# Millimeter Wave and sub-THz Indoor Radio Propagation Channel Measurements,
Models, and Comparisons in an Office Environment

Yunchou Xing1, Theodore S. Rappaport1, and Amitava Ghosh2 (Invited paper) This
research is supported by the NYU WIRELESS Industrial Affiliates Program and
NSF Research Grants: 1909206 and 2037845. 1NYU WIRELESS, NYU Tandon School of
Engineering, Brooklyn, NY, 11201, {ychou<EMAIL_ADDRESS>
2NOKIA Bell Labs, Naperville, IL, 60563<EMAIL_ADDRESS>
###### Abstract

This letter provides a comparison of indoor radio propagation measurements and
corresponding channel statistics at 28, 73, and 140 GHz, based on extensive
measurements from 2014-2020 in an indoor office environment. Side-by-side
comparisons of propagation characteristics (e.g., large-scale path loss and
multipath time dispersion) across a wide range of frequencies from the low
millimeter wave band of 28 GHz to the sub-THz band of 140 GHz illustrate the
key similarities and differences in indoor wireless channels. The measurements
and models show remarkably similar path loss exponents over frequencies in
both line-of-sight (LOS) and non-LOS (NLOS) scenarios, when using a one meter
free space reference distance, while the multipath time dispersion becomes
smaller at higher frequencies. The 3GPP indoor channel model overestimates the
large-scale path loss and has unrealistic large numbers of clusters and
multipath components per cluster compared to the measured channel statistics
in this letter.

###### Index Terms:

mmWave; THz; channel models; multipath time dispersion; 5G; 6G; large-scale
path loss; 3GPP; InH.

##  I Introduction

The use of wide bandwidths (e.g., $\geq$100 MHz) in 5G and future wireless
communication systems will enable multi-Gbps data rates for mobile devices and
will usher in many new applications such as wireless cognition and centimeter-
level positioning [1, 2, 3]. Highly directional electronically-steered antenna
arrays will be used by both handset terminals and base stations, resulting in
directional wireless channels at mmWave, a significant departure from sub-6
GHz frequencies that use less directional antennas but undergo less
penetration and diffraction loss from obstacles [1, 2, 4]. As shown in our
companion paper [1], the FCC has recently opened up spectrum above 95 GHz with
four new unlicensed bands from 116 GHz to 246 GHz, ushering in a new era of
wireless networks that will have hundreds of Gbps of throughput.

Propagation channels at mmWave frequencies in the 24-73 GHz range are somewhat
different from sub-6 GHz channels in transmission properties yet are viable
through the use of directional antennas on both ends of the link [4, 5, 6, 7,
8, 9]. However, there is very little known about indoor channels above 100 GHz
that have a coverage range of 30-40 m. Furthermore, it is currently unknown
whether sub-THz wireless networks (i.e., 100-300 GHz) have similar or
contrasting propagation characteristics compared to radio channels at lower
frequencies. Knowledge of channel characteristics at frequencies above 100
GHz, as well as key differences from lower frequencies, is vital for the
creation of frequency-dependent channel models that can be applied over vast
frequency ranges (e.g., from 1-300 GHz) to support the design of multi-band
indoor wireless modems in global standard bodies such as 3GPP and IEEE [10].
Also, knowledge of how channel characteristics vary over wide frequency ranges
can be useful for futuristic applications such as intelligent reflecting
surfaces [11] and precise position location [3]. Therefore, this work provides
needed insights on indoor wireless channels from mmWave to sub-THz frequencies
for futuristic wireless system designs.

Previous channel measurements in frequency bands from 28 to 380 GHz [12, 13,
14] have mostly focused on very close ranges (less than 10 m) due to the
difficulty in achieving sufficient transmit power and large measurable path
loss range [2]. Prior measurements at frequencies above 100 GHz [15, 16, 17]
focused on line-of-sight (LOS) propagation using either reflective materials
[15] or an RF-over-fiber extension [16, 17] of a VNA based system achieving
over 100 m distance. However, no prior work has performed extensive indoor
measurements wherein the same environment is used over a vast frequency range
with commercially relevant distances of many tens of meters to provide
comparisons of channel characteristics.

This paper presents the first comparison of the radio propagation
characteristics from mmWave to sub-THz frequencies in an identical indoor
office setting (the NYU WIRELESS center) based three different indoor
propagation measurement campaigns at 28, 73, and 142 GHz with coverage
distances up to 40 m. Section II shows both indoor directional and
omnidirectional large-scale path loss models at 142 GHz and compares the data
to 28 and 73 GHz using identical locations in the same office. Section III
provides indoor channel statistics including delay spread, and the number of
clusters and multipath components (MPCs) at 28, 73, and 142 GHz. Finally,
concluding remarks in Section IV show that, after normalizing over antenna
gains and using a 1 meter free space reference distance, how remarkably
similar the path loss exponents are for both LOS and non-LOS (NLOS) channels
over 28-142 GHz. The multipath time dispersion becomes less prominent at
higher frequencies meaning that it may be more difficult to exploit multipath
diversity at THz frequencies without additive reflecting surfaces.

##  II Indoor Radio Propagation Measurements and Path Loss models at 28, 73,
and 142 GHz

Above 100 GHz, high phase noise and Doppler spread, limited output power due
to device limitations, and the need to fabricate more compact directional
phased arrays present challenges for the deployment of future wireless
networks above 100 GHz [2, 15, 18, 19]. To overcome these challenges, it is
critical that designers first understand the channel characteristics for radio
frequencies above 100 GHz.

###  II-A Measurement Systems and Environments

To evaluate the similarity and differences of incumbent 5G bands with
futuristic sub-THz bands above 100 GHz, extensive radio propagation
measurements were conducted in an indoor hotspot (InH) office environment at
28, 73, and most recently 142 GHz over a period of 6 years since 2014 with TX-
to-RX (TR) separation distances up to 40 m [6].

Sliding correlation-based wideband channel sounder systems with identical
steerable horn antennas at both the TX and RX for each frequency were used
during the measurements [20, 4, 21], as shown in Table I. The channel sounders
provide power delay profiles (PDPs) with dual-directional angular information.
Five TX locations and 20 RX locations were selected across the entire floor,
with the same locations used at 28, 73, and 142 GHz, resulting in 20 TX-RX
combinations for each measured band (8 LOS and 12 NLOS locations) ranging from
3.9 m to 40.0 m. The measured office environment and measurement procedures
are described in [6, 22].

