A Comparative Study of Long and Short GRBs. II. A Multiwavelength Method to Distinguish Type II (Massive Star) and Type I (Compact Star) GRBs

Our method makes use of the multiwavelength data of GRBs. In order to come up with a set of sound classification criteria, one needs to first look into the statistical distributions of the observational properties for different types of GRBs. We use a sample based on the catalog presented in Li et al. (2016), in which the prompt emission and host galaxy parameters of 407 GRBs were compiled. In total, there are 16 parameters, including the redshift, 7 prompt emission parameters, and 8 host galaxy parameters. The prompt emission parameters include the duration T₉₀, two spectral parameters (the peak energy ${E}_{{\rm{p}}}$ and the low energy photon index α) of the best-fitting Band function (Band et al. 1993), the isotropic γ-ray energy E_iso, the isotropic γ-ray peak luminosity L_iso, the amplitude f parameter, and the effective amplitude parameter f_eff (the f parameter when the background is shifted to make the duration T₉₀ be 2 s, see Lü et al. 2014 for a detailed discussion of f and f_eff parameters). The host galaxy parameters include stellar mass of the host M_*, star formation rate (SFR), specific star formation rate (sSFR; the ratio between SFR and M_*), metallicity of the host [X/H], half-light radius of the host R₅₀, offset of the GRB from the center of the host galaxy in units of kpc R_off, normalized offset r₅₀ = R_off/R₅₀, and fraction of the host light in the area fainter than the GRB position, F_light.

The "consensus" LGRBs and SGRBs are defined based on the online GRB catalog maintained by Jochen Greiner.¹² The LGRBs and SGRBs in this catalog are defined based on the published results in the literature, including GCNs. Such definitions usually take into account the multiwavelength observational data, and the definitions of "long" and "short" already implies their possible physical origins. There are no well-defined criteria regarding how the category of one particular burst is assigned, but the classification presented on the website usually reflects the consensus in the GRB community regarding each GRB with redshift and multiwavelength information. We consider this "consensus" LGRB sample as the preliminary sample of Type II GRBs, and the "consensus" SGRB sample as the preliminary sample of Type I GRBs. The classification of some bursts are subject to debate, e.g., GRB 060505, GRB 060614, GRB090426, and GRB 060121. These bursts are excluded from our control sample. Altogether, there are 403 GRBs in our control sample.

For each parameter except F_light, we use a Gaussian function

$$\begin{eqnarray}&&P({x}_{i}| \mathrm{Type})=\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^{2}}}\exp \left(-\displaystyle \frac{{\left({x}_{i}-\mu \right)}^{2}}{2{\sigma }^{2}}\right)\end{eqnarray} \tag{ 2 }$$

to fit the distribution of each of these two classes of GRBs. For F_light, an exponential distribution

$$\begin{eqnarray}&&P({x}_{i}| \mathrm{Type})=\displaystyle \frac{\gamma }{\exp (\gamma )-1}{e}^{\gamma {x}_{i}},\end{eqnarray} \tag{ 3 }$$

is used to fit the distribution. The normalization is obtained by requiring the integration in [0,1] to be unity.

For most of the parameters, not all GRBs have the measured values. We properly select a smaller sample to perform the fittings. For the redshift parameter, we only consider the precise, spectrally identified redshifts. The original definition of f_eff (Lü et al. 2014) is slightly larger than 1. In this work, we use log (f_eff − 1) instead to perform the statistical analysis. The stellar mass of the host galaxy M_* can be estimated by either the spectral energy density (SED) fitting or the infrared (IR) luminosity. The IR luminosity method assumes a 70 Myr old stellar population, which gives a very large uncertainty. Therefore, we only select the M_* values that are derived using the SED method. The star formation rate (SFR) can be estimated by emission lines such as Hα, Hβ,[O iii], [O ii], and Lyα, as well as continuum such as UV, IR, and SED fitting. Emission lines indicate the SFR with age 0–10 Myr, around the life of stars with mass >30 M_⊙. The continuum, on the other hand, indicates the SFR with age 0–100 Myr (Kennicutt & Evans 2012). In our analysis, we only adopt the values obtained from the emission lines to perform the fitting. We further exclude the upper and lower limits of the SFR. The metallicity [X/H] estimated with ${R}_{23}=([{\rm{O}}\,{\rm{II}}]\lambda 3727+[{\rm{O}}\,{\rm{III}}]\lambda 4959,5007)/{\rm{H}}\beta$ is double-valued (Kewley & Ellison 2008; Savaglio et al. 2009). Following the suggestions of Kobulnicky & Kewley (2004) and Berger (2009), we use the larger of the two values as the metallicity of a particular GRB.

