# Planar near-field measurements of specular and diffuse reflection of
millimeter-wave absorbers

Fumiya Miura1,2  Hayato Takakura2  Yutaro Sekimoto2  Junji Inatani2  Frederick
Matsuda2  Shugo Oguri2  Shogo Nakamura1  1Department of Physics and
Engineering, Graduate School of Science and Engineering, Yokohama National
University, 79-1, Tokiwadai, Hodogaya-ku, Yokohama, Kanagawa 240-8501, Japan
2Institute of Space and Astronautical Science (ISAS), Japan Aerospace
Exploration Agency (JAXA), 3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa
252-5210, Japan<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

Mitigating the far sidelobes of a wide field-of-view telescope is one of the
critical issues for polarization observation of the cosmic microwave
background. Since even small reflections of stray light at the millimeter-wave
absorbers inside the telescope may create nonnegligible far sidelobes, we have
developed a method to measure the reflectance of millimeter-wave absorbers,
including diffuse reflections. By applying the planar near-field measurement
method to the absorbers, we have enabled two-dimensional diffuse-reflection
measurements, in addition to characterizing specular reflection. We have
measured the reflectance of five samples (TK RAM Large and Small Tiles and
Eccosorb AN-72, HR-10, and LS-22) at two angles of incidence in the frequency
range from 70 GHz to 110 GHz. Compared with conventional horn-to-horn
measurements, we obtained a consistent specular reflectance with a higher
precision, less affected by standing waves. We have demonstrated that the
angular response and diffuse-to-specular reflectance ratio differ among
various materials. The measurements also imply that some absorbers may affect
the polarization direction when reflecting the incident waves.

##  1 Introduction

The primordial gravitational waves due to inflation are believed to have left
a characteristic polarization pattern called $B$ modes in the cosmic microwave
background (CMB)[1]. The $B$-mode polarization is expected to peak at large
angular scales of several degrees or more, and the stray light contamination
due to emission from the galactic plane entering the telescope is expected to
be one of the largest systematic errors[2, 3, 4]. Stray light can enter the
telescope through the far sidelobes. Next-generation CMB telescopes such as
LiteBIRD need to know sidelobe to an accuracy of -56 dB[5].

To achieve such a requirement, it is essential to characterize reflections
from millimeter-wave absorbers[6], for example, at the baffling walls, in
addition to minimizing stray light in the optical design optimization[7]. In
recent years, novel millimeter-wave absorbers have been reported, such as
metamaterial absorbers[8, 9], 3D printed absorbers[10, 11, 12], and new
carbon-based absorbers[13, 14].

The reflectance properties of millimeter-wave absorbers in free space have
been made using two horns and focusing optics, such as mirrors[9], in such a
way that the attenuation of the coupling constant between the horns is
measured (horn-to-horn method). Since stray light enters the absorber from
various angles inside the telescope, the specular reflectance is measured at
multiple angles of incidence[15, 16]. Also, by rotating one of the horn
antennas around the absorbers, one-dimensional distributions of diffuse
reflection have been measured[17, 18, 9]. However, if the scattered waves from
the absorber are only partially coupled to the receiving horn, the measured
reflectance is underestimated.

In this paper, we report a new method to measure the reflectance of
millimeter-wave absorbers. Applying the planar near-field antenna measurement
method[19, 20], we measure the amplitude and phase of the reflected waves from
absorbers on a plane. By applying plane wave expansion to the measurements, we
characterize the two-dimensional angular patterns of diffuse reflection, from
which the specular reflectance is also evaluated.

We characterized the specular and diffuse reflectance for five absorbers
(AN-72, HR-10, LS-22, TK RAM Large, and TK RAM Small) in the 70 – 110 GHz
range. Since the reflectance characteristics of the absorber depend on the
polarization direction[21, 17], we performed measurements for both incident
polarizations (S-pol and P-pol). We set the polarization direction of the
probe not only parallel but also perpendicular to the polarization direction
of the feed horn.

The structure of this paper is as follows. Section 2 describes the principles
of the new measurement method proposed in this paper. Section 3 describes the
measurement setup constructed to demonstrate the measurement method. Section 4
reports the measurement results of millimeter-wave absorbers and discusses the
specular and diffuse reflectance. Section 5 states the conclusion.

##  2 Reflection measurement method

###  2.1 Horn-to-horn method

Conventional millimeter-wave absorber reflection measurements have been made
using two horns and focusing optics, such as mirrors[9] in such a way that the
attenuation of the coupling constant between the horns is measured (horn-to-
horn method; Fig. 1(a))[16, 9]. This method has been extended to one-
dimensional diffuse reflection characterization by changing the angle of the
receiving horn ($\theta$ in Fig. 1(a))[17, 18]. However, if the scattered
waves from the absorber are only partially coupled to the receiving horn, the
measured reflectance is underestimated. It has been reported that the measured
reflectance depends on the distance between the absorber and the receiving
horn[17].

Figure 1: Schematic of reflection measurements of millimeter-wave absorbers.
(a) Conventional[17, 16, 9] measurement method (horn-to-horn method). A horn
antenna illuminates the absorber, and the coupling between the horns is
measured. One-dimensional diffuse reflection is also measured by rotating the
receiving horn concerning the reflection point. The reflection point and the
center of rotation must coincide. Point c indicates the center of rotation.
The blue lines show specular reflection, and the red lines show diffuse
reflection. (b) Planar near-field measurement method. We illuminate the test
absorber and measure the amplitude and phase of the reflected waves by moving
the receiver with the probe on a two-dimensional plane. The measured amplitude
and phase distributions can be separated into angular plane wave components.
The polarization direction of the S- and P-polarized feed horn are parallel to
the $y$ and $x$ axes, respectively.

###  2.2 Planar near-field method

In order to accurately characterize absorber reflection, including diffuse
reflection, we have applied the planar near-field measurement method[19, 22]
to the reflection measurement of millimeter-wave absorbers. Figure 1(b) shows
a schematic diagram of the proposed method. In this method, we illuminate the
test absorber with a feed horn and measure the waves reflected from the test
absorber by moving the receiver with the probe on a two-dimensional plane. The
measured amplitude and phase distributions are decomposed into plane-wave
components with different angular directions $\varepsilon$ by plane wave
expansion (two-dimensional Fourier transform) as[23, 24]

$\bm{I}(\theta,\phi)=|\bm{\varepsilon}|^{2};\quad\bm{\varepsilon}(k_{l},k_{m})\propto\iint\bm{E}_{\text{r}}(x,y)e^{-i(k_{l}x+k_{m}y)}dxdy.$
(1)

Here, $\bm{E}_{\text{r}}(x,y)=|E_{\text{r}}(x,y)|\exp(i\Phi(x,y))$ is the
complex electric field reflected by the test absorber; $(k_{l},k_{m})$ denote
the spatial frequency in the $(x,y)$ plane, which relate the angle
$(\theta,\phi)$ as $k_{l}=k\sin{\theta}\cos{\phi}$ and
$k_{m}=k\sin{\theta}\sin{\phi}$ with $k,\lambda,c$ being the wave number,
wavelength and speed of light, respectively. $\phi$ is the azimuth angle from
the $x$-axis.