TABLE I: Summary of channel sounder systems and antennas used in measurements at 28, 73, and 142 GHz [4, 20, 21]. The overall received power does not change with the signal bandwidth but the RX noise power and noise figure change [23]. Thus, higher gain (more directional) antennas were used at both the TX and RX to compensate for noise and greater path loss in the first meter at higher frequencies. RF Freq. | RF Null-to-Null | Antenna | Ant. Gain
---|---|---|---
(GHz) | Bandwidth | HPBW | (dBi)
28 [4] | 0.8 GHz | 30° | 15.0
73 [20] | 0.8 GHz | 15° | 20.0
142 [21] | 1.0 GHz | 8° | 27.0

###  II-B Path Loss Models at 28, 73, and 142 GHz

Radio propagation in mmWave and THz bands is range limited due to the severe
path loss in the first meter and the limited output power of amplifiers, which
are compensated by using directional antennas/arrays [1, 2]. The measured path
loss in this letter is calculated by:

$\begin{split}PL~{}\text{[dB]}&=P_{t}+G_{t}+G_{r}-P_{r}+G_{sym},\end{split}$
(1)

where $P_{t}$ is the output power fed into the TX antenna in dBm, $G_{t}$ and
$G_{r}$ are TX and RX antennas gains in dBi, respectively, $P_{r}$ is the
measured received power in dBm, and $G_{sym}$ is the processing gain of the
channel sounder system in dB.

The 1 m close-in (CI) free space reference distance model (2) is one of the
most commonly used large-scale path loss models to predict the signal strength
over distance, and is valid over different frequency bands [22, 24, 25, 10,
6]:

$\small\begin{split}PL^{CI}(f_{c},d_{\text{3D}})\ &=\text{FSPL}(f_{c},\text{1
m})+10n\log_{10}\left(\dfrac{d_{3D}}{\text{1 m}}\right)+\chi_{\sigma},\\\
\text{FSPL}(f_{c},\text{1
m})&=32.4+20\log_{10}(\dfrac{f_{c}}{1\;\text{GHz}}),\end{split}$ (2)

where  $d_{\text{3D}}$ is the 3D separation distance between the TX and RX
antennas, FSPL$(f_{c},1\;\text{m})$ is the free space path loss at carrier
frequency $f_{c}$ in GHz at 1 m in dB, $n$ is the path loss exponent (PLE),
and $\chi_{\sigma}$ is the shadow fading in dB (a zero mean Gaussian random
variable with a standard deviation $\sigma$ in dB) [24, 9, 25, 23, 26].

TABLE II: Directional InH-Office channel parameters of CI and CIF path loss models, RMS delay spread, the number of clusters, and the number of MPCs per cluster in both LOS and NLOS environments at 28, 73, and 142 GHz [22, 24, 10, 9, 27, 6]. Environment type |  LOS |  $\textbf{NLOS}_{\textbf{Best}}$ |  NLOS
---|---|---|---
Frequency [GHz] | 28 | 73 |  142 | 28 | 73 |  142 | 28 | 73 |  142
Directional Antenna HPBW | 30° | 15° | 8°  | 30° | 15° | 8°  | 30° | 15° | 8°
Single-Band | n  | 1.90 | 1.63 | 2.05  | 2.75 | 3.30 | 3.21  | 4.39 | 5.51 | 4.60
Path Loss CI [9, 27, 22] |  $\sigma^{CI}$ [dB]  | 3.38 | 3.06 | 2.89  | 7.00 | 8.76 | 6.03  | 7.30 | 8.94 | 13.80
Multi-Band |  $n$ | 1.86  | 3.07  | 5.02
Path Loss CI [9, 27, 22] |  $\sigma^{CI}$ [dB]  | 3.45  | 7.67  | 13.97
Multi-Band |  $n,b$ |  $n=1.86,b=0.07$ |  $n=3.07,b=0.05$ |  $n=5.02,b=0.03$
Path Loss CIF [9, 27, 22] |  $\sigma^{CIF}$ [dB]  | 3.45  | 7.67  | 13.85
RMS Delay Spread [22, 24] |  $\min_{DS}$ [ns]  | 0.87 | 0.76 | 0.69  | 0.92 | 3.74 | 0.60  | 0.57 | 0.51 | 0.28
|  $\max_{DS}$ [ns]  | 5.50 | 5.34 | 11.94  | 44.49 | 31.37 | 10.76  | 198.55 | 141.97 | 92.45
|  $\mu_{DS}$ [ns]  | 3.85 | 3.53 | 2.71  | 10.23 | 7.39 | 5.65  | 17.64 | 12.50 | 8.86
NumCluster [6] |  $\mu_{NC}$ | 1.41 | 1.32 | 1.25  | 1.65 | 1.48 | 1.16  | 3.41 | 2.60 | 2.39
|  $\sigma_{NC}$ | 0.85 | 0.96 | 0.94  | 0.78 | 0.89 | 0.69  | 1.96 | 1.70 | 1.48
NumMPCperCluster [6] |  $\mu_{MPC}$ | 2.45 | 2.53 | 2.11  | 2.56 | 2.44 | 1.98  | 3.16 | 2.80 | 1.18
|  $\sigma_{MPC}$ | 2.19 | 2.27 | 1.43  | 1.54 | 2.18 | 2.26  | 4.56 | 5.20 | 2.21

The indoor directional LOS PLE for the best-fit CI path loss model is $n=$
1.90 at 28 GHz, $n=$ 1.63 at 73 GHz, and is $n=$ 2.05 at 142 GHz with a shadow
fading standard deviation of $\sigma=$ 3.38 dB, 3.06 dB, and 2.89 dB,
respectively, as presented in Table II. These values all are close to the
theoretical free space PLE value of 2.0, with lower mmWave frequencies
experiencing waveguiding effects and a PLE less than 2.0. This hints at the
fact that higher sub-THz frequencies have fewer reflections in LOS directional
channels (e.g., less waveguiding) due to narrower beam antennas at both ends
of the link that attenuate energy toward the reflecting surfaces of walls,
ceiling, and floor (as shown in Table I). The LOS measurements show that there
is 2-4 dB more average loss at 10 m and 3-7 dB more average loss at 40 m at
142 GHz compared to the average loss at 28 and 73 GHz when referenced to a one
meter free space reference distance [9, 2, 28].

The $\text{NLOS}_{\text{Best}}$ is defined as the best pointing direction of
both the TX and RX antennas for which the maximum power is received at the RX
at each of the NLOS measurement locations. The best-fit
$\text{NLOS}_{\text{Best}}$ PLEs are $n=$ 2.75, 3.30, and 3.21 at 28, 73, and
142 GHz, respectively, with shadow fading standard deviations of $\sigma=$
7.00, 8.76, and 6.03 dB, respectively, showing that the indoor
$\text{NLOS}_{\text{Best}}$ channels at 73 and 142 GHz are more lossy (higher
PLEs) than the channels at 28 GHz.

Two transmission properties, reflections and penetrations, appear to cause the
slight difference of the NLOS PLEs at 28 and 142 GHz [2, 8], since it is known
that as frequencies increase, there is less transmissivity through reflecting
objects and more power is reflected (i.e., greater power in the reflection
direction but less penetrated power is observed at 142 GHz compared to 28 and
73 GHz)[8, 29]. However, the surfaces are relatively rough at higher
frequencies, and indoor reflection and scattering measurements at 28, 73, and
142 GHz show that the scattered power is negligible (more than 20 dB less)
compared to the signal power in the reflection direction for most indoor
materials (e.g., drywall and clear glass) [8, 29].