The maximum likelihood estimation is used to fit the distributions. For Gaussian distribution, there are analytical solutions of μ and σ². (See the Appendix for details.) The histogram of each parameter and the best-fitting results are shown in Figure 1, with red lines for Type II GRBs and blue lines for Type I GRBs. The fitting results are given in Table 1, with errors estimated by 100 bootstraps. The number of samples N, mean μ, and standard deviation σ for Type II and Type I GRBs are listed in Columns 2–4 and 7–9 of Table 1, for all parameters except F_light. For F_light, Columns 3 and 8 give the exponential index γ. In column 12, the values with the same probability in Type I and Type II are presented as the "imputer," which are the values to fill in if the corresponding values are missing. In order to test the goodness of the fit, we applied the Kolmogorov–Smirnov (KS) test, because it is valid for most functions and more sensitive to the center (see more discussions about the Anderson–Darling test in Section 4.4). The KS test results D_KS and the corresponding null probability P_KS are listed in columns 4, 5, 8, and 9. For the F_light of Type I GRB, the large D_KS = 0.56 is due to many Type I GRBs with F_light = 0.0. However, it is highly influenced by the rounding in the literature. We thus try to randomly assign F_light = 0.0 to be [0.0, 0.05]. It results in D_KS = 0.34 and P_KS = 0.02. If we assign F_light = 0.0 to be [0.0, 0.1], D_KS = 0.25 and P_KS = 0.19. The classification results are not affected by the rounding effect correction.

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Table 1.  Fitting Results of Each Parameter for the Preliminary Type II and I Samples of Li et al. (2016)

Parameter	Type II					Type I
	N	Mean μ(γ)	σ	${D}_{\mathrm{KS}}$	${P}_{\mathrm{KS}}$	N	Mean μ(γ)	σ	${D}_{\mathrm{KS}}$	${P}_{\mathrm{KS}}$	Imputera
log T90	371	1.68 ± 0.03	0.59 ± 0.03	0.05	0.41	32	−0.36 ± 0.10	0.57 ± 0.08	0.07	1.00	0.65
log (1+z)	349	0.43 ± 0.01	0.18 ± 0.01	0.03	0.83	24	0.17 ± 0.02	0.08 ± 0.01	0.13	0.77	0.30
log ${E}_{\mathrm{iso}}$ (erg)	371	52.62 ± 0.05	1.01 ± 0.05	0.08	0.03	31	50.92 ± 0.16	0.94 ± 0.11	0.11	0.84	51.78
log ${L}_{\mathrm{iso}}$ (erg s−1)	368	52.00 ± 0.06	1.13 ± 0.07	0.08	0.03	31	51.34 ± 0.19	1.07 ± 0.11	0.08	0.99	51.76
α	197	−1.02 ± 0.03	0.34 ± 0.02	0.06	0.55	15	−0.57 ± 0.06	0.26 ± 0.04	0.17	0.72	−0.81
log ${E}_{{\rm{p}}}$ (keV)	203	2.16 ± 0.04	0.48 ± 0.03	0.05	0.69	17	2.63 ± 0.10	0.42 ± 0.08	0.14	0.89	2.36
log $(f-1)$	222	−0.37 ± 0.03	0.46 ± 0.02	0.09	0.04	28	0.17 ± 0.07	0.36 ± 0.03	0.11	0.84	−0.14
log (${f}_{\mathrm{eff}}-1$)	212	−0.91 ± 0.02	0.34 ± 0.02	0.07	0.25	28	0.13 ± 0.07	0.37 ± 0.03	0.13	0.71	−0.40
log SFR (${M}_{\odot }$ yr−1)	200	0.67 ± 0.05	0.82 ± 0.04	0.07	0.29	20	0.00 ± 0.19	0.87 ± 0.12	0.11	0.95	0.28
log sSFR (Gyr−1)	92	0.06 ± 0.08	0.72 ± 0.06	0.08	0.56	15	−0.53 ± 0.23	1.05 ± 0.15	0.12	0.98	−0.57
log M* (M⊙)	98	9.52 ± 0.08	0.81 ± 0.06	0.04	0.99	22	9.90 ± 0.18	0.84 ± 0.09	0.11	0.96	9.76
$[{\rm{X}}/{\rm{H}}]$	131	−0.70 ± 0.06	0.62 ± 0.03	0.11	0.09	9	−0.03 ± 0.05	0.16 ± 0.02	0.19	0.87	−0.32
log R50 (kpc)	126	0.26 ± 0.03	0.32 ± 0.02	0.07	0.50	22	0.56 ± 0.05	0.29 ± 0.05	0.13	0.81	0.39
log offset (kpc)	134	0.20 ± 0.05	0.55 ± 0.03	0.05	0.88	26	0.96 ± 0.11	0.55 ± 0.05	0.11	0.87	0.58
log offset (R50)	115	−0.12 ± 0.04	0.44 ± 0.03	0.06	0.75	22	0.34 ± 0.08	0.45 ± 0.06	0.12	0.88	0.11
${F}_{\mathrm{light}}$	97	1.37 ± 0.36b	⋯	0.09	0.45	18	−5.65 ± 1.88	⋯	0.56	0.00	0.36
${F}_{\mathrm{light}}$	97	1.33 ± 0.36c	⋯	0.09	0.39	18	−4.70 ± 1.88	⋯	0.25	0.19	0.38

