering results aresub-optimal in majority of the cases and hence we have not includedthose results in this paper. Reasonable outcomes with Hierarchicalclustering are obtained only for 2016 outburst of IGR J17091-3624(see §4.3 and Figure 8). Besides this, unlike K-Means clustering,Hierarchical clustering cannot correct itself as it proceeds in an ag-glomerative way (Chattopadhyay et al. 2007). In Table 3, we provideonly the results from the quantitatively better (i.e. more accurate) ofthe two methods for each outburst.Results obtained from K-means clustering method are consistentwith the standard results (see Table 3) and so it can be considered asan alternative to the ‘standard’ approach. However, pure observableslike flux and hardness ratios are not enough for a good classification.As mentioned before, the presence of QPOs and spectral parameterslike Γ and kT in are also to be considered. For instance, Type-C QPOsare usually found only in the LHS and HIMS states. Similarly, Γ haslow values in the harder states, while it is higher in the soft states asmentioned in the §1. The weightage of these parameters are evidentfrom the EVRs mentioned in Table 2. In most of the cases the Type-CQPOs are found to have higher weightage than the other parameters.Anyway the values indicate that all the tabulated parameters aresignificant. The minimum weightage is for photon index with a value MNRAS000 , 1–10 (2020)
L approach for state classification of 4.5% for the 2010 outburst of GX 339-4. An alternative approachis to carry out principal component analysis (PCA) to generate newparameters called principal components from the existing parametersin the order of decreasing EVRs. However, we have not employedPCA in this work.From the confusion matrix in Figure 6, it is clear that most of theconfusion (6 / = . • K-Means clustering when compared to standard results givesbetter accuracy than Hierarchical clustering. • Most of the discrepancy between standard results and K-Meansoutcomes occur during the SIMS and HSS states. • K-Means clustering algorithm helps us to categorise data moreaccurately than the standard method, because of its ability to club to- gether data with similar properties in a multi-dimensional parameterspace.
ACKNOWLEDGMENTS
The authors thank the reviewer for the valuable suggestions that haveimproved the quality of the manuscript. AN thank GH, SAG; DD,PDMSA and Director, URSC for encouragement and continuoussupport to carry out this research. Authors thank Ravishankar B.T.for careful reading and comments on the manuscript. This researchmade use of the data obtained from HEASARC (NASA) and AstroSatarchive (Astrobrowse) of Indian Space Science Data Center (ISSDC).
DATA AVAILABILITY
Data used for this work is obtained from the HEASARC( https://heasarc.gsfc.nasa.gov/cgi-bin/W3Browse/w3browse.pl )archive of NASA and the AstroSat archive( https://astrobrowse.issdc.gov.in/astro_archive/archive )of the Indian Space Science Data Center (ISSDC).
REFERENCES
Achen C., 1982, Interpreting and Using Regression. Quantitative applicationsin the social sciences. SAGE Publications, Inc., Thousand Oaks, CA,doi:10.4135/9781412984560Agrawal V. K., Nandi A., 2020, MNRAS, 497, 3726Agrawal V. K., Nandi A., Girish V., Ramadevi M. C., 2018, MNRAS, 477, 5437Altamirano D., Belloni T., 2012, ApJ, 747, L4Aneesha U., Mandal S., Sreehari H., 2019, MNRAS, 486, 2705Baby B. E., Agrawal V. K., Ramadevi M. C., Katoch T., Antia H. M., Mandal S., Nandi A., 2020, MNRAS, 497, 1197Belloni T. M., 2010, States and Transitions in Black Hole Binaries. Springer BerlinHeidelberg, p. 53, doi:10.1007/978-3-540-76937-8_3Belloni T. M., Altamirano D., 2013, MNRAS, 432, 10Belloni T. M., Stella L., 2014, Space Sci. Rev., 183, 43Belloni T., Klein-Wolt M., Méndez M., van der Klis M., van Paradijs J., 2000, A&A,355, 271Belloni T., Homan J., Casella P., van der Klis M., Nespoli E., Lewin W. H. G., MillerJ. M., Méndez M., 2005, A&A, 440, 207Belloni T., et al., 2006, MNRAS, 367, 1113Belloni T. M., Bhattacharya D., Caccese P., Bhalerao V., Vadawale S., Yadav J. S., 2019,MNRAS, 489, 1037Carruba V., Aljbaae S., Lucchini A., 2019, MNRAS, 488, 1377Casella P., Belloni T., Homan J., Stella L., 2004, A&A, 426, 587Casella P., Belloni T., Stella L., 2005, ApJ, 629, 403Chakrabarti S. K., Debnath D., Nandi A., Pal P. S., 2008, A&A, 489, L41Chattopadhyay T., Misra R., Chattopadhyay A. K., Naskar M., 2007, ApJ, 667, 1017Dhillon I. S., Guan Y., Kulis B., 2004, in Proceedings of the TenthACM SIGKDD International Conference on Knowledge Discoveryand Data Mining. KDD ’04. Association for Computing Machin-ery, New York, NY, USA, p. 551–556, doi:10.1145/1014052.1014118, https://doi.org/10.1145/1014052.1014118