The planar near-field measurement has three advantages. First, by measuring
the reflected wave of the absorber, we can evaluate the reflectance
characteristics in more detail. Compared with the conventional horn-to-horn
method, the information obtained from the two-dimensional distribution of
reflected waves, including diffuse reflections, is much greater. Second, since
there is no need to know the reflection point of the absorber, it is less
susceptible to alignment adjustments in optical measurements. Third, compared
with horn-to-horn measurements, measurements with a small-aperture probe
enable more accurate measurements less affected by standing waves.

We evaluate the specular and total reflectance of the absorbers by integrating
the measured two-dimensional angular reflectance into different integration
ranges. The former is obtained by the integration of the reflected waves in
the Half Width at Half Maximum (HWHM) of the beams of the feed horn,

$R^{\text{s}}=\frac{\int^{\theta=\text{HWHM}}_{\theta=0^{\circ}}I_{\text{absorber}}(\theta,\phi)d\Omega}{\int^{\theta=\text{HWHM}}_{\theta=0^{\circ}}I_{\text{Al}}(\theta,\phi)d\Omega},$
(2)

which is considered to be similar to the conventional horn-to-horn
measurement. The latter is obtained by the integration of the entire two-
dimensional angular reflectance ($0\leq\theta\leq\theta_{\text{max}}$),

$R^{\text{t}}=\frac{\int^{\theta=\theta_{\text{max}}}_{\theta=0^{\circ}}I_{\text{absorber}}(\theta,\phi)d\Omega}{\int^{\theta=\theta_{\text{max}}}_{\theta=0^{\circ}}I_{\text{Al}}(\theta,\phi)d\Omega}.$
(3)

Here, the angular response $I(\theta,\phi)$ is calculated for each frequency
using Eq. (1).

##  3 Experimental verification

###  3.1 Experimental setup

Figure 2 shows the measurement setup developed in this study. We measured the
amplitude and phase of the electric field reflected by the test absorbers
using a vector network analyzer (VNA: Keysight Technologies N5222B) and
frequency extenders (Virginia Diodes WR10VNAX). Based on the reciprocity
theorem, the coupling between the two horns is equivalently measured when the
VNA transmitter and receiver are reversed[25]. For the convenience of the
setup, we connected the probe and lens horns to the transmitter and receiver,
respectively, in this experiment.

The measurements were performed with a 1 GHz interval over a frequency range
of 70 GHz – 110 GHz. Signals from 11.6 to 18.3 GHz were output from the VNA
and converted to six times higher frequencies. Then, the 70 – 110 GHz signal
is output from the probe. We measured the reflected waves from the millimeter-
wave absorber by moving the probe in 1.5 mm increments in a 75 mm $\times$ 75
mm square area. The distance between the probe and test absorbers is set to 60
mm, considering mechanical clearance. This configuration measures the specular
and diffuse reflection in $0\leq\theta\leq 30^{\circ}$. The reflected waves
are received by a conical horn (aperture diameter 23.0 mm, opening angle
10.0∘) with a polymethyl pentene (TPX) lens (diameter 25.4 mm, focal length 50
mm) placed in the aperture. The distance between the lens horn and test
absorbers is 50 mm.

The millimeter-wave absorber under test is placed on two rotating and one-axis
stages. Since the movable range of the measurement stage is finite, in
addition to the angle of incidence of the lens horn (AoI in Fig. 1(b)), the
angle of the absorber relative to the measurement stage ($\theta$ in Fig.
1(b)) was also rotated so that the measurement plane and the signal optical
path are vertical. In addition, using a one-axis stage, we aligned the surface
of the absorbers at a common plane for all the measurements. Considering the
situation in telescopes, we measured the absorbers fixed on a 5 mm thick
aluminum plate. The absorbers and the aluminum plate are 200 mm $\times$ 200
mm in size. An acrylic frame held the absorber in place, leaving no space
between it and the aluminum plate.

Table 1 summarizes the measurement parameters for this experiment. We measure
the reflectance for two polarizations; the polarization direction of the lens
horn is set parallel to the $y$ and $x$ axes in Fig. 1 (S- and P-pol,
respectively). In addition, the polarization direction of the probe is set not
only parallel but also perpendicular to that of the lens horn.

Figure 2: A photograph of the measurement setup corresponds to Fig. 1(b). Table 1: Measurement parameters. The polarization direction of the S- and P-polarized lens horn are parallel to the $y$ and $x$ axes in Fig. 1, respectively. The “sp” polarization represents the S-polarized lens horn and the P-polarized probe, and vice versa for “ps”.  Frequency range | 70 - 110 GHz
---|---
Frequency step | 1 GHz
Angle of incidence (AoI) | 30 deg, 45 deg
Co-polarization of the lens horn | S-pol, P-pol
Probe polarization | ss, pp (parallel to the lens horn)
| sp, ps (perpendicular to the lens horn)

###  3.2 Millimeter-wave absorbers

Table 2: Millimeter-wave absorber samples. An aluminum plate is placed on the back side of the absorbers during measurement. Samples | Material | Form | Structure | Thickness | Mass | Ref.
---|---|---|---|---|---|---
|  |  |  | [mm] | [kg/m2] |
AN-72 | polyurethane | sheet | multi-layer | 6 | 1.0 | [26]
HR-10 | polyurethane | sheet | single-layer | 10 | 0.6 | [27]
LS-22 | polyurethane | sheet | single-layer | 9.8 | 0.5 | [28]
TK-L | polypropylene | tile | pyramid | 14.9 | 10.6 | [29]
TK-S | polypropylene | tile | pyramid | 7.6 | 5.9 | [29]

Table 2 summarizes the test millimeter-wave absorbers measured in this
experiment. Eccosorb AN-72[26], HR-10[27], and LS-22[28] are commercially
available millimeter-wave absorbers by Laird. They are made of polyurethane-
based foams with different structures, which are expected to have different
diffuse reflection characteristics: LS-22 has a single-layer structure; AN-72
has a multi-layer structure; HR-10 has a single-layer structure and contains
carbon powder. LS-22 is available in several thicknesses, and we measured a
3/8-inch one. Eccosorb AN-72 and Eccosorb HR-10 are used in CMB ground-based
experiments[30, 31, 32]. TK RAM is a radar absorber developed by Thomas
Keating for the terahertz and millimeter-wave bands, and there are two types
of TK RAM[29]. TK RAM Small tile (abbreviated as TK-S) is a small tile of 25
mm $\times$ 25 mm and is recommended for 200 – 600 GHz. TK RAM Large tile
(abbreviated as TK-L) is a large tile of 100 mm $\times$ 100 mm and
recommended for 50 – 200 GHz. TK-L and TK-S have different-sized surface
structures, 5.0 mm, and 2.5 mm, respectively.

###  3.3 Reference measurement with an aluminum plate

Assuming that the aluminum plate without the absorber is a perfect
conductor[33], we measured it for intensity calibration.