Notes. All parameters but ${F}_{\mathrm{light}}$ are fitted with a Gaussian distribution, and the distribution of ${F}_{\mathrm{light}}$ is fitted with an exponential distribution.

^aThe values to replace the missing values. ^bFor ${F}_{\mathrm{light}}$, this is the index γ of the exponential distribution. ^cAfter considering the rounding effect modification.

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One of the most important problems in machine learning is how to deal with the missing values. Here we choose to impute the missing values with the equal-likelihood values, that is, the value to have the same likelihood to be Type I and Type II GRBs. The values are listed in the last column of Table 1. It is equivalent to multiplying the likelihood $P({x}_{i}| \mathrm{Type})$ with observed values only in Equation (1), since $P({x}_{i}| \mathrm{II})=P({x}_{i}| {\rm{I}})$ for missing values.

Naive Bayes assumes no correlation among parameters. However, there are some obviously correlated parameters in our sample, e.g., the subsets {SFR, sSFR and M_*}, {L_iso, E_iso, and T₉₀}, {R₅₀, R_off, and r_off}, and redshift z. In order to eliminate significant correlations, we would like to exclude one parameter in each subset. Since we aim to distinguish the two types of GRBs, it is better to exclude the parameter with the least separations in each subset. Table 7 of Li et al. (2016) tested the difference between each parameter of Type II and Type I by KS test, with P_KS indicating the difference between each parameter of Type II and Type I GRB. Thus, we exclude those parameters with the largest P_KS, i.e., the least difference, in each parameter subset. For example, the isotropic peak luminosity L_iso is excluded, since it could be roughly reproduced by T₉₀ and E_iso, and its P_KS value for Type II and Type I GRBs is larger than those of T₉₀ and E_iso. Similarly, we exclude f, SFR, and R_off. Finally, as presented in Li et al. (2016), the redshift z is subject to a selection effect and is correlated with many parameters, such as E_iso, SFR, and [X/H]. We therefore do not include z in the analysis. In summary, we take the parameters $\{x\}=\{{T}_{90},{E}_{\mathrm{iso}}$, α, ${E}_{{\rm{p}}},{f}_{\mathrm{eff}}$, sSFR, ${M}_{* },[{\rm{X}}/{\rm{H}}]$, ${R}_{50},{r}_{\mathrm{off}}={R}_{\mathrm{off}}/{R}_{50}$, ${F}_{\mathrm{light}}\}$ to implement the naive Bayes analysis.

