# The High Energy X-ray Probe (HEX-P): Magnetars and Other Isolated Neutron
Stars

J. A. J. Alford 1,∗, G. A. Younes 2, Z. Wadiasingh 3,4,5, M. Abdelmaguid 1, H.
An6, M. Bachetti,7, M. Baring8, A. Beloborodov9, A. Y. Chen10, T. Enoto11, J.
A. García 12, J. D. Gelfand 1, E. V. Gotthelf9, A. Harding13, C.-P. Hu 14,
A.D. Jaodand 15, V. Kaspi 16, C. Kim 6, C. Kouveliotou 2, L. Kuiper 17 K. Mori
8, M. Nynka 18, J. Park 6, D. Stern 19, J. Valverde 20, D. Walton 21

###### Abstract

The hard X-ray emission from magnetars and other isolated neutron stars
remains under-explored. An instrument with higher sensitivity to hard X-rays
is critical to understanding the physics of neutron star magnetospheres and
also the relationship between magnetars and Fast Radio Bursts (FRBs). High
sensitivity to hard X-rays is required to determine the number of magnetars
with hard X-ray tails, and to track transient non-thermal emission from these
sources for years post-outburst. This sensitivity would also enable previously
impossible studies of the faint non-thermal emission from middle-aged
rotation-powered pulsars (RPPs), and detailed phase-resolved spectroscopic
studies of younger, bright RPPs. The High Energy X-ray Probe (HEX-P) is a
probe-class mission concept that will combine high spatial resolution X-ray
imaging ($<5$ arcsec half-power diameter (HPD) at 0.2–25 keV) and broad
spectral coverage (0.2–80 keV) with a sensitivity superior to current
facilities (including XMM-Newton and NuSTAR). HEX-P has the required timing
resolution to perform follow-up observations of sources identified by other
facilities and positively identify candidate pulsating neutron stars. Here we
discuss how HEX-P is ideally suited to address important questions about the
physics of magnetars and other isolated neutron stars.

## 1

##  2 Keywords:

X-ray sources, HEX-P, pulsars, magnetars, neutron stars, spectra

##  3 Introduction

Massive stars ($\gtrsim 8$ $M_{\odot}$) end their lives in spectacular
supernovae, leaving behind either a neutron star (NS) or a black hole (BH)
(Heger et al., 2003). NSs are very compact (having masses of $\approx 1.4$
$M_{\odot}$ and radii of $\approx 10$ km), and isolated NSs typically have
strong magnetic fields ($B\gtrsim 10^{12}$ G, though see the discussion of
central compact objects (CCOs) below). The telltale sign for the existence of
an NS is the detection of their spin ephemerides, most notably their spin
periods $P$ and, if multi-epoch observations exist, their period derivatives
$\dot{P}$. These two observational properties can be used to estimate three
key fundamental physical scales under the assumption of a rotating dipole
magnet in vacuum (Figure 1): (1) rotational energy loss
$|\dot{E}|\propto\dot{P}/P^{3}$, (2) characteristic spin-down age $\tau_{\rm
char}=P/2\dot{P}$, and (3) surface dipolar magnetic field strength $B_{\rm
dip}\propto\sqrt{P\dot{P}}$.

Since their discovery, a number of unique classes of isolated NSs (INSs) have
been identified through differences in their broadband emission
characteristics, location/environment, and timing behavior. The most common of
the INS population are the rotation-powered pulsars (RPPs), aptly named given
that their large spin-down power $\dot{E}$ far exceeds their total radiative
luminosity. Then, there are the magnetars, which possess periods typically in
the range of about $1$-$15$ seconds and large spin-down rates ($\dot{P}\sim
10^{-11}$ s s-1) , thus occupying a unique space in the $P$-$\dot{P}$ diagram
(Figure 1, red squares). Assuming magnetic dipole braking, the magnetar timing
properties translate to dipole field strengths of the order of $10^{14}$ G
(two orders of magnitude larger than RPPs, Kouveliotou et al. 1998, 1999;
Kaspi et al. 2003), average spin-down ages of $10^{4}$ years (confirmed
through the association of a few with young, X-ray bright supernova remnants
(SNRs), see, e.g., Vasisht and Gotthelf 1997), and low spin-down power
$|\dot{E}|\sim 10^{32}$ erg s-1 (Olausen and Kaspi, 2014). Despite the latter,
magnetars are observed as hot thermal emitters with surface thermal
temperatures $kT\approx 0.5$ keV (factors of $\sim$2 larger than young RPPs;
see, e.g., Figure 14 of Olausen and Kaspi 2014), and X-ray luminosities
$\gtrsim 10^{33}$ erg s-1. Hence, magnetar radiative power is attributed to
the decay of their extreme external and internal $B$-fields (Thompson and
Duncan, 1995; Kaspi and Beloborodov, 2017).

Then there are the dozen known central compact objects (CCOs), which are
perhaps the least understood class of INSs (De Luca, 2017). The three CCOs
with measured spin ephemerides are indicated with blue diamonds in Figure 1.
These point-like X-ray sources are found in young X-ray bright SNRs, and lack
both emission at other wavelengths and any associated pulsar wind nebula.
While there are relatively few identified CCOs, their locations in young SNRs
suggests that they may represent a significant fraction of NS births. CCOs are
also known as ‘anti-magnetars’, since some of them have the smallest spin-
down-measured dipole magnetic field strengths among all known young NSs
(Gotthelf et al., 2013), at odds with their relatively bright thermal X-ray
emission.

X-ray dim isolated NSs (XDINSs, yellow crosses in Figure 1) constitute a
population with observational characteristics that do not quite fit the above
classes. They are nearby (within a few hundred parsecs, Kaplan 2008), radio-
quiet (but see Rigoselli et al., 2019), yet thermally-emitting with surface
temperatures in the range of $45$-$110$ eV and luminosities $L_{\rm
X}<10^{32}$ erg s-1. This emission is pulsed at a spin-period of a few
seconds, while slowing down with a rate of $\dot{P}\approx 10^{-14}$ s s-1,
implying $B_{\rm dip}\sim 10^{13}$ G, $\tau\gtrsim 1$ Myr, and an average
$\dot{E}\sim 5\times 10^{30}$ erg s-1. Given their small $|\dot{E}|$ compared
to their surface thermal emission, XDINS are thought to be powered by their
relatively large $B$-fields, akin to their younger counterparts, magnetars.

Recently, a new class of long-period radio-emitting NSs have been discovered
with spin periods of tens of seconds and relatively large spin-down rates (Tan
et al. 2018; Caleb et al. 2022; green stars in Figure 1). Their dipole-
inferred fields are comparable to those of magnetars and XDINs. They are also
relatively nearby with distances of the order of 1 kpc, yet deep X-ray
observations have not detected their high energy counterpart with upper-limits
comparable to those of XDINs (Rea et al., 2022; Beniamini et al., 2023). These
limits are consistent with the old magnetar interpretation, though how they
maintain dipolar-fields of $10^{14}$ G is less clear (Caleb et al., 2022;
Beniamini et al., 2023).

Our understanding of the young INS population has been predominantly shaped by
radio and X-ray observations. Radio surveys of large areas of the Galaxy
provide understanding of the birth rate and properties of NSs (e.g.,
Arzoumanian et al., 2002; Faucher-Giguère and Kaspi, 2006), while X-ray
observations established several of the above sub-classes, and highlighted the
connections among them; e.g., RPPs and CCOs showing magnetar-like bursting and
outburst abilities (Gavriil et al., 2008; Archibald et al., 2016b; Rea et al.,
2016), the existence of low $B$-field magnetars (Rea et al., 2010), and bona-
fide magnetars with RPP characteristics (Camilo et al., 2006, 2007; Younes et
al., 2016). All of these X-ray discoveries have led to several attempts at
unifying these sub-classes (Kaspi and Boydstun, 2010), mainly through magneto-
thermal evolution models (Viganò et al., 2013; Gourgouliatos et al., 2016).
Lastly, in a breakthrough discovery, the long-held view that Fast Radio Bursts
(FRBs) might be powered by magnetar activity was confirmed after the detection
of an FRB (CHIME/FRB Collaboration et al., 2020; Bochenek et al., 2020)
simultaneous with a hard X-ray burst (e.g., Mereghetti et al., 2020; Li et
al., 2021) from a Galactic magnetar in outburst.

