# Thermal and Nonthermal Emission from a Peculiar Long-duration GRB 211211A

Xue-Zhao Chang11affiliation: Guangxi Key Laboratory for Relativistic
Astrophysics, School of Physical Science and Technology, Guangxi University,
Nanning 530004, China<EMAIL_ADDRESS>, Hou-Jun Lü11affiliation: Guangxi Key
Laboratory for Relativistic Astrophysics, School of Physical Science and
Technology, Guangxi University, Nanning 530004, China<EMAIL_ADDRESS>, Xing
Yang11affiliation: Guangxi Key Laboratory for Relativistic Astrophysics,
School of Physical Science and Technology, Guangxi University, Nanning 530004,
China<EMAIL_ADDRESS>, Jia-Ming Chen22affiliation: School of Physics and
Astronomy, Yunnan University, Kunming 650500, China , and En-Wei
Liang11affiliation: Guangxi Key Laboratory for Relativistic Astrophysics,
School of Physical Science and Technology, Guangxi University, Nanning 530004,
China<EMAIL_ADDRESS>
###### Abstract

Long-duration GRB 211211A that lacks a supernova emission even down to very
stringent limits at such a low redshift $z=0.076$ and is associated with
kilonova emission, suggests that its physical origin is from a binary compact
star merger. By reanalyzing its data observed with the Gamma-Ray Burst Monitor
on board the Fermi mission, we find that both time-integrated and time-
resolved spectra can be fitted well by using a 2SBPL plus blackbody (2SBPL+BB)
model in the prompt emission. The bulk Lorentz factors ($\Gamma_{\rm ph}$) of
the outflow can be inferred by invoking the observed thermal emission at the
photosphere radius within a pure fireball model, and we find out that the
temporal evolution of $\Gamma_{\rm ph}$ seems to be tracking with the light
curve. The derived values of $\Gamma_{\rm ph}$ are also consistent with the
$\Gamma_{\rm ph}$-$L_{\gamma,\rm iso}$/$E_{\gamma,\rm iso}$ correlations that
had been found in other bursts. Moreover, we also calculate the magnetization
factor $\sigma_{0}$ in the central engine and $\sigma_{\rm ph}$ at the
photosphere radius within the framework of a hybrid jet model, and find that
the values of both $1+\sigma_{\rm 0}$ and $1+\sigma_{\rm ph}$ are larger than
1 for different time slices. It suggests that at least the Poynting-flux
component is indeed existent in the outflow. If this is the case, one possible
physical interpretation of thermal and nonthermal emissions in GRB 211211A is
from the contributions of both $\nu\bar{\nu}$ annihilation and the Blandford-
Znajek mechanisms in the relativistic jet when a stellar mass black hole
resides in the central engine.

###### Subject headings:

Gamma-ray burst: general

##  1\. Introduction

The field that studies of gamma-ray bursts (GRBs) has made a great progress in
both observations and theories (see Kumar & Zhang, 2015, for a review). In
general, GRBs are thought to be from a catastrophic event (such as massive
star core collapse or the merger of two compact stars) to release its
gravitational energy that is converted in the form of thermal energy, and a
central engine (black hole or neutron star) is formed after the catastrophic
event (Eichler et al., 1989; Usov, 1992; Thompson, 1994; Dai & Lu, 1998a, b;
Popham et al., 1999; Narayan et al., 2001; Zhang & Mészáros, 2001; Lei et al.,
2009; Metzger et al., 2011; Bucciantini et al., 2012; Lü & Zhang, 2014;
Berger, 2014; Lü et al., 2015). The fireball model is the most popular one to
interpret both $\gamma$-ray emission and broadband afterglow emission of GRB
phenomena (Mészáros & Rees, 1997; Sari et al., 1998; Mészáros, 2002; Zhang &
Mészáros, 2004; Zhang et al., 2016). The observed prompt emission can be
explained by the photosphere with a quasi-thermal spectrum (Paczynski, 1986;
Goodman, 1986), or internal shocks with a nonthermal electromagnetic radiation
(Rees & Meszaros, 1994), or dissipation of the magnetic energy with a
nonthermal spectrum at a large radius (Zhang & Yan, 2011).

Within the above scenario, the photons that are produced in the thermalized
outflow can escape near the photospheric radius when the optical depth is
close to 1, and a purely quasi-thermal component should be observed in the
prompt emission of a GRB, such as GRB 090902B (Abdo et al., 2009; Ryde et al.,
2010; Zhang et al., 2011). On the contrary, if the outflow is Poynting-flux
dominated, it should produce a purely nonthermal emission, such as GRB 080916C
(Zhang & Pe’er, 2009). In general, the dissipated radius of the photospheric
and internal shock cannot be distinguished very well, and the observed
spectrum of GRB prompt emission should be the superposition of thermal and
nonthermal components (Ryde, 2005; Gao & Zhang, 2015). Such evidence of
thermal and nonthermal emission is already found in several solid cases, e.g.,
GRB 100724B (Guiriec et al., 2011); GRB 110721A (Axelsson et al., 2012); GRB
120323A (Guiriec et al., 2013); GBR 160625B (Lü et al., 2017); GRB 081221 (Hou
et al., 2018).

Most recently, a peculiar and nearby long-duration GRB 211211A that triggered
the Fermi Gamma-Ray Burst Monitor (GBM; Mangan et al. 2021), Swift Burst Alert
Telescope (BAT; D’Ai et al. 2021), as well as Insight-HXMT (Zhang et al.,
2021), is very excited for attention with redshift $z$=0.076. The light curve
of prompt emission is composed of an initial hard-main emission (with a
duration $\sim 13$ s) followed by a series of soft gamma-ray extended emission
(EE) with a duration $\sim 55$ s, and the structure of the light curve is
similar to the particularly interesting case GRB 060614(Yang et al., 2022;
Xiao et al., 2022) and GRB 211227A (Lü et al., 2022). More interestingly, no
associated supernova signature is detected for GRB 211211A, even down to very
stringent limits at such a low redshift, but associated with kilonova is
observed by several optical telescopes (Rastinejad et al., 2022). That
observed evidence suggests that GRB 211211A is originated from a binary
compact star merger (Rastinejad et al., 2022; Yang et al., 2022; Xiao et al.,
2022; Gompertz et al., 2022). However, how to produce such long-duration
emission within the compact star merger scenario remains an open question. Gao
et al. (2022) proposed that the black hole central engine surrounded by a
strong magnetic flux can well interpret the behavior of long-duration emission
of GRB 211211A. Gompertz et al. (2022) found that the spectrum can be fitted
well with a double smoothly broken power-law model SBPL, which is interpreted
as synchrotron emission, including both characteristic synchrotron frequency
($\nu_{m}$) and the cooling frequency ($\nu_{c}$). So that identifying the
composition of the jet in such a binary system will play an important role in
understanding the physical process and mechanism (Zhang et al., 2011; Kumar &
Zhang, 2015; Zhang, 2018).

In this paper, by analyzing the data observed with the GBM on board the Fermi
mission, we find that a blackbody emission with a nonthermal component 2SBPL
function can be fitted well to the spectra in the prompt emission of GRB
211211A, especially in the initial hard-main emission phase. It means that the
thermal component should be indeed existent, and it is different from previous
studies that claimed the single nonthermal component is dominated in the
prompt emission phase (Yang et al., 2022; Xiao et al., 2022). The data
analysis and spectral fitting are presented in §2. In §3, we derive the
Lorentz factor of the jet, its photospheric radius, magnetization
$\sigma_{0}$, and the dimensionless entropy $\eta$ based on the observed
thermal and nonthermal emissions of GRB 211211A. The conclusions are drawn in
§4 with some additional discussion. Throughout the paper, a concordant
cosmology with parameters $H_{0}=70~{}\rm km~{}s^{-1}~{}Mpc^{-1}$,
$\Omega_{M}=0.30$, and $\Omega_{\Lambda}=0.70$ is adopted.

##  2\. Fermi/GBM data analysis

###  2.1. Light curve and spectral fits

GRB 211211A triggered the Fermi/GBM at 13:09:59.651 UT on 2021 December 11
(Mangan et al., 2021). This GRB was also detected by Swift/BAT (D’Ai et al.,
2021) and Insight-HXMT (Zhang et al., 2021). We downloaded the corresponding
time-tagged-event (TTE) data of GRB 211211A from the public science support
center at the official Fermi
website111https://heasarc.gsfc.nasa.gov/FTP/fermi/data/gbm/daily/. The GBM has
12 sodium iodide (NaI) detectors covering an energy range from 8 keV to 1 MeV,
and two bismuth germanate (BGO) scintillation detectors sensitive to higher
energies between 200 keV and 40 MeV (Meegan et al., 2009). We select the
brightest NaI and BGO detectors for the analyses, namely n2, na and b0. For
more details of data reduction of the light curve, please refer to Lü et al.
(2017) and Zhang et al. (2018).

We extract the light curves with a 128 ms time bin (Figure 1) by running
gtbin. The light curve shows a complex structure with a total duration of
about $T_{90}\sim$43 s, an initially main emission (with a duration $\sim 13$
s) followed by a series of soft gamma-ray extended emission with a duration
$\sim 55$ s (also see Yang et al., 2022).

Both time-integrated and time-resolved spectra of this source are extracted
from the TTE data. The background spectrum from the GBM data is extracted from
the continuous spectroscopy (CSPEC) format data with two time intervals before
and after the prompt emission phase and are modeled with a polynomial
function. We perform the spectral fit with the multimission maximum likelihood
framework package (Vianello et al. 2017), which adopts the Markov Chain Monte
Carlo (MCMC) technique to perform time-resolved spectral fitting. Also, we
evaluate the goodness of our fits with the maximum likelihood based
statistics, the so-called PGSTAT. Yang et al. (2022) invoked a cutoff power-
law model to do the spectral fitting in both the time-averaged and time-
resolved spectra of GRB 211211A (also see Gompertz et al., 2022). In our
analysis, several spectral models can be selected to test the spectral fitting
of the burst, such as cutoff power-law (CPL), Band function (Band), a smoothly
broken power-law model (SBPL), a double smoothly broken power-law model
(2SBPL) and blackbody (BB), as well as combinations of any two models. The
Band function (Band et al., 1993) and blackbody function, and CPL models are
written as follows:

$\displaystyle
N_{\textrm{Band}}(E)=A\left\\{\begin{array}[]{clcc}(\frac{E}{100~{}\mathrm{keV}})^{\alpha}\mathrm{exp}\left[-\frac{(\alpha+2)E}{E_{p}}\right],E<E_{c},\\\
(\frac{E}{100~{}\mathrm{keV}})^{\beta}\mathrm{exp}(\beta-\alpha)(\frac{E_{c}}{100~{}\mathrm{keV}})^{\alpha-\beta},E\geq
E_{c}\end{array}\right.$ (3)

where $A$ is the normalization of the spectrum, $\alpha$ and $\beta$ are the
low and high-energy photon spectral indices, respectively,and
$E_{p}=(2+\alpha)E_{c}$ is the peak energy;

$N_{\textrm{BB}}(E)=A(t)\frac{E^{2}}{\exp[E/kT]-1}$ (4)

where $k$ and $T$ are Boltzmann constant and temperature, respectively;

$\displaystyle N_{\rm CPL}(E)=A\cdot E^{-\alpha}\rm exp(-\frac{E}{E_{\rm
p}}).$ (5)

The SBPL function (Ravasio et al., 2018) function is defined as follows:

$\displaystyle
N_{\mathrm{SBPL}}(E)=A\left(\frac{E}{100~{}\mathrm{keV}}\right)^{b}10^{\left(a-a_{\mathrm{piv}}\right)}$
(6)

where $a=m\Lambda\ln\left(\frac{e^{q}+e^{-q}}{2}\right)$,
$a_{\mathrm{piv}}=m\Lambda\ln\left(\frac{e^{q_{\mathrm{piv}}}+e^{-q_{\mathrm{piv}}}}{2}\right)$,
$q=\frac{\log\left(E/E_{b}\right)}{\Lambda}$,
$q_{\mathrm{piv}}=\frac{\log\left(100~{}\mathrm{keV}/E_{b}\right)}{\Lambda}$,
$m=\frac{\beta-\alpha}{2}$, $b=\frac{\alpha+\beta}{2}$. Here, $\alpha$,
$\beta$ and $E_{b}$ are the lower power-law index, upper power-law index, and
a break energy, respectively; $\Lambda$ is the break scale, which is fixed at
0.3.

The 2SBPL function (Kaneko et al., 2006) function is defined as follows:

$\displaystyle\begin{aligned}
N_{\mathrm{2SBPL}}=AE_{\mathrm{b}}^{\alpha_{1}}[f_{1}(E)+f2(E)]^{-\frac{1}{n_{2}}}\end{aligned}$
(7)

where$f_{1}(E)=[(\frac{E}{E_{\mathrm{b}}})^{-\alpha_{1}n_{1}}+(\frac{E}{E_{\mathrm{b}}})^{-\alpha_{2}n_{2}}]^{n_{2}/n_{1}}$,
$f_{2}(E)=(\frac{E}{E_{\mathrm{j}}})^{-\beta
n_{2}}[(\frac{E_{\mathrm{j}}}{E_{\mathrm{b}}})^{-\alpha_{1}n_{1}}+(\frac{E_{\mathrm{j}}}{E_{\mathrm{b}}})^{-\alpha_{2}n_{2}}]^{n_{2}/n_{1}}$,
and $E_{\mathrm{j}}=E_{\text{pk
}}\cdot\left(-\frac{\alpha_{2}+2}{\beta+2}\right)^{\frac{1}{(\left.\beta-\alpha_{2}\right)n_{2}}}$;
$\alpha_{1}$ and $\alpha_{2}$ are the photon index below and above the break
energy, respectively. $E_{b}$ and $E_{pk}$ are the break energy and peak
energy, respectively; $\beta$ is the high-energy photon index above the peak
energy; $n_{1}$ (for the break) and $n_{2}$ (for the peak) are the smoothness
parameters. In this work, we fix the $n_{1}=5.38$ and $n_{2}=2.69$,
respectively (Ravasio et al., 2018).

