A Comparative Study of Host Galaxy Properties between Fast Radio Bursts and Stellar Transients

The most important global properties of the host galaxies are stellar mass M_*, star formation rate SFR, specific star formation rate sSFR (SFR/M_*), metallicity 12 + log(O/H), as well as the half-light radius R₅₀ of the host galaxies. The stellar mass M_* of a galaxy is usually estimated by fitting the broadband spectral energy distribution (SED) to the stellar population synthesis (Arnouts et al. 1999; Kauffmann et al. 2003; Boquien et al. 2019). Star formation rate is estimated from emission lines such as Hα, or ultraviolet (UV) luminosity (see Kennicutt & Evans 2012 for a review). sSFR is estimated as SFR/M_* when SFR and M_* are both available. The transients related to younger populations (LGRBs and SLSNe) are more likely to reside in the galaxies with smaller stellar masses and more intense SFR and sSFR (and sometimes less metallicity) than those related to old populations (Kelly et al. 2008; Schulze et al. 2018). The metallicity of a galaxy is usually estimated from its emission (Kewley & Dopita 2002; Kobulnicky & Kewley 2004; Pettini & Pagel 2004; Kewley & Ellison 2008; Dopita et al. 2016) or absorption line ratios (Draine 2011). The emission line method gives metallicity in the form of 12 + log(O/H), with the solar metallicity being 12 + log(O/H)_⊙ = 8.69 (Asplund et al. 2009). The absorption line method, on the other hand, gives metallicity in the form of [X/H] = log(N_X/N_H) − log(N_X/N_H)_⊙, where N_X indicates the column density of element X. To be consistent, we convert 12 + log(O/H) to [X/H] in this paper. For metallicities estimated from emission lines, we choose to use those estimated based on Dopita et al. (2016) when available, to be consistent with that estimated for FRB 180916.J0158+65.

The half-light radius R₅₀ is the radius that encloses 50% of the total light of the galaxy. It is usually estimated by fitting the surface brightness of a galaxy with the Sérsic profile

$$\begin{eqnarray*}&&{\rm{\Sigma }}(r)={{\rm{\Sigma }}}_{{\rm{e}}}\exp \{-{k}_{{\rm{n}}}[{\left(r/{r}_{{\rm{e}}}\right)}^{1/n}-1]\},\end{eqnarray*}$$

where the effective radius r_e represents R₅₀. Another way is to fit the brightness profiles with ellipses centered around the galaxy and identify the one whose enclosed flux is half of the total flux. R₅₀ is defined as the semimajor axis of the ellipse. In general, R₅₀ scales with stellar mass M_*. For the same stellar mass, a star-forming galaxy usually has a larger R₅₀ than a passive galaxy.

The same galaxy may host different types of transients. Thus, local properties at the sub-galactic level can provide more precise diagnostics to the environment of a certain transient. One important property is the offset of the transient from the center of the host galaxy. It can be measured in physical units (kiloparsecs) as R_off, or normalized to the characteristic radius of the host r_off = R_off/R₅₀. The larger R_off, the farther away the transient is from the center of the host, and the fainter and more quiescent the local environment is. However, R_off is misleading for irregular galaxies since the center of the galaxy is hard to define and usually does not mark the region with most intense star formation. For these cases, F_light is a more efficient parameter. It is defined as the total light emitted in the region fainter than the transient position, within the host. By definition, a transient within the brightest region would have F_light ∼ 1, and one within the faintest region would have F_light ∼ 0 (Fruchter et al. 2006; Anderson et al. 2015a, 2015b).

For a nearby transient, surface brightness Σ, local color, and local star formation rate density Σ_SFR would give more precise information. However, these parameters are not available for most objects at larger distances. Since the redshifts of the localized FRBs are in the range 0–1, a valid local star formation rate density Σ_SFR is hard to obtain for most of them. We therefore use galaxy-scale properties M_*, SFR, sSFR, [X/H], and R₅₀, and sub-galactic scale properties R_off, r_off = R_off/R₅₀, and F_light in this study.