The priors P(I) and P(II) are unknown. We use the ratio of their observed numbers, i.e., P(II) = 371/403 = 0.92, and P(I) = 32/403 = 0.08 as the priors.

In order to examine the results in more detail, we calculate the posterior odds

$$\begin{eqnarray}&&O(\mathrm{II}:{\rm{I}}| \{x\})=\displaystyle \frac{P(\mathrm{II}| \{x\})}{P({\rm{I}}| \{x\})}=\displaystyle \frac{P(\mathrm{II})}{P({\rm{I}})}\displaystyle \frac{P(\{x\}| \mathrm{II})}{P(\{x\}| {\rm{I}})},\end{eqnarray} \tag{ 4 }$$

which gives the degree that the observed parameter set $\{x\}$ supports the Type II against the Type I hypothesis. The first term P(II)/P(I) is the prior odds, and the second term $P(\{x\}| \mathrm{II})/P(\{x\}| {\rm{I}})$ is the Bayes factor (or likelihood ratio) (Currell & Dowman 2009 and their updates).¹³

By definition, a positive log $O(\mathrm{II}:{\rm{I}})$ indicates a preference to Type II GRB, and a negative log $O(\mathrm{II}:{\rm{I}})$ indicates a preference to Type I.

We plot the logarithmic posterior odds log $O(\mathrm{II}:{\rm{I}})$ in the left panel of Figure 2. Red and blue histograms show the distribution of log $O(\mathrm{II}:{\rm{I}})$ for preliminary Type II GRBs (LGRBs) and Type I GRBs (SGRBs). The logarithmic value $\mathrm{log}O(\mathrm{II}:{\rm{I}})$ of preliminary Type I GRBs are all negative, with the largest value −1.2. Most of preliminary Type II GRBs have positive log $O(\mathrm{II}:{\rm{I}})$, with two exceptions. They are GRB 131004A and GRB 090927. GRB 131004A has a duration ${T}_{90}=1.54\,{\rm{s}}$, and f_eff = 1.94, similar to a Type I GRB. However, the spectrum of it is quite soft, with a α = −1.4 in Fermi/GBM detector, similar to a Type II GRB. And there is no more host galaxy information. Thus, the naive Bayes classifier does not give a strong preference. Its log $O(\mathrm{II}:{\rm{I}})=-0.16$, the smallest one of the preliminary Type II GRBs. GRB 090927A has T₉₀ = 2.16 s, z = 1.37, and $\mathrm{log}O(\mathrm{II}:{\rm{I}})=-0.14$, showing no strong preference to either GRB type.

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The debated GRBs, GRB 060505, GRB 060614, GRB 090426, and GRB 060121 are not included in the histogram. They are presented as green and orange rectangles. They are all located near zero, consistent with the fact that their origins are under debate. The ironic neutron star–neutron star merger event GRB 170817A is overplotted as the dark blue rectangle or star, which shows the smallest log $O(\mathrm{II}:{\rm{I}})$, indicating that it is a prototype of Type I GRBs.

Note that there is no overlap between the two preliminary samples of Type I and Type II GRBs. The gap is between −1.2 and −0.16. It means that an additional modification

$$\begin{eqnarray}&&{ \mathcal O }=\mathrm{log}\ O(\mathrm{II}:{\rm{I}})+0.7\end{eqnarray} \tag{ 5 }$$

would be a better criterion to classify GRBs physically. With it, every predicted type matches the preliminary type.

In order to examine the effect of prompt emission and host galaxy properties, we further study them separately. The posterior odds are

$$\begin{eqnarray*}&&\begin{array}{l}\mathrm{log}\ O{(\mathrm{II}:{\rm{I}})}_{\mathrm{prompt}}=\mathrm{log}\ O{(\mathrm{II}:{\rm{I}}| \{x\})}_{\mathrm{prompt}}\\ \quad =\,\mathrm{log}\ O(\mathrm{II}:{\rm{I}}| \{{T}_{90},{E}_{\mathrm{iso}},\alpha ,{E}_{\mathrm{peak}},{f}_{\mathrm{eff}}\})\end{array}\end{eqnarray*}$$