Figure 3 shows reflection measurements of the reference aluminum plate. The
upper panels show the angular response of the electric field parallel and
perpendicular to the co-polarization of the S and P-polarized lens horn at 90
GHz. The angular response $I(\theta,\phi)$ are calculated using Eq. (1), and
$I(\theta,\phi)$ is normalized by the peak value of $I(\theta,\phi)$. The HWHM
of the lens horn was measured to be 4.34 deg at 90 GHz, consistent with the
prediction by a Gaussian-beam approximation[34]. The cross-polarization level
of the lens horn was measured to be $-20$ dB at 90 GHz. The lower panel shows
the profile in a cut-plane. We confirm that this setup can be used to measure
for two polarizations since the patterns measured with both polarizations are
consistent at the $-40$ dB level over a range of $0\leq\theta\leq 30^{\circ}$.

Figure 3: Reflection measurements of the reference aluminum plate at 90 GHz
for AoI = 30∘. The upper panels show the angular response of the electric
field parallel and perpendicular to the co-polarization of the S- and
P-polarized lens horn. The lower panel shows $\phi$-cut profiles for parallel
field ($\phi=90^{\circ}$ for ss $\phi=0^{\circ}$ for pp), and perpendicular
field ($\phi=45^{\circ}$). The angular response $I(\theta,\phi)$ are
calculated using Eq. (1), and $I(\theta,\phi)$ is normalized by the peak value
of $I(\theta,\phi)$.

##  4 Results and discussion

###  4.1 Planar near-field reflection measurements

We have measured the reflectance of the millimeter-wave absorbers using the
planar near-field measurement with the setup described in Sec. 3.1. Figure 4
shows the amplitude and phase distributions of the measured millimeter-wave
absorber reflections with ss polarization (S-polarized lens horn; probe
polarization parallel to the lens horn) at 90 GHz for AoI = 30∘. The amplitude
distributions are normalized by the peak value of the amplitude distribution
of the reference aluminum plate for each polarization. Compared with AN-72,
HR-10 sample has a more spread pattern.

Figure 4: Amplitude and phase distributions of the reflected waves from the
millimeter wave absorbers with ss polarization at 90 GHz for AoI = 30∘. The
upper panel shows amplitude, and the lower panel shows phase. Amplitudes are
normalized by the peak amplitude of the reference aluminum plate.

###  4.2 Angular responses and reflectance of the absorbers

Figure 5: Two-dimensional reflection patterns of reflectance of millimeter-
wave absorbers at 90 GHz for AoI = 30∘. This pattern was calculated using the
amplitude and phase of the reflected wave for each absorber by Eq. (1). The
upper panel shows the measurements with ss polarization and the lower panel
shows the measurements with pp polarization.

Figure 5 shows the angular responses of the reflected waves from the absorbers
at 90 GHz for AoI = 30∘. The results are normalized by the peak value of the
reference aluminum plate for each polarization. AN-72 has an almost isotropic
attenuation from the reference aluminum plate pattern, while HR-10 has a
spread pattern. LS-22 has similar patterns to AN-72, but the difference in
S-pol and P-pol is smaller than that of AN-72. TK-S also shows a pronounced
diffuse reflection pattern, indicating an upward tilt. This may be caused by
the surface structure and errors in attaching TK-S to the reference aluminum
plate.

Figure 6 shows the angular responses at 90 GHz for AoI = 45∘. It is expected
that a small but direct signal from the transmitter to the receiver is
present. Such a direct path between the lens horn and probe was also measured
without placing a millimeter-wave absorber. As the incident angle increases,
the direct path becomes more extensive, and stripe patterns due to
interference with the direct path are observed more apparently than those of
S-pol in the case of P-pol. This is because the sidelobe level of the lens
horn is higher in the polarization direction. This may be improved in future
measurements by using horns with small sidelobes, such as a corrugated horn.

Figure 7 is the cross-sectional profiles of Fig. 5 for $\phi=0^{\circ}$,
$\phi=\pm 90^{\circ}$ and $\phi=45^{\circ}$. While AN-72 and LS-22 have a
constant attenuation from the reference aluminum plate pattern, the intensity
of the reflected waves from HR-10 ranges between -40 and -20 dB regardless of
the angle $\theta$. Notably, in the angular range of 20
${}^{\circ}\leq\theta\leq$ 30∘, the level of the reflected waves from HR-10 is
higher than that of the reference aluminum plate.

Figure 6: Same as Fig. 5, but shows the measurements for AoI = 45∘. Figure 7:
Cross-sectional profiles of Fig. 5 for $\phi=0^{\circ}$, $\phi=\pm
90^{\circ}$, $\phi=45^{\circ}$. The gray line shows the direct path. Figure 8:
Total and specular reflectance of millimeter-wave absorbers for AoI = 30∘. The
solid lines represent the measurements with ss polarization, and the dashed
lines represent the measurements with pp polarization. The green line shows
the specular reflectance defined by Eq. (2), and the red line shows the total
reflectance defined by Eq. (3). The gray line shows the total reflectance of
the direct path. The vertical axis is different between the upper and lower
panels.

Figure 8 compares the specular and total reflectance of the absorbers in the
S- and P-polarization, calculated using Eqs. (2) and (3). Here,
$\theta_{\text{max}}$ in Eq. (3) is set to 30 degrees. The gray line shows
$R^{\text{t}}$ of the direct path from the probe to the lens horn measured
without placing a millimeter-wave absorber. The effects of direct path is
about 10 dB larger in P-pol than in S-pol. The specular reflectance of AN-72
is about -12 dB for S-pol and -17 dB for P-pol, which is about 5 dB different
for the two polarizations. On the other hand, the specular reflectance of
HR-10 is about -20 dB for both polarizations, which is smaller than that of
AN-72. In general, dielectric materials have more significant reflectance for
the P-pol than the S-pol for oblique incident wave[21]. The measurements of
AN-72 show a consistent tendency. By contrast, the reflectance of LS-22 shows
little difference between the two polarizations. The reflectance of TK-L was
about -40 dB for both polarizations, the lowest among the five absorbers
measured in this experiment.

The specular reflectance (green line) of HR-10 for the S-polarization is about
-20 dB, while it rises to about -15 dB for total reflectance (red line). This
shows that HR-10 has larger diffuse reflection than other absorber, consistent
with the isotropic spreading feature of angular patterns in Fig. 5. The total
reflectance of TK-S is also higher than the specular reflectance by around 5
dB. For other absorbers, the two reflectances have little difference,
indicating that diffuse reflection is small.

We also compared the measured specular reflectance discussed in Sec. 2.1 with
conventional horn-to-horn measurements. We performed the horn-to-horn
measurements using the same experimental setup in Fig. 1[Sec. 3.1] by
replacing the probe with the lens horn. The two lens horns for the transmitter
and receiver are identical. The results demonstrate that the planar near-field
measurement and the horn-to-horn reflection measurement agree within a few dB
of accuracy and that the near-field measurements are less affected by standing
waves (Fig. 9).