In this paper, we present the case for the probe-class mission HEX-P as the
next generation X-ray satellite to build upon the legacy of all past and
current X-ray satellites, and open up avenues for new discoveries in the
timely field of high energy studies of young INSs. A companion paper, Ludlam
et al. (in prep.), discusses the power of HEX-P to study neutron stars in low-
and high-mass binary systems.

Figure 1: Pulsar period period-derivative diagram highlighting the range of
neutron star classes. Data are from the ATNF pulsar catalog (Manchester et
al., 2005).

##  4 MISSION DESIGN

The High-Energy X-ray Probe (HEX-P; Madsen et al. 2023) is a probe-class
mission concept that offers sensitive broad-band coverage ($0.2-80$ keV) of
the X-ray spectrum with exceptional spectral, timing and angular capabilities.
It features two high-energy telescopes (HETs) that focus hard X-rays and one
low-energy telescope (LET) that focuses lower-energy X-rays.

The LET consists of a segmented mirror assembly coated with Ir on
monocrystalline silicon that achieves a half power diameter of 3.5”, and a
low-energy DEPFET detector, of the same type as the Wide Field Imager (WFI;
Meidinger et al., 2020) onboard Athena (Nandra et al., 2013). It has 512
$\times$ 512 pixels that cover a field of view of 11.3’ $\times$ 11.3’ (1.32"
/ pixel). It has an effective passband of $0.2-25$ keV, and a full frame
readout time of 2 ms, which can be operated in a 128 and 64 channel window
mode for higher count-rates to mitigate pile-up and faster readout. Pile-up
effects remain below an acceptable limit of $\sim 1\%$ for fluxes up to $\sim
100$ mCrab in the smallest window configuration. Excising the core of the
point spread function (PSF), a common practice in X-ray astronomy, will allow
for observations of brighter sources, with a typical loss of up to $\sim 60\%$
of the total photon counts.

The HET consists of two co-aligned telescopes and detector modules. The optics
are made of Ni-electroformed full shell mirror substrates, leveraging the
heritage of XMM-Newton (Jansen et al., 2001), and coated with Pt/C and W/Si
multilayers for an effective passband of $2-80$ keV. The high-energy detectors
are of the same type as flown on NuSTAR (Harrison et al., 2013), and they
consist of 16 CZT sensors per focal plane, tiled 4 $\times$ 4, for a total of
128 $\times$ 128 pixel spanning a field of view of 13.4’ $\times$ 13.4’.

The broad X-ray passband and superior sensitivity will provide a unique
opportunity to study INSs across a wide range of energies, luminosity, and
dynamical regimes.

##  5 SIMULATIONS

All the simulations presented here were produced with a set of response files
that represent the observatory performance based on current best estimates as
of Spring 2023 (see Madsen+23). The effective area is derived from a ray-trace
of the mirror design including obscuration by all known structures. The
detector responses are based on simulations performed by the respective
hardware groups, with an optical blocking filter for the LET and a Be window
and thermal insulation for the HET. The LET background was derived from a
GEANT4 simulation (Eraerds et al., 2021) of the WFI instrument, and the HET
background was derived from a GEANT4 simulation of the NuSTAR instrument, with
both simulations adopting a Lagrange point L1 orbit for HEX-P.

We utilize Xspec version 12.13.0c to simulate point source spectra,
implementing HEX-P LET and HET response matrices, ancillary, and background
files version v07 (Arnaud, 1996). All simulated spectra are binned to have 5
counts per energy channel. For spectral fitting, we utilize the Cash-statistic
(C-stat in Xspec, Cash 1979), to derive the best fit model parameters and
corresponding uncertainties. To assess the goodness of fit, we utilize the
goodness command which simulates 1000 spectral realizations from a given model
and compares their fit statistic to that of the data; if the data is drawn
from the model, or, in other words, the model is a good fit to the data, the
fit statistic should lie around the $50\%$ mark.

##  6 Magnetars

###  6.1 Persistent broadband X-ray emission

Magnetar persistent emission consists of two components, a thermal emission
likely emanating from surface hot-spots with temperatures of $\sim$ 0.4 – 0.5
keV, and a non-thermal component with photon index $\Gamma=0.0-1.5$, i.e.,
rising in $\nu F_{\rm\nu}$, known as the hard X-ray tail. The latter likely
originates from inverse Compton scattering of soft photons by relativistic
electrons in non-potential magnetospheric loops energized by twists and
currents, tied to footpoints whose evolution is driven internally by the crust
of the magnetar. Due to the large magnetic fields, the scattering in the
magnetosphere is resonant at the electron cyclotron frequency, which is
efficient at boosting the photon energies by orders of magnitude relative to
the non-resonant case, a process known as Resonant Inverse Compton Scattering
(RICS).

Figure 2: Left panel. Magnetar $\log~{}N-\log~{}S$ distribution in the hard
(10 – 100 kev) X-ray band. NuSTAR doubled the number of magnetar detected at
$>10$ keV, which currently stands at 10. The faintest hard X-ray tail known
has a flux of the order $10^{-12}$ erg $s^{-1}$ cm-2 (SGR 0526$-$66, Park et
al., 2020). HEX-P, with its superior sensitivity, will triple the current
number of hard X-ray tails detected from magnetars (gray shaded area). Right
panel. 100 ks HEX-P simulation of the SGR 0526$-$66 broad-band spectrum and of
a generic magnetar with a hard X-ray flux in the 10-100 keV band an order of
magnitude weaker, i.e., $10^{-13}$ erg s-1 cm-2 (see main text for details).

Hard X-ray tails were first discovered in 2004 with INTEGRAL and RXTE (Kuiper
et al., 2004). Prior to the NuSTAR launch, only a handful of magnetars were
detected at energies $>10$ keV (Kuiper et al., 2006; den Hartog et al., 2008b,
a; Enoto et al., 2010). NuSTAR, with its superior sensitivity at hard X-rays,
has doubled the pool of detected sources, which is currently standing at 10
(e.g., Enoto et al., 2017). The faintest known magnetar hard X-ray tail has a
10-100 keV flux of $\sim 10^{-12}$ erg s-1 cm-2; an order of magnitude fainter
than detections by NuSTAR predecessors. HEX-P, owing to its superior
sensitivity, simultaneous broad-band coverage, and more efficient orbit, will
detect magnetar hard X-ray tails an order of magnitude fainter again, and
characterize their spectral curvature (i.e. a possible energy dependence of
the photon index $\Gamma$). This is demonstrated through a 100 ks HEX-P
simulation of a typical magnetar spectrum, i.e., $kT=0.45$ keV, $\Gamma=1.0$,
and $N_{\rm H}=10^{22}$ cm-2, with absorption-corrected fluxes in the 1–10 keV
and 10–100 keV band of $5.0\times 10^{-14}$ erg s-1 cm-2 and $10^{-13}$ erg
s-1 cm-2, respectively (Figure 2). The former flux level represents the faint
end of the soft X-ray flux level from the currently known magnetar population
while the latter is an order of magnitude fainter than the faintest magnetar
hard X-ray tail known (SGR 0526$-$66; Park et al., 2020). The hard tail is
detected up to 35 keV with a count rate of $(2.3\pm 0.2)\times 10^{-3}$ counts
s-1 (i.e., a $10\sigma$ detection significance), and the spectral curvature of
the hard tail is constrained to $30\%$ ($\Gamma=1.0\pm 0.3$). For comparison,
we also perform a 100 ks simulated observation of the broad-band X-ray
spectrum of the magnetar SGR 0526$-$66 in the Large Magellanic Cloud based on
results from Chandra and NuSTAR (Park et al., 2020). HEX-P observations will
produce a 10-80 keV count rate of $(44.3\pm 0.2)\times 10^{-3}$ counts s-1 and
a 0.5-10 keV rate of $(11.6\pm 0.1)\times 10^{-2}$ counts s-1, providing
excellent quality data for a detailed spectral and temporal analysis.

According to the current magnetar $\log{N}$-$\log{S}$ distribution, where $S$
is the 10-100 keV flux, the HEX-P sensitivity limit will enable hard X-ray
tail detection and characterization in about 30 magnetars, tripling the
current number. Consequently, this will permit a far more comprehensive
population-wide correlation analysis between the soft and hard X-ray
properties (Marsden and White, 2001; Kaspi and Boydstun, 2010; Enoto et al.,
2010, 2017; Seo et al., 2023), in turn informing on the evolution of internal
and external $B$-field and globally or locally twisted magnetospheres
(Beloborodov, 2009; Parfrey et al., 2013; Viganò et al., 2013; Chen and
Beloborodov, 2017).