and

$$\begin{eqnarray*}&&\begin{array}{l}\mathrm{log}\ O{(\mathrm{II}:{\rm{I}})}_{\mathrm{host}}=\mathrm{log}\ O{(\mathrm{II}:{\rm{I}}| \{x\})}_{\mathrm{host}}\\ \quad =\,\mathrm{log}\ O(\mathrm{II}:{\rm{I}}| \{\mathrm{sSFR},{M}_{* },[{\rm{X}}/{\rm{H}}],{R}_{50},{r}_{\mathrm{off}},{F}_{\mathrm{light}}\}).\end{array}\end{eqnarray*}$$

The results are presented in the right panel of Figure 2. The preliminary Type II GRBs are shown with red points, and the preliminary Type I GRBs are shown with blue squares. The GRBs without host galaxy information have $\mathrm{log}\ O{(\mathrm{II}:{\rm{I}})}_{\mathrm{host}}=\mathrm{log}\ P(\mathrm{II})/P({\rm{I}})=1.06$, and cluster in the figure. By definition, a positive log $O{(\mathrm{II}:{\rm{I}})}_{\mathrm{prompt}}$ or log $O{(\mathrm{II}:{\rm{I}})}_{\mathrm{host}}$ indicates the preference of a Type II GRB over a Type I GRB. It can be seen that most Type II GRB candidates (red dots) are indeed located in the first quadrant, with log $O{(\mathrm{II}:{\rm{I}})}_{\mathrm{prompt}}\gt 0$ and log $O{(\mathrm{II}:{\rm{I}})}_{\mathrm{host}}\gt 0$, and lots of Type I candidates (blue dots) are located in the third quadrant. Also, there are a few preliminary Type II GRBs in the second and forth quadrant, and there are some preliminary Type I GRBs in the second quadrant. These may be results of the correlations among parameters, or the simple estimation of the priors by the detected Type II/Type I GRBs. Still, in the $\mathrm{log}\ O{(\mathrm{II}:{\rm{I}})}_{\mathrm{host}}$ versus $\mathrm{log}\ O{(\mathrm{II}:{\rm{I}})}_{\mathrm{prompt}}$ diagram, we find that a straight line

$$\begin{eqnarray}&&\mathrm{log}\ O{(\mathrm{II}:{\rm{I}})}_{\mathrm{host}}=-\mathrm{log}\ O{(\mathrm{II}:{\rm{I}})}_{\mathrm{prompt}},\end{eqnarray} \tag{ 6 }$$

is able to separate these two types of GRBs, as shown by the solid line.

All of the four debated GRBs are quite close to the Type I region in the right panel of Figure 2, although they are located in the Type II region. GRB 060121 has a T₉₀ of 2.61 s, longer than the conventional separation line for SGRBs and LGRBs. The low energy photon index α is −0.5, around the typical value of Type I GRBs. The size of its host is ${R}_{50}={10}^{0.6}$ kpc, which is also a typical value of Type I GRBs. On the other hand, its offset and fraction of light F_light suggest that it belongs to Type II. This object is somewhat between Type II and Type I GRBs. Note that there is no convincing redshift measurement of GRB 060121A, and a redshift of z = 0.5 has been assumed (Li et al. 2016).

GRB 170817A, the only ironclad Type I GRB firmly associated with a binary neutron star merger from the gravitational wave observations, is presented as a dark blue star. The prompt emission properties are taken from Zhang et al. (2018), and the host galaxy properties are taken from Blanchard et al. (2017). Its log O(II:I)_prompt and log O(II:I)_host are −5.2 and −3.8, respectively. Both are well below the separation line, well consistent with our Type I GRB classification criterion.

The SN association and observational limits of the existence of SN provide important information about the physical origin of GRBs. We add the SN information to posterior odds calculations and they are presented in Figure 3.