In order to test which model is the best fit with the data, we compare the
goodness of the fits by invoking the Bayesian information criteria (BIC)222BIC
is a criterion to evaluate the best model fit among a finite set of models,
and the lowest BIC of a model is preferred (Neath & Cavanaugh, 2012). The
definition of BIC can be written as: $BIC=\rm-2lnL+k\cdot ln(n)$, where $k$ is
the number of model parameters, $n$ is the number of data points, and $L$ is
the maximum value of the likelihood function of the estimated model. (1) if
$0<\Delta BIC<2$, the evidence against the model with higher BIC is not worth
more than a bare mention; (2) if $2<\Delta BIC<6$, the evidence against the
model with higher BIC is positive; (3) if $6<\Delta BIC<10$, the evidence
against the model with higher BIC is strong; (4) if $10<\Delta BIC$, the
evidence against the model with higher BIC is very strong.. For the specific
definition of the goodness of data fitting by the empirical model, please
refer to Li (2019). We find that the 2SBPL+BB model is the best one to
adequately describe the observed data, and it means that the thermal emission
component in GRB 211211A is a significant presence (see Table 1). The spectral
fitting result of the time-integrated spectra is shown in Figure 2. Moreover,
we find that the photon indices of the time-integrated spectra are
$\alpha_{1}=-0.6$ and $\alpha_{2}=-1.62$, respectively. These values are
consistent with the results in Gompertz et al. (2022), and it suggests that
the nonthermal component may be originated from the synchrotron emission in
the fast regime.

We also extract time-resolved spectral analyses of GRB 211211A between $T_{0}$
\+ 0.5 and $T_{0}$ \+ 70\. We divide the time interval into 31 slices, and fit
those slices by invoking 2SBPL and 2SBPL+BB models. The fitting results are
shown in Table 2. Figure 1 shows the evolution of $\Delta$BIC, which is
defined as $\rm\Delta BIC=BIC_{2SBPL}-BIC_{2SBPL+BB}$. We find that the
$\Delta$BIC of all time slices is larger than zero, especially, the
$\Delta$BIC at the peak of light curve in both the main emission and extended
emission is much larger than 10. Even during the late period of the extended
emission phase, the $\Delta$BIC of most time slices remains in the range of
[6-10]. By comparing the BIC of 2SBPL and 2SBPL+BB models, we find that it is
strong to support the 2SBPL+BB model, which is better than the 2SBPL model to
describe the observed data during the time-resolved spectra of GRB 211211A. In
other words, the thermal emission component in GRB 211211A remains a
significant presence in the time-resolved spectrum. By adopting $z=0.076$ of
GRB 211211A, the total isotropic-equivalent energy ($E_{\rm\gamma,iso}$) and
luminosity ($L_{\rm\gamma,iso}$) can be as high as $7.6\times 10^{51}~{}\rm
erg$ and $1.9\times 10^{51}~{}\rm erg~{}s^{-1}$, respectively.

###  2.2. Fitting results

In order to test the behavior of temporal evolution of main parameters for
thermal and nonthermal emissions, we present the temporal evolution of
$E_{b}$, $E_{\rm pk}$, $kT$, the flux of BB emission ($F_{\rm BB}$), and the
ratio of $F_{\rm BB}$ and total observed flux ($F_{\rm obs}$) in Figure 3. By
comparing the $E_{b}$ and ${E_{\rm pk}}$ evolution of both 2SBPL and 2SBPL+BB
models, we find that the $E_{\rm b}$ and $E_{\rm pk}$ evolution of 2SBPL+BB
model are similar to that of the 2SBPL model, and the behavior of its
evolution seems to be tracking with pulses of the light curve. The temperature
($kT$) and the flux ($F_{\rm BB}$) of BB emission also exhibits the tracking
behavior with its pulses. The $F_{\rm BB}$ can reach to as high as $8.16\times
10^{-6}\rm~{}erg~{}cm^{-2}~{}s^{-1}$. We also calculate the total observed
flux, which includes both thermal and nonthermal emission and find that the
fraction of thermal emission flux ($F_{\rm BB}/F_{\rm obs}$) can reach as high
as $\sim 0.2$. The significant thermal component in the prompt emission is
also independent to support the BB emission, which should indeed be presented.

##  3\. Derivation of the physical parameters within the Fireball Models

The initial Lorentz factor of a GRB jet is a very important parameter for
understanding GRB physics, and is also very difficult to measure for most GRBs
(Zhang, 2018). In general, there are three methods that have been proposed to
estimate the Lorentz factor. The first one is to use the high-energy cutoff of
the prompt gamma-ray spectrum, which is from the pair production when the
absorption optical depth is close to one (Fenimore et al., 1993; Lithwick &
Sari, 2001; Zhang & Pe’er, 2009). The second approach to estimating the
Lorentz factor is using the early afterglow light curves with a smooth onset
bump that shows the signal of fireball deceleration (Sari & Piran, 1999;
Kobayashi & Zhang, 2007; Liang et al., 2010). The third one is using the
blackbody component detected in some GRB spectra (Pe’er et al., 2007; Ryde et
al., 2010; Gao & Zhang, 2015). In this section, we derive the physical
parameters based on the observed thermal component in prompt emission of GRB
211211A, such as the Lorentz factor ($\Gamma_{\rm ph}$) and the radius of the
photosphere ($R_{\rm ph}$). We also calculate the magnetization factor
($\sigma_{0}$) and dimensionless entropy ($\eta$) by assuming the hybrid jet
of GRB 211211A.

###  3.1. Lorentz Factor and Photosphere Radius

In our analyses, the time-resolved spectra of the GRB 211211A prompt emission
is composed of a thermal component (BB component) and a nonthermal component
2SBPL component). Following the method of Pe’er et al. (2007), we estimate the
$\Gamma_{\rm ph}$ and $R_{\rm ph}$ with the BB component derived from our
spectral fits in different time slices;

$\Gamma_{\mathrm{ph}}=\left[(1.06)(1+z)^{2}D_{\mathrm{L}}\frac{Y\sigma_{\mathrm{T}}F_{\mathrm{obs}}}{2m_{\mathrm{p}}c^{3}\Re}\right]^{1/4}$
(8)

where $D_{\rm L}$ is the luminosity distance, $m_{\rm p}$ is the proton mass,
$\sigma_{\mathrm{T}}$ is the Thomson scattering cross section, and $F_{\rm
obs}$ is the observed total flux. In our calculation, we fix the $Y=1$, which
is the ratio between the total fireball energy and the energy emitted in the
$\gamma$-ray. The definition of $\Re$ is written as

$\Re\equiv(\frac{F_{BB}}{\sigma T^{4}})^{1/2}$ (9)

where $\sigma$ and $F_{\rm BB}$ are the Stefan’s constant and the observed
blackbody component flux, respectively. On the other hand, the $R_{\rm ph}$
can be expressed as

$R_{ph}=\left(L\sigma_{T}/8\pi\Gamma^{3}_{ph}m_{p}c^{3}\right)$ (10)

where $L=4\pi D_{L}^{2}F_{\rm obs}$ is the luminosity that is measured for
bursts with known redshift. The calculation results are shown in Table 3.

Figure 4 shows the temporal evolution of $\Gamma_{\rm ph}$ and $R_{\rm ph}$.
During main emission phase, the evolution of $\Gamma_{\rm ph}$ is initially
tracking with the light curve, and maximum value can be reached as high as 311
at the peak of the light curve. In the extended emission phase, the
$\Gamma_{\rm ph}$ seems to be also tracking with the light curve, and it
peaked at $\Gamma_{\rm ph}=197$, then, it is gradually going down to 91 until
it reaches the end of extended emission. As for the $R_{\rm ph}$, the highest
value of $R_{\rm ph}$ is around $\sim 2.71\times 10^{10}$ cm and it keeps
fluctuating around $\sim 10^{10}$ cm.

###  3.2. Magnetization parameter and dimensionless entropy

One is different from the method of Pe’er et al. (2007) who inferred the
central engine parameters by using the observed data within the framework of a
pure fireball shock model. Gao & Zhang (2015) proposed a hybrid relativistic
outflow of GRB (e.g., hot fireball component and Poynting-flux component), and
developed a theory of its photosphere emission. One interesting question is
that the observed nonthermal component of GRB 211211A is from the internal
shock of a fireball or another cold Poynting-flux component. Here, we infer
the magnetization factor ($\sigma_{0}$) and dimensionless entropy ($\eta$) by
assuming the hybrid jet of GRB 211211A.

The magnetization factor $\sigma_{0}$ is defined as $\sigma_{0}=L_{\rm
c}/L_{\rm h}$, where the $L_{\rm h}$ and $L_{\rm c}$ are the luminosity of the
hot fireball component and cold Poynting-flux component, respectively. The
dimensionless entropy $\eta$ can be defined as $\eta=L_{\rm h}/\dot{M}c^{2}$,
where $\dot{M}$ is the accretion rate. The time varying of the ($\eta$,
$\sigma_{0}$) pair at the central engine can result in the evolution of the
photosphere emission proprieties. Based on the results of derivation in Gao &
Zhang (2015), several situations of different ($\eta$, $\sigma_{0}$) pairs are
considered; (1) ${\eta\gg 1,\sigma_{0}\ll 1}$: it means that the photosphere
emission is dominated by a pure fireball component; (2) ${\eta\sim
1,1+\sigma_{0}\gg 1}$: a Poynting-flux-dominated outflow; no detection of any
thermal component in the GRB spectrum. By invoking the ’top-down’ approach and
based on the judgment criteria proposed by Gao & Zhang (2015), we can infer
the parameters of the central engine by using the observed quasi-thermal
photosphere emission parameters (such as $F_{\rm BB}$, $F_{\rm obs}$, and
$kT$). However, the inferred parameters of the central engine are sensitively
dependent on the selected initial radius ($r_{0}$) of the central engine. For
convenience, we adopt the initial radius $r_{0}=10^{7}\rm~{}cm$, and one can
obtain all the photosphere characteristic parameters of the hybrid model
(e.g., $\eta$, $1+\sigma_{0}$, $R_{\rm ph}$, $\Gamma_{\rm ph}$, and
$1+\sigma_{\rm ph}$). Here, $1+\sigma_{\rm ph}$ is the magnetization parameter
at $R_{\rm ph}$. Table 4 shows the derived parameters of GRB 211211A in a
hybrid jet model.

Figure 4 also shows the comparisons of $\Gamma_{\rm ph}$ and $R_{\rm ph}$
evolution for both the pure fireball model and hybrid jet model with fixed
$r_{0}=10^{7}\rm~{}cm$. It is interesting that the $\Gamma_{\rm ph}$ evolution
behavior of a pure fireball model is similar to that of the hybrid jet model,
and the evolution of $R_{\rm ph}$ is matched very well between those two
models. Figure 5 presents the temporal evolution of $1+\sigma_{0}$ and $\eta$
in the central engine with $r=r_{0}$, as well as $1+\sigma_{\rm ph}$ at
$R_{\rm ph}$. The values of $1+\sigma_{\rm 0}$ and $\eta$ are larger than 1
and 10 for different time slices, respectively. It means that at least the
Poynting-flux component is indeed existent in the central engine. Meanwhile,
it is clear to see that the values of $1+\sigma_{\rm ph}$ are larger than 1
for different time slices, and range in [1-50]. So that, those results suggest
that the Poynting-flux component should always be a presence at the position
of the central engine and photosphere radius. The observed thermal and
nonthermal components in GRB 211211A seem to be from the contributions of hot
fireball and Poynting-flux outflow, respectively.

###  3.3. $\Gamma_{\rm ph}$ \- $E_{\gamma,\rm iso}$/$L_{\gamma,\rm iso}$
correlations

Liang et al. (2010) discovered a tight correlation between initial Lorentz
factor $\Gamma_{0}$ and isotropic $\gamma$-ray energy $E_{\rm\gamma,iso}$,
namely $\Gamma_{0}\propto E^{0.25}_{\rm\gamma,iso}$. Lü et al. (2012) also
found another tight correlation of $\Gamma_{0}$ and isotropic $\gamma$-ray
luminosity $L_{\rm\gamma,iso}$, e.g., $\Gamma_{0}\propto
L_{\rm\gamma,iso}^{0.3}$. Those two correlations can be interpreted well by
using a neutrino-cooled hyperaccretion disk around a stellar mass black hole
as the GRB central engine (Lü et al., 2012).

In order to test whether the $\Gamma_{\rm ph}$ from time-resolved BB emission
and $E_{\rm\gamma,iso}$/$L_{\rm\gamma,iso}$ of GRB 211211A are tracking
similar correlations to the above, Figure 6 shows the relationship between
$\Gamma_{\rm ph}$ and $E_{\rm\gamma,iso}$/$L_{\rm\gamma,iso}$. We caution the
reader that we do not adopt the data from Liang et al. (2010) and Lü et al.
(2012) to do the joint fitting, because they adopt different methods to
estimate the Lorentz factor. However, we collect the data of time-resolved
spectra in GRB 160625B, which have the thermal emission and use the same
method (e.g., BB emission) to infer the Lorentz factor. So, we used
$\Gamma_{\rm ph}$ in our fitting to replace $\Gamma_{0}$ in Liang et al.
(2010), and the data of GRB 160625B are taken from Wang et al. (2017). For the
diagram of $\Gamma_{\rm ph}-L_{\gamma,\rm iso}$, we make a joint fitting of
the two groups (GRB 160626B and GRB 211211A) with a power-law model and find
$\Gamma_{\rm ph}\propto L_{\rm\gamma,iso}^{0.24\pm 0.01}$ with a Pearsons
correlation coefficient of 0.96 and p $<10^{-4}$. Also, applying the power-law
fitting to $\Gamma_{\rm ph}-E_{\rm\gamma,iso}$, we find $\Gamma_{\rm
ph}\propto E_{\rm\gamma,iso}^{0.26\pm 0.02}$ with a Pearsons correlation
coefficient of 0.91 and p $<10^{-4}$. Those two results are similar to that of
Lü et al. (2012) and Liang et al. (2010), respectively. It suggests that the
central engine of GRB 211211A may be a stellar mass black hole with a
neutrino-cooled hyperaccretion disk.