Figure 3: Phase-resolved RICS spectra overlaid on INTEGRAL data for 4U 0142+61
(orange data points, Kuiper et al. 2006), along with a power-law with
exponential cutoff at 350 keV (dotted green). The model emission is computed
for surface photons of temperature $5\times 10^{6}$ K scattered by
$\gamma_{e}=10$-$10^{1.5}$ electrons uniformly populating field bundle from
magnetic footpoint colatitudes 12-45∘. The assumed dipole field strength is
$4\times 10^{14}$ G. Brown and dark blue lines show total intensities, while
red and light blue curves show the case with QED attenuation. Left panel.
Observer angle to magnetic axis $\theta_{v}=30^{\circ}$. Right panel.
$\theta_{v}=90^{\circ}$. The RICS emission is predicted to be highly phase
dependent, as shown here. Adapted from Wadiasingh et al. (2019a).

For the brightest magnetars, HEX-P will also provide the most detailed look at
the 0.3-80 keV spectro-temporal properties that are crucial to guide the
theoretical development of radiation transport in the high B-field ($>10^{14}$
G) regime, inaccessible to terrestrial laboratories, such as photon splitting.
Early predictions for the phase-resolved spectra and energy dependent profiles
in the 10–80 keV band are presented in Wadiasingh et al. (2018, also see: ),
and more sophisticated models are in development (e.g., Wadiasingh et al.,
2019b, 2022). Yet, the current data quality above 10 keV for even the
brightest magnetars (i.e., 4U 0142+61 and 1RXS J170849.0$-$400910), is
inadequate for the detailed phase-resolved spectroscopy required to confront
these models. HEX-P, providing far superior broad-band data for the brightest
magnetars, will allow us to answer fundamental questions which currently
remain open: (1) Where are the locales of particle acceleration within
magnetar magnetospheres? (2) What are the Lorentz factors of the energetic
particles? (3) How do the physical properties governing the hard X-ray tails
evolve with age and field strength? (4) Is the hard X-ray emission for
persistently emitting magnetars dominated by the dipolar field, or do higher
order, crustal fields dominate?

RICS emission is highly anisotropic and sensitive to where cyclotron resonance
in the magnetosphere is sampled by an observer. Moreover, beginning around 30
keV, photon splitting can begin to impact spectra depending in the viewing
angle or, equivalently, pulse phase (Hu et al., 2019; Wadiasingh et al., 2022;
Hu et al., 2022). As such, RICS radiation models may obtain a variety of
phase-dependent spectral energy distributions (or equivalently, energy
dependent pulse profiles) depending on viewing geometry and zones of
activation of relativistic particle populations (Figure 3). Detailed fitting
of phase-resolved spectra to models of RICS emission is sensitive to the
activated zones and observer viewing geometry, providing answers to the open
questions laid out above. Furthermore, a comparison of viewing and field
geometries will test related constraints obtained for soft thermal emission
hot spot modeling from IXPE observations of bright magnetars (Taverna et al.,
2022; Zane et al., 2023).

RICS constraints on the relativistic electron population, along with 0.2–80
keV broadband spectroscopy provided by HEX-P will also test if return currents
and particle bombardment play a significant role in heating surface layers of
magnetars. Moreover, HEX-P will provide a detailed population-level phase-
resolved spectral survey of magnetars. This will inform evolutionary traits in
the RICS parameters with age, and determine if beaming is compatible with the
lack of observed hard X-ray emission in some moderately bright X-ray
magnetars. We note that fitting phase-resolved spectra with RICS models has
not yet been attempted due to the paucity of pulsed counts at high energies
($>10$ keV). For instance, the brightest magnetar at hard X-rays, 1E
1841$-$045, has a NuSTAR count rate in the 10–79 keV band of $0.16$ counts
s-1, which, for the 350 ks existing observation (An et al., 2015), results in
56,000 phase-averaged counts, and 11,200 pulsed counts. For a modest phase-
resolved spectroscopic analysis with 10 phase bins, the 1100 counts in each
bin were able to constrain the hard X-ray photon index to about $\approx 20\%$
(An et al., 2015). In contrast, the HEX-P count rate for 1E 1841$-$045 in the
same energy range is predicted to be 0.62 counts s-1, which, for the same
considerations above, would result in $\sim$ 4350 counts per phase bin,
allowing us to search for phase variability in the hard X-ray tail down to the
$\lesssim 5$ percent level, notably aided by the LET instrument which will
constrain the soft thermal part of the spectrum. Furthermore, observing from
L1 rather than from low-Earth orbit like NuSTAR, HEX-P has nearly twice the
observing efficiency of NuSTAR.

Lastly, we note that magnetars have attracted interest from the dark matter
community as testbeds for certain axion-like particle (ALP) models. ALPs
produced in the magnetar core are predicted to convert into photons in the
magnetosphere. ALP models generally predict an opposite hard X-ray phase
dependence to RICS, thereby enabling HEX-P to provide important constraints to
the dark matter community (e.g., Maruyama et al., 2018; Fortin et al., 2021).

###  6.2 Transient behavior - crustal and magnetospheric dynamics

Figure 4: Simulated HEX-P broadband X-ray spectra of two transient magnetars
near the peak of their outbursts: XTE J1810$-$197 (left) and SGR J1745$-$2900
(right) based on data from their 2018 and 2013 outbursts, respectively. LET
and HET data are shown, with the LET data distinguished by its higher flux at
low energies. Note that SGR J1745$-$2900 is highly absorbed due to its
location in the Galactic center region.  Figure 5: Simulated HEX-P
measurements of the photon index $\Gamma$ following the 2013 outburst of SGR
1745$-$2900 based on NuSTAR and Chandra observations. The upper panel of the
left figure shows the evolution of the blackbody temperature, where the
evolution of the 0.3$-$10.0 keV X-ray luminosity is shown in the lower panel
of the left figure. The right figure shows the evolution of $\Gamma$ obtained
with NuSTAR and HEX-P. HEX-P’s improved sensitivity will enable monitoring of
these outbursts significantly longer than NuSTAR.

In addition to the persistent magnetars, with X-ray luminosities reaching
$\sim 10^{35}$ erg s-1, there are also ‘transient’ magnetars, which only reach
$\sim 10^{35}$ erg s-1 during outbursts, when their X-ray fluxes increase by
over two orders of magnitude (Coti Zelati et al., 2018). Transient magnetar
outbursts decay on exponential timescales ranging from months to years (Rea et
al., 2009; Kaspi et al., 2014; Coti Zelati et al., 2018). The outburst spectra
feature both surface thermal emission and non-thermal magnetospheric emission.

During the decay phase, thermal hot spots are observed to cool and shrink in
size. This behaviour is consistent with currents circulating through a twisted
magnetosphere, depositing heat at the surface footpoints of magnetic current
loops Beloborodov (2009). Alternatively, this behavior might be due to
internal heating of the crust through magnetic stresses associated with
evolving toroidal fields (Lander et al., 2015; Lander and Gourgouliatos, 2019;
Lander, 2023). While the decay of the thermal emission has been well
observationally constrained, the hard, non-thermal X-ray emission fades beyond
current detection limits fairly quickly (over the course of weeks). It is
therefore unknown whether the non-thermal emission decays in tandem with the
soft X-ray emission, as expected in the case of surface bombardment by
accelerated particles in the magnetosphere, or whether the two evolve
independently. This might be the case if the surface heating is indeed induced
internally (e.g., Kouveliotou et al., 2004; Pons and Rea, 2012; Deibel et al.,
2017), independent of the external magnetospheric emission. This is a major
open question in transient magnetars given its potential to investigate
crustal micro- and macro-physical properties, which are poorly known, and are
highly relevant to the NS equation of state. HEX-P’s sensitivity and broadband
X-ray coverage are uniquely capable of addressing this fundamental open
question.