We summarize the host galaxy properties of the FRBs studied in this paper in Table 1. For FRB 180916.J0158+65, the global SFR is scaled from that at the FRB position as SFR = 0.016 ⨯ 6.57/1.002 = 0.1 M_⊙ yr⁻¹. The metallicity of FRB 180916.J0158+65 is estimated based on Dopita et al. (2016). We then choose to use the metallicity values based on the same reference for FRB 121102 as well as for other transients when available. The half-light radius R₅₀ of FRB 180916.J0158+65 is not available from the paper. We estimate R₅₀ with the Petrosian half-light radius R_50,petro = 466 and the 90% radius as R_90,petro = 853 from the Sloan Digital Sky Survey (SDSS) catalog, using the formula R₅₀ = R_50,petro/(1 – 8 × 10⁻⁶(R_90,petro/R_50,petro)^8.47) = 47 (Graham et al. 2005). This gives a physical distance of 3.3 kpc. We estimate the offset of FRB 190523 with the coordinates (J2000) of the FRB, right ascension (R.A.) 13:48:15.6(2), declination (decl.) +72:28:11(2), and the host coordinates from the PanSTARRS, stack R.A. 13:48:15.426 (20706427), decl. +72:28:14.6 (7247072). The offset is estimated to be ${3.7}_{-2.1}^{+2.2}$ arcsec, corresponding to ${26.5}_{-14.9}^{+15.5}$ kpc for its redshift. Note that the astrometric registration between FRB 190523 and the PanSTARRS image is not available here, which may result in an additional ∼(03–05) uncertainty.

Table 1.  Host Galaxy Properties of FRBs with Host Galaxies Identified

	Instrument	z	log SFR	log sSFR	log M*	[X/H]	R50	Offset	Offset	Flight	References
			(M⊙ yr−1)	(Gyr−1)	(M⊙)		(kpc)	(kpc)	(R50)
FRB 121102	Arecibo	0.19273	−0.40	0.86	7.7	−0.59	1.4	0.82	0.60	1.0	1
FRB 180916.J0158+65	CHIME/FRB	0.0337	−1.0	−2.0	10.0	0.13	3.3	4.7	1.5	⋯	5, 7
FRB 180924	ASKAP	0.3214	<0.30	<−2.0	10.3	⋯	2.8	3.8	1.4	0.08	3
FRB 181112	ASKAP	0.4755	−0.22	−0.62	9.4	⋯	3.9	${3.1}_{-3.1}^{+15.7}$	0.79	⋯	4, 6
FRB 190102	ASKAP	0.2913	0.18	−0.33	9.5	−0.25	5.3	${1.5}_{-1.5}^{+3.4}$	0.28	⋯	6
FRB 190523	DSA-10	0.66	<0.11	<−2.0	11.1 ± 0.1	−0.52	⋯	${26.5}_{-14.9}^{+15.5}$	⋯	⋯	2
FRB 190608	ASKAP	0.11778	0.079	−1.3	10.4	−0.34	⋯	6.8 ± 1.3	⋯	⋯	6
FRB 190611	ASKAP	0.378	⋯	⋯	⋯	⋯	⋯	17.2 ± 4.9	⋯	⋯	8
FRB 190711	ASKAP	0.522	⋯	⋯	⋯	⋯	⋯	${1.5}_{-1.5}^{+3.6}$	⋯	⋯	8

References. (1) Tendulkar et al. (2017); (2) Ravi et al. (2019); (3) Bannister et al. (2019); (4) Prochaska et al. (2019); (5) Ahumada et al. (2020); (6) Bhandari et al. (2020); (7) Marcote et al. (2020); (8) Macquart et al. (2020).

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The metallicities from Bhandari et al. (2020) are provided as Z. They are converted to [X/H] by using [X/H] = log₁₀(Z/Z_⊙), where Z_⊙ = 0.0196 is the solar metallicity.

In summary, seven well localized FRBs have stellar mass, SFR, and sSFR available. All of the nine FRBs have offset information available. Note that while FRB 121102, FRB 180916.J0158+65, FRB 180924, FRB 190608, and FRB 190611 have well defined offsets, the offsets of FRB 181112, FRB 190102, and FRB 190711are consistent with being zero. Although the localization of FRB 190523 is relatively poor, a relatively large offset is favored.