Figure 9: Comparison of planar near-field reflection measurements and specular
reflectance measured by the horn-to-horn method. The left panel shows the
reflectance of Eccosorb AN-72 with ss polarization for AoI = 45∘. The right
panel shows the reflectance of TK-S with ss polarization for AoI = 30∘. The
red lines represent reflectance derived from Eq. 2 using planar near-field
measurements, and the blue lines represent reflectance measured by the horn-
to-horn method. The blue line is normalized by the reference aluminum plate
measured by the horn-to-horn method. For the reflectance of TK-S, the green
line shows the horn-to-horn measurements of Z. Xu et al. 2021[9] at 90 GHz –
110 GHz.

###  4.3 Electric field measurements perpendicular to the co-polarization of
the lens horn

Figure 10: Two-dimensional reflection pattern of the electric field
perpendicular to co-polarization of the lens horn at 90 GHz for AoI = 30∘. The
label “sp” represents the measurements with the S-polarized lens horn and the
P-polarized probe horn, and vice versa for “ps”. The “sp” and “ps” patterns
are normalized by the peak value of the “ss” and “pp” measurements of the
reference aluminum plate, respectively. Figure 11: Total reflectance of the
electric field perpendicular to co-polarization of the lens horn for AoI =
30∘. The solid red line shows the reflectance with sp polarization, and the
solid green line shows the reflectance with ps polarization. $R^{t}_{sp}$ and
$R^{t}_{ps}$ shown by solid lines include $\eta_{sp}R_{pp}$ and
$\eta_{ps}R_{ss}$ shown by dashed lines.

Using the planar near-field measurement method, we also measured the electric
field perpendicular to the co-polarization of the lens horn. Figure 10 shows
the angular responses of the electric field perpendicular to the co-
polarization of the lens horn at 90 GHz for AoI = 30∘. The patterns of AN-72,
LS-22, and TK-S measured with the S- and P-polarized lens horn (labeled as
“sp” and “ps” respectively) are similar to those of the reference aluminum
plate (“sp” and “ps” in Fig. 3) with attenuated levels, showing a four-divided
pattern. This indicates that the patterns are determined mainly by the cross-
polarization of the lens horn. On the other hand, the patterns of HR-10 and
TK-L do not resemble that of the reference aluminum plate.

Such trends are also apparent in total reflectance defined by Eq. (3), as
summarized in Fig. 11. For AN-72, LS-22, and TK-S, the total reflectance
measured with the S- and P-polarized lens horn ($R_{sp}^{t}$ and $R_{ps}^{t}$
respectively) are mostly consistent with $\eta_{sp}R_{pp}$ and
$\eta_{ps}R_{ss}$. Here, $\eta_{sp}$ and $\eta_{ps}$ indicate the cross-
polarization components of the lens horn, which are obtained from the
reference aluminum plate measurements as,

$\eta_{sp}=\frac{\int^{\theta=30^{\circ}}_{\theta=0^{\circ}}I^{\text{Al}}_{sp}(\theta,\phi)d\Omega}{\int^{\theta=30^{\circ}}_{\theta=0^{\circ}}I^{\text{Al}}_{ss}(\theta,\phi)d\Omega},\quad\eta_{ps}=\frac{\int^{\theta=30^{\circ}}_{\theta=0^{\circ}}I^{\text{Al}}_{ps}(\theta,\phi)d\Omega}{\int^{\theta=30^{\circ}}_{\theta=0^{\circ}}I^{\text{Al}}_{pp}(\theta,\phi)d\Omega}.$
(4)

However, the total reflectance $R_{sp}$ and $R_{ps}$ of HR-10 and TK-L are
$\sim 5$ dB larger than $\eta_{sp}R_{pp}$ and $\eta_{ps}R_{ss}$. The
measurements imply that HR-10 and TK-L may affect the polarization direction
when reflecting the incident waves.

##  5 Conclusion

By applying the planar near-field antenna measurement method, we have
developed a method to measure the reflectance of millimeter-wave absorbers,
including the two-dimensional distribution of diffuse reflections. We have
measured five absorbers (TK-L, TK-S, and Eccosorb AN-72, HR-10, LS-22) at
$30^{\circ}$ and $45^{\circ}$ angles of incidence for two polarizations (S-pol
and P-pol) in the frequency range from 70 GHz to 110 GHz. From these
measurements, we evaluate the specular and total reflectance of each absorber.
The obtained specular reflectance is less affected by standing waves and
agrees with the conventional measurement. The specular reflectance of AN-72
and HR-10 are higher for S-pol, while that of LS-22, TK-L, and TK-S does not
depend on incident polarization. We also evaluate the total reflectance,
including diffuse reflection. The diffuse-to-specular reflectance ratio is
significantly higher for HR-10 and TK-S compared with AN-72 and LS-22. We also
measured the electric field perpendicular to the co-polarization of the lens
horn. The measurements imply that HR-10 and TK-L may affect the polarization
direction when reflecting the incident waves.

While the conventional horn-to-horn method measures the power coupling
efficiency between the two horns, the present method allows the detailed near-
field mapping of the wave reflected from the absorber. Thus, the latter can
investigate the nature of the reflection more extensively. We have
demonstrated in this paper that the diffuse-to-specular reflectance ratio is
different among various materials. It is also a merit of the new method that
the measurement is not sensitive to the alignment of the absorber in the setup
optics. Practically, it will help us conduct similar reflection measurements
in cryogenic conditions. The method is expected to contribute to the
development of millimeter-wave absorbers by achieving accurate measurements,
including diffuse reflections, and to make a better design against the stray
light in CMB observation telescopes.

Funding This work was supported by MEXT/JSPS KAKENHI Grant Numbers 23K25889,
JP23K17309, JP23K25889 and JP24K17078 and by JSPS Core-to-core Program Number
JPJSCCA20200003.

Acknowledgments We are grateful to Luca Lamagna, Andrea Occhiuzzi, Haruaki
Hirose and Takuro Fujino for their helpful discussions on the measurements. We
are also grateful to Oshima Prototype Engineering Co. for fabricating the horn
antennas used in the measurements.

Disclosures The authors declare no conflicts of interest.

Data Availability Data underlying the results presented in this paper are not
publicly available at this time but may be obtained from the authors upon
reasonable request.

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See pages 1-last of main.pdf See pages 1-last of supplement.pdf


# Challenges and opportunities in applying Neural Temporal Point Processes to
large scale industry data

Dominykas Šeputis<EMAIL_ADDRESS>Vilnius
UniversityNaugarduko g. 24VilniusLithuania03225 ,  Jevgenij Gamper
<EMAIL_ADDRESS>VintedŠvitrigailos g. 13-1-4
floorVilniusLithuania03228 and  Remigijus Paulavičius
<EMAIL_ADDRESS>Vilnius UniversityNaugarduko g.
24VilniusLithuania03225

(2022)

###### Abstract.

In this work, we identify open research opportunities in applying Neural
Temporal Point Process (NTPP) models to industry scale customer behavior data
by carefully reproducing NTPP models published up to date on known literature
benchmarks as well as applying NTPP models to a novel, real world consumer
behavior dataset that is twice as large as the largest publicly available NTPP
benchmark. We identify the following challenges. First, NTPP models, albeit
their generative nature, remain vulnerable to dataset imbalances and cannot
forecast rare events. Second, NTPP models based on stochastic differential
equations, despite their theoretical appeal and leading performance on
literature benchmarks, do not scale easily to large industry-scale data. The
former is in light of previously made observations on deep generative models.
Additionally, to combat a cold-start problem, we explore a novel addition to
NTPP models - a parametrization based on static user features.

generative modeling;neural temporal point process;user behavior modeling;

††copyright: acmcopyright††journalyear: 2022††doi:
XXXXXXX.XXXXXXX††conference:  KDD Undergraduate Consortium; August 14–18,
2022; Washington, D.C.