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For some GRBs, stringent limits of SNe were set, which give a strong indication of their origins. The inclusion of SN limits leads to a correction of the posterior odds. Assuming a Gaussian distribution of the peak absolute magnitudes of the associated SNe, we estimate the probability to have an associated SN fainter than the upper limit P(SN) with the Gaussian distribution and modify the odds to be log ${O}_{\mathrm{sn}}(\mathrm{II}:{\rm{I}})=\mathrm{log}\ O(\mathrm{II}:{\rm{I}})+\mathrm{log}\ P(\mathrm{SN})$. The peak absolute magnitudes of the observed GRB-associated SNe have a narrow distribution, −18.52 ± 0.45 mag (Hjorth & Bloom 2012; Berger 2014). However, it is possible that fainter associated SNe are not detected. Since most of the LGRB-associated SNe are Type Ic with broad lines, we adopt the peak absolute magnitude distribution of SN Ic-BL as the distribution of that of LGRB-SNe. It is estimated to be −18.3 ± 1.6 mag from the open SN catalog.¹⁴ The results of GRB classifications after including the SN criterion are presented in Figure 3. Because the GRBs nearest to the gaps usually do not have SN information, the position of the gap between Type I and Type does not change.

With the SN limit correction, two of the highly debated objects, Type II candidates, GRB 060614 and 060505, are now shifted to the Type I region. These two GRBs are also widely discussed to be long-duration Type I candidates (Della Valle et al. 2006; Gehrels et al. 2006; Fynbo et al. 2006; Zhang et al. 2007). Their prompt emission properties are not that typical compared with other Type II GRBs, and their host galaxy properties are similar to Type II GRBs. However, their SN limits strongly favor the merger origin of these two events.

The SN limits are also presented in the log $O{(\mathrm{II}:{\rm{I}})}_{\mathrm{host}}$-log $O{(\mathrm{II}:{\rm{I}})}_{\mathrm{prompt}}$ diagram (the right panel of Figure 3) as dark cyan lines.

GRBs with spectrally confirmed SN associations are definitely Type II GRBs from core collapse of massive stars. In the right panel of Figure 3, GRBs with spectrally confirmed SN associations are shown as triangles with solid arrows, and those with SN spectral features are presented as triangles with dashed arrows (Hjorth & Bloom 2012).

Machine-learning studies reveal that the errors in the training sample, the sample to estimate the parameter distributions, are usually smaller than the true error. A common method to estimate the true error is cross validation. By randomly dividing the total sample into a training sample and a test sample, which does not contribute to the parameter estimation, the true error is around the error of the test sample. However, the sample size of the preliminary Type I GRBs is only 32. An equal division would significantly reduce the sample size, and overestimate the test error. In order to eliminate this issue, we utilize the "leave one out cross validation" (LOOCV) method. This method runs the naive Bayes method 403 times, since we have 403 GRBs in the controlled sample. For each time, one GRB is left out as a test sample, while the other 402 GRBs are used as a training samples to fit the distributions of the parameters. The sum of the errors of the 403 realizations is considered as the test error. As a result, for Type II GRBs are misclassified as Type I GRBs, and no Type I GRBs are misclassified as Type II GRBs. The test error rate is thus 4/403 = 1%.

The selection of the control sample might affect our results. We, therefore, test the case with the four highly debated GRBs in the analysis. With the naive Bayes method, there are three preliminary Type II GRBs misclassifed as Type I GRBs, and two preliminary Type I GRB misclassified as Type II GRBs. The two misclassified Type I GRBs are GRB 090426 and GRB 060121, two of the highly debated GRBs. With the log $O^{\prime} =$ log $O(\mathrm{II}:{\rm{I}})+0.7$ as a criterion, only the two highly debated GRB 090426 and GRB 060121 are misclassified.

There is a slight overlap between the two candidate populations. The overlap fraction is 2% (8) for Type II candidates and 6% (2) for Type I candidates. Even in this case, the overlap fraction is much smaller than previous methods. For example, for the duration classification method, there are 7% Type II and 20% Type I candidates in the overlapping region. For the f_eff parameter method, the corresponding fraction is 7% (48%) for Type II (Type I) candidates, respectively.

The naive Bayes method assumes no correlation among parameters. To eliminate strong correlations, we select $\{x\}=\{{T}_{90}$, ${E}_{{iso}}$, α, ${E}_{{\rm{p}}},{f}_{\mathrm{eff}}$, sSFR, ${M}_{* },[{\rm{X}}/{\rm{H}}],{R}_{50}$, ${r}_{\mathrm{off}},{F}_{\mathrm{light}}\}$ only. Since naive Bayes performs amazingly well in mildly correlated cases, we also wonder what the result would be if we used different parameter groups.