##  4\. Conclusion and discussion

GRB 211211A was observed by Fermi/GBM, Swift/BAT, and Insight-HXMT to have a
duration of $\sim 84$ seconds at redshift $z=$0.076, but the light curve is
characterized by an initial hard-main emission (with a duration $\sim 13$ s)
followed by a series of soft gamma-ray extended emission with a duration $\sim
55$ s. The structure of the light curve is similar to the cases of GRB 060614
and GRB 211227A, which are believed to be from the compact star merger (Yang
et al., 2022; Xiao et al., 2022; Lü et al., 2022). The total isotropic-
equivalent energy ($E_{\rm\gamma,iso}$) and luminosity ($L_{\rm\gamma,iso}$)
are as high as $7.6\times 10^{51}~{}\rm{erg}$ and $1.9\times
10^{51}~{}\rm{erg~{}s^{-1}}$, respectively. At such low redshift, it is
surprising that deep searches of an underlying SN give null results, but being
associated with a kilonova is observed by several optical telescopes
(Rastinejad et al., 2022). That observed evidence suggests that GRB 211211A is
originated from a binary compact star merger (Rastinejad et al., 2022; Yang et
al., 2022; Xiao et al., 2022; Gompertz et al., 2022).

By reanalyzing the data observed with the GBM on board the Fermi mission, we
find the following interesting results:

  * •

We find that the 2SBPL+BB model is the best one to adequately describe the
observed data in both time-integrated and time-resolved spectra in the prompt
emission of GRB 211211A based on the BIC criterion, and it means that both
thermal and nonthermal components in GRB 211211A should be a significant
presence.

  * •

The behavior of temporal evolution of $E_{\rm b}$, $E_{\rm pk}$ for the 2SBPL
function and $kT$ for BB emission seem to be tracking with pulses of the light
curve.

  * •

By inferring the Lorentz factor $\Gamma_{\rm ph}$ and photosphere radius
$R_{\rm ph}$ based on the observed BB emission within the framework of a pure
fireball model, we find that the temporal evolution of $\Gamma_{\rm ph}$ seems
to be tracking with the light curve, and its range from 87 to 311. However,
the highest value of $R_{\rm ph}$ is around $\sim 2.7\times 10^{10}$ cm and it
keeps fluctuating around $\sim 10^{10}$ cm.

  * •

By calculating the magnetization factor $\sigma_{0}$ in the central engine and
$\sigma_{\rm ph}$ at the photosphere radius within the framework of the hybrid
jet model, we find that the values of both $1+\sigma_{\rm 0}$ and
$1+\sigma_{\rm ph}$ are larger than 1 for different time slices for fixed
initial $r_{0}=10^{7}\rm~{}cm$, and its range of [1,120]. It means that at
least the Poynting-flux component is indeed existent in both the central
engine and photosphere radius.

  * •

Moreover, we also plot the diagrams of $\Gamma_{\rm ph}$\-
$E_{\rm\gamma,iso}$/$L_{\rm\gamma,iso}$, and find the relationships as
$\Gamma_{\rm ph}\propto E_{\rm\gamma,iso}^{0.26\pm 0.02}$ and $\Gamma_{\rm
ph}\propto L_{\rm\gamma,iso}^{0.24\pm 0.01}$ for the time-resolved spectral
data of GRB 211211A. Those two correlations are consistent with that of in
Liang et al. (2010) and Lü et al. (2012), respectively. It suggests that the
central of GRB 211211A may be a stellar mass black hole with a neutrino-cooled
hyperaccretion disk.

If the central engine of GRB 211211A is a stellar mass black hole that is
formed from a binary compact star merger (Gao et al., 2022). Lei et al. (2017)
proposed that two jet launching mechanisms, i.e., $\nu\bar{\nu}$ annihilation
and the Blandford-Znajek (BZ; Blandford & Znajek 1977) process, are indeed
considered to power thermal and nonthermal components in the relativistic jet,
respectively; the $\nu\bar{\nu}$ annihilation mechanism liberates the
gravitational energy from the accretion disk, and the BZ mechanism extracts
the spin energy from the Kerr black hole (Popham et al., 1999; Lee et al.,
2000; Li & Paczyński, 2000; Gu et al., 2006; Lei et al., 2009, 2013). In this
scenario, the observed thermal and nonthermal emissions in GRB 211211A can be
interpreted as follows. The jet of GRB 211211A can be launched from a
hyperaccreting black hole via $\nu\bar{\nu}$ annihilation and the BZ
mechanism. Initially, due to a very high accretion rate of the black hole, the
$\nu\bar{\nu}$ annihilation should be dominated. It can produce thermal
emission when the photons escape the system at the photosphere radius, and the
internal shock or Poynting-flux outflow is used to produce the nonthermal
component. After tens of seconds, the $\nu\bar{\nu}$ annihilation is not
strong enough along with the decreasing of accretion rate. But the magnetic
field of the black hole becomes gradually stronger due to storage time, and
the BZ mechanism will be dominated. If this is the case, it is natural to
explain the observed gradually decreased thermal emission and gradually
increased nonthermal emission of GRB 211211A.

Alternatively, what we discussed above about the evidence of Poynting-flux
existence (e.g., magnetization factor $\sigma_{0}$) in our results is
dependent on the selected $R_{0}$ in the calculations of the hybrid model, and
we fixed $R_{0}=10^{7}$ cm in our calculations. For different selected
$R_{0}$, such as $R_{0}=10^{8}$ cm or $R_{0}=10^{9}$ cm, it corresponds to
different values of $\sigma_{0}$. So, it is difficult to judge whether it is
accompanied by the Poynting-flux component in the jet. If this is the case,
only $\nu\bar{\nu}$ annihilation mechanism can interpret the observed thermal
emission (from the photosphere) and nonthermal emission (from the internal
shock).

To find out such a uniform model to explain all observed characteristics of
GRB 211211A, of course, is not an easy task. We therefore encourage intense
multi-band follow-up observations for GRB 211211A-like events in the future.

We are very grateful to thank the referee for helpful comments and suggestions
to improve this manuscript. We acknowledge the use of the public data from the
Fermi/GBM data archive. This work is supported by the National Natural Science
Foundation of China (grant Nos. 11922301, and 12133003), the Program of Bagui
Young Scholars Program (LHJ), and the Guangxi Science Foundation (grant No.
2017GXNSFFA198008).

## References

  * Abdo et al. (2009) Abdo, A. A., Ackermann, M., Ajello, M., et al. 2009, ApJ, 706, L138. doi:10.1088/0004-637X/706/1/L138
  * Axelsson et al. (2012) Axelsson, M., Baldini, L., Barbiellini, G., et al. 2012, ApJ, 757, L31. doi:10.1088/2041-8205/757/2/L31
  * Band et al. (1993) Band, D., Matteson, J., Ford, L., et al. 1993, ApJ, 413, 281. doi:10.1086/172995
  * Berger (2014) Berger, E. 2014, ARA&A, 52, 43.
  * Blandford & Znajek (1977) Blandford, R. D. & Znajek, R. L. 1977, MNRAS, 179, 433. doi:10.1093/mnras/179.3.433
  * Bucciantini et al. (2012) Bucciantini, N., Metzger, B. D., Thompson, T. A., et al. 2012, MNRAS, 419, 1537. doi:10.1111/j.1365-2966.2011.19810.x
  * D’Ai et al. (2021) D’Ai, A., Ambrosi, E., D’Elia, V., et al. 2021, GRB Coordinates Network, Circular Service, No. 31202, 1
  * Dai & Lu (1998a) Dai, Z. G. & Lu, T. 1998a, MNRAS, 298, 87. doi:10.1046/j.1365-8711.1998.01681.x
  * Dai & Lu (1998b) Dai, Z. G. & Lu, T. 1998b, A&A, 333, L87
  * Eichler et al. (1989) Eichler, D., Livio, M., Piran, T., et al. 1989, Nature, 340, 126. doi:10.1038/340126a0
  * Fenimore et al. (1993) Fenimore, E. E., Epstein, R. I., & Ho, C. 1993, A&AS, 97, 59
  * Gao & Zhang (2015) Gao, H. & Zhang B et al. 2015, ApJ, 801, 5. doi:0.1088/0004-637X/801/2/103
  * Gao et al. (2022) Gao, H., Lei, W.-H., & Zhu, Z.-P. 2022, ApJ, 934, L12. doi:10.3847/2041-8213/ac80c7
  * Gompertz et al. (2022) Gompertz, B. P., Ravasio, M. E., Nicholl, M., et al. 2022, Nature Astronomy. doi:10.1038/s41550-022-01819-4
  * Goodman (1986) Goodman, J. 1986, ApJ, 308, L47. doi:10.1086/184741
  * Guiriec et al. (2011) Guiriec, S., Connaughton, V., Briggs, M. S., et al. 2011, ApJ, 727, L33. doi:10.1088/2041-8205/727/2/L33
  * Guiriec et al. (2013) Guiriec, S., Daigne, F., Hascoët, R., et al. 2013, ApJ, 770, 32. doi:10.1088/0004-637X/770/1/32
  * Gu et al. (2006) Gu, W.-M., Liu, T., & Lu, J.-F. 2006, ApJ, 643, L87. doi:10.1086/505140
  * Hou et al. (2018) Hou, S.-J., Zhang, B.-B., Meng, Y.-Z., et al. 2018, ApJ, 866, 13. doi:10.3847/1538-4357/aadc07
  * Kumar & Zhang (2015) Kumar, P. & Zhang, B. 2015, Phys. Rep., 561, 1. doi:10.1016/j.physrep.2014.09.008
  * Kobayashi & Zhang (2007) Kobayashi, S. & Zhang, B. 2007, ApJ, 655, 973. doi:10.1086/510203
  * Lee et al. (2000) Lee, H. K., Wijers, R. A. M. J., & Brown, G. E. 2000, Phys. Rep., 325, 83. doi:10.1016/S0370-1573(99)00084-8
  * Lei et al. (2009) Lei, W. H., Wang, D. X., Zhang, L., et al. 2009, ApJ, 700, 1970\. doi:10.1088/0004-637X/700/2/1970
  * Lei et al. (2013) Lei, W.-H., Zhang, B., & Liang, E.-W. 2013, ApJ, 765, 125. doi:10.1088/0004-637X/765/2/125
  * Lei et al. (2017) Lei, W.-H., Zhang, B., Wu, X.-F., et al. 2017, ApJ, 849, 47\. doi:10.3847/1538-4357/aa9074
  * Li & Paczyński (2000) Li, L.-X. & Paczyński, B. 2000, ApJ, 534, L197. doi:10.1086/312678
  * Li (2019) Li, L. 2019, ApJS, 242, 16.
  * Liang et al. (2010) Liang, E.-W., Yi, S.-X., Zhang, J., et al. 2010, ApJ, 725, 2209. doi:10.1088/0004-637X/725/2/2209
  * Lithwick & Sari (2001) Lithwick, Y. & Sari, R. 2001, ApJ, 555, 540. doi:10.1086/321455
  * Lü et al. (2017) Lü, H.-J., Lü, J., Zhong, S.-Q., et al. 2017, ApJ, 849, 71. doi:10.3847/1538-4357/aa8f99
  * Lü & Zhang (2014) Lü, H.-J. & Zhang, B. 2014, ApJ, 785, 74\. doi:10.1088/0004-637X/785/1/74
  * Lü et al. (2015) Lü, H.-J., Zhang, B., Lei, W.-H., et al. 2015, ApJ, 805, 89. doi:10.1088/0004-637X/805/2/89
  * Lü et al. (2022) Lü, H.-J., Yuan, H.-Y., Yi, T.-F., et al. 2022, ApJ, 931, L23. doi:10.3847/2041-8213/ac6e3a
  * Lü et al. (2012) Lü, J., Zou, Y.-C., Lei, W.-H., et al. 2012, ApJ, 751, 49. doi:10.1088/0004-637X/751/1/49
  * Kaneko et al. (2006) Kaneko, Y., Preece, R. D., Briggs, M. S., et al. 2006, ApJS, 166, 298.
  * Mangan et al. (2021) Mangan, J., Dunwoody, R., Meegan, C., et al. 2021, GRB Coordinates Network, Circular Service, No. 31210, 1
  * Meegan et al. (2009) Meegan, C., Lichti, G., Bhat, P. N., et al. 2009, ApJ, 702, 791. doi:10.1088/0004-637X/702/1/791
  * Metzger et al. (2011) Metzger, B. D., Giannios, D., Thompson, T. A., et al. 2011, MNRAS, 413, 2031. doi:10.1111/j.1365-2966.2011.18280.x
  * Mészáros & Rees (1997) Mészáros, P. & Rees, M. J. 1997, ApJ, 476, 232. doi:10.1086/303625
  * Mészáros (2002) Mészáros, P. 2002, ARA&A, 40, 137. doi:10.1146/annurev.astro.40.060401.093821
  * Narayan et al. (2001) Narayan, R., Piran, T., & Kumar, P. 2001, ApJ, 557, 949. doi:10.1086/322267
  * Neath & Cavanaugh (2012) Neath, A. A. & Cavanaugh, J. E. 2012, WIREs Comput. Stat., 4, 199. doi: 10.1002/wics.199
  * Paczynski (1986) Paczynski, B. 1986, ApJ, 308, L43. doi:10.1086/184740
  * Pe’er et al. (2007) Pe’er, A., Ryde, F., Wijers, R. A. M. J., et al. 2007, ApJ, 664, L1. doi:10.1086/520534
  * Popham et al. (1999) Popham, R., Woosley, S. E., & Fryer, C. 1999, ApJ, 518, 356. doi:10.1086/307259
  * Ravasio et al. (2018) Ravasio, M. E., Oganesyan, G., Ghirlanda, G., et al. 2018, A&A, 613, A16.
  * Rastinejad et al. (2022) Rastinejad, J. C., Gompertz, B. P., Levan, A. J., et al. 2022, Nature, 612, 223. doi:10.1038/s41586-022-05390-w
  * Rees & Meszaros (1994) Rees, M. J. & Meszaros, P. 1994, ApJ, 430, L93. doi:10.1086/187446
  * Ryde (2005) Ryde, F. 2005, ApJ, 625, L95. doi:10.1086/431239
  * Ryde et al. (2010) Ryde, F., Axelsson, M., Zhang, B. B., et al. 2010, ApJ, 709, L172. doi:10.1088/2041-8205/709/2/L172
  * Sari et al. (1998) Sari, R., Piran, T., & Narayan, R. 1998, ApJ, 497, L17. doi:10.1086/311269
  * Sari & Piran (1999) Sari, R. & Piran, T. 1999, A&AS, 138, 537. doi:10.1051/aas:1999342
  * Thompson (1994) Thompson, C. 1994, MNRAS, 270, 480. doi:10.1093/mnras/270.3.480
  * Usov (1992) Usov, V. V. 1992, Nature, 357, 472. doi:10.1038/357472a0
  * Vianello et al. (2017) Vianello, G., Lauer, R. J., Burgess, J. M., et al. 2017, Proceedings of the 7th International Fermi Symposium, 130
  * Wang et al. (2017) Wang, Y.-Z., Wang, H., Zhang, S., et al. 2017, ApJ, 836, 81\. doi:10.3847/1538-4357/aa56c6
  * Xiao et al. (2022) Xiao, S., Zhang, Y,-Q., Zhu, Z,-P., et al.2022, arXiv:2205.02186
  * Yang et al. (2022) Yang, J., Ai, S., Zhang, B.-B., et al. 2022, Nature, 612, 232. doi:10.1038/s41586-022-05403-8
  * Zhang (2018) Zhang, B. 2018, The Physics of Gamma-Ray Bursts by Bing Zhang. ISBN: 978-1-139-22653-0. Cambridge Univeristy Press, 2018. doi:10.1017/9781139226530
  * Zhang & Pe’er (2009) Zhang, B. & Pe’er, A. 2009, ApJ, 700, L65. doi:10.1088/0004-637X/700/2/L65
  * Zhang & Mészáros (2001) Zhang, B. & Mészáros, P. 2001, ApJ, 552, L35. doi:10.1086/320255
  * Zhang & Mészáros (2004) Zhang, B. & Mészáros, P. 2004, International Journal of Modern Physics A, 19, 2385. doi:10.1142/S0217751X0401746X
  * Zhang & Yan (2011) Zhang, B. & Yan, H. 2011, ApJ, 726, 90. doi:10.1088/0004-637X/726/2/90
  * Zhang et al. (2016) Zhang, B., Lü, H.-J., & Liang, E.-W. 2016, Space Sci. Rev., 202, 3. doi:10.1007/s11214-016-0305-9
  * Zhang et al. (2011) Zhang, B.-B., Zhang, B., Liang, E.-W., et al. 2011, ApJ, 730, 141. doi:10.1088/0004-637X/730/2/141
  * Zhang et al. (2018) Zhang, B.-B., Zhang, B., Castro-Tirado, A. J., et al. 2018, Nature Astronomy, 2, 69. doi:10.1038/s41550-017-0309-8
  * Zhang et al. (2021) Zhang, Y. Q., Xiong, S. L., Li, X. B., et al. 2021, GRB Coordinates Network, Circular Service, No. 31236, 1