##  1\. Introduction

Considering online platforms, most use cases of generative modeling
applications are motivated by the need to capture user behavior. When in need
to answer questions such as ”How long do we need to wait for the next user’s
visit?” or ”What is the optimal time of recommending winter coats?” prediction
type tasks where the goal is to predict times and marks of the subsequent
events are formed. With the rapid growth of online platforms, it is crucial to
estimate how its users will behave in the near future.

Commonly referred to as Buy-till-You-Die models, Pareto/NBD (Schmittlein et
al., 1987), BG/NBD (Fader et al., 2005), and other generative models alike
(Jasek et al., 2019) might be too simplistic and lack flexibility in modeling
heterogeneity of customer behavior. Nonetheless, the generative approach
towards customer behavior modeling described by (Fader and Hardie, 2009) might
be useful, albeit in a different form.

For example, in Pareto/NBD model, the number of purchases that an
$i’\text{th}$ customer makes in their lifetime is described by a Poisson
distribution parameterized by $\lambda_{i}$. Given the inferred parameters for
Poisson and Exponential distributions representing a customer’s purchase
number and lifetime, one can estimate the probability of a customer being
”alive.” Buy-till-You-Die models assume a fixed parameter $\lambda$ that does
not depend on the time, and the models are static. A customer who had a
positive experience with a service is more likely to come back and keep
returning if the experience continues to be positive and vice versa - these
models do not capture such behavior.

An alternative would be to use a self-exciting point process like Hawkes,
which implies that $\lambda$ parameter is a function of time, positively
influenced by past events. Lately, a new class of temporal point processes
(TPPs) has emerged - Neural Temporal Point Process (NTPP) (Mei and Eisner,
2017b; Zhang et al., 2019; Chen et al., 2018). The NTPPs are based on the
self-exciting nature of the Hawkes process but are more versatile and can
capture more diverse user behavior. In this work, we identify open research
opportunities in applying neural temporal point process models to industry
scale customer behavior data.

We explore and compare different approaches to NTPP: Neural Ordinary
Differential Equations (Chen et al., 2018) and Attention (Vaswani et al.,
2017) based temporal point processes (Zhang et al., 2019; Enguehard et al.,
2020). We apply these models to various synthetic and real-life datasets. To
further test these models, we introduce online second-hand marketplace users
behavior dataset. The dataset consists of 4,886,657 events and is 10 times
larger than the commonly used Stack Overflow dataset (Du et al., 2016; Jia and
Benson, 2019; Enguehard et al., 2020; Mei and Eisner, 2017a). We attempt to
use selected models to describe users’ actions in the marketplace and explore
the benefits of parameterizing Self-Attention based NTPP models with static
features. Finally, we discuss explored models’ scalability and their ability
to capture rare events.

This paper is organized as follows. In section 2, we explore different ways
temporal point processes are used in the industry. In section 3, we give a
formal overview of traditional and neural temporal point processes. In section
4, we describe the data and our experimental approach. Section 5 reports the
main results of the document, and the final section concludes.

##  2\. Related work

With the rapid growth of online platforms, it is crucial to estimate how its
users will behave in the near future. Multivariate point processes like
FastPoint (Türkmen et al., 2020) prove that optimal results are achieved by
combining neural networks with traditional generative temporal point process
models. The approach scales to millions of correlated marks with superior
predictive accuracy by fusing deep recurrent neural networks with Hawkes
processes. Combining traditional temporal point process models with neural
networks enables the industry to solve diverse problems: from predicting when
customers will leave an online platform to simulating A/B tests.

#### Churn prediction

Customer churn prediction is one of the most crucial problems to solve in
subscription and non-subscription-based online platforms. Traditionally churn
prediction is handled as a supervised learning problem (Wangperawong et al.,
2016; Merchie and Ernst, 2022; Mena et al., 2019) where the goal is to predict
if a user will be leaving the platform after a fixed time period. However,
platform usage habits can differ from one user to another, and a decrease in
an activity does not always is a sign of churn. (Khodadadi et al., 2019) is
one of the solutions that tackle churn prediction via generative modeling. The
solution interprets user-generated events in time scope of sessions and
proposes using user return times and session durations to predict user churn.

#### Time-Sensitive Recommendations

When considering item recommendations for a user, we are thinking about the
optimal suggestion and what time of the year we should suggest. (Du et al.,
2015) introduces a novel convex formulation of the problems by establishing an
under-explored connection between self-exciting point processes and low-rank
models. The proposed solution expands its applicability by offering time-
sensitive recommendations and users’ returning time predictions that can
determine previously discussed churn or optimize marketing strategies. The
item recommendation part is accomplished by calculating intensity
$\lambda_{u,i}$ for each item $i$ and user $u$. After, items are sorted by
descending order of calculated intensity, and top-k items are returned.

#### Interactive Systems Simulation

It is common to see modern systems using numerous different machine learning
models interacting with each other. As a result, user experience is often
defined by various machine learning systems layered iteratively atop each
other. The previously discussed item-recommendation problem is one example of
such complex system (Gomez-Uribe and Hunt, 2016). When there is a need to
change one or multiple models in the recommendation system, one might ask
themselves what effect a particular change might have on the overall system
performance. Usually, A/B testing answers these types of questions. Although
when different system modules are changing rapidly and systems are getting
more complex, it becomes more practical to simulate the effect than test it
using traditional methods. (McInerney et al., 2021) presents one of many ways
to simulate interactive systems. The paper proposes Accordion, a fully
trainable simulator for interactive systems based on inhomogeneous Poisson
processes. While combining multiple intensity functions, the Accordion enables
a comparison between realistic simulation settings and their effect on the
total number of visits, positive interactions, impressions, and any other
empirical quantity derived from a sampled dataset.

Figure 1. Architecture of parameterized Self Attention encoder.

Architecture of parameterized Self Attention encoder.

##  3\. Methods

The base of our explored generative user behavior modeling methods is Poisson
process - a continuous-time version of a Bernoulli process where event
arrivals are happening at any point of the time (Bertsekas and Tsitsiklis,
2000). The homogeneous Poisson process describes intensity of event arrival by
$\lambda=\lim_{\delta\rightarrow 0}=\frac{P(1,\delta)}{\delta}$, where
$\delta$ is a minimal interval between events time. If needed to model events
that are influenced by the time dimension, we can use non-homogeneous version
of the Poisson process. Simply instead of using a static intensity function
$\lambda$ to represent the intensity of the events, we transform it into a
function of time $\lambda=\lambda(t)$ (Pishro-Nik, 2014).