Li et al. (2016) compiles 407 GRBs with redshifts or host galaxy properties from the literature (e.g., Fruchter et al. 2006; Fong et al. 2010; Berger 2014; Blanchard et al. 2016 and references therein), with all the estimation methods labeled. The stellar mass (M_*) values of Li et al. (2016) are mainly obtained from SED fitting by Savaglio et al. (2009) and Leibler & Berger (2010), but some are estimated from the K-band or infrared (IR) magnitudes. Due to the large uncertainty of the latter method, only those estimated with SED fitting are included in this study. The SFR values collected in Li et al. (2016) are mainly estimated with emission lines from Savaglio et al. (2009), Krühler et al. (2015), and Berger (2009), with some estimated using UV fluxes. The emission lines usually trace recent star formation (<10 Myr), related to LGRBs, while the UV fluxes trace 10–200 Myr star formation, related to core-collapse SNe (see Kennicutt & Evans 2012 for a review). Thus, we use the SFR values estimated using both methods. The metallicity values collected in Li et al. (2016) are estimated with both emission line ratios (Kobulnicky & Kewley 2004; Pettini & Pagel 2004; Kewley & Ellison 2008; Savaglio et al. 2009; Krühler et al. 2015) and absorption line ratios (Draine 2011; Cucchiara et al. 2015). If the metallicity [X/H] is estimated with R₂₃ = ([O ii] λ3727 + [O iii] λλ4959, 5007)/Hβ, the results are double-valued from Kewley & Ellison (2008) and from Savaglio et al. (2009). In this case, we only select the larger value of the two, following Kobulnicky & Kewley (2004) and Berger (2009). The half-light radius R₅₀ of the host galaxy and the offset of the GRB from the center of the host are mainly derived from Hubble Space Telescope (HST) images (Bloom et al. 2002; Wainwright et al. 2007; Fong et al. 2010; Fong & Berger 2013; Blanchard et al. 2016). Usually galaxies are inclined. The offsets are sometimes corrected for the inclination. However, the statistical results are not influenced significantly by the inclinations of the objects (Japelj et al. 2018). All the FRB and GRB offsets are projected offsets. To be consistent, we also use projected offsets in the SN section. See Li et al. (2016) for more details.

Here we only use the well measured parameters, with upper and lower limits excluded. Following Li et al. (2020), here we do not include the four GRBs whose physical categories are subject to debate, i.e., GRB 060505, GRB 060614, GRB 090426, and GRB 060121. The numbers of LGRBs and SGRBs with host galaxy properties are 263 and 31, respectively. To make the GRB sample more consistent with that of FRBs, we only use LGRBs and SGRBs with redshifts z < 1 for comparison. This results in smaller sample sizes, i.e., 72 LGRBs and 22 SGRBs.

For SNe, we use the data from the Open Supernovae Catalog (OSC)⁶ as the starting point to build our samples. The OSC includes the coordinates (R.A. and decl.) of the SNe and the names and coordinates of their hosts. Some host galaxies in the OSC do not have coordinates labeled. For these, we search for their names in SIMBAD⁷ to collect their coordinates. To be consistent, all available host galaxy names and coordinates are calibrated to SIMBAD IDs and coordinates. Some host galaxies are not available in SIMBAD. They are calibrated to NED⁸ instead. In order to reduce the misidentification of the host galaxies, we exclude those SNe with host galaxy distances larger than 1°. We then collect the host galaxy information from the papers exploring SNe and galaxy catalogs.

Properties from SN papers—We first supplement the host galaxy properties from the papers exploring the SN host galaxy properties. We match their SN names with the OSC names or aliases during this process.

For the stellar mass, we use the values estimated with SED fitting only. For SFR, we prefer those estimated by emission lines, especially Hα. If this is not available, we use the value estimated using the far-ultraviolet (FUV) method. When no value from the above two methods is available, we use the SFR value derived from the SED fitting. Kelly & Kirshner (2012) estimated the stellar mass using SED fitting, sSFR using fiber spectra, and metallicity using emission lines and the method of Pettini & Pagel (2004, hereafter PP04) for different types of transients. Taggart & Perley (2019) and Schulze et al. (2018) estimated the stellar mass and SFR of core-collapse supernovae and SLSNe using SED fitting. For SLSNe only, Perley et al. (2016) estimated log M_* and SFR using SED fitting and estimated the SFR with emission lines when spectra are available. For SNe Ia, more than 600 log M_*, SFRs, and metallicities are produced using SED fitting by Kim et al. (2018).