In order to test this, we include all 16 available parameters in Table 1. It turns out that five preliminary Type II GRBs are misclassifed as Type I GRBs, and no preliminary Type I GRBs are misclassified as Type II GRBs. There is a small overlap region between [−2.84, −2.68], which covers one preliminary Type II GRB and one preliminary Type I GRB. It is clear that, even with the strong correlated parameters, the naive Bayes method still performs better than the one-parameter criteria, although not as well as what we used in Section 2.

In Section 2.1, we use a Gaussian function to fit the parameter distribution of each GRB class except ${F}_{\mathrm{light}}$. The fitting results of all parameters are acceptable with a significance level of 0.01, when the goodness of fit is examined with the Kolmogorov–Smirnov (KS) test.

On the other hand, the Anderson–Darling (AD) test is usually considered as a better proxy of Gaussianity, especially the tails. When the AD test is applied to the examination, the null probability for ${E}_{\mathrm{iso}}$, ${L}_{\mathrm{peak}}$, f, and [X/H] of Type II GRBs is smaller than 0.01. This indicates that they may not be well described as Gaussian distributions. For E_iso and L_peak, the low-luminosity GRBs contribute as a low-luminosity tail (Liang et al. 2007). For [X/H], this is a result of different metallicity estimation methods or the metallicity dependence on redshift. For low redshift objects, the metallicities are estimated with emission lines such as Hα, O ii, while for high redshift objects, they are mainly estimated with absorption lines. We thus tried to use two Gaussians for the above four parameters of Type II GRBs. With two Gaussians, the resulting AD test parameters A² (and null probabilities P) are 0.38(0.56), 0.98(0.14), 0.29(0.69), and 0.26(0.70), respectively. The classification results are nearly the same. One more preliminary Type II GRB, GRB 070714A, has a log O smaller than 0. The gap between preliminary Type II and Type I GRBs still exists, just change to [−1.23, −0.45].

Theoretically, the AD test is more sensitive to the tail, the KS test is more sensitive to the center, and the Cramer-von Mises test is somewhat in between. The additional Gaussian contributes 0.05, 0.10, 0.27, and 0.19 to E_iso, L_peak, f, and [X/H], respectively, and significantly changes the Anderson–Darling test results. The choice of goodness of fit methods depends on the aim of the test. For our purpose, the centers of Type I and Type II GRBs are more important than the tails. Since the results do not change significantly, to have a clear and consistent picture, we use one Gaussian in the main part of the paper. Two-Gaussian fitting reveals more details and may be needed if more detailed classifications, including subclasses, are to be operated.

The purpose of physical classification of GRBs is to identify physically distinct classes using (typically multiple) observational criteria. The most important physical question in GRB physics is the progenitor systems. Our Type I/II classification scheme developed in this paper (proposed in Zhang et al. 2009) is dedicated to addressing this problem. Notice that such a classification scheme does not specify the detailed progenitor system. While the leading progenitor model of Type II GRBs is the collapse of single stars (collapsars; Woosley 1993; MacFadyen & Woosley 1999), these GRBs may be also produced by massive stars in binary systems (e.g., Izzard et al. 2004; Fryer & Heger 2005). The leading Type I progenitor model is NS–NS mergers (Paczynski 1986; Eichler et al. 1989; Narayan et al. 1992), but BH–NS mergers are quite possible to be another type of progenitor system (Paczynski 1991; Faber et al. 2006). It is possible that there exist subclasses within each broad class (e.g., Zhang 2018 for a detailed discussion). For example, within the Type I population, those with extended emission or internal plateau may comprise a subclass of NS–NS merger events that have a rapidly spinning neutron star post-merger product (Lazzati et al. 2001; Norris & Bonnell 2006; Norris et al. 2010; Kaneko et al. 2015). Within the Type II category, the so-called low-luminosity GRBs (Liang et al. 2007) and ultra-long GRBs (Gendre et al. 2013; Levan et al. 2014; Zhang et al. 2014) may signify different subclasses. Also, if one is interested in radiation mechanisms and jet compositions, GRBs may be classified as thermally dominated fireballs and Poynting-flux-dominated jets. It is possible to further identify physically distinct subclasses or even new classes with the multiwavelength data. This is beyond the scope of the current paper.