To capture more complex behavior, for example item recommendation, where one
successful transaction mostly reinforces members to buy more items, self-
exciting processes are used. The self-exciting process is a point process with
a property that the arrival of the event causes the conditional intensity
function to increase. One of the most known self-exciting processes is Hawkes
process (HAWKES, 1971). The key feature of the process is a conditional
intensity function, influenced by the events in the past
$\lambda(t|\mathcal{H}_{t})=\lambda_{0}(t)+\sum_{i:t>T_{j}}\phi(t-T_{j}),$
where $\mathcal{H}_{t}$ is the history of the events that happened before time
$t$. From the defined Hawkes function we can see that there is two major parts
in it: base intensity function $\lambda_{0}$ and a summation of kernel
$\phi(t-T_{i})$. The base intensity function $\lambda_{0}$ keeps the
independence property from the Poisson process, and past events do not
influence it. The influence of the past arrivals in the Hawkes is defined with
a kernel function $\phi(\cdot)$, which usually takes form of exponential
function $\phi(x)=\alpha e^{-\beta x}\mathcal{,}$ where $\alpha\geq
0\mathcal{,}\beta>0$ and $\alpha<\beta$. The kernel function ensures that more
recent events would have a greater influence on the current intensity than
events in the past.

###  3.1. Neural Hawkes Process

While Hawkes process brings a significant improvement in capturing the self-
exciting type of events, the process is not fully sufficient at representing a
more complex type of events. Hawkes process’s intensity function dictates
additive and decaying influence of the past events. While this behavior can be
true for some events, event streams like item recommendations do not benefit
from the Hawkes process’s additive nature. The more similar items are shown,
the less influence they bring and even can discourage people from buying the
product. Neural Hawkes Process (NHP) (Mei and Eisner, 2017b) is better suited
to battle more complex TPP problems. The Neural Hawkes Process introduces
methods that can define processes where a past event’s influence: is not
always additive, is not always positive, and does not always decay. The Neural
Hawkes process brings improvements in two parts. First, NHP introduces non-
linear transfer function $f:\mathbb{R}\rightarrow\mathbb{R}^{+}$, which
usually takes form of non-linear softplus function
$f(x)=\alpha\log(1+e^{x/\alpha})$. Second, rather than predicting
$Hawkes\>\lambda(t)$ as a simple summation, NHP uses recurrent neural network
(RNN) (Hopfield, 1982) to determine the intensity function $\lambda=f(h(x)),$
where $h$ is a hidden state of the RNN. This change allows learning a complex
dependence of the intensities on past events’ number, order, and timing. This
architecture lets to improve the base Hawkes process shortcomings: inability
to represent events where past event’s influence is not always additive is not
always positive, and does not always decay; however, the neural Hawkes process
has a downside - weaknesses of RNNs are inherited; namely the model’s
inability to capture long-term dependencies (also called as ”forgetting”) and
is difficult to train (Bengio et al., 1994).

###  3.2. Self-Attention Hawkes process

Despite not being directly applicable to model point process, recently,
Transformer architecture was successfully generalized to continuous-time
domain (Zuo et al., 2020; Zhang et al., 2019). Self-Attentive Hawkes Process
(SHAP) is an self-attention based approach to a Hawkes process proposed by
(Zhang et al., 2019). SHAP employs self-attention to measure the influence of
historical events to the next event. The SHAP approach to the temporal point
processing field also brings a time-shifted positional embedding method where
time intervals act as phase shifts of sinusoidal functions. Transformers
Hawkes Process (THP) presented by (Zuo et al., 2020) is another self-attention
based approach to the Hawkes process. THP improves recurrent neural network-
based point process models that fail to capture short-term and long-term
temporal dependencies (Informatik et al., 2003). The key ingredient of the
proposed THP model is the self-attention module. Different from RNNs, the
attention mechanism discards recurrent structures. However, the THP model
still needs to be aware of the temporal information of inputs, such as
timestamps. To achieve this temporal encoding procedure is proposed that is
used to compute the positional embedding.

###  3.3. Neural Jump Stochastic Differential Equations

Sometimes user’s interest is built up continuously over time but may also be
interrupted by stochastic events. To simultaneously model these continuous and
discrete dynamics, we can use the Neural Jump Stochastic Differential
Equations (NJSDEs) (Jia and Benson, 2019). The NJSDE uses latent vector $z(t)$
to encode the system’s state, which continuously flows over time until an
event at random, introducing an abrupt jump and changing its trajectory. The
event conditional intensity function and the influence of the jump are
parameterized with neural networks as a function of $z(t)$, while the
continuous flow is described by Neural Ordinary Differential Equations (Chen
et al., 2018). The advantage of Neural NJSDEs is that they can be used to
model a diverse selection of marked point processes, where a discrete value
can compliment events (for example, a class of purchased item) or a vector of
real-valued features (e.g., spatial locations).

###  3.4. Parameterised Self-Attention Hawkes process

To combat a cold-start problem, we explore a novel addition to an encoder-
decoder architecture based NTPP models - a parametrization based on static
user features. Specifically, we experiment with self-attention type encoders.
We concatenate outputs of the Transformer model $m$ with processed static user
features $p$ and pass them thought multilayer perceptron (MLP):
$f(X)=MLP(m||p)$. The change differs from the approach proposed by (Enguehard
et al., 2020), where outputs of the Transformer model are directly passed to
the MLP. We present the illustrated architecture of our approach in Figure 1.

##  4\. Experimental setup

###  4.1. Data

(a) Comparison between common (left) and rare (right) event type distribution
within Stack Overflow platform.

(b) Comparison between common (left) and rare (right) event type distribution
within Vinted platform.

(c) Vinted inter-event time distribution.

Figure 2. Different properties of Stack Overflow and Vinted datasets.

Different properties of Stack Overflow and Vinted datasets.

The first dataset we use is a simulated Hawkes process. The synthetic Hawkes
data serves as a starting point while evaluating selected models’ performance.
As the data is generated, we can have unlimited samples. Also, as the
intensity values at any point in time are known, we are able to compare them
with values provided by the trained models. We designed the synthetic dataset
consisting of two independent processes.

The second and the third datasets are real user behavior datasets. Stack
Overflow is a question-answering website that awards users various badges
based on their activity on the website. The website’s users activity is
commonly used when benchmarking various temporal point process models (Du et
al., 2016; Jia and Benson, 2019; Enguehard et al., 2020; Mei and Eisner,
2017a). Novel customer behavior dataset comes from a second hand marketplace
Vinted, where users sell and buy various goods.

Figure 3. Comparison between passive (top) and active (bottom) Stack Overflow
user’s activity within the website. Passive website user tends to acquire less
diverse badges’ set than an active one.

Comparison between active (top) and passive (bottom) Stack Overflow user’s
activity within the website. Passive website user tends to acquire less
diverse badges’ set than an active one.