For metallicity, we prefer those values estimated using the emission line method, especially those based on PP04 and Dopita et al. (2016, hereafter D16). To be consistent with FRB 180916.J0158+65, whose host metallicity was estimated with D16, we adopt those from D16 whenever available. Otherwise, the values from PP04 are adopted. Graham (2019) estimated the SN host metallicities and provided the largest SN sample estimated with D16. Kelly & Kirshner (2012), Schady et al. (2019), and Anderson et al. (2016) estimated metallicities with the PP04 method for various transients. For SNe Ia located in passive galaxies, Kang et al. (2016, 2020) estimated the host metallicities using absorption line ratios. We use those [M/H] values based on the Yonsei evolutionary population synthesis models (Chung et al. 2013).

We prefer the half-light radius R₅₀ estimated using 2D Sérsic fitting. Lunnan et al. (2015) and Japelj et al. (2018) provided R₅₀ in the r band for SLSNe and SNe Ic-BL, respectively. Other R₅₀ values of SN hosts are obtained from galaxy catalogs.

The offsets R_off between SNe and their hosts are usually available from the OSC. However, the offsets are sometimes not trustworthy, because the R.A. of the host galaxies in the Asiago SN catalog is accurate to seconds, with an uncertainty of 15''. These are not suitable for offset calculations. We thus extract the host galaxy coordinates from the SDSS-II catalog (Sako et al. 2018), the ASAS-SN catalog,⁹ and the bright SN catalog¹⁰ by matching the OSC SN names with the names in other catalogs. In our analysis, the offsets for the FRB sample as well as the OSC and GRB samples are not corrected for inclination. We therefore use the projected offsets whenever available. We also update the offset values from the detailed papers exploring SN offsets. In particular, Anderson et al. (2016) and Japelj et al. (2018) provided the projected R_off. Kelly & Kirshner (2012) gave the deprojected r_off.

The F_light values are compiled from the papers exploring SN properties. Anderson et al. (2012) calculated F_light in both Hα and near-UV bands for various SNe transients. Hα traces on-going star formation (0–16 Myr), which is more relevant to LGRBs, and near-UV traces recent star formation (16–100 Myr) (Gogarten et al. 2009), which is more relevant to core-collapse supernovae. However, F_light values for Hα are usually not available for relatively high redshifts. We thus employ the F_light for near-UV whenever available. Anderson et al. (2015a) and Lunnan et al. (2015) provided F_light values from UV for SNe Ia and SLSNe, respectively. Kelly et al. (2008) estimated F_light from the g band.

Properties from galaxy catalogs—We then supplement the host galaxy properties from the catalogs of galaxies. During this process we match galaxies with SN hosts both by coordinates with a 3'' precision and by names.

For the galaxies within the coverage of SDSS, we use the parameters derived from the SDSS spectrum and broadband photometrics. The MPA-JHU group provided the stellar mass M_*, SFR, and metallicity of SDSS DR8 galaxies by taking both spectrum and photometrics into account (Kauffmann et al. 2003). We use their results when available. However, they did not provide the results for galaxies later than DR8. The flexible stellar population synthesis (SPS) used the SPS method to estimate the galaxy properties for both DR8 and DR12 galaxies (Conroy et al. 2009). We use their results for those not available in the MPA-JHU catalog. Karachentsev et al. (2013)¹¹ collected the SFR values estimated from Hα and the FUV of galaxies in the Local Volume. Up to 2020 January 1, this includes 1212 galaxies. We calibrate the galaxy names to their SIMBAD IDs, and match the names to the OSC SN hosts. Similarly, the SFR and stellar mass M_* from Vaddi et al. (2016) as well as the age and metallicity from Terlevich & Forbes (2002) are appended to the OSC SN hosts.

For the galaxies within the coverage of SDSS, we use the Sérsic half-light radius R_50,s in the NASA-SDSS Atlas catalog (NSA)¹² when available, which photometered 640,000 SDSS galaxies within z < 0.15. For large galaxies, e.g., R₅₀ ∼ 1'–15, Jarrett et al. (2003) estimated the radii of the largest 656 galaxies with images from the Two Micron All Sky Survey, which also include the galaxies outside the coverage of SDSS. We use the J-band R_e in Jarrett et al. (2003) for large galaxies instead. In addition, some R₅₀ of SLSN hosts are provided in Lunnan et al. (2015). Otherwise, we use the the half-light radius within the SDSS catalog¹³ by matching the SN host galaxies with SDSS galaxies within 3''. In the SDSS catalog, the half-light radius is estimated from the r-band Petrosian half-light radius R_50,petro and the Petrosian 90% radius R_50,petro following R₅₀ = R_50,petro/(1 – 8 × 10⁻⁶(R_90,petro/R_50,petro)^8.47) (Graham et al. 2005).