Table 1Time-integral spectral analysis results of GRB 211211A. $t_{1}-t_{2}$ | $\rm Model$ | $\alpha(\alpha_{1})$ | $E_{\rm b}$ | $E_{\rm p}/E_{\rm c}$ | $\beta$ | $\alpha_{2}$ | kT | BIC | favorite Model
---|---|---|---|---|---|---|---|---|---
(s) |  |  | (keV) | (keV) |  |  | (keV) |  |
0.5-70 | Band | $-1.22^{+0.01}_{-0.01}$ | … | $497^{+15}_{-15}$ | $-2.21^{+0.03}_{-0.03}$ | … | … | 8633 |
| CPL | $-1.27^{+0.01}_{-0.01}$ | … | $916^{+19}_{-19}$ | … | … | … | 8964 |
| BB | … | … | … | … | … | $~{}53$ | 125651 |
| Band+BB | $-1^{+0.01}_{-0.01}$ | … | $175^{+6}_{-6}$ | $-2.06^{+0.02}_{-0.02}$ | … | $201^{+7}_{-7}$ | 7864 |
| CPL+BB | $-1.08^{+0.01}_{-0.01}$ | … | $250^{+5}_{-5}$ | … | … | $312^{+5}_{-5}$ | 8149 |
| SBPL | $-1.19^{+0.01}_{-0.01}$ | $210^{+10}_{-11}$ | … | ${-2.11^{+0.02}_{-0.02}}$ | … | … | 8523 |
| SBPL+BB | $-0.96^{+0.02}_{-0.02}$ | $81^{+4}_{-4}$ | … | ${-2.08^{+0.02}_{-0.02}}$ | … | $199^{+6}_{-6}$ | 7732 |
| 2SBPL | ${-0.6^{+0.04}_{-0.04}}$ | $26.2^{+1.3}_{-1.3}$ | $1020^{+40}_{-50}$ | $-2.93^{+0.07}_{-0.07}$ | $-1.59^{+0.01}_{-0.01}$ | … | 7461 |
| 2SBPL+BB | ${-0.6^{+0.04}_{-0.04}}$ | $27^{+1.9}_{-1.8}$ | $1200^{+110}_{-110}$ | $-3.14^{+0.14}_{-0.13}$ | $-1.62^{+0.02}_{-0.02}$ | $77^{+16}_{-15}$ | 7443 | $\surd$