The left panel of Figure 4 shows simulated HEX-P and NuSTAR spectra based on
the X-ray spectrum of the transient magnetar XTE J1810-197 during its late
2018 outburst (Gotthelf et al., 2019). This highlights the essential role that
the LET plays in constraining the soft $<2$ keV spectrum, especially in the
case of less absorbed sources such as for XTE J1810-197.

The right panel of Figure 4 shows HEX-P simulations of the hard X-ray spectrum
of decaying flux from SGR J1745$-$2900 following an outburst, based on NuSTAR
and Chandra observations of the 2013 outburst (Kaspi et al., 2014). The soft
thermal emission follows the multi-year evolution observed by Chandra (Rea et
al., 2020). The luminosity of the non-thermal emission is set equal to what
was observed by NuSTAR at the beginning of the outburst, and is assumed to
decay with the thermal luminosity. While NuSTAR tracked the SGR J1745$-$2900
non-thermal emission for only four months post-outburst, HEX-P will be capable
of tracking a similar magnetar outburst for two years (Figure 5). This will
provide data critical to our understanding of NS crustal and magnetospheric
dynamics, unique observational characteristics of magnetars following an
outburst.

###  6.3 Fast Radio Bursts

Fast Radio Bursts (FRBs) are millisecond, bright radio bursts (fluence $\sim$
few Jy ms) observed over a broad range of frequencies, from $\sim$120 MHz to a
few GHz. They were first reported in 2007 (Lorimer et al., 2007), and since
then several hundred have been detected by a suite of radio dishes across the
Earth, e.g., Parkes, Arecibo, ASKAP, FAST, and CHIME (Petroff et al., 2022).
FRBs are distributed nearly isotropically across the sky and show very large
dispersion measures, indicating an extragalactic origin. Hence, their large
fluences translate into very bright luminosities, $\sim$9 orders of magnitude
brighter than the Crab’s giant pulses. While most FRBs appear as single
events, a few have associated with a single position on the sky, i.e.,
repeating FRBs (Fonseca et al., 2020). The origin of FRBs is currently a
matter of intense debate, and while many theoretical possibilities exist
(Platts et al., 2019), one of the leading models is a NS or magnetar central
engine.

In a breakthrough discovery, observational evidence for the magnetar model as
a source of FRBs occurred on 2020 April 28, when an FRB-like radio burst was
detected from the Galactic magnetar SGR 1935+2154 (CHIME/FRB Collaboration et
al., 2020; Bochenek et al., 2020), in the winding hours of a major burst storm
(Younes et al., 2020); it had a fluence rivaling those of the faint end of
extragalactic FRBs. Moreover, the FRB occurred simultaneously with a bright,
short X-ray burst, connecting it to magnetar activity and providing crucial
evidence for its triggering mechanism (e.g., Mereghetti et al., 2020; Li et
al., 2021; Ridnaia et al., 2021). In the following years, SGR 1935+2154 has
shown several more radio bursts (Kirsten et al., 2020), most notably at times
of bursting activity. Though most radio bursts occur simultaneous to X-ray
bursts, the majority of X-ray bursts occur without a simultaneous radio signal
(Lin et al., 2020b; Bailes et al., 2021) suggesting special circumstances for
the emission of FRB-like bursts from magnetars. Indeed, a comparison of the
X-ray burst associated with the FRB and NICER+Fermi bursts belonging to the
same burst storm of April 2020 reveal the former to have a distinctive
spectrum. This is a clue to either its emission mechanism or triggering locale
(Younes et al., 2021), and has been seen in data from INTEGRAL (Mereghetti et
al., 2020), Fermi/GBM (Lin et al., 2020a), Konus (Ridnaia et al., 2021), and
HXMT (Li et al., 2021), among others.

Figure 6: Left panel. $\log N-\log E$ distribution of magnetar short bursts,
where $E$ is the burst total energy. The light and dark shaded areas are the
sensitivity limits ($>5\sigma$ detection) of Swift/BAT and Fermi/GBM,
respectively, assuming a source distance of 20 kpc. HEX-P is sensitive to the
full burst energy distribution for every magnetar in the Galaxy. Right panel.
Panel (a) shows a HEX-P Xspec simulation of the FRB200428-associated X-ray
burst spectrum (gray diamonds) assuming a non-thermal cutoff power-law as
measured with HXMT (Li et al., 2021). The solid line is the best-fit thermal
double blackbody (2BB) model. Black squares are a simulated spectrum with the
same assumptions, but with a fluence that is a factor 15 smaller. Panel (b)
shows the residuals from the thermal fit in units of 1$\sigma$, with the same
grey and black symbols. Panel (c) shows the residuals from the non-thermal fit
in units of 1$\sigma$, again with the same grey and black symbols. HEX-P will
distinguish the thermal vs non-thermal nature of short X-ray bursts for
fluences that are over an order of magnitude weaker than FRB-X (see text).

The discovery of FRB-like bursts from magnetars opened up new avenues for the
study of extragalactic FRBs (e.g., Wadiasingh and Timokhin, 2019), radio
emission from young INSs (Philippov and Kramer, 2022), and, more generally,
plasma and pair production in magnetar magnetospheres (e.g. Mahlmann et al.,
2022; Beloborodov, 2020; Yuan et al., 2020). Yet, as is the case with every
new fundamental discovery, more questions arise in its aftermath, e.g., (1)
What is unique about the FRB-associated X-ray bursts, and why do the majority
of X-ray bursts lack a radio counterpart? (2) What is the distribution of the
spectral properties for FRB-associated X-ray bursts? Are their distinctive
spectral properties universal across radio fluence? (3) Is the radio to X-ray
flux ratio ($L_{\rm R}/L_{\rm X}$) constant for all FRB-like radio bursts? (4)
What is the radio-X-ray time-lag across burst fluence?

The answers to these open questions are critical for improving our
understanding of the FRB phenomenon, both Galactic and extragalactic.
Answering these questions will require (1) a broad X-ray coverage given that
the spectral energy distributions of magnetar bursts peak in the 20-30 keV
range, (2) high timing resolution ($\lesssim 1$ ms) for accurately measuring
the radio-X-ray lag in burst arrival time (Mereghetti et al., 2020), (3)
sensitivity to faint X-ray bursts, i.e., fluence $<10^{-7}$ erg cm-2, to
sample a large fraction of the X-ray and radio burst fluence distribution
given their steep shapes ($N\propto S^{-0.6}$, Figure 6, Younes et al. 2020).
HEX-P is the only facility to satisfy all the above criteria. Its only
limitation is the small field-of-view, yet, most radio FRB bursts detected
from the Galactic magnetar SGR 1935+2154 occurred at the time of major burst
storms, which last up to a few days. This is sufficient time for HEX-P to slew
to the target. We also note that NuSTAR was observing SGR 1935+2154 at the
time of its 2022 October FRB-like burst, though the source was Earth-occulted
(Dong and Chime/Frb Collaboration, 2022; Enoto et al., 2022). Due to the L1
orbit of HEX-P, such misfortune is naturally avoided.

The left panel of Figure 6 shows a simulated $\log N-\log S$ magnetar burst
energy distribution, which follows a power-law of the form $dN/dE\propto
E^{-1.6}$ (GöǧüŞ et al., 1999; Göǧüş et al., 2000; Gavriil et al., 2004; van
der Horst et al., 2012; Younes et al., 2020). The Fermi/GBM $>5\sigma$
sensitivity to typical short bursts from a magnetar at a distance of 20 kpc is
shown in dark gray (Meegan et al., 2009) while that for Swift/BAT is shown in
light gray (Lien et al., 2016). At that distance, HEX-P will detect bursts
with energies comparable to the persistent emission, i.e., $\sim 10^{37}$ erg
(in a 1-second interval), covering significantly more of the short burst
energy distribution. This will ensure the detection of X-ray bursts associated
with faint radio bursts and provide answers to questions (3) and (4).