We also update the offset parameters from the literatures. Lunnan et al. (2015) calculated the projected R_off from HST images for SLSNe. Anderson et al. (2016) reported the projected R_off from SNe II to the nearest H ii regions. Kelly & Kirshner (2012) provided the deprojected r_off = R_off/R₅₀ for SNe II and SNe Ibc. Although projected offsets and deprojected offsets are statistically consistent, they are different for specific objects. Since most offsets in the literature and catalogs are projected, we use these deprojected offsets only when projected offsets are not available. We also convert them to r_off when R₅₀ is available in our catalog.

The sample sizes for SLSNe, SNe Ibc, SNe II, and SNe Ia with host galaxy properties are 107, 900, 4251, and 7221, respectively. To have a consistent z range as the FRB sample, we screen the samples with z < 1. This gives sample sizes of 95, 874, 2829, and 7003 for the four types, respectively. Most SNe Ibc, SNe II, and SNe Ia have z < 1. The excluded objects are mostly those without redshift information.

The sample size for each parameter for each type of stellar transient is presented in Table 2.

The classical KS test compares the cumulative distribution function (CDF) of two distributions P(<x) and P(<x'), which could be one data sample and one model sample, or two data samples. The largest distance between the two CDFs is defined as D_KS = max(P(<x) − P(<x')), representing the difference between the two distributions x and x'. The distribution of D_KS is free from the distribution of x and x' and independent of the direction of data ordering, i.e., D_KS calculated from P(<x) is the same as that calculated from P(>x). This method is widely applied in defining the goodness of a fit or comparing two samples.

The key problem in generalizing the classical KS test to multiple dimensions is the direction of the data ordering. Peacock (1983) suggested using the maximum absolute difference between the two samples D_DKS when all possible directions along the axes are considered. For a two-dimensional problem, the difference D_DKS values are calculated for four quadrants of n² origins,

$$\begin{eqnarray*}&&\left\{\begin{array}{l}(x\lt {X}_{i},y\lt {Y}_{j})\\ (x\lt {X}_{i},y\gt {Y}_{j})\\ (x\gt {X}_{i},y\lt {Y}_{j})\\ (x\gt {X}_{i},y\gt {Y}_{j})\end{array}\right.,\quad (i,j=1,\ \ldots ,\ n)\end{eqnarray*}$$

for all possible i and j values. Here n is the sample size, and the method applies to comparison of a data sample with a model. D_DKS is confirmed to be efficient for correlated samples. For D dimensions, the number of quadrants to be calculated would be D²n^D, which is computationally expensive for dimensions larger than 2 and/or n > 100.

Fasano & Franceschini (1987, hereafter FF87) proposed to use a simpler and faster method. For a two-dimensional problem, the difference D_DKS is calculated for four quadrants of n origins,

$$\begin{eqnarray*}&&\left\{\begin{array}{l}(x\lt {X}_{i},y\lt {Y}_{i})\\ (x\lt {X}_{i},y\gt {Y}_{i})\\ (x\gt {X}_{i},y\lt {Y}_{i})\\ (x\gt {X}_{i},y\gt {Y}_{i})\end{array}\right.,\quad (i=1,\ \ldots ,\ n).\end{eqnarray*}$$

This method is proved not to compromise the power of the test. With this method, only D²n quadrants should be considered for a D-dimensional comparison of size n, which is much more efficient than the method of Peacock (1983).

Moreover, FF87 generalized the method to two-sample multi-dimensional KS tests by proposing to use the average ${\bar{D}}_{\mathrm{DKS}}$ of the two D_DKS estimated according to the data sample 1 and sample 2. We use the two-sample method of FF87 in our analysis. However, the two-sample test in FF87 requires that the correlations among parameters are similar to each other in the two samples in order to use the probability P distribution presented in their paper. This may not be the case in our problem. We thus estimate the null probability P with Monte Carlo simulations. Since our FRB sample is much smaller than our stellar transient samples, in each trial we randomly extract the transient samples to have the same number of events as FRBs and calculate D_mcDKS. We calculate D_mcDKS 1000 times and obtain the distribution of D_mcDKS. The null probability P_DKS between the FRB host galaxy sample and the transient host galaxy samples is estimated by interpolating D_DKS in the simulated D_mcDKS distribution. Notice that such P_DKS values from this MC simulation may have large uncertainties when both the FRB sample and the transient sample are small.