Table 2The fitting results of time-resolved spectra with 2SBPL and 2SBPL+BB in GRB 211211A prompt emission. $t_{1}$ | $t_{2}$ |  |  | 2SBPL |  |  | BIC (PGSTAT) |  |  | 2SBPL + BB |  |  |  | BIC (PGSTAT) |  $\Delta$BIC
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
s | s | $\alpha_{1}$ | $\alpha_{2}$ |  $E_{\rm b}$(keV) |  $E_{\rm pk}$(keV) | $\beta$ |  | $\alpha_{1}$ | $\alpha_{2}$ |  $E_{\rm b}$(keV) |  $E_{\rm pk}$(keV) | $\beta$ | kT(keV) |  |
0.5 | 1.5 | $-0.9^{+0.5}_{-0.5}$ | $-1.1^{+0.6}_{-0.5}$ | $13^{+10}_{-9}$ | $151^{+24}_{-60}$ | $-2.9^{+0.5}_{-0.5}$ | 1640.4(1605.2) | $-0.9^{+0.5}_{-0.4}$ | $-1.1^{+0.5}_{-0.5}$ | $13^{+9}_{-9}$ | $121^{+24}_{-40}$ | $-3.1^{+0.5}_{-\par\par\par 0.5}$ | $46^{+19}_{-19}$ | 1632.6(1585.7) | 7.8
1.5 | 2.5 | $-0.9^{+0.4}_{-0.5}$ | $-1.07^{+0.4}_{-0.33}$ | $26^{+15}_{-13}$ | $820^{+120}_{-120}$ | $-2.52^{+0.14\par\par\par}_{-0.15}$ | 2001.2(1966.1) | $-0.83^{+0.34}_{-0.5}$ | $-1.17^{+0.5}_{-0.3}$ | $27^{+12}_{-12}$ | $980^{+210}_{-\par\par\par 210}$ | $-2.66^{+0.22}_{-0.22}$ | $65^{+18}_{-19}$ | 1999.9(1953.1) | 1.3
2.5 | 3.5 | $-0.73^{+0.9}_{-0.01}$ | $-1.48^{+0.1}_{-0.1}$ | $98^{+18}_{-19}$ | $1370^{+130}_{-130}$ | $-3.2^{+0.16\par\par\par}_{-0.17}$ | 2594.7(2559.5) | $-0.67^{+0.04}_{-0.04}$ | $-1.64^{+0.04}_{-0.04}$ | $92^{+12}_{-12}$ | $2100^{+\par\par\par 210}_{-210}$ | $-3.99^{+0.31}_{-0.31}$ | $168^{+17}_{-16}$ | 2569.5(2522.6) | 25.2
3.5 | 4.5 | $-0.6^{+0.4}_{-0.8}$ | $-1.09^{+0.8}_{-0.34}$ | $38^{+5}_{-5}$ | $840^{+150}_{-100}$ | $-2.88^{+0.09\par\par\par}_{-0.09}$ | 2327.8(2292.6) | $-0.61^{+0.33}_{-0.8}$ | $-1.21^{+0.8}_{-0.35}$ | $42^{+7}_{-7}$ | $960^{+190}_{-190}$ | $-2.99^{+0.18}_{-0.14}$ | $100^{+40}_{-40}$ | 2326(2279.1) | 1.8
4.5 | 5.5 | $-0.68^{+0.13}_{-0.12}$ | $-1.66^{+0.06}_{-0.08}$ | $38^{+9}_{-9}$ | $530^{+80}_{-80}$ | $-3.2^{+0.4}_{-0.4}$ | 2102.9(2067.7) | $-0.69^{+0.12}_{-0.11}$ | $-1.71^{+0.08}_{-0.08}$ | $39^{+9}_{-9}$ | $\par\par\par 570^{+110}_{-110}$ | $-3.3^{+0.4}_{-0.4}$ | $73^{+26}_{-30}$ | 2095.1(2048.2) | 7.8
5.5 | 6.5 | $-0.62^{+0.04}_{-0.03}$ | $-1.55^{+0.02}_{-0.03}$ | $81^{+9}_{-9}$ | $1700^{+100}_{-100}$ | $-3.36^{+0.14}_{-0.14}$ | 2592.3(2557.2) | $-0.59^{+0.05}_{-0.04}$ | $-1.55^{+0.24}_{-0.04}$ | $72^{+9}_{-\par\par\par 9}$ | $1860^{+170}_{-160}$ | $-3.48^{+0.17}_{-0.18}$ | $126^{+40}_{-35}$ | 2579.9(2533.1) | 12.4
6.5 | 7.5 | $-0.57^{+0.12}_{-0.03}$ | $-1.35^{+0.05}_{-0.1}$ | $70^{+7}_{-7}$ | $1630^{+70}_{-110}$ | $-3.09^{+0.09}_{-0.09}$ | 2751.6(2716.4) | $-0.69^{+0.25}_{-0.7}$ | $-1.22^{+0.7}_{-0.24}$ | $67^{+7}_{-7\par}$ | $1750^{+280}_{-190}$ | $-3.12^{+0.1}_{-0.1}$ | $100^{+60}_{-50}$ | 2743.6(2696.7) | 8
7.5 | 8.5 | $0.68^{+0.29}_{-0.8}$ | $-1.21^{+0.8}_{-0.28}$ | $57^{+5}_{-5}$ | $1440^{+270}_{-160}$ | $-3.34^{+0.13}_{-0.31}$ | 2539.6(2504.4) | $-0.56^{+0.2}_{-0.05}$ | $-1.35^{+0.09}_{-0.2}$ | $55^{+6}_{-6}$ | $1670^{+\par\par\par 180}_{-180}$ | $-3.64^{+0.21}_{-0.22}$ | $136^{+24}_{-23}$ | 2524.6(2477.7) | 15
8.5 | 9.5 | $-0.5^{+0.11}_{-0.11}$ | $-1.67^{+0.04}_{-0.05}$ | $34^{+5}_{-5}$ | $660^{+90}_{-90}$ | $-3.2^{+\par\par\par 0.35}_{-0.4}$ | 2090.2(2055.1) | $-0.51^{+0.11}_{-0.11}$ | $-1.69^{+0.05}_{-0.06}$ | $35^{+5}_{-5}$ | $650^{+120}_{-100}$ | $-3.2^{+0.4}_{-0.4}$ | $80^{+50}_{-40}$ | 2080.5(2033.7) | 9.7
9.5 | 10.5 | $-0.33^{+0.11}_{-0.11}$ | $-1.75^{+0.04}_{-0.04}$ | $33.4^{+3.2}_{-3.2}$ | $680^{+120}_{-130}$ | $\par\par-3.2^{+0.4}_{-0.4}$ | 2072.5(2037.3) | $-0.34^{+0.11}_{-0.11}$ | $-1.76^{+0.04}_{-0.04}$ | $33.8^{+3.4}_{-3.4}$ | $\par\par\par 690^{+160}_{-140}$ | $-3.2^{+0.4}_{-0.4}$ | $64^{+31}_{-33}$ | 2066.6(2019.8) | 5.9
10.5 | 11.5 | $-0.41^{+0.12}_{-0.12}$ | $-1.8^{+0.04}_{-0.04}$ | $31^{+3.1}_{-3}$ | $590^{+90}_{-100}$ | $\par\par-3.9^{+0.5}_{-0.5}$ | 1991.1(1955.9) | $-0.44^{+0.12}_{-0.12}$ | $-1.85^{+0.07}_{-0.08}$ | $32^{+4}_{-4}$ | $\par\par\par 530^{+150}_{-170}$ | $-3.8^{+0.5}_{-0.5}$ | $90^{+50}_{-50}$ | 1981.1(1934.3) | 10
11.5 | 12.5 | $-0.68^{+0.21}_{-0.19}$ | $-1.82^{+0.07}_{-0.14}$ | $26^{+6}_{-5}$ | $410^{+170}_{-220}$ | $-3^{+0.7}_{-0.7}$ | 1850.3(1815.1) | $-0.68^{+0.16}_{-0.17}$ | $-1.89^{+0.06}_{-0.08}$ | $27^{+5}_{-5\par\par\par}$ | $450^{+220}_{-250}$ | $-3.1^{+0.6}_{-0.6}$ | $50^{+23}_{-22}$ | 1843.1(1796.2) | 7.2
12.5 | 13.5 | $-0.56^{+0.32}_{-0.32}$ | $-1.83^{+0.1}_{-0.12}$ | $21^{+5}_{-5}$ | $540^{+280}_{-350}$ | $\par\par\par-2.9^{+0.6}_{-0.6}$ | 1711.8(1676.6) | $-0.53^{+0.3}_{-0.29}$ | $-1.86^{+0.09}_{-0.11}$ | $20^{+5}_{-4}$ | $570^{+300}_{-400}$ | $-2.9^{+0.6}_{-0.6}$ | $45^{+18}_{-18}$ | 1703.9(1657.1) | 7.9
13.5 | 15 | $-0.63^{+0.23}_{-0.25}$ | $-1.88^{+0.07}_{-0.08}$ | $22^{+4}_{-4}$ | $500^{+270}_{-310}$ | $\par\par\par-3^{+0.6}_{-0.6}$ | 2154.1(2118.8) | $-0.65^{+0.24}_{-0.23}$ | $-1.86^{+0.05}_{-0.12}$ | $21^{+4}_{-4}$ | $\par\par\par 530^{+320}_{-350}$ | $-3^{+0.6}_{-0.6}$ | $43^{+18}_{-18}$ | 2152.1(2105.3) | 2
15 | 17 | $-0.53^{+0.19}_{-0.17}$ | $-1.81^{+0.05}_{-0.13}$ | $26^{+4}_{-4}$ | $600^{+270}_{-310}$ | $-2.8^{+0.5}_{-0.6}$ | 2624.1(2588.8) | $-0.52^{+0.15}_{-0.16}$ | $-1.88^{+0.06}_{-0.07}$ | $26.1^{+3.5}_{-3.3}$ | $\par\par\par 670^{+310}_{-350}$ | $-2.9^{+0.6}_{-0.6}$ | $46^{+21}_{-20}$ | 2618.4(2571.6) | 5.7
17 | 19 | $-0.5^{+0.1}_{-0.1}$ | $-1.71^{+0.04}_{-0.04}$ | $32^{+4}_{-4}$ | $560^{+70}_{-70}$ | $-3.31^{+0.34}_{-0.35}$ | 2860.4(2825.2) | $-0.52^{+0.1}_{-0.1}$ | $-1.74^{+0.05}_{-0.05}$ | $33^{+4}_{-4}$ | $570^{+\par\par\par 90}_{-80}$ | $-3.33^{+0.4}_{-0.35}$ | $67^{+40}_{-35}$ | 2853.4(2806.6) | 7
19 | 21 | $-0.63^{+0.13}_{-0.13}$ | $-1.75^{+0.08}_{-0.09}$ | $33^{+7}_{-6}$ | $450^{+90}_{-90}$ | $-3^{+0.4}_{-0.4}$ | 2781.7(2746.5) | $-0.66^{+0.1}_{-0.1}$ | $-1.81^{+0.08}_{-0.08}$ | $35^{+6}_{-6}$ | $530^{+\par\par\par 150}_{-140}$ | $-3.2^{+0.5}_{-0.5}$ | $60^{+22}_{-23}$ | 2774.5(2727.7) | 7.2
21 | 23 | $-0.57^{+0.1}_{-0.1}$ | $-1.79^{+0.05}_{-0.05}$ | $31^{+4}_{-4}$ | $350^{+50}_{-50}$ | $-3.2^{+0.31}_{-0.32}$ | 2822.7(2787.5) | $-0.59^{+0.09}_{-0.1}$ | $-1.84^{+0.07}_{-0.08}$ | $33^{+4}_{-4}$ | $350^{+80}_{-70}$ | $-3.2^{+0.4}_{-0.4}$ | $68^{+23}_{-27}$ | 2814.3(2767.5) | 8.4
23 | 25 | $-0.52^{+0.1}_{-0.1}$ | $-1.75^{+0.05}_{-0.05}$ | $31^{+4}_{-4}$ | $390^{+50}_{-50}$ | $-3.25^{+\par\par\par 0.32}_{-0.31}$ | 2836.4(2801.2) | $-0.53^{+0.11}_{-0.11}$ | $-1.78^{+0.06}_{-0.06}$ | $32^{+4}_{-4}$ | $\par\par\par 410^{+80}_{-70}$ | $-3.3^{+0.4}_{-0.4}$ | $62^{+22}_{-27}$ | 2829.1(2782.2) | 7.3
25 | 27 | $-0.53^{+0.12}_{-0.12}$ | $-1.86^{+0.05}_{-0.05}$ | $28.1^{+3.3}_{-3.4}$ | $330^{+70}_{-70}$ | $\par-3.6^{+0.5}_{-0.5}$ | 2718.9(2683.7) | $-0.54^{+0.12}_{-0.12}$ | $-1.87^{+0.06}_{-0.06}$ | $28.1^{+3.4}_{-3.4}$ | $\par\par\par 290^{+90}_{-100}$ | $-3.5^{+0.6}_{-0.6}$ | $63^{+40}_{-32}$ | 2706.8(2660.1) | 12.1
27 | 29 | $-0.56^{+0.17}_{-0.18}$ | $-1.78^{+0.2}_{-0.17}$ | $22^{+4}_{-5}$ | $108^{+30}_{-27}$ | $-3.1^{+\par\par\par 0.6}_{-0.6}$ | 2643.6(2608.4) | $-0.7^{+0.5}_{-0.6}$ | $-1.1^{+0.7}_{-0.6}$ | $14^{+8}_{-7}$ | $76^{+11}_{-15}$ | $\par\par\par-2.69^{+0.25}_{-0.19}$ | $69^{+14}_{-13}$ | 2628.7(2581.9) | 14.9
29 | 31 | $-0.81^{+0.32}_{-0.3}$ | $-0.9^{+0.4}_{-0.4}$ | $15^{+9}_{-9}$ | $64^{+5}_{-5}$ | $-2.29^{+0.06\par\par\par}_{-0.06}$ | 2601.8(2566.6) | $-0.88^{+0.4}_{-0.31}$ | $-0.8^{+0.4}_{-0.4}$ | $13^{+7}_{-7}$ | $62^{+5}_{-5\par\par\par}$ | $-2.31^{+0.07}_{-0.06}$ | $45^{+17}_{-19}$ | 2594.1(2547.2) | 7.7
31 | 33 | $-0.37^{+0.24}_{-0.24}$ | $-1.81^{+0.18}_{-0.15}$ | $15.5^{+2.4}_{-2.5}$ | $75^{+19}_{-18}$ | $\par-2.8^{+0.4}_{-0.4}$ | 2616.2(2581) | $-0.8^{+0.5}_{-0.5}$ | $-1.2^{+0.7}_{-0.7}$ | $12^{+5}_{-5}$ | $58^{+9}_{-10}$ | $-2.6^{+0.23}_{-0.22}$ | $63^{+21}_{-20}$ | 2605.6(2558.8) | 10.6
33 | 35 | $-0.45^{+0.27}_{-0.23}$ | $-1.72^{+0.14}_{-0.21}$ | $21^{+5}_{-5}$ | $116^{+28}_{-27}$ | $-3^{+0.5}_{-0.5}$ | 2633.9(2598.7) | $-0.51^{+0.32}_{-0.24}$ | $-1.59^{+0.19}_{-0.34}$ | $20^{+5}_{-5}$ | $\par\par\par 105^{+28}_{-29}$ | $-2.9^{+0.4}_{-0.4}$ | $49^{+21}_{-21}$ | 2627.4(2580.5) | 6.5
35 | 37 | $-0.56^{+0.15}_{-0.15}$ | $-1.89^{+0.05}_{-0.06}$ | $24.9^{+3.3}_{-3.1}$ | $230^{+50}_{-60}$ | $\par-3.6^{+0.6}_{-0.6}$ | 2640.5(2605.3) | $-0.55^{+0.13}_{-0.14}$ | $-1.91^{+0.06}_{-0.06}$ | $24.9^{+2.8}_{-2.8}$ | $\par\par\par 190^{+70}_{-80}$ | $-3.4^{+0.6}_{-0.6}$ | $60^{+28}_{-28}$ | 2631.5(2584.6) | 9
37 | 39 | $-0.53^{+0.23}_{-0.23}$ | $-1.83^{+0.08}_{-0.09}$ | $19^{+4}_{-4}$ | $135^{+24}_{-24}$ | $-3.5^{+0.5}_{-0.5}$ | 2606.6(2571.4) | $-0.55^{+0.23}_{-0.22}$ | $-1.82^{+0.08}_{-0.11}$ | $18.9^{+4}_{-3.4}$ | $\par\par\par 119^{+29}_{-30}$ | $-3.5^{+0.5}_{-0.5}$ | $52^{+22}_{-22}$ | 2597.9(2551.1) | 8.7
39 | 41 | $-0.29^{+0.3}_{-0.25}$ | $-1.8^{+0.07}_{-0.15}$ | $17.5^{+0.26}_{-0.26}$ | $99^{+26}_{-24}$ | $-3.2^{+0.5}_{-0.6}$ | 2557.7(2522.5) | $-0.23^{+0.3}_{-0.28}$ | $-1.81^{+0.14}_{-0.13}$ | $17.1^{+0.26}_{-0.26}$ | $84^{+24}_{-24}$ | $-3.2^{+0.5}_{-0.5}$ | $51^{+25}_{-24}$ | 2547.1(2500.3) | 10.6
41 | 43 | $-1^{+0.5}_{-0.4}$ | $-1.1^{+0.6}_{-0.4}$ | $9^{+5}_{-5}$ | $66^{+8}_{-7}$ | $-2.44^{+0.08}_{-\par\par\par 0.15}$ | 2578.5(2543.3) | $-1.1^{+0.5}_{-0.4}$ | $-1^{+0.8}_{-0.6}$ | $7^{+4}_{-4}$ | $65^{+11}_{-9}$ | $-3.4^{+\par\par\par 0.4}_{-0.4}$ | $93^{+13}_{-11}$ | 2563.5(2516.7) | 15
43 | 45 | $-0.4^{+0.4}_{-0.4}$ | $-1.2^{+0.4}_{-0.4}$ | $9^{+5}_{-4}$ | $38.9^{+3}_{-3}$ | $-2.31^{+0.18}_{-\par\par 0.24}$ | 2511.5(2476.3) | $-0.48^{+0.35}_{-0.4}$ | $-1.26^{+0.27}_{-0.27}$ | $13^{+6}_{-8}$ | $37^{+3.2}_{-\par\par\par 3.1}$ | $-2.67^{+0.21}_{-0.2}$ | $58^{+18}_{-19}$ | 2504(2457.1) | 7.5
45 | 50 | $-0.4^{+0.5}_{-0.8}$ | $-1.3^{+0.8}_{-0.5}$ | $12.2^{+3.4}_{-4}$ | $49^{+7}_{-6}$ | $-2.47^{+0.16}_{-\par\par\par 0.17}$ | 3520.4(3485.1) | $-0.3^{+0.28}_{-0.29}$ | $-1.54^{+0.29}_{-0.28}$ | $12.9^{+3.2}_{-4}$ | $44^{+5}_{-\par\par\par 6}$ | $-2.53^{+0.19}_{-0.18}$ | $49^{+17}_{-18}$ | 3512.7(3465.8) | 7.7
50 | 70 | $-0.2^{+0.5}_{-0.5}$ | $-1.64^{+0.11}_{-0.27}$ | $11.2^{+1.5}_{-1.8}$ | $51^{+8}_{-9}$ | $-2.66^{+\par 0.28}_{-0.3}$ | 4977.5(4942.3) | $-0.1^{+0.5}_{-0.5}$ | $-1.66^{+0.21}_{-0.19}$ | $10.9^{+1.6}_{-2.4}$ | $43^{+6}_{-7}$ | $-2.64^{+0.25}_{-0.25}$ | $42^{+16}_{-17}$ | 4968.1(4921.2) | 9.4