Omnidirectional path loss models were developed from the directional
measurements by synthesizing omnidirectional antenna patterns, received power,
and path loss from directional antennas [30, 24]. The LOS omnidirectional PLE
at 142 GHz is 1.74 with a shadow fading standard deviation of 3.62 dB, which,
as expected, is less than the LOS directional PLE of 2.05 at the same 142 GHz
due to the capture of energy over the entire horizon using an omnidirectional
pattern [30, 24]. The LOS omnidirectional channel offers 3.1 dB less average
loss at 10 m and 5.0 dB less average loss at 40 m than the LOS directional
channel at 142 GHz (with antenna gains removed). However, in practice,
omnidirectional antennas cover a shorter link range due to the lower antenna
gain. The NLOS omnidirectional PLE at 142 GHz is 2.83 with a shadow fading
standard deviation of 6.07 dB which is a much lossier channel than LOS, yet
offers better link coverage than the $\text{NLOS}_{\text{Best}}$ PLE of 3.21
and the arbitrary directional NLOS PLE of 4.60 at 142 GHz as shown in Tables
II and III. The higher PLE in NLOS at higher frequencies suggest accurate
beamforming algorithms will be needed to find, capture, and combine the most
dominant multipath energy to maintain indoor NLOS communication links above
100 GHz [2, 28, 7].

Figure 1: InH-Office 28, 73, and 142 GHz multi-band omnidirectional CI and CIF
path loss models for both LOS and NLOS scenarios with antenna gains removed
and with respect to a 1 m free space reference [9, 22, 27]. The diamonds and
circles represent measured omnidirectional path loss in NLOS and LOS
locations, respectively [6, 22, 8]. The fixed reference frequency $f_{0}$ for
the CIF path loss model is 81 GHz for both LOS and NLOS conditions. Results
here and Table III show the 142 GHz band is lossier in NLOS conditions, with
the CI and CIF models being remarkably similar [9, 22].

The omnidirectional PLE and shadowing parameters of the CI path loss model
with 1 m free space reference distance for both LOS and NLOS over each of the
three bands and for all bands combined are summarized in Table III. The indoor
omnidirectional LOS PLEs and shadow fading standard deviations $(n,\sigma)$
are (1.17, 2.72 dB), (1.36, 2.30 dB), and (1.74, 3.62 dB) at 28, 73, and 142
GHz bands, respectively, and the higher LOS PLE at 142 GHz indicates that the
MPCs undergo penetration, reflection, and absorption with more loss at 142 GHz
than at 28 and 73 GHz [26, 8]. In the NLOS omnidirectional case, the PLEs and
shadow fading standard deviations $(n,\sigma)$ are (2.37, 7.22 dB), (2.81,
8.71 dB), and (2.83, 6.07 dB) at 28, 73, and 142 GHz, respectively, showing
that 28 GHz is less lossy than both 73 and 142 GHz, with the two higher bands
behaving very similarly, except for the first meter loss.

The CI model with a frequency-weighted PLE (CIF) path loss model [22, 9] was
proposed as a viable multi-band model for indoor path loss and can be
considered as an extension of the CI path loss model to offer an extra degree
of freedom for more accurate statistical modeling over a wide range of
frequencies:

$\small\begin{split}PL^{CIF}&(f_{c},d_{\text{3D}})\;\text{[dB]}=\text{FSPL}(f_{c},\text{1
m})+\\\
&10n\left(1+b\left(\dfrac{f-f_{0}}{f_{0}}\right)\right)\log_{10}\left(\dfrac{d}{\text{1
m}}\right)+\chi^{CIF}_{\sigma},\end{split}$ (3)

where $n$ is the PLE at the weighted frequency average $f_{0}$ of all
measurements for each specific environments, $b$ is a model fitting parameter
that presents the slope of linear frequency dependency of path loss, and
$n\left(1+b({f-f_{0}})/{f_{0}}\right)$ represents the frequency-dependent PLE
at frequency $f$. The weighted average frequency $f_{0}$ is computed over $K$
frequency bands as $f_{0}=\sum_{k=1}^{K}f_{k}N_{k}/\sum_{k=1}^{K}N_{k}$, where
$N_{k}$ is the number of measurements at a particular frequency $f$ [22, 9].

In this paper, we used the data to calculate $f_{0}=$ 81 GHz for both LOS and
NLOS scenarios. In LOS omnidirectional scenarios, $n^{CIF}_{LOS}=1.42$ and
$b^{CIF}_{LOS}=0.29$ are the best-fit parameters of the omnidirectional CIF
path loss model with a shadow fading standard deviation of
$\sigma^{CIF}_{LOS}=2.94$ dB, as shown in Table III, when applied to all
measurements across 28, 73, and 142 GHz. In NLOS omnidirectional scenarios,
$n^{CIF}_{NLOS}=2.66$ and $b^{CIF}_{NLOS}=0.11$ are the best-fit parameters of
the omnidirectional CIF path loss model with a shadow fading standard
deviation of $\sigma^{CIF}_{NLOS}=7.53$ dB.

Note that the CI path loss model (2) can also be used as a multi-band path
loss model, using only a single parameter PLE [22, 27, 9]. Fig. 1 shows the
indoor office 28, 73, and 142 GHz multi-band omnidirectional CI and CIF path
loss models for both LOS and NLOS scenarios with antenna gains removed at both
ends of the link and with respect to a 1 m free space reference. CI and CIF
path loss models show remarkable similarity in terms of path loss exponents
over 28, 73, and 142 GHz bands, when referenced to the first meter free space
reference distance [9, 22, 27], implying the extra parameter in the CIF model
may not be necessary over wide frequency bands. This illuminates the fact that
an identical PLE may accurately model path loss over a vast range of
frequencies, with the only frequency-dependent effect being the path loss in
the first meter of propagation as energy spreads into the far field [1, 22,
27, 9].