For a Gaussian distribution

$$\begin{eqnarray*}&&f(x| \mu ,{\sigma }^{2})=\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^{2}}}\exp \left(-\displaystyle \frac{{\left(x-\mu \right)}^{2}}{2{\sigma }^{2}}\right),\end{eqnarray*}$$

the logarithmic likelihood for a sample with n variables is

$$\begin{eqnarray*}\begin{array}{rcl}\mathrm{ln}\ { \mathcal L } & = & \mathrm{ln}\prod _{i=1}^{n}f({x}_{i}| \mu ,{\sigma }^{2})\\ & = & \mathrm{ln}\prod _{i=1}^{n}\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^{2}}}\exp \left(-\displaystyle \frac{{\left({x}_{i}-\mu \right)}^{2}}{2{\sigma }^{2}}\right)\\ & = & -\displaystyle \frac{n}{2}\mathrm{ln}(2\pi {\sigma }^{2})-\displaystyle \frac{1}{2{\sigma }^{2}}\sum _{i=1}^{n}{\left({x}_{i}-\mu \right)}^{2}.\end{array}\end{eqnarray*}$$

To obtain the extreme of the logarithmic likelihood, thus the extreme of the likelihood, we require the derivatives of this log-likelihood to be zero

$$\begin{eqnarray*}\begin{array}{rcl}\displaystyle \frac{\partial }{\partial \mu }(\mathrm{ln}{ \mathcal L }) & = & -\displaystyle \frac{2({\sum }_{i=1}^{n}{x}_{i}-n\mu )}{2{\sigma }^{2}}\\ & = & 0\Rightarrow \mu =\displaystyle \frac{{\sum }_{i=1}^{n}{x}_{i}}{n}=\bar{x}\\ \displaystyle \frac{\partial }{\partial \sigma }(\mathrm{ln}{ \mathcal L }) & = & -\displaystyle \frac{n}{\sigma }+\displaystyle \frac{{\sum }_{i=1}^{n}{\left({x}_{i}-\mu \right)}^{2}}{{\sigma }^{3}}\\ & \Rightarrow & {\sigma }^{2}=\displaystyle \frac{{\sum }_{i=1}^{n}{\left({x}_{i}-\mu \right)}^{2}}{n}.\end{array}\end{eqnarray*}$$

The exponential distribution we used for the parameter light fraction F_light is $f(x| \gamma )=A\exp (\gamma x)$. To have the integrated value within [0, 1] normalized to be 1, one has

$$\begin{eqnarray*}&&f(x| \gamma )=A\exp (\gamma x),A=\displaystyle \frac{\gamma }{\exp (\gamma )-1}.\end{eqnarray*}$$

The logarithmic likelihood for a sample with n variables is

$$\begin{eqnarray*}&&\mathrm{ln}\ { \mathcal L }=\mathrm{ln}\prod _{i=1}^{n}f({x}_{i}| \gamma )=\mathrm{ln}\prod _{i=1}^{n}A\exp (\gamma {x}_{i})=n\mathrm{ln}A+\gamma \sum _{i=1}^{n}{x}_{i}.\end{eqnarray*}$$

To obtain the extreme of the logarithmic likelihood, thus the extreme of the likelihood, we require the derivatives of this log-likelihood to be zero

$$\begin{eqnarray*}\begin{array}{rcl}\displaystyle \frac{\partial }{\partial \gamma }(\mathrm{ln}{ \mathcal L }) & = & n\displaystyle \frac{{e}^{\gamma }-1-\gamma {e}^{\gamma }}{\gamma ({e}^{\gamma }-1)}+\sum _{i=1}^{n}{x}_{i}\\ & = & 0\Rightarrow \bar{x}=\displaystyle \frac{1}{1-{e}^{-\gamma }}-\displaystyle \frac{1}{\gamma }.\end{array}\end{eqnarray*}$$