To address questions (1) and (2), we performed HEX-P HET simulations of an
X-ray burst with spectral properties similar to that of the FRB-associated
X-ray burst (which we call FRB-X) as determined by HXMT (Li et al., 2021);
i.e., a cutoff power-law with $\Gamma=1.6$ and $E_{\rm cut}=80$ keV, and a
fluence of $5.0\times 10^{-7}$ erg cm-2 (Figure 6, gray diamonds in panel
(a)). We then fit this spectrum with a thermal 2 blackbody (2BB) model (shown
as a solid black line). The thermal model fails to provide a statistically
acceptable fit to the data (panel (b)), unlike the non-thermal model (panel
(c)). This is confirmed through Xspec simulations which show that $\gg 99\%$
of simulated spectra drawn from the thermal model have better fit statistics.
This indicates that HET alone will discern the non-thermal nature of any
bursts similar to FRB-X. We then performed a set of simulations assuming the
same spectral model as FRB-X, while decreasing the fluence by increments of
factor 2. We then fit each spectrum to a thermal 2BB model, and assess the fit
quality through simulations. We find that we can discern (at the $\approx
3\sigma$ level) between the thermal and non-thermal model down to a fluence of
$\sim 3\times 10^{-8}$ erg cm-2, i.e., a factor 15 fainter than FRB-X (Figure
6, black squares).

##  7 Central Compact Objects

The CCO class of NSs are defined by the following observational
characteristics: steady, soft thermal X-ray emission, lack of a surrounding
pulsar wind nebula, and non-detection at all other wavelengths. X-ray pulses
have been detected from only three of the dozen known CCOs (Gotthelf et al.,
2013). Two of these three, the CCOs in the Puppis A and Kes 79 SNRs, have the
lowest spin down measured magnetic fields among young neutron stars, with
$B_{s}\sim 3\times 10^{10}$ G.

While only a dozen CCOs are currently known, their locations in young SNRs
indicates they may represent a significant fraction of all NS births. Hence,
understanding how young NSs are born with such small dipole magnetic fields is
important to address how CCOs fit within the broad INS family. Increasing the
number of CCO spin period and period derivative measurements is critical.
Because CCOs are only detected at X-ray wavelengths, these spin period
searches can only be done in the X-ray band. This is an area where HEX-P can
shine as a follow-up observatory, capable of both searching for X-ray
pulsations and, after identifying a spin period, making the required phase
resolved spectroscopic measurements.

Figure 7: HEX-P pulsed fraction upper limits as a function of exposure for the
CCO in G330.2+1.0.

As an example, the CCO in G330.2$+$1.0 is a promising target for HEX-P thermal
pulse searches. Previous searches with XMM-Newton were limited by the high
background from the surrounding SNR thermal emission (Alford and Halpern,
2023). This is highlighted in Equation 1 which relates the calculated pulsed
fraction upper limit $f_{p}^{\rm max}$ to the number of source counts $N_{s}$
and background counts $N_{b}$:

$f_{p}^{\rm max}=2(1+N_{b}/N_{s})\sqrt{P_{s}/N},$ (1)

Since CCOs are found in young supernova remnants, in many cases the thermal
background emission from the remnant can be significant, hindering the ability
to detect the underlying NS pulsations. To overcome these obstacles, both high
X-ray timing and angular resolution are required. HEX-P’s 3.5 arcsec PSF (for
the LET detector) will allow for a significantly reduced background. Detailed
comparisons of the performance of HEX-P in pulsar searches compared to XMM-
Newton and NuSTAR can be found in Bachetti et al., in prep. (ULXs and
extragalactic pulsars) and Mori et al., in prep. (the Galactic Center). Figure
7 presents the pulsed fraction upper limits as a function of exposure time for
the CCO in G330.2$+$1.0. HEX-P will significantly reduce the current pulsed
fraction upper limit, likely leading to a secure determination of the spin
period.

Once an X-ray pulse period is found, a measurement of the period derivative
can easily follow, as well as the characterization of the timing properties of
these sources, e.g., $\dot{E}$, $B$-field strength, and spin down luminosity
$\dot{E}$. Moreover, such observations will allow us to study the thermal
pulse profiles of new CCOs. Energy-dependent pulse profile modeling is a
powerful tool to map the surface thermal emission (Bogdanov, 2014; Alford et
al., 2022). The HEX-P LET has the required effective area, timing resolution,
low background, and angular resolution to produce more detailed maps, while
increasing the pool of studied sources.

##  8 Rotation-Powered Pulsars (RPPs)

In contrast to the magnetically-powered magnetars and passively cooling CCOs,
many pulsars are powered by the loss of their rotational kinetic energy. The
33 ms Crab pulsar is perhaps the best known pulsar in this class, with
significant rotationally powered emission extending into the hard X-ray band
(Madsen et al., 2015). HEX-P observations of the Crab pulsar will be a
significant improvement over NuSTAR. For instance, HEX-P will enable low
background phase-resolved studies of the Crab pulsar, by resolving the Crab
pulsar from the bright pulsar wind nebula (PWN) background emission.

HEX-P will allow us to observe in detail the faint, middle-aged ($\sim 10^{5}$
yr) pulsars which offer an opportunity to study how pulsars evolve and
eventually “die”, ceasing as X-ray and radio emitting sources. As RPPs age,
their X-ray luminosity will decrease with their spin down power, making
observations more challenging compared to younger RPPs.

There are three well-known nearby middle-aged pulsars that have been dubbed
the “three musketeers”: PSR B0656$+$14, PSR B1055$-$52 and Geminga. These
three pulsars all exhibit thermal surface emission and non-thermal
magnetospheric emission. They have similar $\sim 10^{12}$ G spin down magnetic
fields and $10^{34}$ erg s-1 spin down luminosities (De Luca et al., 2005).
Despite their relative proximity, open questions remain regarding the physics
of their surface thermal emission, and the extent of their non-thermal
emission.

Figure 8: Simulated HEX-P observations of the Geminga pulsar, PSR B0656+14,
and PSR B1055-52. HEX-P will be capable of extending the current hard X-ray
detection limit for Geminga from 20 keV to 50 keV.

Open questions about Geminga are particularly interesting given its
potentially large contribution to the local leptonic cosmic ray flux, and its
status as the second brightest gamma-ray source in the sky. Mori et al. (2014)
reported on a 150 ks NuSTAR observation of Geminga and found several spectral
models were consistent with the data. Figure 8 shows a simulated 200 ks HEX-P
observation of Geminga based on the two blackbody plus powerlaw model, with
$\Gamma=1.7$, $kT_{1}=44$ eV, and $kT_{2}=195$ eV. The cooler thermal
component corresponds to emission from the whole NS surface, and the hot
component, if its existence is confirmed, may correspond to emission from a
hot polar cap. We find that, if the powerlaw extends to higher energies, HEX-P
will extend the detection of non-thermal emission from $\sim$20 keV to
$\sim$50 keV. The X-ray emission of Geminga above 20 keV is unexplored
territory, and the detection of changes in the spectrum could be important
clues to the physics of its magnetosphere.

Figure 8 also shows simulated HEX-P observations of PSR B0656$+$14 and PSR
B1055$-$52, performed using the spectral parameters from De Luca et al.
(2005). We find that 150 ks HEX-P observations will measure the photon indices
$\Gamma$ of all three of these faint pulsars to better than $10\%$.

HEX-P will also potentially address a fundamental mystery regarding PSR
B0656$+$14, PSR B1055$-$52 and Geminga. B0656$+$14, and PSR B1055$-$52 clearly
have small surface thermal hot spots, presumably corresponding to the heated
pulsar polar caps. If Geminga has a similar hot spot, then its luminosity is
at least two orders of magnitude dimmer (Jackson and Halpern, 2005). HEX-P’s
high throughput and broad coverage of Geminga’s X-ray spectrum will allow us
to answer this question.

##  9 Potential discovery space

Present and future large field-of-view facilities at all wavelengths from
radio to PeV energies will result in a large number of unidentified sources,
especially within the Galactic plane. This is already evident at GeV energies
and beyond, i.e., Fermi/LAT, H.E.S.S, and LHASSO, where the number of unknown
sources outweigh the number of identified ones, a problem that will only be
exacerbated with the Cherenkov Telescope Array Observatory (CTAO). These high
energy sources represent the most efficient particle accelerators in the
universe, and for the Galactic ones, their most likely counterpart involves a
pulsar (Mori et al. 2023 (in prep.)). At X-ray energies, eROSITA (Predehl et
al. 2021, and potentially STAR-X Zhang et al. 2022) will provide some of the
deepest wide-field X-ray surveys of our Galaxy, with a $>10\times$ increase in
the number of X-ray sources compared to ROSAT. A large fraction of these X-ray
sources will be of unknown origin, and a non-negligible fraction should be
INSs (Pires et al., 2017). Deeper, targeted exposures, as possible with HEX-P,
will be required to identify them. Finally, in the radio, the Square Kilometer
Array and Deep Synoptic Array 2000 are expected to increase the number of
currently known pulsars by a factor of 10 (i.e., to $\sim$20,000 pulsars), as
well as detect a large number of new SNR shells and candidate wind nebulae.