Figure 4. Comparison between passive (top) and active (bottom) Vinted user’s
activity within the platform. Active Vinted user completes same actions as the
passive user, only in a more frequent manner.

Comparison between passive (top) and active (bottom) Vinted user’s activity
within the platform. Active Vinted user completes same actions as the passive
user, only in a more frequent manner.

We collect Vinted users’ activity data by sampling all of the recorded
activity (from user’s registration to the time of collecting the data) of
5,082 Vinted users. All of the selected users were active in the last year
(2021/01/01 - 2022/01/01) and completed at least 40 actions. Inside the
dataset, four different types of events can be found: Purchase, Listing,
Search, and Sale. Frequent event occurrences and long recorded activity
periods result in the dataset being ten times larger than Stack Overflow
(4,886,667 vs. 480,414 events). While inspecting the event count distributions
(see Figure 2(c)), we notice that users’ activities within Vinted platform are
similar to Stack Overflow, where common events’ distributions are shaped
similarly. Events collected from Vinted platform as well as from the Stack
Overflow website suffer from data imbalance. Events like Publicist, Populist,
or Booster are given to the website users only when he accomplishes unique
action. These actions can be tricky to capture for temporal point process
models, as the data imbalance is quite significant within the dataset. The
most common events are Search and Listing. Purchases and Sales are happening
much rarer - as they are influenced not only by users’ behavior but also by
external factors such as the listed item’s appeal or Vinted’s recommendation
system’s quality.

Where the differences between Vinted and Stack Overflow platforms begins is
the measured inter-event times (see Figure 2(c)). Vinted users are returning
to the platform more frequently than Stack Overflow ones. This difference is
caused by the fact that Stack Overflow browsing activity is not recorded.
Activity tendencies between active and passive platform users are also
different between mentioned platforms. Active Vinted user completes the same
actions as the passive user, only in a more systematic manner (see Figure 4).

Besides user activity records, we collect static Vinted users’ features. We
use the features to explore a solution to the cold start problem in modeling
user behavior. We parameterize temporal point process models in the hope of
more accurate event type classification results.

We split each dataset into training, test and validation parts. Dataset
statistics are summarized in Table 1.

Table 1. Properties of datasets used for experimentation. |  |  |  |  | Size
---|---|---|---|---|---
Dataset | # classes | Task type | # events | Avg. length | Train | Valid | Test | Batch
Hawkes | 2 | Multi-class | 350,281 | 14 | 233,537 | 58,451 | 58,293 | 512
Stack Overflow | 22 | Multi-class | 480,414 | 72 | 335,870 | 47,966 | 96,578 | 32
Vinted | 4 | Multi-class | 4,886,657 | 962 | 3,428,380 | 482,512 | 975,765 | 4

###  4.2. Training

We base our model selection on the model’s performance in capturing the type
and time of the next event based on the time it occurred. Also, the
accessibility of a source code is a significant deciding factor. We select
(Jia and Benson, 2019) and (Enguehard et al., 2020) as the primary source of
models used for our experimentation. (Enguehard et al., 2020) demonstrates
that models based on Encoder-decoder architectures are effective not only for
Natural Language Processing (Cho et al., 2014; Vaswani et al., 2017) but also
work well in a temporal point processing field. In the paper, the usage of
Self-Attention as a building block of the model’s architecture proves to be
beneficial while modeling healthcare records. Where (Jia and Benson, 2019)
showcases a Neural Jump Stochastic Differential Equations - model that
captures temporal point processes with a piecewise-continuous latent
trajectory. The model demonstrates its predictive capabilities on various
synthetic and real-world marked point process datasets.

We conclude the final list of models from six different ones. Five of them are
selected from (Enguehard et al., 2020): SA-COND-POISSON, SA-LNM, SA-MLP-MC,
SA-RMTPP-POISSON, SA-SA-MC. Where the beginning of the model’s name ”SA-”
means the type of encoder (Self-Attention), and the rest of the name marks the
type of encoder. The sixth model used in experimentation is Neural Jump
Stochastic Differential Equations (NJSDE).

For all models, we use the hyperparameters specified in the original
literature, with an exception of batch size and training epochs. Experiments
on Vinted data use the same training configuration as experiments on Stack
Overflow users’ activity.

Furthermore, we explore different solutions to the cold start problem in
modeling behavior of new Vinted users, we experiment with parameterizing self-
attention based NTPPs. Similarly to (Gu and Budhkar, 2021), we join processed
tabular user features to the output of Encoder’s transformer network. We train
all models on a workstation with a Intel 32 core 3rd gen Xeon CPU and 120 GB
memory. The complete implementation of our algorithms and experiments are
available at https://github.com/dqmis/ntpps.

###  4.3. Model evaluation

All trained models are trained and evaluated using a five-fold cross-
validation strategy. As in (Jia and Benson, 2019), we use a weighted F1 score
as one of the primary evaluation metrics. This allows us to verify model’s
capabilities of predicting the type of the next event given the time it
occurred. As a secondary metric, we use classification accuracy, mainly to
compare our results with (Enguehard et al., 2020). Despite its popularity in
quantifying the predictive performance of TPPs, we do not use MAE/MSE metric
in this paper. As the metric compares actual intensity values with the model
predicted ones, the lack of such data (only synthetic Hawkes dataset intensity
values are known) is the main reason that determines our choice. Additionally,
we assess each model’s predictions via a classification report. As the data
imbalance problem is a concern, the metric enables us to validate model’s
performance in predicting low-samples label.

##  5\. Results

Table 2. Evaluation on Hawkes and Stack Overflow datasets. Best performances are boldened. Where comparable, results from (Jia and Benson, 2019) and (Enguehard et al., 2020) are displayed in italic font style.  | Hawkes |  | Stack Overflow
---|---|---|---
Model | Accuracy | F1 score |  | Accuracy | F1 score
NJSDE | .541 | .552 |  |  .548 (.527) | .363
SA-COND-POISSON | .538 | .538 |  | .501 |  .332 (.326)
SA-LNM | .537 | .536 |  | .369 | .319 (.305)
SA-MLP-MC | .536 | .531 |  | .216 | .194 (.327)
SA-RMTPP-POISSON | .526 | .474 |  | .515 | .316 (.288)
SA-SA-MC | .507 | .467 |  | .382 | .305 (.324)

We report our results on Hawkes and Stack Overflow data (table 2) and on
Vinted data (table 3). First, we note that replication of the models’
performance results provided by (Jia and Benson, 2019) and (Enguehard et al.,
2020) are mostly successful. Notably, the results of SA-MLP-MC model trained
on the Stack Overflow dataset are not similar to the ones presented in (Jia
and Benson, 2019). This could be caused by a mismatch in training environments
or errors in our or (Enguehard et al., 2020) evaluation process. Notably, the
best performing models were NJSDE and SA-COND-POISSON. Both presented great
results while classifying synthetic Hawkes events and Stack Overflow users’
actions. However, NJSDE model required much more resources and took longer to
train to reach high accuracy (15 hours to train NJSDE model vs. 4 hours to
train SA-COND-POISSON). NJSDE implementation proposed by (Jia and Benson,
2019) requires time series conversion into grid later used for modeling. This
action consumes more memory as the sequence count in the batch grows. Because
of this, we could not validate the NJSDE model’s performance on Vinted data.