TABLE III: Omnidirectional InH-Office channel parameters of CI and CIF path loss models, RMS delay spread, the number of clusters, and the number of MPCs per cluster at 28, 73, and 142 GHz compared to 3GPP InH-Office standard [22, 24, 10, 9, 27, 6]. Environment type |  InH-Office LOS |  InH-Office NLOS
---|---|---
omnidirectional NYU & 3GPP | NYU WIRELESS |  3GPP InH-Office | NYU WIRELESS |  3GPP InH-Office
Frequency [GHz] | 28 | 73 | 142 | 28 |  73 | 28 | 73 | 142 | 28 |  73
Single-Band | n  | 1.17 | 1.36 | 1.74 | N/A | N/A  | 2.37 | 2.81 | 2.83 | N/A | N/A
Path Loss CI [9, 27, 22] |  $\sigma^{CI}$ [dB]  | 2.72 | 2.30 | 3.62 | N/A | N/A  | 7.22 | 8.71 | 6.07 | N/A | N/A
Multi-Band |  $n$ | $1.42$ | 1.73  | 2.66 | 3.19
Path Loss CI [9, 27, 22] |  $\sigma^{CI}$ [dB]  | 3.71 | 3.00  | 7.82 | 8.29
Multi-Band |  $n,b$ | $n=1.42,b=0.29$ | N/A | N/A  | $n=2.66,b=0.11$ | N/A | N/A
Path Loss CIF [9, 27, 22] |  $\sigma^{CIF}$ [dB]  | 2.94 | N/A | N/A  | 7.53 | N/A | N/A
RMS Delay Spread [22, 24] |  $\min_{DS}$ [ns]  | 0.70 | 0.60 | 0.71 | N/A | N/A  | 0.60 | 0.50 | 0.60 | N/A | N/A
|  $\max_{DS}$ [ns]  | 134.40 | 101.90 | 11.94 | N/A | N/A  | 198.50 | 142.00 | 60.87 | N/A | N/A
|  $\mu_{DS}$ [ns]  | 10.80 | 6.24 | 3.00 | 20.40 | 20.21  | 17.10 | 12.30 | 9.20 | 27.40 | 21.52
NumCluster [6] |  $\mu_{NC}$ | 4.60 | 2.76 | 1.90 | 15 | 15  | 5.40 | 3.20 | 2.80 | 19 | 19
|  $\sigma_{NC}$ | 1.94 | 2.32 | 1.30 | N/A | N/A  | 1.96 | 1.70 | 1.65 | N/A | N/A
NumMPCperCluster [6] |  $\mu_{MPC}$ | 4.70 | 3.43 | 2.40 | 20 | 20  | 6.40 | 3.20 | 2.20 | 20 | 20
|  $\sigma_{MPC}$ | 3.65 | 2.86 | 2.20 | N/A | N/A  | 4.58 | 5.20 | 2.47 | N/A | N/A

##  III Multipath Statistics at 28, 73, and 142 GHz

The indoor wireless channel statistics at 28, 73, and 142 GHz bands are
summarized in Tables II and III for directional and omnidirectional channels,
respectively.

###  III-A RMS Delay Spread

RMS Delay spread characterizes the multipath richness, time dispersion, and
coherence bandwidth of a radio propagation channel, depending on the signal’s
bandwidth [23, 26]. A 5 dB SNR threshold relative to the mean thermal noise
floor of a raw PDP was used for detecting and keeping the MPCs in each PDP
[24]. The number of MPCs are computed by peak tracking algorithms as
illustrated in [6] that one peak in a PDP is counted as a valid MPC if the
time delay between its next adjacent peak (above the 5 dB SNR threshold) is
larger than the channel sounder time resolution (the inverse of the channel
bandwidth).

As shown in Table II, there is negligible RMS DS in LOS directional scenarios
with a mean of 3-4 ns, which is close to the width of the channel sounder’s
probing signal across these three frequency bands when directional antennas
are used to line up antenna boresights (our probe had a 3 dB power width of
2.5 ns at 28 and 73 GHz and 2 ns at 142 GHz as shown in Table I). The excess
delays of the multipath components reflected from the walls, ceiling, and
floor would be within (and cause a spread of) the original pulse width.

In $\text{NLOS}_{\text{Best}}$ scenarios, the minimum RMS DS ($\min_{DS}$) is
extremely small (e.g., 0.5 ns) and similar to the minimum RMS DS in LOS
directional environments across 28, 73, and 142 GHz bands, however, the mean
RMS DS ($\mu_{DS}$) increases by a few ns (e.g., 3-6 ns) at all frequencies
compared to LOS directional scenarios, and the maximum RMS DS ($\max_{DS}$)
was observed to be 44 ns at 28 GHz, 31 ns at 73 GHz, and 11 ns at 142 GHz,
with 90% of the RMS DS less than 40 ns, 25 ns, and 9 ns at 28, 73, and 142 GHz
bands, respectively. The data clearly show maximum observable RMS DS is
frequency dependent, with much less maximum delay spread at higher
frequencies.

The omnidirectional RMS DS obtained from measurements for both the LOS and
NLOS scenarios at 28, 73, and 142 GHz bands [6, 22] are presented in Table III
and are compared with mean $\mu_{DS}$ and standard deviation $\sigma_{DS}$
values from 3GPP TR 38.901 Release 16 (see Table 7.5-6 in [10]). Note that
3GPP does not specify the RMS DS for channels above 100 GHz.

In LOS omnidirectional scenarios, the mean RMS DS decreases with increasing
frequency and is 11, 6, and 3 ns at 28, 73, and 142 GHz bands, respectively.
By contrast, the mean RMS DS predicted by the 3GPP InH channel model is
virtually identical at 28 and 73 GHz (20.40 ns and 20.21 ns, respectively),
however, in our work the measured mean RMS DS at 73 GHz decreases by 40%
(about 5 ns lower) than the measured mean RMS DS at 28 GHz, and the mean RMS
DS decreases by 50% further to a very small 3 ns at 142 GHz.

In NLOS omnidirectional scenarios, when compared to LOS omnidirectional
channels, the mean RMS DS increases by 60%, 100%, and 200% at 28, 73, and 142
GHz, respectively, with mean values of 17, 12, and 9 ns at 28, 73, and 142 GHz
bands, respectively, indicating a wider time spread of multipath energy that
is frequency dependent in NLOS environments. The free space path loss in the
first meter and partition losses are larger at higher frequencies, such that
the signal power of MPCs with large time delays (longer propagation distances
and are more likely to be blocked, and hence have weaker power) may be below
the noise floor and not detected by the channel sounder RX [24]. This is one
cause for why the RMS DS at 142 GHz is lower than the RMS DSs at 73 and 28
GHz.

###  III-B The Numbers of Time Clusters and MPCs per Cluster

The numbers of time clusters and MPCs per cluster depends on the minimum
inter-cluster time void interval (MTI), which is the minimum time interval
between two separate time clusters [6, 31, 4]. There are fewer time clusters
but more MPCs per cluster when the MTI is larger. An MTI of 6 ns [6]
(determined by the width of corridors, 2 m) is used in this letter, based on
the idea that the physical environment helps delineate the observed temporal
cluster partitions of multipath energy arriving at a receiver [31, 4].

The number of time clusters follows a Poisson distribution at all three
frequencies as shown in [6]. The omnidirectional mean numbers of time clusters
are found to decrease with increasing frequency and are 4.60, 2.76, and 1.90
at 28, 73, and 142 GHz, respectively, in indoor LOS scenarios, and are 5.40,
3.20, and 2.80 at 28, 73, and 142 GHz in indoor NLOS scenarios, again showing
the frequency dependent nature of multipath. The channels are more sparse in
both LOS and NLOS omnidirectional scenarios at higher frequencies due to the
larger partition loss (e.g., 4-8 dB higher loss at 142 GHz than 28 GHz for
different materials [8]) which cause MPCs with longer propagation times and
weaker power to fall below the noise floor. Using smart antennas to capture
the most dominant multipath energy for beam combining to achieve range
extension is needed as wireless moves into the THz regime [2, 28, 27, 7].