The most common parameters available for transient host galaxies are stellar mass log M_* and log sSFR, regardless of whether the transients are well located on the sub-galactic scale. Thus, log M_* and log sSFR present the largest samples, both for FRBs and for other stellar transients. Seven FRB host galaxies have the information for log M_* and log sSFR. The numbers for each type transient with both log M_* and log sSFR are listed in the second column of Table 3. The FRBs (stars) are compared with various stellar transients in the log M_*–log sSFR plane in Figure 1. The stellar transient hosts and SDSS galaxies are presented as solid and dotted contours, with 1σ (68%) and 3σ (99.7%) confidence levels, respectively. The 1σ contours seem to occupy three regions in the log sSFR−log M_* diagram. (1) 7 < log M_* < 10.5 and −1 < log sSFR < 1: LGRB (magenta) and SLSNe (orange) hosts occupy the strongest star formation region, while SLSN hosts have smaller stellar mass log M_* than LGRB hosts. (2) 8.5 < log M_* < 11.5 and −3 < log sSFR < 0: SN Ibc (green) and SN II (dark green) hosts reside in the middle region in the log sSFR–log M_* plane. This region also contains the star-forming galaxy of SDSS and SN Ia (cyan) hosts. (3) 10 < log M_* < 12 and log sSFR < −2: There are also passive galaxy components in SN Ia hosts and SDSS (gray) galaxies. SGRB (blue) hosts occupy all three regions in the log sSFR−log M_* diagram. For FRBs, the host of FRB 121102 is located in the region with the smallest stellar mass and strongest star formation. The hosts of FRB 181112, FRB 190102, and FRB 190608 reside in the middle region. The hosts of FRB 180924, FRB 190523, and FRB 180916.J0158+65 are in the joint region between the star formation galaxies and the passive galaxy component of SDSS galaxies, SN Ia hosts, and SGRB hosts. All FRB hosts except that for FRB 121102 are consistent with the hosts of SNe Ibc, SNe II, SNe Ia, and SGRBs within the 68% region.

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We perform a two-dimensional KS test between FRBs and other transients in the log M_*–log sSFR space. D_2KS and the null probability values are presented in the third and fourth columns of Table 3. It turns out that the whole FRB sample rejects the origin similar to LGRBs and SLSNe at a significance level of 0.02, while other origins similar to SN Ibc, SN II, SN Ia, and SGRBs are still consistent with the FRB sample. Although the SGRB sample is small and the relation between its D_DKS and P_DKS has significant uncertainties, D_DKS of SGRBs is the smallest among all the transients, indicating similarity or more consistency of FRBs with SGRBs.

We also examine the differences among different types of transients in the log M_*–log SFR space (i.e., sSFR is replaced by SFR). The results are similar: that the FRB hosts and LGRB/SLSN hosts are significantly different, suggesting that the FRB host galaxy as a whole disfavors the LGRB and SLSNe origin. Similarly, the FRB hosts are also consistent with those of other transients in the log M_*–log SFR space.

We note that different SFR indicators represents stars with different ages (Kennicutt & Evans 2012), which introduces uncertainties to the SFR and sSFR in our sample. Moreover, SFR indicators, including emission lines, UV and IR emission, are usually influenced by the emission from active galactic nuclei. Many FRBs, including FRB 190102, FRB 180924, and FRB 190608, are located in galaxies with active nuclei or LINER emissions (Bhandari et al. 2020). In these cases, the log SFR and log sSFR values should be considered as the upper limits. As a result, to compare their environments more accurately, sub-galactic parameters should be taken into account.

The positions of the transients within their host galaxies also provide important and usually more accurate information about the origin of the transients. The offset is available for all nine FRBs. However, the positional uncertainty of FRB 181112 is larger than the size of the galaxy. We do not include it in our analysis here. Although the positional uncertainty of FRB 190523 is also large, its offset is even more significant. We thus keep it in the sample. There are six FRBs with log M_*, log sSFR, and log R_off in our analysis. We perform a three-dimensional KS test between these six FRBs and other transients in the log M_*–log sSFR–log R_off space. The results are presented in columns 5–7 of Table 3. The conclusion is similar to what we found in the previous section: the FRB hosts as a whole disfavor the LGRB and SNSL origin.