Duda R. O., Hart P. E., Stork D. G., 2000, Pattern Classification (2nd Edition). Wiley-Interscience, USADunn R. J. H., Fender R. P., Körding E. G., Belloni T., Cabanac C., 2010, MNRAS,403, 61Fawcett T., 2006, Pattern Recognition Letters, 27, 861Fender R. P., Belloni T. M., Gallo E., 2004, MNRAS, 355, 1105Fender R. P., Homan J., Belloni T. M., 2009, MNRAS, 396, 1370
MNRAS , 1–10 (2020) Sreehari & Anuj Nandi
George S. V., Misra R., Ambika G., 2019, Chaos, 29, 113112Giannios D., 2005, A&A, 437, 1007Hasinger G., van der Klis M., 1989, A&A, 225, 79Heil L. M., Uttley P., Klein-Wolt M., 2015, MNRAS, 448, 3339Homan J., Belloni T., 2005, Ap&SS, 300, 107Homan J., Wijnands R., van der Klis M., Belloni T., van Paradijs J., Klein-Wolt M.,Fender R., Méndez M., 2001, ApJS, 132, 377Huppenkothen D., Heil L. M., Hogg D. W., Mueller A., 2017, MNRAS, 466, 2364Ingram A. R., Motta S. E., 2019, New Astron. Rev., 85, 101524Iyer N., Nandi A., Mandal S., 2015, ApJ, 807, 108Kennea J. A., Evans P. A., Beardmore A. P., Krimm H. A., Romano P., Yamaoka K.,Serino M., Negoro H., 2017, The Astronomer’s Telegram, 10700, 1Kuulkers E., Lutovinov A., Parmar A., Capitanio F., Mowlavi N., Hermsen W., 2003,The Astronomer’s Telegram, 149MacQueen J., 1967, in Proceedings of the Fifth Berkeley Sympo-sium on Mathematical Statistics and Probability, Volume 1: Statis-tics. University of California Press, Berkeley, Calif., pp 281–297, https://projecteuclid.org/euclid.bsmsp/1200512992
Markert T. H., Canizares C. R., Clark G. W., Lewin W. H. G., Schnopper H. W., SprottG. F., 1973, ApJ, 184, L67McKinney W., 2010, in van der Walt S., Millman J., eds, Proceedings of the 9th Pythonin Science Conference. pp 51 – 56Miller J. M., et al., 2018, ApJ, 860, L28Mislis D., Pyrzas S., Alsubai K. A., 2018, MNRAS, 481, 1624Morgan E. H., Remillard R. A., Greiner J., 1997, ApJ, 482, 993Motta S., Muñoz-Darias T., Casella P., Belloni T., Homan J., 2011, MNRAS, 418, 2292Nandi A., Debnath D., Mandal S., Chakrabarti S. K., 2012, A&A, 542, A56Nandi A., et al., 2018, Ap&SS, 363, 90Negoro H., et al., 2017, The Astronomer’s Telegram, 10699, 1Ng A. Y., Jordan M. I., Weiss Y., 2001, in Proceedings of the 14th International Confer-ence on Neural Information Processing Systems: Natural and Synthetic. NIPS’01.MIT Press, Cambridge, MA, USA, p. 849–856Pedregosa F., et al., 2011, Journal of Machine Learning Research, 12, 2825Radhika D., Nandi A., 2014, Advances in Space Research, 54, 1678Radhika D., Nandi A., Agrawal V. K., Seetha S., 2016a, MNRAS, 460, 4403Radhika D., Nandi A., Agrawal V. K., Mandal S., 2016b, MNRAS, 462, 1834Radhika D., Sreehari H., Nandi A., Iyer N., Mand al S., 2018, Ap&SS, 363, 189Remillard R. A., McClintock J. E., 2006, ARA&A, 44, 49Remillard R. A., Morgan E. H., McClintock J. E., Bailyn C. D., Orosz J. A., 1999, ApJ,522, 397Rodriguez J., Durouchoux P., Mirabel I. F., Ueda Y., Tagger M., Yamaoka K., 2002,A&A, 386, 271Rubin A., Gal-Yam A., 2016, ApJ, 828, 111Sánchez Almeida J., Aguerri J. A. L., Muñoz-Tuñón C., de Vicente A., 2010, ApJ,714, 487Shakura N. I., Sunyaev R. A., 1973, A&A, 500, 33Sreehari H., Nandi A., Radhika D., Iyer N., Mandal S., 2018,Journal of Astrophysics and Astronomy, 39, 5Sreehari H., Iyer N., Radhika D., Nandi A., Mandal S., 2019a,Advances in Space Research, 63, 1374Sreehari H., Ravishankar B. T., Iyer N., Agrawal V. K., Katoch T. B., Mandal S., Nand iA., 2019b, MNRAS, 487, 928Sreehari H., Nandi A., Das S., Agrawal V. K., Mandal S., Ramadevi M. C., Katoch T.,2020, MNRAS, 499, 5891Strohmayer T. E., 2001, ApJ, 554, L169Tanaka Y., Lewin W. H. G., 1995, in X-ray Binaries. pp 126–174Tao L., et al., 2018, MNRAS, 480, 4443Teimoorinia H., Keown J., 2018, MNRAS, 478, 3177Tetarenko B. E., Sivakoff G. R., Heinke C. O., Gladstone J. C., 2016, ApJS, 222, 15Titarchuk L., 1994, ApJ, 434, 570Tomsick J. A., Kaaret P., Kroeger R. A., Remillard R. A., 1999, ApJ, 512, 892Ward J. H., 1963, Journal of the American Statistical Association, 58, 236 Xu Y., et al., 2017, ApJ, 851, 103Yadav J. S., et al., 2016, ApJ, 833, 27Zdziarski A. A., Johnson W. N., Magdziarz P., 1996, MNRAS, 283, 193van der Klis M., Jansen F., van Paradijs J., Lewin W. H. G., van den Heuvel E. P. J.,Trumper J. E., Szatjno M., 1985, Nature, 316, 225
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