Simply detecting X-ray point sources in the error region of unidentified
gamma-ray and radio sources will not yield a secure identification.
Furthermore, providing a high quality soft X-ray spectrum of eROSITA/STAR-X
sources will not be enough to firmly distinguish their origin. High timing
resolution is required to discern the pulsar nature from other types of X-ray
emitters, such as low-mass X-ray binaries, cataclysmic variables, background
active galaxies, etc. Moreover, being limited to a soft X-ray detector will
hamper our ability to probe deep into the Galaxy due to absorption, stressing
the need for hard X-ray coverage. Last but not least, high spatial resolution
is required for crowded regions such as Sgr A⋆ (Mori et al. 2023b (in prep.)),
and to isolate the pulsar emission from any surrounding SNR and/or nebula (as
noted in §5 with regards to candidate CCOs). These issues have already been
demonstrated with NuSTAR. For instance, although a candidate soft X-ray
counterpart to the TeV $\gamma$-ray source HESS J1640-465 had been proposed,
only NuSTAR was able to detect pulsed emission confirming its pulsar nature
(Gotthelf et al., 2019). This was mainly due to the combination of heavy
absorption ($10^{23}$ cm-2) in the source direction and non-negligible
contamination by the PWN (Gotthelf et al., 2019). Continued NuSTAR follow-up
of this pulsar found the braking index $n>3$, possibly pointing to a magnetic
quadrupole in the source (Archibald et al., 2016a).

HEX-P presents the ideal satellite to follow-up Galactic gamma-ray sources,
INS candidates from wide-area X-ray surveys, and pulsar/pulsar candidates from
deep radio surveys. Increasing the number of isolated X-ray pulsars, such as
XDINs, CCOs, RPPs, and magnetars, will enable a deeper understanding of their
physics, X-ray properties, environment (e.g., wind nebulae, SNRs and nearby
cosmic ray acceleration sites), and progenitors, in turn teaching us about NS
formation and evolutionary tracks through, e.g., population synthesis modeling
(e.g., Gullón et al., 2015; Dirson et al., 2022) and magneto-thermal evolution
models (e.g., Viganò et al., 2013; Gourgouliatos et al., 2016).

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

This paper is the result of the work of the HEX- P pulsar/magnetar working
group, led by JA and GY. JA and GY designed and performed simulated HEX-P
observations. JA, GY, and ZW wrote the manuscript. All authors have provided
useful advice during working group discussions and comments on the manuscript.

## Acknowledgments

The work of D. S. was carried out at the Jet Propulsion Laboratory, California
Institute of Technology, under a contract with NASA. Z. W. acknowledges
support by NASA under award number 80GSFC21M0002. This work has made use of
the NASA Astrophysics Data System.

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# Priority prediction of Asian Hornet sighting report using machine learning
methods

Yixin Liu  School of Software Engineering
South China University of Technology
Guangzhou 510006, PR China  Jiaxin Guo  School of Computer Science and
Engineering
South China University of Technology
Guangzhou 510006, PR China  Jieyang Dong  School of Mathematics
South China University of Technology
Guangzhou 510006, PR China  Luoqian Jiang  School of Software Engineering
South China University of Technology
Guangzhou 510006, PR China  Haoyuan Ouyang  Yixin Liu, Luoqian Jiang and
Haoyuan Ouyang are with the School of Software Engineering, South China
University of Technology. Jiaxin Guo is with the School of Computer Science
and Engineering, South China University of Technology. Jieyang Dong is with
the School of Mathematics, South China University of Technology. E-mail:
{seyixinliu<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>School of Software Engineering
South China University of Technology
Guangzhou 510006, PR China

###### Abstract

As infamous invaders to the North American ecosystem, the Asian giant hornet
(Vespa mandarinia) is devastating not only to native bee colonies, but also to
local apiculture. One of the most effective way to combat the harmful species
is to locate and destroy their nests. By mobilizing the public to actively
report possible sightings of the Asian giant hornet , the government could
timely send inspectors to confirm and possibly destroy the nests. However,
such confirmation requires lab expertise, where manually checking the reports
one by one is extremely consuming of human resources. Further given the
limited knowledge of the public about the Asian giant hornet and the
randomness of report submission, only few of the numerous reports proved
positive, i.e. existing nests. How to classify or prioritize the reports
efficiently and automatically, so as to determine the dispatch of personnel,
is of great significance to the control of the Asian giant hornet.  In this
paper, we propose a method to predict the priority of sighting reports based
on machine learning.  We model the problem of optimal prioritization of
sighting reports as a problem of classification and prediction. We extracted a
variety of rich features in the report: location, time, image(s), and textual
description. Based on these characteristics, we propose a classification model
based on logistic regression to predict the credibility of a certain report.
Furthermore, our model quantifies the impact between reports to get the
priority ranking of the reports.  Extensive experiments on the public dataset
from the WSDA (the Washington State Department of Agriculture) have proved the
effectiveness of our method.

###### Index Terms:

Sighting Report, Priority Prediction, Machine Learning

##  I Introduction

Bees are the medium for pollination of plants, and they have made great
contributions to human society and ecosystem. Effective pollination by bees
can increase the quantity and quality of crops, plant diversity and plant
resistance to pests [1]. These crops account for one third of the world’s food
production, which is an immense contribution. In addition, the products
produced by bees can also be further processed into daily necessities for
cleaning, beauty and other purposes. Therefore, as contributors to the
ecosystem and human society, bees are worthy of protection.

However, the bee ecosystem has been severely threatened in recent years[2].
The Asian giant hornet is a kind of bumblebee with bright yellow feet. As an
alien species, they could kill an average of 40 other kinds of bees in their
lifetime, which has caused great damage to the local bee ecological
environment. At the same time, the hornet is also very aggressive when it
comes into contact with humans, with venom deadly enough to kill. Therefore,
the prevention and control of the Asian giant hornet is of great significance
to the protection of local apiculture and public safety.

Locating the nests of the hornets and destroying them is one of the most
effective ways to control the Asian giant hornet [3]. Existing methods for
identifying their nests are mainly based on remote tracking [4] with image
recognition [3]. Although these methods are able to locate nests
automatically, they require the equipment of drones and face the problem of
great search cost. In actual situations, the government often mobilizes the
public to submit sighting reports actively, and then sends inspectors to
identify and destroy nests in a targeted manner [5]. This method can greatly
reduce the cost of searching by leveraging the power of citizens.

However, given that citizens often lack related knowledge of the Asian giant
hornet and the randomness of citizens’ report submission, the report
collection is actually full of “noise”, and only a small part of the reports
inspected are finally confirmed as positive. At the same time, the current
traditional methods require manual credibility assessment of the reports,
which is time-consuming and inefficient [6]. To solve these problems, under
the limitation of human and financial resources, the automatic assessment and
prioritization of the reports is an important way to improve the efficiency of
government resource utilization and the prevention and control of pests.

The field of credibility evaluation has used multi-factor [7], natural
language processing (NLP) and other technologies to evaluate the confidence of
the report. There are currently multiple application scenarios, e.g. bug
report [8] and vehicle data [9] . However, there is little research and
application of these technologies in the field of sighting report credibility
evaluation.

In this paper, we combine the related technologies of report credibility
assessment, and propose a method based on machine learning (ML) to implement
the automatic report prioritization. Different from existing methods of
identifying hornet nests, we locate them from a new perspective based on the
credibility prediction of public sighting reports. We abstracted this problem
into a two-classification problem, and extracted a variety of factors in a
report to construct rich features. Finally, we built a classification model
based on logistic regression to implement automated prediction.

Our contributions are summarized as follows:

  * •

We have proposed an automated prediction method for the confidence level of
sighting reports of hornets, which provides a new idea for the identification
and precise control of pests.

  * •

We abstract the modeling of the prioritization of sightings reports as a two-
classification problem. By utilizing a variety of data from the sighting
report, abundant features have been extracted about the credibility of textual
notes, sighting location, and attached image(s).