Table 3The calculated flux and derived parameters of GRB 211211A prompt emission. $t_{1}$ | $t_{2}$ | $F_{\rm BB}$ | $F_{\rm obs}$ | $F_{\rm BB}/F_{\rm obs}$ | $\Gamma_{\rm ph}$ | $R_{\rm ph}$
---|---|---|---|---|---|---
$\rm(s)$ | $\rm(s)$ | $\rm(10^{-6}~{}erg~{}cm^{-2}~{}s^{-1})$ | $\rm(10^{-6}~{}erg~{}cm^{-2}~{}s^{-1})$ |  |  | $(10^{10}~{}\rm cm)$
0.5 | 1.5 | $0.03^{+0.07}_{-0.02}$ | $0.48^{+0.06}_{-0.06}$ | $0.06^{+0.15}_{-0.05}$ | $86.93^{+31.69}_{-\par\par\par 20.19}$ | $0.61^{+0.66}_{-0.42}$
1.5 | 2.5 | $0.32^{+0.62}_{-0.28}$ | $13.30^{+0.30}_{-0.20}$ | $0.02^{+0.05}_{-0.02}$ | $176.23^{+49.35}_{-\par\par\par 31.14}$ | $2.02^{+1.69}_{-1.07}$
2.5 | 3.5 | $8.16^{+1.54}_{-1.66}$ | $54.40^{+0.50}_{-0.40}$ | $0.15^{+0.03}_{-0.03}$ | $268.67^{+15.01}_{-\par\par\par 15.22}$ | $2.33^{+0.39}_{-0.40}$
3.5 | 4.5 | $0.91^{+3.69}_{-0.85}$ | $39.70^{+0.30}_{-0.70}$ | $0.02^{+0.09}_{-0.02}$ | $252.02^{+137.33}_{-\par\par\par 58.45}$ | $2.06^{+3.36}_{-1.43}$
4.5 | 5.5 | $0.28^{+0.65}_{-0.25}$ | $13.90^{+0.30}_{-0.10}$ | $0.02^{+0.05}_{-0.02}$ | $192.10^{+65.90}_{-\par\par\par 40.46}$ | $1.63^{+1.67}_{-1.03}$
5.5 | 6.5 | $2.98^{+1.82}_{-1.99}$ | $63.00^{+1.00}_{-0.30}$ | $0.05^{+0.03}_{-0.03}$ | $273.76^{+48.23}_{-\par\par\par 49.10}$ | $2.55^{+1.35}_{-1.37}$
6.5 | 7.5 | $0.70^{+2.20}_{-0.66}$ | $81.50^{+0.50}_{-0.50}$ | $0.01^{+0.03}_{-0.01}$ | $311.73^{+154.09}_{-\par\par\par 100.38}$ | $2.23^{+3.31}_{-2.16}$
7.5 | 8.5 | $4.80^{+2.00}_{-2.10}$ | $62.20^{+0.50}_{-0.40}$ | $0.08^{+0.03}_{-0.03}$ | $267.11^{+27.37}_{-\par\par\par 27.73}$ | $2.71^{+0.83}_{-0.84}$
8.5 | 9.5 | $0.15^{+0.88}_{-0.14}$ | $15.10^{+0.30}_{-0.20}$ | $0.01^{+0.06}_{-0.01}$ | $221.40^{+174.20}_{-\par\par\par 73.68}$ | $1.15^{+2.72}_{-1.15}$
9.5 | 10.5 | $0.08^{+0.35}_{-0.07}$ | $12.70^{+0.20}_{-0.20}$ | $0.01^{+0.03}_{-0.01}$ | $204.54^{+117.81\par}_{-54.60}$ | $1.23^{+2.13}_{-0.99}$
10.5 | 11.5 | $0.28^{+0.96}_{-0.27}$ | $9.24^{+0.19}_{-0.17}$ | $0.03^{+0.10}_{-0.03}$ | $192.26^{+97.56\par\par}_{-58.06}$ | $1.08^{+1.64}_{-0.98}$
11.5 | 12.5 | $0.04^{+0.12}_{-0.03}$ | $3.85^{+0.11}_{-0.10}$ | $0.01^{+0.03}_{-0.01}$ | $148.01^{+66.21\par\par}_{-37.63}$ | $0.98^{+1.32}_{-0.75}$
12.5 | 13.5 | $0.03^{+0.07}_{-0.02}$ | $2.03^{+0.08}_{-0.08}$ | $0.01^{+0.03}_{-0.01}$ | $125.75^{+49.89\par\par}_{-28.54}$ | $0.85^{+1.01}_{-0.58}$
13.5 | 15 | $0.02^{+0.06}_{-0.02}$ | $2.20^{+0.08}_{-0.06}$ | $0.01^{+0.03}_{-0.01}$ | $127.76^{+52.29\par}_{-30.06}$ | $0.87^{+1.07}_{-0.62}$
15 | 17 | $0.03^{+0.09}_{-0.03}$ | $3.90^{+0.10}_{-0.04}$ | $0.01^{+0.02}_{-0.01}$ | $145.94^{+61.89}_{-\par\par\par 36.96}$ | $1.04^{+1.32}_{-0.79}$
17 | 19 | $0.10^{+0.47}_{-0.09}$ | $10.90^{+0.10}_{-0.20}$ | $0.01^{+0.04}_{-0.01}$ | $197.65^{+133.22}_{-\par\par\par 63.08}$ | $1.17^{+2.37}_{-1.12}$
19 | 21 | $0.14^{+0.30}_{-0.12}$ | $9.57^{+0.13}_{-0.14}$ | $0.01^{+0.03}_{-0.01}$ | $173.16^{+57.02}_{-\par\par\par 37.15}$ | $1.53^{+1.51}_{-0.98}$
21 | 23 | $0.25^{+0.42}_{-0.22}$ | $9.22^{+0.12}_{-0.13}$ | $0.03^{+0.05}_{-0.02}$ | $169.90^{+46.13}_{-\par\par\par 34.54}$ | $1.56^{+1.27}_{-0.95}$
23 | 25 | $0.13^{+0.35}_{-0.12}$ | $10.70^{+0.13}_{-0.17}$ | $0.01^{+0.03}_{-0.01}$ | $182.36^{+69.38}_{-\par\par\par 38.22}$ | $1.46^{+1.67}_{-0.92}$
25 | 27 | $0.07^{+0.33}_{-0.06}$ | $6.40^{+0.12}_{-0.12}$ | $0.01^{+0.05}_{-0.01}$ | $175.75^{+121.45}_{-\par\par\par 59.16}$ | $0.98^{+2.03}_{-0.99}$
27 | 29 | $0.38^{+0.26}_{-0.31}$ | $3.36^{+0.08}_{-0.08}$ | $0.11^{+0.08}_{-0.09}$ | $125.82^{+16.68}_{-\par\par\par 17.90}$ | $1.40^{+0.56}_{-0.60}$
29 | 31 | $0.02^{+0.06}_{-0.02}$ | $2.81^{+0.07}_{-0.06}$ | $0.01^{+0.02}_{-0.01}$ | $138.25^{+51.92}_{-\par\par\par 29.94}$ | $0.88^{+0.99}_{-0.57}$
31 | 33 | $0.17^{+0.17}_{-0.15}$ | $2.43^{+0.06}_{-0.07}$ | $0.07^{+0.07}_{-0.06}$ | $122.80^{+25.81}_{-\par\par\par 24.50}$ | $1.09^{+0.69}_{-0.65}$
33 | 35 | $0.04^{+0.12}_{-0.03}$ | $2.28^{+0.07}_{-0.08}$ | $0.02^{+0.05}_{-0.02}$ | $128.03^{+56.02}_{-\par\par\par 30.89}$ | $0.90^{+1.18}_{-0.65}$
35 | 37 | $0.08^{+0.19}_{-0.07}$ | $3.36^{+0.10}_{-0.08}$ | $0.02^{+0.06}_{-0.02}$ | $143.44^{+55.25}_{-\par\par\par 37.13}$ | $0.94^{+1.09}_{-0.73}$
37 | 39 | $0.06^{+0.17}_{-0.05}$ | $2.46^{+0.08}_{-0.08}$ | $0.02^{+0.07}_{-0.02}$ | $128.31^{+54.07}_{-\par\par\par 30.66}$ | $0.97^{+1.22}_{-0.69}$
39 | 41 | $0.05^{+0.16}_{-0.04}$ | $1.95^{+0.06}_{-0.07}$ | $0.02^{+0.08}_{-0.02}$ | $123.11^{+60.78}_{-\par\par\par 33.25}$ | $0.87^{+1.28}_{-0.70}$
41 | 43 | $0.48^{+0.11}_{-0.14}$ | $1.74^{+0.07}_{-0.07}$ | $0.28^{+0.06}_{-0.08}$ | $120.34^{+9.17}_{-\par\par\par 9.59}$ | $0.83^{+0.19}_{-0.20}$
43 | 45 | $0.08^{+0.10}_{-0.07}$ | $1.05^{+0.05}_{-0.04}$ | $0.08^{+0.09}_{-0.06}$ | $104.86^{+22.62}_{-\par\par\par 19.58}$ | $0.76^{+0.49}_{-0.42}$
45 | 50 | $0.04^{+0.08}_{-0.03}$ | $1.16^{+0.03}_{-0.04}$ | $0.03^{+0.07}_{-0.03}$ | $108.06^{+32.72}_{-\par\par\par 22.00}$ | $0.76^{+0.69}_{-0.47}$
50 | 70 | $0.02^{+0.04}_{-0.01}$ | $0.52^{+0.02}_{-0.02}$ | $0.03^{+0.08}_{-0.02}$ | $91.12^{+33.68}_{-\par\par\par 18.95}$ | $0.57^{+0.63}_{-0.36}$
Table 4Derived parameters of GRB 211211A in a hybrid jet model. $t_{1}$ | $t_{2}$ | $1+\sigma_{0}$ | $\eta$ | $R_{\rm ph}$ | $\Gamma_{\rm ph}$ | $1+\sigma_{\rm ph}$
---|---|---|---|---|---|---
$(\rm s)$ | $(\rm s)$ |  |  | ($10^{10}$ cm) |  |
0.5 | 1.5 | 11.57 | 26.46 | 0.51 | 71.17 | 4.31
1.5 | 2.5 | 21.71 | 38.16 | 1.54 | 131.05 | 6.34
2.5 | 3.5 | 4.18 | 97.49 | 2.59 | 291.33 | 1.40
3.5 | 4.5 | 28.45 | 57.89 | 1.41 | 167.94 | 9.83
4.5 | 5.5 | 31.82 | 42.32 | 1.09 | 125.05 | 10.79
5.5 | 6.5 | 12.63 | 73.34 | 2.17 | 227.15 | 4.09
6.5 | 7.5 | 82.86 | 57.57 | 1.15 | 156.42 | 30.56
7.5 | 8.5 | 7.31 | 79.44 | 2.68 | 256.82 | 2.27
8.5 | 9.5 | 87.35 | 45.45 | 0.56 | 105.38 | 37.76
9.5 | 10.5 | 122.25 | 36.58 | 0.56 | 91.13 | 49.18
10.5 | 11.5 | 27.43 | 51.29 | 0.71 | 123.36 | 11.43
11.5 | 12.5 | 76.10 | 28.70 | 0.51 | 75.17 | 29.12
12.5 | 13.5 | 59.24 | 25.84 | 0.47 | 68.10 | 22.53
13.5 | 15 | 73.64 | 24.71 | 0.46 | 65.66 | 27.77
15 | 17 | 89.47 | 26.48 | 0.53 | 71.80 | 33.07
17 | 19 | 91.10 | 38.27 | 0.58 | 94.59 | 36.94
19 | 21 | 42.91 | 34.85 | 0.95 | 105.14 | 14.25
21 | 23 | 22.60 | 39.56 | 1.15 | 121.69 | 7.36
23 | 25 | 54.28 | 35.86 | 0.85 | 103.18 | 18.91
25 | 27 | 80.90 | 35.89 | 0.49 | 85.97 | 33.85
27 | 29 | 4.65 | 40.46 | 1.57 | 137.02 | 1.38
29 | 31 | 94.71 | 25.78 | 0.43 | 66.13 | 37.00
31 | 33 | 8.88 | 36.59 | 1.01 | 110.63 | 2.94
33 | 35 | 42.08 | 28.19 | 0.55 | 75.94 | 15.66
35 | 37 | 32.97 | 34.42 | 0.61 | 89.68 | 12.68
37 | 39 | 29.91 | 30.00 | 0.65 | 83.59 | 10.76
39 | 41 | 30.64 | 29.34 | 0.57 | 79.06 | 11.40
41 | 43 | 2.63 | 54.13 | 1.00 | 141.83 | $\cdots$
43 | 45 | 9.28 | 33.40 | 0.67 | 91.03 | 3.41
45 | 50 | 21.26 | 28.19 | 0.55 | 76.05 | 7.90
50 | 70 | 24.26 | 24.05 | 0.39 | 61.14 | 9.56

Figure 1.— $\Delta BIC$ (solid red circles) between 2SBPL and 2SBPL+BB models
in the time-resolved spectra of GRB 211211A. The gray line is the light curve
of prompt emission, and the shaded area indicates the range of [6-10].

Figure 2.— Time-integrated spectrum measured from $T_{0}+0.5$ to $T_{0}+70$ s
is fitted by 2SBPL+BB model. Left: observed and modeled photon count spectra.
Right: the parameter constraints of the spectral fit.

Figure 3.— Left: temporal evolution of $E_{\rm b}$, $E_{\rm pk}$ and kT of
2SBPL+BB model. Right: similar to left panels, but for the flux ($F_{\rm BB}$)
of BB emission and $F_{\rm BB}/F_{\rm obs}$. Figure 4.— Temporal evolution of
derived parameters $\Gamma_{\rm ph}$ and $R_{\rm ph}$ by using the Pe’er et
al. (2007) model and hybrid jet (Gao & Zhang 2015). Figure 5.— Temporal
evolution of $1+\sigma_{0}$, $\eta$, and $1+\sigma_{\rm ph}$ in the hybrid jet
model for fixed $r_{0}=10^{7}$ cm.

Figure 6.— Lorentz Factor ($\Gamma_{\rm ph}$) as a function of $L_{\gamma,\rm
iso}$ (a) and $E_{\gamma,\rm iso}$ of GRB 211211A (black solid circles). The
black solid lines are the best fit with power-law model. The red triangles are
the data of GRB 160625B from Wang et al. (2017).


Paris1Univ Gustave Eiffel Paris2ESIEE Paris Paris3LIGM TokyoThe University of
Tokyo

Paris1, Paris2<EMAIL_ADDRESS>Paris1, Paris2,
<EMAIL_ADDRESS>Paris1, Paris2,
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# New metrics for analyzing continual learners

Nicolas MICHEL  Giovanni CHIERCHIA  Romain NEGREL  Jean-François BERCHER
Toshihiko YAMASAKI

## Abstract

Deep neural networks have shown remarkable performance when trained on
independent and identically distributed data from a fixed set of classes.
However, in real-world scenarios, it can be desirable to train models on a
continuous stream of data where multiple classification tasks are presented
sequentially. This scenario, known as Continual Learning (CL) poses challenges
to standard learning algorithms which struggle to maintain knowledge of old
tasks while learning new ones. This stability-plasticity dilemma remains
central to CL and multiple metrics have been proposed to adequately measure
stability and plasticity separately. However, none considers the increasing
difficulty of the classification task, which inherently results in performance
loss for any model. In that sense, we analyze some limitations of current
metrics and identify the presence of setup-induced forgetting. Therefore, we
propose new metrics that account for the task’s increasing difficulty. Through
experiments on benchmark datasets, we demonstrate that our proposed metrics
can provide new insights into the stability-plasticity trade-off achieved by
models in the continual learning environment.