The number of MPCs per cluster follows a composite of a delta function
$\delta(n-1)$ and a discrete exponential distribution at all three frequencies
[6]. At 28 GHz, the mean number of MPCs per cluster in the NLOS
omnidirectional scenarios is larger than in the LOS omnidirectional scenarios.
However, there are fewer MPCs per cluster in the NLOS scenarios than in the
LOS scenarios at 73 and 142 GHz, since the larger partition loss in NLOS
scenarios makes the channels more sparse at higher frequencies [8, 6].

##  IV Conclusion

Indoor mmWave and sub-THz wireless channels at 28, 73, and 142 GHz are
compared based on extensive radio propagation measurements in a typical indoor
office environment. The indoor office large-scale path loss results (CI and
CIF path loss models) show that there is remarkable similarity in terms of
path loss exponents over 28, 73, and 142 GHz for both LOS and NLOS scenarios,
when referenced to the first meter free-space reference distance [9, 22, 4].
The results imply that THz channels are similar to today’s mmWave wireless
propagation channels except for the path loss in the first meter of
propagation when energy spreads into the far field. Our results differ from
existing 3GPP InH prediction models, and show strong frequency dependence on
multipath time dispersion, with much less time dispersion at higher
frequencies. Mathematical distributions of the number of multipath clusters,
RMS delay spread, the number of multipath components or subpaths per cluster
can be applied for frequencies above and below 100 GHz, although the
statistical means of those distributions decrease with increasing frequency.
The measurements and models presented here will help in the development of
frequency-dependent indoor office channel models that can be applied over vast
mmWave and THz frequency ranges.

## References

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# Analyzing Astronomical Data with Machine Learning Techniques

Mohammad H. Zhoolideh Haghighi E-mail<EMAIL_ADDRESS>
(Department of Physics, K.N. Toosi University of Technology, Tehran P.O. Box
15875-4416, Iran,
School of Astronomy, Institute for Research in Fundamental Sciences (IPM),
Tehran, 19395-5746, Iran, )

###### Abstract

Classification is a popular task in the field of Machine Learning (ML) and
Artificial Intelligence (AI), and it happens when outputs are categorical
variables. There are a wide variety of models that attempts to draw some
conclusions from observed values, so classification algorithms predict
categorical class labels and uses it in classifying new data. Popular
classification models including logistic regression, decision tree, random
forest, Support Vector Machine (SVM), multilayer perceptron, naive bayes,
neural networks have proven to be efficient and accurate applied on many
industrial and scientific problems. Particularly, application of ML to
astronomy has shown to be very useful for classification, clustering and data
cleaning. It is because after learning computers, these tasks can be done
automatically by them in a more precise and more rapid way than human
operators. In view of this, in this paper, we will review some of these
popular classification algorithms, and then we apply some of them to the
observational data of nonvariable and the RR Lyrae variable stars that come
from the SDSS survey. For the sake of comparison, we calculate the accuracy
and $F_{1}$-score of the applied models.

Key words: Machine learning, Classification, Sloan Digital Sky Survey,
Nonvariable stars, RR Lyrae variables.

##  1 Introduction

Machine learning is a branch of artificial intelligence (AI) and computer
science which focuses on the use of data and algorithms to imitate the way
that humans learn. One of the most popular branches of ML is Supervised
Learning (Jain, 1999). It is a machine learning approach defined by its use of
labeled datasets to train algorithms to classify data and predict outcomes.
The labeled dataset, which is called training set, has output labeled
corresponding to input data. The machine is supposed to understand what to
search for, in the unseen data and provide some predictions based on what it
has already learned from the labeled data. There are two main areas where
supervised machine learning comes in handy, classification problems and
regression problems. Classification refers to taking an input value and
mapping it to a discrete value. In classification problems, our output
typically consists of classes or categories, but regression is related to
continuous data and in that the predicted output values are real numbers
(Dutton and Conroy, 1996).

As astronomy is experiencing a rapid growth in the size and complexity of
data, these changes foster the development of data-driven science as a useful
companion to the common model-driven data analysis paradigm, where astronomers
develop automatic tools to mine datasets and extract novel information from
them. In recent years, machine learning algorithms have become increasingly
popular among astronomers, and ML is now used for a wide variety of tasks.
Machine learning algorithms can be used in the field of astronomy and
cosmology for astronomical object detection and object classification
(Richards et al., 2004), clustering (Campana et al., 2007), anomaly detection
(Nun et al., 2014), feature extraction (Sen et al., 2022) and other related
domains.

##  2 Review; Popular Classification Algorithms

###  2.1 Linear regression

Simple linear regression is a type of regression analysis where there is one
independent variable (x) with a linear relationship with another dependent
variable (y). Based on the given data points, the algorithm try to find a line
that models the points in the best possible way. The line can be modeled based
on a linear equation (Efron & Tibshirani, 1991)

$y=a*x+b$ (1)

The motivation of the linear regression algorithm is to find the best values
for $a$ and $b$. The cost function helps us to figure out the best possible
values for $a$ and $b$ which would provide the best fit line for the data
points. Since the best values for $a$ and $b$ are desired, we convert this
search problem into a minimization problem where we would like to minimize the
error between the predicted value and the actual value.

$\operatorname{Minimize}\frac{1}{n}\sum_{i=1}^{n}\left(y_{pred}-y_{i}\right)^{2}$
(2)

$J=\frac{1}{n}\sum_{i=1}^{n}\left(y_{pred}-y_{i}\right)^{2}$ (3)

We choose the above function for minimization. The difference between the
predicted values and ground truth measures the error difference. We square the
error difference and sum over all data points and divide that value by the
total number of data points. This provides the average squared error over all
the data points. Now, using this cost function we can change the values of $a$
and $b$ such that the cost function value settles at the minima. To minimize
the cost function, one can use gradient descent which is a method of updating
$a$ and $b$ to reduce the cost function. The idea is that we start with some
values for $a$ and $b$ and then we change these values iteratively to reduce
the cost. Gradient descent helps us on how to change the values. To update $a$
and $b$, we take gradients from the cost function. To find these gradients, we
take partial derivatives with respect to $a$ and $b$ as follows:

$\begin{gathered}J=\frac{1}{n}\sum_{i=1}^{n}\left(y_{pred}-y_{i}\right)^{2}\\\
J=\frac{1}{n}\sum_{i=1}^{n}\left(a+b\cdot x_{i}-y_{i}\right)^{2}\\\
\frac{\partial J}{\partial a}=\frac{2}{n}\sum_{i=1}^{n}\left(a+b\cdot
x_{i}-y_{i}\right)\Longrightarrow\frac{\partial J}{\partial
a}=\frac{2}{n}\sum_{i=1}^{n}\left(y_{pred}-y_{i}\right)\\\ \frac{\partial
J}{\partial b}=\frac{2}{n}\sum_{i=1}^{n}\left(a+b\cdot x_{i}-y_{i}\right)\cdot
x_{i}\Longrightarrow\frac{\partial J}{\partial
b}=\frac{2}{n}\sum_{i=1}^{n}\left(y_{pred}-y_{i}\right)\cdot
x_{i}\end{gathered}$ (4)

$\begin{gathered}a=a-\alpha\cdot\frac{2}{n}\sum_{i=1}^{n}\left(y_{pred}-y_{i}\right)\\\
b=b-\alpha\cdot\frac{2}{n}\sum_{i=1}^{n}\left(y_{pred}-y_{i}\right)\cdot
x_{i}\end{gathered}$ (5)

The partial derivates are the gradients and they are used to update the values
of $a$ and $b$. $\alpha$ is the learning rate which is a hyper-parameter that
must be specified. A smaller learning rate could help to be closer to the
minima but takes more time to reach the minima, a larger learning rate
converges sooner but there is a chance that one could overshoot the minima.

###  2.2 Logistic regression

Logistic regression is another powerful supervised ML algorithm used for
binary classification problems (when target is categorical). The best way to
think about logistic regression is that it is a linear regression but for
classification problems. Logistic regression essentially uses a logistic
function defined below to model a binary output variable (Sur & Candès, 2019).
The primary difference between linear regression and logistic regression is
that logistic regression’s range is bounded between 0 and 1. In addition, as
opposed to linear regression, logistic regression does not require a linear
relationship between inputs and output variables. In this approach, instead of
defining cost function based on $y$, the sigmoid of $y$ is used.

$\begin{array}[]{l}Output=0~{}or~{}1\\\ \\\ y=a*x+b\\\ \\\
\mathrm{h}\Theta(\mathrm{x})=\operatorname{sigmoid}(\mathrm{y})\\\ \\\
\operatorname{sigmoid}(y)=\frac{1}{1+e^{-y}}\\\ \\\ \end{array}$ (6)

$\displaystyle\operatorname{Cost}(h\ominus(x),y(\text{ actual }))=$
$\displaystyle-\log(h\ominus(x))\text{ if }y=1$ (7)
$\displaystyle-\log(1-h\ominus(x))\text{ if }y=0$

By having the cost function of logistic regression as defined in equation (7),
one can easily find $a$ and $b$ via minimizing this cost function; as a result
$y$ will be determined.

###  2.3 Naive Bayesian (NB)

Using Bayes theorem, we can find the probability of A, happening, given that B
has occurred. Here, B is the evidence and A is the hypothesis. The assumption
made here is that the predictors/features are independent. Bayes theorem can
be rewritten as:

$P(y\mid X)=\frac{P(X\mid y)P(y)}{P(X)}$ (8)

Where X is the feature vector and by substituting for X and expanding using
the chain rule we get,

$P\left(y\mid x_{1},\ldots,x_{n}\right)=\frac{P\left(x_{1}\mid
y\right)P\left(x_{2}\mid y\right)\ldots P\left(x_{n}\mid
y\right)P(y)}{P\left(x_{1}\right)P\left(x_{2}\right)\ldots
P\left(x_{n}\right)}$ (9)

Now, you can obtain the values for each by looking at the dataset and
substitute them into the equation. For all entries in the dataset, the
denominator does not change, it remain static. Therefore, the denominator can
be removed and a proportionality can be introduced.

$P(y\mid x_{1},...,x_{n})\propto P(y)~{}\Pi_{i}^{n}~{}P(x_{i}\mid y)$ (10)

One popular type of Naive Bayes Classifier is Gaussian Naive Bayes. When the
predictors take up a continuous value and are not discrete, we assume that
these values are sampled from a gaussian distribution. Since the way the
values are present in the dataset changes, the formula for conditional
probability changes to,

$P\left(x_{i}\mid
y\right)=\frac{1}{\sqrt{2\pi\sigma_{y}^{2}}}\exp\left(-\frac{\left(x_{i}-\mu_{y}\right)^{2}}{2\sigma_{y}^{2}}\right)$
(11)

In our case, the class variable(y) has only two outcomes, yes or no. There
could be cases where the classification could be multivariate. Therefore, we
need to find the class y with maximum probability.

$y=\text{argmax}P(y)~{}\Pi_{i}^{n}~{}P(x_{i}\mid y)$ (12)

Naive Bayes algorithms are mostly used in sentiment analysis, spam filtering,
recommendation systems etc. They are fast and easy to implement but their
biggest disadvantage is that the requirement of predictors to be independent
(du Buisson et al., 2015). In most of the real cases, the predictors are
dependent, this hinders the performance of the classifier.

###  2.4 Support Vector Machines (SVMs)

Support vector machine is highly preferred by many as it produces significant
accuracy with less computation power. Support Vector Machine, abbreviated as
SVM can be used for both regression and classification tasks. But, it is
widely used in classification objectives. The objective of the support vector
machine algorithm is to find a hyperplane in an N-dimensional space(N — the
number of features) that distinctly classifies the data points (Vapnik, 1995).
To separate the two classes of data points, there are many possible
hyperplanes that could be chosen. Our objective is to find a plane that has
the maximum margin, i.e the maximum distance between data points of both
classes. Maximizing the margin distance provides some reinforcement so that
future data points can be classified with more confidence. Hyperplanes are
decision boundaries that help classify the data points. Data points falling on
either side of the hyperplane can be attributed to different classes. Also,
the dimension of the hyperplane depends upon the number of features. In
logistic regression, we take the output of the linear function and squash the
value within the range of [0,1] using the sigmoid function. If the squashed
value is greater than a threshold value(0.5) we assign it a label 1, else we
assign it a label 0. In SVM, we take the output of the linear function and if
that output is greater than 1, we identify it with one class and if the output
is -1, we identify is with another class. Since the threshold values are
changed to 1 and -1 in SVM, we obtain this reinforcement range of
values([-1,1]) which acts as margin.

In the SVM algorithm, we are looking to maximize the margin between the data
points and the hyperplane. The loss function that helps maximize the margin is
Hinge loss. The cost is 0 if the predicted value and the actual value are of
the same sign. If they are not, we then calculate the loss value.
$c(x,y,f(x))=\left\\{\begin{array}[]{ll}0,~{}~{}\text{ if }y*f(x)\geq 1&\\\
1-y*f(x),\quad\text{ else }\end{array}\quad\right.$

We also add a regularization parameter the cost function. The objective of the
regularization parameter is to balance the margin maximization and loss. After
adding the regularization parameter, the cost functions looks as below.