  * •

We build a machine learning framework based on logistic regression algorithms,
which can take into account multiple factors of the report and predict the
report credibility. At the same time, for the extreme imbalance problem in the
sample, we propose a weighted loss function. On the public data set from the
Washington Department of Agriculture, a large number of experiments have
proved the effectiveness of our method.

##  II Methodology

In this section, we first formalized the problem of priority prediction of
sighting reports into a two-category problem. Next, we used multi-factor
analysis to construct rich features for the various types of data in the
sighting report: location factors, time factors, image factors, and text
factors. Finally, we built a machine learning model based on logistic
regression to predict the certainty of the report, and further determined the
priority of the report based on the confirmation relationship among the
reports. We show the overall architecture of the model in Fig. 1.

Figure 1: Given a sighting report, we first extract four features: locational,
temporal, graphical, and textual. After constructing the feature vector of the
report, we use a logistic regression model to predict and classify the
credibility of the report. Finally, we quantitatively consider the interaction
between the reports and determine the priority of the reports.

###  II-A Problem Definition

To maximize the use of limited government resources, it is essential to
accurately predict the credibility of an sighting report so as to prioritize
the report, and thus efficiently complete the elimination of the Asian Hornet
nest. For a sighting report $x^{\left(i\right)}\sim u\left(\cdot\right)$,
where $u\left(\cdot\right)$ denotes a certain distribution that the report
data is sampled from. Due to the possibility of mistaken, there will be two
different categories. Let the report category label be $y\in{\\{0,1\\}}$,
where $y=1$ denotes that the report confirms the existence of a nest, and
$y=0$ means that the report is a false positive. We denotes the probability
that report $x^{\left(i\right)}$ belongs to the true positive as
$p^{(i)}\in[0,1]$. Given multiple sighting reports, we can be prioritized
according to their confidence probability $p^{\left(i\right)}$. For credible
reports, we seek to get a higher probability $p^{(i)}$ and vice versa.
Formally, we need to solve the following classification optimization problem,
that is, to minimize the Cross Entropy in Equation 1:

$\max\mathbb{E}_{x^{\left(i\right)}\sim
u\left(\cdot\right)}\left[y^{\left(i\right)}p^{\left(i\right)}+\left(1-y^{\left(i\right)}\right)\left(1-p^{\left(i\right)}\right)\right]$
(1)

To solve the above optimization problem, we seek to fit a function
$p^{(i)}=f(\cdot|x^{(i)};\theta)$ to predict the confidence probability of the
report, so as to further prioritize reports. In the following section, we
first construct features from multiple dimensions of the report, and then
build a machine learning predictor based on these features to learn an optimal
function.

###  II-B Features Extraction

Location Feature.  For the location factor, we get the probability feature of
reporting the hive found at a certain position by constructing the prior
distribution probability feature of the Asian Hornet in space based on the
mechanism of bee migration. We first briefly explain the habits of the Asian
Hornet. The hornet will only be seen by the public from April to December of a
year since its will hibernate in winter. During migration season, the average
distance between the migration destination and the original nest is about
30km. After arriving at the destination, the bees move around the nest with a
maximum radius of 8km. Based on these habits, given the probability
distribution of the initial hive in space, we aim to construct the observation
probabilities of bees everywhere in the space.

Given the initial distribution $P_{x,y}^{(0)}$ of the nest in space, we assume
that the distance of the new location of nest from the old one follows a
Gaussian distribution with an average value of 30 km. We quantify the relation
between probability values before and after the migration as follow:

$P_{\text{nest}}^{\left(t+1\right)}\left(\lambda,\phi\right)=\sum_{\left(\lambda^{\prime},\phi^{\prime}\right)}{\frac{1}{\sigma\sqrt{2\pi}}e^{\frac{-\left(d-30\right)^{2}}{2\sigma^{2}}}P_{\text{nest}}^{\left(t\right)}\left(\lambda^{\prime},\phi^{\prime}\right)}$
(2)

where $\left(\lambda,\phi\right)$ and
$\left(\lambda^{\prime},\phi^{\prime}\right)$ represent the latitude and
longitude of two different coordinates in space,
$P_{\mathrm{nest}}^{\left(t\right)}\left(\lambda,\phi\right)$ denotes the
probability of a nest existing at position $(\lambda,\phi)$ in month $t$,
$\sigma$ is a shape parameter for Gaussian distribution, which measures the
difference of distances of the newly selected location from the old one, and
thus reflects the migration precision. The larger the $\sigma$, the wider the
range over which the queen builds her new nest.

In non-migratory months, we can spot hornets away from near their nest, as its
possible location is definite. The further away from the nest, the lower the
probability of spotting hornets, and vice versa. Formally, according to the
nest distribution as defined by Equation (2), we can obtain the probability
distribution of spotting hornets in the whole space as:

$P_{\text{observe}}^{\left(t\right)}\left(\lambda^{\prime},\phi^{\prime}\right)=\sum_{\left(\lambda,\phi\right)}{e^{-\beta_{1}d}P_{\text{nest}}^{\left(t\right)}\left(\lambda,\phi\right)}$
(3)

where
$P_{\mathrm{observe}}^{\left(t\right)}\left(\lambda^{\prime},\phi^{\prime}\right)$
denotes the probability of finding hornets at position
$\left(\lambda^{\prime},\phi^{\prime}\right)$ in month $t$ and $\beta_{1}$ is
an impact factor that determines the strength of the influence of distance on
the probability of spotting a hornet.

Time Feature.  For the time factor, we considered the migration habit of the
hornet. Since they are more prevalent amid April-December, the public are more
likely to spot them during this time, and the report is more likely to be
positive. In December-March, the only surviving individual queens hibernate
underground, so it is unlikely to spot a hornet above the ground. The period
from April-December each year is hence defined as the active period, denoted
by $\mathbb{T}$. The temporal feature of each report was characterized as
$s_{\text{T}}$ by the following rules:

$s_{\mathrm{T}}=\begin{cases}c_{T},&T\in\mathbb{T}\\\
0,&\text{otherwise}\end{cases}$ (4)

where $c_{T}$ is the number of positive reports in month $T$ over the
historical data.

Image Feature.  For the image factor, we considered the number of images
attached to a sighting report for the reason that report with graphical proof
improves the credibility of the report. In the process of manual verification,
the existence of image information is also an important factor to consider for
deciding the sending of personnel. We therefore define $s_{\text{I}}$ to
represent the integrity of image information:

$s_{\mathrm{I}}=\begin{cases}n_{\text{I}},&\text{image(s) provided}\\\
0,&\text{otherwise}\end{cases}$ (5)

where $n_{\text{I}}$ denotes the number of image(s) attached to the report.

Text Feature.  For the textual factor, we considered the contribution of the
length of the text and the keywords contained in the text to the credibility
of the report. As for the length of the text, we believe that the length of
the text is proportional to its credibility because it contains more evidence.
In addition, we extracted two types of text keywords. If the note involves
more features closer to V. mandarinia than other species, the report is less
likely to be mistaken. We constructed a hornet feature dictionary (Table I)
from a glossary extracted from attached data, related Wikipedia webpages, and
literatures. For each report, we are then able to calculate the word
frequencies of all the words it contains with respect to this dictionary and
obtain its word frequency feature.

TABLE I: The dictionary of key characteristics of V. mandarinia

| Asian giant hornet | Other confusing hornets
---|---|---
Nest Location |  | ”Underground”,”forests”,
---
”burrows”,”roots”,”trunks”, $\cdots$
| ”Time limbs”, ”house eaves”,
---
”exposed”, ”lawns”, $\cdots$
Body Appearance |  | ”Yellow heads”, ”black thorax”,
---
”striped abdomens”, ”giant”, $\cdots$
| ”Small”,
---
”black and white colored”, $\cdots$

Combining these two points, we define $s_{\text{W}}$ as the text feature to
evaluate a report:

$s_{\mathrm{W}}=\frac{1}{n_{\mathrm{W}}}\sum_{i=1}^{n_{\mathrm{W}}}{\left(q_{i}-k_{\mathrm{i}}\right)}+\beta_{2}\log\left(n_{\mathrm{W}}+1\right)$
(6)

###  II-C Priority Prediction

Classification model.  Logistic regression is a generalized linear model,
which conducts mapping from any real number to a probability values. For a
report $x^{\left(i\right)}$ which is characterized by the feature vector
$\mathbf{s}$, the logistic function can be used to construct its probability
$p$.