##  1 Introduction

Although deep neural networks can exhibit remarkable performances when trained
on independent and identically distributed data drawn from a fixed set of
classes, the practical need arises to train models on a continuous stream of
data. In such real-world scenarios, multiple classification tasks are
presented sequentially, leading to a situation where the data pertaining to
previous tasks becomes inaccessible when learning new ones. This scenario is
known as Continual Learning (CL) and poses many challenges to standard
learning algorithms which can suffer from Catastrophic Forgetting (CF).
Learning algorithms must trade-off between maintaining performances on old
tasks (stability) and achieving competitive performances on the current task
(plasticity). While the main objective remains to maximize accuracy across all
classes at the end of training, it is also essential to define meaningful
metrics to capture individual methods learning behavior and capabilities.

Metrics such as the Average Accuracy (AA) and the Average Forgetting (AF) have
been proposed in past studies. However, none of these metrics takes into
account the increasing difficulty of the classification task, which
automatically induces a loss in performance for any model. In that sense, we
argue that the AF metric is inherently linked to the continual learning setup
and does not fairly represent how the model reacts to the continuous
environment.

In this paper, we analyze the limitations of the current forgetting metric
through simple examples and propose new metrics for CL that take into account
the increasing difficulty of the task being solved by the model. We show
through several experiments on benchmark datasets that our proposed metrics
can shed new light on the stability-plasticity trade-off reached by the model
when training on the continual environment. In that sense, we make the
following contributions:

  * •

We review traditional metrics and show their current limitations;

  * •

We propose new metrics which take into account the increasing difficulty of
the continual setup;

  * •

We experimentally demonstrate the advantages of our metrics compared to
traditional metrics for analyzing continual learners.

The rest of the paper is organized as follows. In section 2 we describe work
related to ours. In section 3.2, we review two classical CL metrics and show
their limitations. In section 4 we defined our proposed metrics. Section 5
presents our experiments and eventually section 6 concludes the paper.

##  2 Related Work

###  2.1 Online Continual Learning (OCL)

OCL addresses the problem of learning from a continuous stream of data.
Formally, we consider a sequential learning setup with a sequence
$\\{\mathcal{T}_{1},\cdots,\mathcal{T}_{K}\\}$ of $K$ tasks, and
$\mathcal{D}_{k}=(X_{k},Y_{k})$ the corresponding data-label pairs. In CL, we
often assume that for any value $k_{1},k_{2}\in\\{1,\cdots,K\\}$, if
$k_{1}\neq k_{2}$ then we have $Y_{k_{1}}\cap Y_{k_{2}}=\emptyset$ and the
number of classes in each task is the same. Contrary to standard CL, in OCL
only one pass over the data is allowed. This setup has been widely studied in
the supervised scenario [2, 1, 9, 13, 10, 12, 3]. It is also referred to as
Online Class Incremental Learning. This setup is the one considered for
experiments in this paper.

###  2.2 Memory Based Method

Memory-based methods consist in using a memory buffer to store a small portion
of past samples. When encountering a new batch coming from the stream, another
batch is retrieved from the memory, and the model is trained on the
combination of both stream and memory batches. Between one stream batch and
the other, the memory is updated using the current stream batch data. Memory-
based methods have been especially popular in Online Continual Learning as
they achieve the best performances [12, 9, 13, 8, 11, 4, 5]. Compared methods
in the experiments section are all memory-based.

##  3 Traditional Metrics

In continual learning for image classification, we are interested in the
accuracy of the held-out test sets of the learned classes, just as in standard
learning. However, as the model is trained on a sequence of tasks
$\\{\mathcal{T}_{1},\cdots,\mathcal{T}_{k}\\}$, specific metrics have been
designed. In this section, we first define traditional metrics used in CL for
measuring plasticity and stability. Second, we study their current
limitations.

###  3.1 Traditional Metrics Definitions

###### Definition 3.1

The Average Accuracy (AA) after training on $\mathcal{T}_{k}$, for a
classifier $g$ in the set $\mathcal{G}$ of all classifiers, is:

$AA_{k}(g)=\frac{1}{k}\sum_{j=1}^{k}{a_{k,j}(g)},$ (1)

with $a_{k,j}(g)$, the accuracy of classifier $g$ on task $j$ after training
on $\\{\mathcal{T}_{1},\cdots,\mathcal{T}_{k}\\}$.

The final average accuracy $AA_{K}(g)$ is the accuracy after training $g$ on
the last task $\mathcal{T}_{K}$ and is the metric of interest for evaluating
the performance of $g$.

###### Definition 3.2

The Average Forgetting (AF) after training on $\mathcal{T}_{k}$, for a
classifier $g\in\mathcal{G}$ is:

$AF_{k}(g)=\frac{1}{k-1}\sum_{j=1}^{k-1}{f_{k,j}(g)},$ (2)

with $f_{k,j}(g)=\max\limits_{l\in\\{j,\cdots,k-1\\}}{a_{l,j}(g)}-a_{k,j}(g)$
and $k\geq 2$.

Other CL metrics also exist but are not covered in this work. Please note that
none of existing metrics take into account task difficulty.

###  3.2 Traditional Metrics Limitations

Here, we point out some current limitations of AA and AF by using the example
of a random classifier and a toy example.

Random classifier case. Looking at Definition 3.2, it can be observed that AF
refers to how poorly the model currently performs when compared to its all-
time-high performance on previous tasks. The main limitation of AF is that it
does not solely measure how much the model forgot, but also how much harder
the current overall problem is compared to a single task. To illustrate this
phenomenon, let us consider a simple example where we learn from a sequence of
5 tasks $\\{\mathcal{T}_{1},\cdots,\mathcal{T}_{5}\\}$, each task being
composed of two classes for a total of 10 distinct classes. We want to use a
random classifier $Rand_{C_{k}}$, where $C_{k}$ is the total number of classes
when training on task $k$. For this example, the accuracy and forgetting are
displayed in Tables 1 and 2 respectively.

$a_{k,j}$ | $\mathcal{T}_{1}$ | $\mathcal{T}_{2}$ | $\mathcal{T}_{3}$ | $\mathcal{T}_{4}$ | $\mathcal{T}_{5}$ | $AA_{k}$
---|---|---|---|---|---|---
$\mathcal{T}_{1}$ | $50$ | - | - | - | - | $50$
$\mathcal{T}_{2}$ | $25$ | $25$ | - | - | - | $25$
$\mathcal{T}_{3}$ | $16.7$ | $16.7$ | $16.7$ | - | - | $16.7$
$\mathcal{T}_{4}$ | $12.5$ | $12.5$ | $12.5$ | $12.5$ | - | $12.5$
$\mathcal{T}_{5}$ | $10$ | $10$ | $10$ | $10$ | $10$ | $10$

Table 1:  Accuracy (%) of a random classifier in a continual setting of 5
tasks with 2 classes per task. Each element at row $k$ and column $j$ is the
accuracy $a_{k,j}(Rand_{2k})$

$f_{k,j}$ | $\mathcal{T}_{1}$ | $\mathcal{T}_{2}$ | $\mathcal{T}_{3}$ | $\mathcal{T}_{4}$ | $\mathcal{T}_{5}$ | $AF_{k}$
---|---|---|---|---|---|---
$\mathcal{T}_{1}$ | - | - | - | - | - | -
$\mathcal{T}_{2}$ | $25$ | - | - | - | - | $25$
$\mathcal{T}_{3}$ | $33.3$ | $8.33$ | - | - | - | $20.83$
$\mathcal{T}_{4}$ | $37.5$ | $12.5$ | $4.16$ | - | - | $18.06$
$\mathcal{T}_{5}$ | $40$ | $15$ | $6.66$ | $2.5$ | - | $16.04$

Table 2: Forgetting (%) of a random classifier in a continual setting of 5
tasks with 2 classes per task. Each element at row $k$ and column $j$ is the
forgetting $f_{k,j}(Rand_{2k})$

As expected, the accuracy decreases during training as the number of classes
becomes larger, therefore the AF is positive. However, a random classifier can
hardly forget knowledge as it does not learn at all. While expanding the
number of classes should make the task harder for the model, the drop in
performance is not due to the model forgetting knowledge but rather to the
fact that the model is not able to learn how to solve the hard task as well as
the easy one. Using AA and AF can demonstrate forgetting for any model, even
if the model cannot possibly forget.

Toy example. Previous example demonstrates a setup-induced forgetting due to
the increasing task difficulty. This increasing task difficulty effect is also
illustrated in Figure 1 with a simple 2 tasks sequence example. In this case,
the classification tasks $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ are simple.
However, the overall task
$\mathcal{T}_{all}=\\{\mathcal{T}_{1},\mathcal{T}_{2}\\}$ is more complex and
it is the task the model is evaluated on after training on $\mathcal{T}_{2}$.
It is unlikely that the model forgot how to solve both binary classification
tasks separately, as they are very similar. Specifically, in this example, the
model might not forget how to differentiate classes 1 and 2 or how to
differentiate classes 3 and 4. Rather, the model might not learn how to
differentiate classes 1 and 3 or how to differentiate classes 2 and 4.

Figure 1: Toy example of a two tasks learning sequence. The model learns from
$\mathcal{T}_{1}$ then $\mathcal{T}_{2}$ and is then evaluated on
$\mathcal{T}_{all}$. Left tile represents learning $\mathcal{T}_{1}$, middle
tile represents learning $\mathcal{T}_{2}$ and right tile represents learning
$\mathcal{T}_{all}=\\{\mathcal{T}_{1},\mathcal{T}_{2}\\}$. Best viewed in
color.

##  4 New Metrics for Continual Learning

Current accuracy and forgetting metrics depend strongly on the continual
learning setup and can mislead forgetting analysis. We propose two metrics
that attempt to dissociate this setup-induced forgetting from the overall
performance by rescaling the original AA and AF using the performances of a
random classifier to account for task difficulty.

###  4.1 Rescaled Average Accuracy and Forgetting

###### Definition 4.1

The unnormalized Rescaled Average Accuracy (uRAA) and unnormalized Rescaled
Average Forgetting (uRAF) after training on $\mathcal{T}_{k}$, for a
classifier $g$ are:

$\displaystyle uRAA_{k}(g)$
$\displaystyle=\frac{AA_{k}(g)}{AA_{k}(Rand_{C_{k}})},$ (3) $\displaystyle
uRAF_{k}(g)$ $\displaystyle=\frac{AF_{k}(g)}{AF_{k}(Rand_{C_{k}})},$

where $C_{k}$ is the total number of classes seen at task $\mathcal{T}_{k}$.

###### Definition 4.2

The Rescaled Average Accuracy (RAA) and Rescaled Average Forgetting (RAF)
after training on $\mathcal{T}_{k}$, for a classifier $g\in\mathcal{G}$ are:

$\displaystyle RAA_{k}(g)$
$\displaystyle=\frac{uRAA_{k}(g)}{\max\limits_{f\in\mathcal{G},k}{uRAA_{k}(f)}},$
(4) $\displaystyle RAF_{k}(g)$
$\displaystyle=\frac{uRAF_{k}(g)}{\max\limits_{f\in\mathcal{G},k}{uRAF_{k}(f)}}.$

###### Proposition 4.3

Let $K$ be the total number of tasks and $C_{K}$ the total number of classes.
The Rescaled Average Accuracy (RAA) after training on $\mathcal{T}_{k}$, for a
classifier $g$ can be expressed as:

$\displaystyle RAA_{k}(g)=\gamma_{k}AA_{k}(g)\ \ \ \text{with}\ \ \
\gamma_{k}=\frac{C_{k}}{C_{K}}.$ (5)

###### Proposition 4.4

If every task has the same number of classes, the Rescaled Average Forgetting
(RAF) after training on $\mathcal{T}_{k}$, for a classifier $g$ can be
expressed as:

$\displaystyle RAF_{k}(g)=\beta_{k}AF_{k}(g),$ (6) $\displaystyle\text{with}\
\ \ \ \beta_{k}$ $\displaystyle=\frac{(H_{K}-1)(k-1)}{(H_{k}-1)(K-1)}\ \
\text{and}\ \ H_{k}=\sum_{i=1}^{k}\frac{1}{i}.$

Figure 2: Visualization of accuracy coefficient $\gamma_{k}$ and forgetting
coefficient $\beta_{k}$ for (a) $K=20$ (b) $K=100$

The coefficients $\beta_{k}$ and $\gamma_{k}$ act as task difficulty
coefficients. Figure 2 shows different values for these coefficients along
training. The harder the task, the higher the coefficient. Indeed,
$(\beta_{k},\gamma_{k})\in\ ]0,1]^{2}$ with one being the maximum difficulty
corresponding to the final task. With such definitions, we have equal values
at the end of training such that $RAA_{K}(g)=AA_{K}(g)$ and
$RAF_{K}(g)=AF_{K}(g)$.

###  4.2 Interpreting RAA and RAF

In subsequent we give insight on how to interpret previously refined metrics.

Comparison to a random classifier. RAA and RAF represent how much the model
learns or forgets when compared to a random classifier. For example, a
constant RAA indicates that the model performances are decreasing exactly like
a random classifier. This situation can happen if the model cannot learn new
tasks efficiently while maintaining high performance on past tasks (high
stability and low plasticity) or oppositely the model perfectly learns new
tasks while forgetting older tasks (low stability and high plasticity). In
other words, a constant RAA translates a failed stability-plasticity trade-
off, while an increasing RAA demonstrates that the model learns more than a
random classifier and hence can still accumulate new knowledge despite the
increasing task difficulty. Both previous situations cannot be easily
differentiated using AA as its value would be decreasing in both cases.