On the other hand, the offset is also controlled by the size R₅₀ and the gravitational well of the host. Also, due to the irregularity of some hosts, such as the host of FRB 121102, F_light better represents the local environments of the transients (Fruchter et al. 2006; Anderson et al. 2012). A more complete comparison of galactic and sub-galactic environments should include five parameters: log M_*, log sSFR, log R₅₀, log R_off, and F_light. The numbers of each type of transient with all five parameters are listed in the eighth column of Table 3, and the D_2KS values and the null probability values of the five-dimensional KS test between FRBs and other transients are presented in the ninth and tenth columns. However, due to the small size, the FRB sample does not show a significant difference with respect to any other type of stellar transient. A larger sample and/or a more detailed analysis of the FRB hosts may help to narrow down the preferred transients, if any, or suggest that FRBs are indeed consistent with a broad range of transient types.

The naive Bayes method is a classification method based on the Bayes theorem and the assumption that parameters are not correlated, i.e.,

$$\begin{eqnarray}&&P(T| \{x\})=\displaystyle \frac{P(\{x\}| T)P(T)}{P(\{x\})},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&P(\{x\}| T)=\displaystyle \prod _{i}P({x}_{i}| T),\end{eqnarray} \tag{ 2 }$$

where $P(T| \{x\})$ is the posterior probability for one object with parameter set {x} to have a type T, which could be LGRBs, SLSNe, SNe Ibc, SNe II, SNe Ia, or SGRBs; $P(\{x\}| T)$ is the likelihood for one object with a type T to have a parameter set {x}, P(T) is the prior probability of one object being in type T, and P({x}) is the probability of one object having a parameter set {x}. Although the assumption that parameters are uncorrelated is "naive," the results are amazingly good (Hand & Yu 2001; Broos et al. 2011).

The implementation of naive Bayes in our problem follows the following steps.

• Estimate the likelihood $P({x}_{i}| T)$ for each parameter x_i and type T with the known stellar transient samples.
• Estimate the prior P(T), usually by the size of the sample. However, for FRBs, we do not have any prior information about the preference to specific types. We thus use equal priors for all types of transients.
• Calculate the posterior probability following Equation (2) for each type of transient T.
• Normalize the posterior probability $P(T| \{x\})$ by requiring ${\sum }_{j}P({T}_{{\text{}}j}| \{x\})=1$ for each FRB, where T_j stands for different transient types.
• The type of the FRB is assigned to the one with the highest posterior probability.

We estimate the likelihood for each parameter x_i and type T, $P({x}_{i}| T)$, with the observed sample in Section 2. Gaussian distributions are assumed for most parameters, while exponential distributions P(x) ∝ exp(−x) are assumed for F_light for LGRBs, SNe Ia, and SGRBs. The fitting results are presented in Table 4 and Figure 2. The distribution of FRBs is also presented in red for comparison.

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The naive Bayes results for the stellar transients are presented in Figure 3. This shows the number of known LGRBs, SLSNe, SNe Ibc, SNe II, SNe Ia, and SGRBs classified as each type with naive Bayes. It turns out that most LGRBs, SLSNe, and SGRBs can be identified with their host galaxy properties. However, about half of the SNe Ibc, SNe II, and SNe Ia may be misclassified as SGRBs, suggesting that the host galaxy properties of these transients are not that different from those of SGRBs.

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We apply the same method to individual FRBs. The results are presented in Table 5 and Figure 4. The following conclusions can be drawn. The host of FRB 121102 has 90% probability of belonging to the SLSN sample and 10% probability of belonging to the LGRB sample, and its probabilities of belonging to SN Ibc, SN II, SN Ia, and SGRB samples are significantly smaller. FRB 180916.J0158+65 has a very low probability of belonging to the LGRB or SLSN samples, but is consistent with belonging to one of the SN Ibc, SN II, SN Ia, or SGRB samples. FRB 180924 also has a very low probability of belonging to the LGRB/SLSN sample but has a fairly high probability of belonging to SN Ia or SGRB samples. FRB 181112 has a reasonable probability of belonging to any sample, due to its mild SFR and log M_* and the lack of sub-galactic information. FRB 190523 also disfavors an LGRB/SLSN origin but is consistent with an SN Ibc, SN II, or SN Ia origin. Its consistency with the SGRB sample is marginal.

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