$\min_{w}[\lambda\|w\|^{2}+\sum_{i=1}^{n}\left(1-y_{i}\left\langle
x_{i},w\right\rangle\right)_{+}]$ (13)

By taking partial derivatives with respect to the weights to find the
gradients. Using the gradients, we can update our weights.

$\frac{\delta}{\delta w_{k}}\lambda\|w\|^{2}=2\lambda w_{k}$ (14)

$\frac{\delta}{\delta w_{k}}\left(1-y_{i}\left\langle
x_{i},w\right\rangle\right)_{+}=\begin{cases}0,&\text{ if }y_{i}\left\langle
x_{i},w\right\rangle\geq 1\\\ -y_{i}x_{ik},&\text{ else }\end{cases}$ (15)

When there is no misclassification, i.e our model correctly predicts the class
of our data point, we only have to update the gradient from the regularization
parameter.

$w=w-\alpha\cdot(2\lambda w)$ (16)

When there is a misclassification, i.e our model make a mistake on the
prediction of the class of our data point, we include the loss along with the
regularization parameter to perform gradient update.

$w=w+\alpha\cdot(y_{i}\cdot x_{i}-2\lambda w)$ (17)

###  2.5 K-means

K-means is an unsupervised classification algorithm, also called
clusterization, that groups objects into k groups based on their
characteristics. The grouping is done by minimizing the sum of the distances
between each object and the group or cluster centroid (MacQueen, 1967). The
distance that is usually used is the quadratic or euclidean distance. The
algorithm has three steps: 1) Initialization: once the number of groups, k has
been chosen, k centroids are established in the data space, for instance,
choosing them randomly, 2) Assignment of objects to the centroids: each object
of the data is assigned to its nearest centroid, 3) Centroids update: The
position of the centroid of each group is updated taking as the new centroid
the average position of the objects belonging to said group. Steps 2 and 3
should be repeated until the centroids do not move, or move below a threshold
distance in each step. The k-means algorithm solves an optimization problem
and the function to be optimized (minimized) is the sum of the quadratic
distances from each object to its cluster centroid. The objects are
represented with d dimension vectors
$\left(\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{n}\right)$ and the
algorithm k-means builds k groups where the sum of the distances of the
objects to its centroid is minimized within each group
$\mathbf{S}=\left\\{S_{1},S_{2},\ldots,S_{k}\right\\}$. The problem can be
formulated as:

$\underset{\mathbf{S}}{\mathrm{min}}\;E\left(\boldsymbol{\mu_{i}}\right)=\underset{\mathbf{S}}{\mathrm{min}}\sum_{i=1}^{k}\sum_{\mathbf{x}_{j}\in
S_{i}}\left\|\mathbf{x}_{j}-\boldsymbol{\mu}_{i}\right\|^{2}$ (18)

where S is the dataset whose elements are the objects $x_{j}$ represented by
vectors, where each of its elements represents a characteristic or attribute.
We will have k groups or clusters with their corresponding centroid
$\boldsymbol{\mu_{i}}$. In each centroid update, from the mathematical point
of view, we impose the extreme (minimum, in this case) necessary condition to
the function $E\left(\boldsymbol{\mu_{i}}\right)$ that, for this quadratic
function (1) is

$\frac{\partial
E}{\partial\boldsymbol{\mu}_{i}}=0\;\Longrightarrow\;\boldsymbol{\mu}_{i}^{(t+1)}=\frac{1}{\left|S_{i}^{(t)}\right|}\sum_{\mathbf{x}_{j}\in
S_{i}^{(t)}}\mathbf{x}_{j}$ (19)

and the solution is to take each group element average as a new centroid. We
have used the gradient descent method. The main advantages of the k-means
method are that it is simple and fast. But it is necessary to decide the value
of k and the final result depends on the initialization of the centroids.
Also, it does not necessarily converge to the global minimum but to a local
minimum.

###  2.6 Decision Trees

Decision Trees are trees that classify objects by sorting them based on
feature values. Each node in a decision tree represents a feature in an
instance to be classified, and each branch represents a value that the node
can assume. Objects are classified starting at the root node and sorted based
on their feature values. As the name goes, it uses a tree-like model of
decisions. In the first split or the root, all attributes/features are
considered and the training data is divided into groups based on this split
(Murthy, 1998). Suppose we have three features, so we will have three
candidate splits. So one can calculate how much accuracy each split will cost,
using a function. The split that costs least is chosen. When there is a large
set of features, it results in large number of split, which in turn gives a
huge tree. Such trees are complex and can lead to overfitting. So, we need to
know when to stop. One way of doing this is to set maximum depth of your
model. Maximum depth refers to the the length of the longest path from a root
to a leaf.

Decision Trees are a type of Supervised Machine Learning (that is you explain
what the input is and what the corresponding output is in the training data)
where the data is continuously split according to a certain parameter. The
tree can be explained by two entities, namely decision nodes and leaves. The
leaves are the decisions or the final outcomes. And the decision nodes are
where the data is split. There are many algorithms out there which construct
Decision Trees, but one of the best is called as ID3 Algorithm. ID3 Stands for
Iterative Dichotomiser 3. Before discussing the ID3 algorithm, we’ll go
through few definitions.

Entropy, also called as Shannon Entropy is denoted by $H(S)$ for a finite set
S, is the measure of the amount of uncertainty or randomness in data.

$H(S)=\sum_{x\in X}p(x)\log_{2}\frac{1}{p(x)}$ (20)

Decision Trees modified Intuitively, it tells us about the predictability of a
certain event. Example, consider a coin toss whose probability of heads is 0.5
and probability of tails is 0.5. Here the entropy is the highest possible,
since there’s no way of determining what the outcome might be. Alternatively,
consider a coin which has heads on both the sides, the entropy of such an
event can be predicted perfectly since we know beforehand that it’ll always be
heads. In other words, this event has no randomness hence it’s entropy is
zero. In particular, lower values imply less uncertainty while higher values
imply high uncertainty.

Information gain is also called as Kullback-Leibler divergence denoted by
IG(S,A) for a set S is the effective change in entropy after deciding on a
particular attribute A. It measures the relative change in entropy with
respect to the independent variables $IG(S,A)=H(S)-H(S,A)$. Alternatively,

$IG(S,A)=H(S)-\sum_{i=0}^{n}P(x)*H(x)$ (21)

Where $IG(S,A)$ is the information gain by applying feature A. $H(S)$ is the
Entropy of the entire set, while the second term calculates the Entropy after
applying the feature A, where $P(x)$ is the probability of event x.

ID3 Algorithm will perform following tasks recursively. a) Create root node
for the tree , b) If all examples are positive, return leaf node ‘positive’ ,
c) Else if all examples are negative, return leaf node ‘negative’ , d)
Calculate the entropy of current state H(S) , e) For each attribute, calculate
the entropy with respect to the attribute ‘x’ denoted by H(S, x) , g) Select
the attribute which has maximum value of IG(S, x) , h) Remove the attribute
that offers highest IG from the set of attributes , i) Repeat until we run out
of all attributes, or the decision tree has all leaf nodes.