Decoupling task difficulty. As explained in section 3.2, the increase in task
difficulty while training often implies a decrease in model accuracy. Current
RAA and RAF expression can be interpreted as the correction of current AA and
AF to compensate for the drop in performance due to task difficulty.
Decoupling the task difficulty allows us to detect if the model’s performances
change due to the increasing number of classes or due to a failed stability-
plasticity trade-off.

To showcase how RAA and RAF help detect learning regimes hardly visible with
AA and AF, we conduct several experiments, detailed in section 5.

##  5 Experiments

In the following, we apply our metric to several Online Continual Learning
methods and compare their values to standard metrics.

###  5.1 Datasets

We use variations of standard image classification datasets [6, 7] to build
continual learning environments. The original datasets are split into several
tasks of non-overlapping classes. Specifically, we experimented on split-
CIFAR100 and split-Tiny ImageNet. In this paper, we omitted the split- suffix
for simplicity. CIFAR100 contains 50,000 32x32 train images as well as 10,000
test images and is split into 10 tasks containing 10 classes each for a total
of 100 distinct classes. Tiny ImageNet is a subset of the ILSVRC- 2012
classification dataset and contains 100,000 $64\times 64$ train images as well
as 10,000 test images and is split into 100 tasks containing two classes each
for a total of 200 distinct classes.

###  5.2 Baselines

In the following, we describe considered baselines. For every memory-based
method, we use reservoir sampling [14] for memory update and random retrieval.
Experience Replay (ER) [13]: ER is a supervised memory based technique using
reservoir sampling [14] for memory update and random retrieval. The model is
trained using cross-entropy.
Finetune: Usual model training with no measure taken to mitigate forgetting.
GDumb [12]: Simple method that stores data from the stream in memory, with the
constraint of having a balanced class selection. At inference time, the model
is trained offline on memory data.

###  5.3 Results

In the following, we analyze our experimental results. Training regimes with
RAA. Comparison of obtained values with AA and RAA are displayed in Figure 3.
It can be observed that for all methods the AA decreases during training,
which is expected. Given AA alone, it can be hard to understand what
dissociates each method’s behavior, as they all follow a similar trend.
However, looking at RAA, distinct training regimes can be observed. Notably,
GDumb RAA reaches a plateau on Tiny ImageNet around task 25 which shows that
this method cannot accumulate knowledge correctly past this point. Similarly,
we can observe that ER also reaches a plateau around task 65 on Tiny ImageNet
while no such behavior is displayed on CIFAR100. Such observation suggests
that in this case, ER reaches maximum learning capabilities on Tiny ImageNet
whereas not on CIFAR100. Interestingly, finetune’s RAA is almost constant and
near zero, which can be expected as such a naive approach shows no plasticity
capabilities.

Increasing forgetting with RAF. Figure 4 shows AF and RAF values for
considered methods. On both datasets, AF is rather constant or slightly
increasing for all methods. However, RAF displays a strong increase throughout
training for finetune and ER. Such an effect is less pronounced for GDumb.
Even if no specific training regime can directly be observed with RAF, its
overall trend highlights the different difficulty gaps between tasks. Indeed,
a $60\%$ loss in accuracy from $\mathcal{T}_{1}$ to $\mathcal{T}_{2}$ should
not be interpreted equally as a $60\%$ loss from $\mathcal{T}_{9}$ to
$\mathcal{T}_{10}$. If we consider $10$ classes per task, the former
corresponds to going from $10$ to $20$ classes to classify while the latter
corresponds to going from $90$ to $100$ classes to classify. The difficulty
gap is therefore much stronger in the former case than in the latter and
should not translate as the same forgetting value.

Figure 3: Values of $AA_{k}$ and $RAA_{k}$ with $k\in[1,K]$ the task id. From
left to write, the performances are shown on CIFAR100 $M=2k$ with $K=5$ and
Tiny ImageNet $M=2k$ with $K=5$. The top row shows the AA while the bottom row
shows the RAA. Comparing AA to RAA, we can observe different training
behavior, notably, GDumb reaches a plateau in CIFAR100 using RAA which is not
the case using AA. Figure 4: Values of $AF_{k}$ and $RAF_{k}$ with $k\in[1,K]$
the task id. From left to write, the performances are shown on CIFAR100 $M=2k$
with $K=5$ and Tiny ImageNet $M=2k$ with $K=5$. The top row shows the AF while
the bottom row shows the RAF. Comparing AF to RAF, we can observe that for
most cases $AF_{k}$ is rather constant while $RAF_{k}$ keeps increasing for
every method. Best viewed in color.

##  6 Conclusions

In this paper, we discussed the limitations of current traditional CL metrics.
After showing that AF and AA suffer from a setup-induced bias which can
deteriorate continual learners analysis, we introduced new metrics for CL
taking into account the increasing difficulty of the setup. We conducted
several experiments on classical methods for the Online Class Incremental
Learning scenario and experimentally demonstrate various advantages of using
RAA and RAF for analyzing continual learners. We believe these metrics can
help design new methods for CL. Extensions of this work might consider
expanding the notion of task difficulty beyond the performance of a random
classifier.

## References

  * [1] Aljundi, R., Belilovsky, E., Tuytelaars, T., Charlin, L., Caccia, M., Lin, M. and Page-Caccia, L.: Online Continual Learning with Maximal Interfered Retrieval, Advances in Neural Information Processing Systems, Vol. 32 (2019).
  * [2] Aljundi, R., Lin, M., Goujaud, B. and Bengio, Y.: Gradient based sample selection for online continual learning, arXiv:1903.08671 [cs, stat] (2019).
  * [3] Guo, Y., Liu, B. and Zhao, D.: Online Continual Learning through Mutual Information Maximization, Proceedings of the 39th International Conference on Machine Learning, pp. 8109–8126 (2022).
  * [4] Guo, Y., Liu, B. and Zhao, D.: Online Continual Learning through Mutual Information Maximization, Proceedings of the 39th International Conference on Machine Learning, pp. 8109–8126 (2022).
  * [5] Jung, D., Lee, D., Hong, S., Jang, H., Bae, H. and Yoon, S.: New Insights for the Stability-Plasticity Dilemma in Online Continual Learning, International Conference on Representation Learning, No. arXiv:2302.08741 (2023).
  * [6] Krizhevsky, A.: Learning Multiple Layers of Features from Tiny Images, Technical report (2012).
  * [7] Le, Y. and Yang, X. S.: Tiny ImageNet Visual Recognition Challenge, Technical report (2015).
  * [8] Lin, H., Zhang, B., Feng, S., Li, X. and Ye, Y.: PCR: Proxy-based Contrastive Replay for Online Class-Incremental Continual Learning, IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), No. arXiv:2304.04408 (2023).
  * [9] Mai, Z., Li, R., Jeong, J., Quispe, D., Kim, H. and Sanner, S.: Online Continual Learning in Image Classification: An Empirical Survey, arXiv:2101.10423 [cs] (2021).
  * [10] Mai, Z., Li, R., Kim, H. and Sanner, S.: Supervised Contrastive Replay: Revisiting the Nearest Class Mean Classifier in Online Class-Incremental Continual Learning, arXiv:2103.13885 [cs] (2021).
  * [11] Michel, N., Negrel, R., Chierchia, G. and Bercher, J.-F.: Contrastive Learning for Online Semi-Supervised General Continual Learning (ICIP), International Conference on Image Processing, pp. 1896–1900 (2022).
  * [12] Prabhu, A., Torr, P. H. S. and Dokania, P. K.: GDumb: A Simple Approach that Questions Our Progress in Continual Learning, Computer Vision â ECCV 2020 (Vedaldi, A., Bischof, H., Brox, T. and Frahm, J.-M., eds.), Vol. 12347, Cham, pp. 524–540 (2020).
  * [13] Rolnick, D., Ahuja, A., Schwarz, J., Lillicrap, T. and Wayne, G.: Experience Replay for Continual Learning, Advances in Neural Information Processing Systems, Vol. 32 (2019).
  * [14] Vitter, J. S.: Random sampling with a reservoir, ACM Transactions on Mathematical Software, Vol. 11, No. 1, pp. 37–57 (1985).

##  Appendix A Proof of proposition 4.3

###### Lemma A.1

$AA_{k}(Rand_{C_{k}})=\frac{1}{C_{k}}$

Proof. Trivial.

###### Lemma A.2

$\max\limits_{f\in\mathcal{G},k}{uRAA_{k}(f)}=C_{K}$. With $K$ the total
number of tasks.

Proof. From definition 4.1 we have:

$\displaystyle\max\limits_{f\in\mathcal{G},k}{uRAA_{k}(f)}=\max\limits_{f\in\mathcal{G},k}\frac{AA_{k}(f)}{AA_{k}(Rand_{C_{k}})}$

With definition 3.1 and lemma A.1:

$\displaystyle\max\limits_{f\in\mathcal{G},k}\frac{AA_{k}(f)}{AA_{k}(Rand_{C_{k}})}=\max\limits_{f\in\mathcal{G},k}\frac{C_{k}}{k}\sum_{j=1}^{k}{a_{k,j}(f)}$

$\forall f\in\mathcal{G}$, $a_{k,j}(f)\in[0,1]$, so for any $k$, $uRAA_{k}(f)$
is maximum when $\forall(k,j)\in[1,K]^{2}$, $a_{k,j}(f)=1$. This corresponds
to having a perfect classifier. Then we have:

$\max\limits_{f\in\mathcal{G},k}{uRAA_{k}(f)}=\max\limits_{k}C_{k}=C_{K}$ (7)

Because the number of classes to classify increases linearly over time in our
setup.
With definition 4.1 and equation 7, we obtained the desired result.

##  Appendix B Proof of proposition 4.4

###### Lemma B.1

$AF_{k}(Rand_{C_{k}})=\frac{1}{C_{k}}(\frac{k}{k-1}H_{k-1}-1)$. With
$H_{k}=\sum_{i=1}^{k}\frac{1}{i}$ the k-th harmonic number.

Proof. From definition 3.2:

$\displaystyle\begin{aligned}
AF_{k}(Rand_{C_{k}})&=\frac{1}{k-1}\sum_{j=1}^{k-1}f_{k,j}(Rand_{C_{k}})\\\
\text{with}\ \ \
f_{k,j}(Rand_{C_{k}})&=\max\limits_{l\in\\{j,\cdots,k-1\\}}{(a_{l,j}(Rand_{C_{l}}))}-a_{k,j}(Rand_{C_{k}})\\\
&=\max\limits_{l\in\\{j,\cdots,k-1\\}}{\left(\frac{1}{C_{l}}\right)}-\frac{1}{C_{k}}\\\
&=\frac{1}{C_{j}}\frac{1}{C_{k}}\\\ &=\frac{C_{k}-C_{j}}{C_{j}C_{k}}\\\
f_{k,j}(Rand_{C_{k}})&=\frac{C_{k-j}}{C_{j}C_{k}}\end{aligned}$

Because $C_{k+1}=C_{k}+n_{C}$ with $n_{C}$ the number of classes per task.
Then we have:

$\displaystyle\begin{aligned}
AF_{k}(Rand{C_{k}})&=\frac{1}{k-1}\sum_{j=1}^{k-1}\frac{C_{k-j}}{C_{j}C_{k}}\\\
&=\frac{1}{(k-1)C_{k}}\sum_{j=1}^{k-1}\frac{k-j}{j}\\\
&=\frac{k}{(k-1)C_{k}}\sum_{j=1}^{k-1}\frac{1}{j}-\frac{1}{C_{k}}\\\
&=\frac{1}{C_{k}}\left(\frac{kH_{k-1}}{k-1}-1\right)\end{aligned}$

With $H_{k}=\sum_{j=1}^{k}\frac{1}{j}$. Falling back to definition 4.1:

$\displaystyle\begin{aligned}
uRAF_{k}(Rand{C_{k}})&=\frac{AF_{k}(g)}{AF_{k}(Rand{C_{k}})}\\\
&=\frac{C_{k}}{\frac{k}{k-1}H_{k-1}-1}\frac{1}{k-1}\sum_{j=1}^{k-1}f_{k,j}(g)\\\
&=\frac{C_{k}}{k(H_{k-1}-1)+1}\sum_{j=1}^{k-1}f_{k,j}(g)\\\
&=\frac{C_{k}}{k(H_{k}-1)}\sum_{j=1}^{k-1}f_{k,j}(g)\end{aligned}$

###### Lemma B.2

$\max\limits_{h\in\mathcal{G},k}{uRAF_{k}(h)}=\frac{C_{k}(K-1)}{K(H_{K}-1)}$.
With $K$ the total number of tasks.

Proof. $\forall h\in\mathcal{G}$, $f_{k,j}(h)\in[0,1]$, so for any $k$,
$uRAF_{k}(h)$ is maximum when $\forall(k,j)\in[1,K]^{2}$, $f_{k,j}(h)=1$. This
corresponds to having the worst classifier. Then we have:

$\displaystyle\max\limits_{h\in\mathcal{G},k}{uRAF_{k}(h)}=\max\limits_{k}\frac{C_{k}(k-1)}{k(H_{k}-1)}=\frac{C_{K}(K-1)}{K(H_{K}-1)}$

Because $C_{k}=kn_{C}$ with $n_{C}$ the number of classes per task and
$\frac{k-1}{H_{k}-1}$ is an increasing function.
Eventually with B.2 we have:

$\displaystyle\begin{aligned}
RAF_{k}(g)&=\frac{uRAF_{k}(g)}{\max\limits_{h\in\mathcal{G},k}{uRAF_{k}(h)}}=\frac{KC_{k}(H_{K}-1)}{kC_{K}(H_{k}-1)(K-1)}\sum_{j=1}^{k-1}f_{k,j}(g)\\\
&=\frac{H_{K}-1}{(H_{k}-1)(K-1)}\sum_{j=1}^{k-1}f_{k,j}(g)\end{aligned}$