Figure 5. Intensity functions on several labels provided by SA-COND-POISSON
model trained on Vinted (top) and Stack Overflow (bottom) datasets. Common
event’s intensity (marked green color) is always higher than rare event’s one
(marked black color). This results in the model omitting rare events while
classifying the next event’s type.

Intensity functions on several labels provided by SA-COND-POISSON model
trained on Vinted (top) and Stack Overflow (bottom) datasets. Dominant class
(samples count in the dataset wise) label always has a higher intensity value
than the recessive one. This results in the model omitting rare events while
classifying the next event’s type.

Notably, all models, including the best-performing ones, suffer from an
inability to classify rare events. This can be seen while comparing the
accuracy and F1 score metrics (also see classification report in Table 4). F1
score and accuracy used in the papers that inspired our research fail to
report this phenomenon. This is caused by a significant data imbalance in the
Stack Overflow and later discussed Vinted datasets. Also, some types of events
could be hard to capture, not only based on their rarity but also their
nature. For example, events inside Stack Overflow dataset like Yearling
(awarded for being an active user for one year time period) or Caucus (awarded
for visiting an election during any phase of an active election and having
enough reputation for casting a vote) can be difficult to identify without a
time dimension. Also, some of the badges like Promoter (given for the first
bounty user offers on his question) are less based on the user’s activity and
more on external factors like the importance of the asked question.

Table 3. Evaluation on Vinted dataset. The scores are provided in two sections: where model was parameterised with user features / and where it was not. Best performances are boldened. Model | Accuracy | F1 score
---|---|---
SA-LNM | .842 / .840 | .771 / .768
SA-COND-POISSON |  .882 / .882 |  .857 / .856
SA-SA-MC | .707 / .839 | .657 / .767
SA-MLP-MC | .654 / .643 | .676 / .670
SA-RMTPP-POISSON | .842 / .839 | .771 / .767

Same tendencies can be seen while inspecting performance results of models
trained on Vinted data. While the best performing SA-COND-POISSON model was
able to reach an F1 score of 0.857, it still failed to predict some event
types like Purchase and Sale (see Table 5). Interestingly Sale event had twice
as many samples as the Listing but failed to be captured by the model. As with
Stack Overflow badges, this can be explained by looking at the nature of the
event. Searching for a new dress to buy or listing some pair of shoes are
events caused mainly by users’ habits. Purchases are motivated more by
Vinted’s search quality or recommendation system. The sale of the uploaded
item is also less caused by its seller’s past activity and more by the item’s
quality and appeal to the buyers. Inspecting differences between intensity
values of the SA-COND-POISSON model trained on Vinted and Stack Overflow data
(see Figure 5) gives a better understanding of the model’s behavior. There is
a significant difference between the intensity values of a frequent and less
frequent event type. The difference tends to stay the same between all time
points. This results in the model capturing overall intensity changes but
lacking the ability to identify rare events. Furthermore, our experimentation
with parameterizing encoder-decoder architecture-based models with user’s
static features did not produce valuable results (see Table 3). There is one
case (SA-SA-MC) where parameterized model performs better than its
counterpart, but this tendency is not identified while comparing the rest of
the models. The lack of models’ performance improvement is probably caused by
the fact that selected user features do not directly impact user’s behavior.
However, to verify if the NTPPs parameterization with tabular features is
beneficial, we need to conduct more detailed experiments with a more diverse
set of features.

##  6\. Conclusion

We discussed broad industry application cases of NTPPs and their ability to
capture online platform users’ behavior. We used synthetic Hawkes dataset,
Stack Overflow users’ badges, and second-hand marketplace Vinted user activity
as our experimentation datasets. Furthermore, we identify that Self-Attention
and Neural Ordinary Differential Equations based NTPP models fail to capture
the time of the next rare event but succeed in identifying overall user
activity intensity. While the NJSDE model is optimal for identifying the next
event’s type and time, we note that it is not scalable to extensive industry
data. Our experimentation with parametrizing Self-Attention-based NTPPs did
not show any significant improvements. However, we identify that we need to
conduct more experiments to verify if the method is beneficial.

###  6.1. Broader Impact

While the explored NTPPs are not suitable for predicting the time of the next
rare event, it does not mean they are not beneficial for subscription and non-
subscription-based online platforms. These models can be valuable when solving
problems such as churn prediction. By grouping individual events into user
sessions, we would be able to detect when a general user’s intensity is
decreasing, thus letting us know when to act. Furthermore, we can optimize
recommendation systems or marketing strategies by knowing the time of the next
user’s arrival on the online platform.

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##  Appendix A Classification reports

Table 4. Classification report of NJSDE model predictions on Stack Overflow dataset. Event type |  | Precision | Recall | F1 Score | # samples
---|---|---|---|---|---
Nice Question |  | 0 | 0 | 0 | 6,016
Good Answer |  | 0 | 0 | 0 | 1.939
Guru |  | 0 | 0 | 0 | 653
Popular Question |  | .471 | .975 | .634 | 42,599
Famous Question |  | 0 | 0 | 0 | 5,544
Nice Answer |  | .317 | .511 | .392 | 6,032
Good Question |  | 0 | 0 | 0 | 1,846
Caucus |  | 0 | 0 | 0 | 1,788
Notable Question |  | 0 | 0 | 0 | 21,956
Necromancer |  | 0 | 0 | 0 | 1,308
Promoter |  | 0 | 0 | 0 | 207
Yearling |  | 0 | 0 | 0 | 2,400
Revival |  | 0 | 0 | 0 | 832
Enlightened |  | 0 | 0 | 0 | 1,859
Great Answer |  | 0 | 0 | 0 | 251
Populist |  | 0 | 0 | 0 | 99
Great Question |  | 0 | 0 | 0 | 269
Constituent |  | 0 | 0 | 0 | 526
Announcer |  | 0 | 0 | 0 | 364
Stellar Question |  | 0 | 0 | 0 | 65
Booster |  | 0 | 0 | 0 | 17
Publicist |  | 0 | 0 | 0 | 8
Table 5. Classification report of SA-COND-POISSON model predictions on Vinted dataset. Event type |  | Precision | Recall | F1 score | # samples
---|---|---|---|---|---
Listing |  | .685 | .654 | .669 | 102,297
Purchase |  | 0 | 0 | 0 | 23,642
Sale |  | 0 | 0 | 0 | 293,68
Search |  | .904 | .968 | .935 | 820,458

##  Appendix B Hawkes datasets

The parameters of our Hawkes datasets are:

$\mu=\left[{\begin{array}[]{c}0.1\\\ 0.05\\\
\end{array}}\right],\alpha=\left[{\begin{array}[]{cc}0.2&0.0\\\ 0.0&0.4\\\
\end{array}}\right],\beta=\left[{\begin{array}[]{cc}1.0&1.0\\\ 1.0&1.0\\\
\end{array}}\right]$