###  2.7 Neural Networks

Neural networks are multi-layer networks of neurons that are used to classify
things and make predictions. A neural network is made up of densely connected
processing nodes (Rumelhart et al., 1986), similar to neurons in the brain.
Each node may be connected to different nodes in multiple layers above and
below it. These nodes move data through the network in a feed-forward fashion,
meaning the data moves in only one direction. The node “fires” like a neuron
when it passes information to the next node. A simple neural network has an
input layer, output layer and one hidden layer between them. A network with
more than three layers, including the input and output, is known as a deep
learning network. In a deep learning network, each layer of nodes trains on
data based on the output from the previous layer. Having more layers leads to,
the greater the ability to recognize more complex information, based on data
from the previous layers. The network makes decisions by assigning each
connected node to a number known as a “weight.” The weight represents the
value of information assigned to an individual node (i.e., how helpful it is
in correctly classifying information). When a node receives information from
other nodes, it calculates the total weight or value of the information. If
the number exceeds a certain threshold, the information is passed onto the
next layer. If the weight is below the threshold, the information is not
passed on. In a newly formed neural network, all weights and thresholds are
set to random numbers. As training data is fed into the input layer, the
weights and thresholds refine to consistently yield correct outputs. In Figure
(1), one can see the different parts of a Neural Network.

Figure 1: A schematic view of a Neural Network. As can be seen, there are
three main parts, input layer, hidden layers, and output layers.

##  3 Data

Here we are going to introduce a real sample of astronomical data. RR Lyrae
variables are periodic variable stars, commonly found in globular clusters
(Ivezic et al., 2005). They are used as standard candles to measure galactic
distances, assisting with the cosmic distance ladder. They are pulsating
horizontal branch stars of spectral class A or F, with a mass of around half
the Sun’s. They are thought to have shed mass during the red-giant branch
phase and were once stars of similar or slightly less mass than the Sun,
around 0.8 solar masses. In contemporary astronomy, a period-luminosity
relation makes them good standard candles for relatively nearby targets,
especially within the Milky Way and Local Group. They are also frequent
subjects in the studies of globular clusters. We use the set of photometric
observations of RR Lyrae stars in the SDSS as our data. The data set comes
from SDSS Stripe 82, and combines the Stripe 82 standard stars, which
represent observations of non-variable stars; and the RR Lyrae variables
pulled from the same observations as the standard stars, and selected based on
their variability using supplemental data. The sample is further constrained
to a smaller region of the overall color–color space following
$0.7<u-g<1.35,-0.15<g-r<0.4,-0.15<r-i<0.22,and-0.21<i-z<0.25$ (Ivezic et al.,
2005). These selection criteria lead to a sample of 92,658 non-variable stars,
and 483 RR Lyraes. Two features of this combined data set make it a good
candidate for testing classification algorithms:

##  4 Classification metrics

When performing classification predictions, there are four types of outcomes
that could occur. True positives, which are when you predict an observation
belongs to a class and it actually does belong to that class. True negatives,
which are when you predict an observation does not belong to a class and it
actually does not belong to that class. False positives, which occur when you
predict an observation belongs to a class when in reality it does not. False
negatives, which occur when you predict an observation does not belong to a
class when in fact it does. These four outcomes are often plotted on a
confusion matrix. This matrix can be generated after making predictions on the
test data and then identifying each prediction as one of the four possible
outcomes described above.

The four main metrics used to evaluate a classification model are accuracy,
precision, recall, and $F_{1}$ score.

Accuracy is defined as the percentage of correct predictions for the test
data. It can be calculated easily by dividing the number of correct
predictions by the number of total predictions.

$\text{accuracy}=\frac{\text{correct predictions}}{\text{all predictions}}$
(22)

Precision is defined as the fraction of relevant examples (true positives)
among all of the examples which were predicted to belong in a certain class.

$\text{precision}=\frac{\text{true positives}}{\text{true positives+false
positives}}$ (23)

Recall is defined as the fraction of examples which were predicted to belong
to a class with respect to all of the examples that truly belong in the class.

$\text{recall}=\frac{\text{true positives}}{\text{true positives+false
negatives}}$ (24)

The $F_{1}$-score combines the precision and recall of a classifier into a
single metric by taking their harmonic mean. It is primarily used to compare
the performance of two classifiers. Suppose that classifier A has a higher
recall, and classifier B has higher precision. In this case, the $F1$-scores
for both the classifiers can be used to determine which one produces better
results.

$F_{1}=\frac{2\text{ precision * recall}}{\text{precision + recall}}$ (25)

##  5 Results

In this section, we apply a collection of popular classifier on our data set
to train a set of machine learning models and compare the different models
with each other. In Figures (2, 3), we have plotted our sample data and the
classification boundary for a Logistic Regression and a Gaussian naive Bayes
classifier. In table (1) the accuracy and $F_{1}$-score for different models
are reported which can be used to compare the proposed models.

Figure 2: Logistic Regression classification method is used to separate variable stars from nonvariable stars. The light blue points show nonvariable sources, while the violet show variable sources. The classification boundary is shown by the black line. Logistic Regression attains an accuracy of 0.969 and a $F_{1}$-score of 0.628. Figure 3: Gaussian naive Bayes classification method is used to separate variable stars from nonvariable stars. The light blue points show nonvariable sources, while the violet show variable sources. The classification boundary is shown by the black line. Naive Bayes attains an accuracy of 0.983 and a $F_{1}$-score of 0.655. Classifier | Accuracy | $F_{1}$
---|---|---
Logistic regression | 0.969 | 0.628
Naive Bayesian | 0.983 | 0.654
SVMs | 0.994 | 0.499
Decision Trees | 0.994 | 0.703
Neural Networks | 0.993 | 0.512
Table 1: Accuracy and $F_{1}$ score for six chosen classifier. Although all
models provide a very good accuracy, $F_{1}$ differs between proposed models
and it is a better metric for evaluation of the models.

##  6 Summary and Conclusion

We reviewed some of the popular classification algorithms including Linear
Regression, Logistic Regression, Naive Bayesian, SVM, Decision Trees and
Neural Networks. Then we applied some of the proposed models on a dataset of
variable and nonvariable stars, extracted from SDSS survey Stripe 82. After
training the models on the proposed dataset, we tested the models and
evaluated their predictions. We used accuracy and $F_{1}$-score as our
indicators, which are reported in Table (1) of the paper. As can be seen, the
decision tree algorithm result in the best accuracy and $F_{1}$-score. In
addition, for the sake of better understanding, we have plotted the decision
boundary of a logistic regression model and a Naive Bayes model in Figures (2,
3)

#### Acknowledgements

The author thanks the organizers of ICRANet – Isfahan Astronomy Meeting for
their kind invitation to present a contribution for this activity.

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