$p\left(x^{\left(i\right)}\right)=\frac{1}{1+e^{-\theta^{T}\phi\left(x^{(i)}\right)}}$
(7)

where $\phi\left(\cdot\right)$ is the feature mapping for reports (yielding
$\mathbf{s}^{(i)}$), and $\theta$ is the parameter of the model to learn.  To
learn the model, we seek to minimize a loss function called _binary cross-
entropy function_ denoted as $H(\cdot)$. However, there are only few positive
report among a large number of reports, which means that dataset is usually
extremely unbalanced. To tackle this problem, we use the weighted binary
cross-entropy function, where, for the correctly classified fraction, viz.
when $y=0$, if our model predicts it as misclassification, we scale up its
loss by a degree of $\tau$ to ensure that our model would not predict all
samples as misclassifications:

$H\left(p,y\right)=\tau
y\log\left(p\right)+\left(1-y\right)\log\left(1-p\right)$ (8)

where $\tau$ is the weighting coefficient of the sample status, set as the
reciprocal of the percentage of non-misclassified reports. On this basis, we
can obtain the overall loss function $J(\cdot)$ of the given dataset
$\mathbf{x}$:

$\min
J\left(\theta\right)=\frac{1}{n}\sum_{i=1}^{n}{H\left(h_{\mathrm{\theta}}\left(x^{\left(i\right)}\right),y^{\left(i\right)}\right)}+\frac{\beta_{3}}{2}\left\|\theta\right\|_{2}^{2}$
(9)

where $h_{\theta}(\cdot)$ is the logistic regression function with parameter
$\theta$, and $\beta_{3}$ is the balancing coefficient for regular terms.

We update the parameter of model using stochastic gradient descent method. Let
$\eta$ be the step of parameter update, we train our model to minimize the
loss $J(\theta)$.

$\theta^{\prime}\leftarrow\theta-\eta\frac{\partial
J\left(\theta\right)}{\partial\theta}.$ (10)

Priority determination.  In the previous section, we predict the credibility
of report based on four dimensions of the report content. However, to predict
the priority of report, we have to consider their connection. Intuitively, if
there are many reports submitted in a certain area at the same time, it highly
indicates that this area is likely to have nest hidden. Therefore, based on
the forecast confidence of a single report, we seek to further consider the
mutual confirmation relationship between the reports, so as to achieve more
accurate report priority ranking.

Specifically, the predicted possibility $p$ in Equation 7 only considers one
report and does not consider its inter-corroboration with reports that are
similar to it in space and time. If the reports are in the same migration
cycle (which means that the reports are between April of a certain year and
March of the next year), within which the hornets do not migrate, we are
interested in whether the reports lead to the same nest; if different, we
consider whether the nests that the reports lead to are related. For this, we
defined the mutual influence factor $F_{i,j}$ between reports $x_{i},x_{j}$
as:

$F_{i,j}=\begin{cases}e^{-\lambda d_{ij}},&\text{if }t_{i}=t_{j}\\\
\frac{1}{\sigma\sqrt{2\pi}}e^{\frac{-1}{2\sigma^{2}}\left(d_{ij}-30\Delta
t_{i,j}\right)^{2}},&\text{otherwise}\end{cases}$ (11)

where $\Delta t_{i,j}$ denotes the difference between the migration cycles of
$i$-th and $j$-th report.  Based on $F_{i,j}$, we construct the priority
evaluation $Z$ of a report:

$Z_{i}=\sum_{j=1}^{n}{F_{i,j}}p_{j}$ (12)

where $Z_{i}$ denotes the priority of $i$-th report and $n$ denotes the total
number of reports involved in ranking. According to Equations (11) and (7),
the priority $Z_{i}$ of each report can be obtained. The greater the $Z$-value
of a report, the more likely it is to be positive, and thus the sooner it
should be investigated.

##  III Experiments

In the experiment, we used the Asian Hornet sighting report dataset111The
dataset is available at https://agr.wa.gov/departments/insects-pests-and-
weeds/insects/hornets/data published by the WSDA (Washington State Department
of Agriculture) to verify our model. This dataset includes the sighting report
ranging from 2019 to 2021, which details of sighting report and the label
after reviewed by the official.

We use accuracy metric to evaluate the classification model and conduct an
ablation experiment to demonstrate the effectiveness of our proposed module.

###  III-A Implementation Details

Data Cleaning.  Among the 4,400 samples, the main informational features of
each report were the coordinates, date, text, (possible) images, and
officially given identification label. We excluded the following parts of the
data: 15 reports with invalid date, 15 with “unprocessed” label, and 56 prior
to 2019. These parts account for only a small fraction, and contribute little
to our later study. After cleaning, we got 4,355 pieces of data. There are
three categories of result labels in the cleansed data: positive, negative,
and unverified. The number of sample categories is highly unbalanced, with
only 14 sightings verified as positive cases, accounting for only 0.3%, and
the remaining half of each case being negative and unverified, respectively.

Parameter Setting.  To determine the initial distribution $P_{x,y}^{(0)}$ in
Equation 2, we use the confirmed sighting report locations of the data set
from March 2019 to April 2020 as the initial nest probability distribution of
Washington State. The probability value of each positive reported location is
set to 1, and the other locations are all set to 0. The parameter $\beta_{1}$
in Equation 3 is set to $0.57$, and the $\sigma$ in Equation 2 is set to $10$.

###  III-B Results of classification prediction

Note that the category of report in this dataset are extremely unbalanced. We
conduct an ablation experiments to prove the effect of the weighted loss
coefficient in Equation 8. To this end, we take the parameter $\tau$ as a
model parameter, and obtain the training and test sets by 5-fold split (we
will ensure the inclusion of samples of both statuses in each set). Then, we
performed grid search for
$\tau\in\left\\{\tau_{0}-\Delta\tau,\tau_{0},\tau_{0}+\Delta\tau\right\\}$,
with a 5-fold cross-validation for each parameter (still ensuring both
inclusions). Since the reports were mostly negative, the naïve use of average
accuracy would not evaluate well the prediction of our model, so we took the
average accuracy on positive reports as the metric.

Figure 2: Under different scaling parameter $\tau$ settings, the cross-
validation accuracy on report samples of imbalanced statuses.

After searching for an optimal parameter setting, a prediction accuracy of
$75\%$ is achieved on only few samples. Considering the prediction difficulty
due to the extreme imbalance of the samples (with
$\text{positive}/\text{negative}\approx 1/400$), the model performs well
enough. Also, comparing results with and without setting the weighting
parameter ($\tau=1$ for the original binary cross-entropy function), we find
that the weighting correction parameter contributes much to improved
performance. With the best parameter settings ($\tau=311$), our classification
model can achieve the average accuracy of 83.5%. However, compared with recent
related works (e.g. [3] achieved detection rate of 93%), there is still room
for improvement in our method.

###  III-C Results of priority prediction

To prove that introducing the mutual influences among reports improves the
priority sorting, we performed ablation experiments to demonstrate the
effectiveness of our prioritization module by comparing to the baseline method
that directly uses the probability computed using Equation 7 to sort the
priority. We construct a small and balanced dataset that includes 28 reports,
half of which are positive. Specifically, the dataset was split as 1:1, with
14 for training and 14 for testing. The results in Fig. 3 show that the
priority of positive sample predicted by our method is much higher compared to
the baseline, which means that our method can better identify high-priority
reports. This demonstrates the effectiveness of the purposed module.

Figure 3: The priority rankings of the $7$ positive samples in the validation
set with different ranking models. For these positive reports, the higher the
priority predicted by the model, the better. Noted that we use the sort
position of the report among $14$ report in testing set to indicate its
priority.

##  IV Conclusion

In this paper, we proposed the priority prediction of Asian Hornet sighting
reports based on Machine Learning, which solved the problem that prioritizing
Asian Hornet sighting reports efficiently and automatically under the
limitation of resources. We formalized the problem of priority prediction of
sighting reports into a two-category problem. To characterize a sighting
report, we construct rich features from four dimensions: location, time, image
and text. Then, we purpose a machine learning model based on logistic
regression to predict the credibility of the report, and determined their
priority based on the relationship among the reports.  Extensive results on
the public dataset demonstrate the effectiveness of the proposed method.

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