ニューラルネットワークを用いた
リングダウン重力波の準固有振動数の推定
山本貴宏（京都大学）

PHYSICAL REVIEW D 99, 124032 (2019)

指導教官：田中貴浩
PRD 99, 124032

Comparison of various methods to extract ringdown frequency
from gravitational wave data
Hiroyuki Nakano,1,* Tatsuya Narikawa,2,3,† Ken-ichi Oohara,4,‡ Kazuki Sakai,5,§
Hisa-aki Shinkai,6,∥ Hirotaka Takahashi,7,8,¶ Takahiro Tanaka,3,9,** Nami Uchikata,2,4,††
Shun Yamamoto,6 and Takahiro S. Yamamoto3,‡‡
1

Faculty of Law, Ryukoku University, Kyoto 612-8577, Japan
Institute for Cosmic Ray Research, The University of Tokyo, Chiba 277-8582, Japan3
Department of2019.12.25-27
Physics, Kyoto University,
Kyoto 606-8502, Japan
理論懇シンポジウム@
国立天文台
4
Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan2

5

Binary BH merger

⇡⇢ ⌧

⇡

!" #"

Black hole ringdown
・Ringdown is emitted when black hole is perturbed.
・Occupied with BH quasi-normal modes (QNMs)
h(t) = A exp [ 2⇡fI (tt0 )] cos [2⇡fR (t<latexit sha1_base64="ZHkimt1/27tdrw2OemLemByT+XM=">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</latexit>

h(t)

t0 )

0]

fR , fI : QNM frequency<latexit sha1_base64="F0t2ptfZKPPDaUT4uGDyYy8knoo=">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</latexit>※In general, ringdown is the linear combination ofvarious modes. Here, we focus on (l,m)=(2,2) mode
which has the longest damping time.

In general relativity,
QNM frequency is determined by BH mass and spin.
-> Useful for the test of GR.

BH = 除夜の鐘
音
東大寺？
知恩院？

BH = 除夜の鐘
重力波
BH or not?
GR or not?

Matched Filter(MF)Calculate the correlation (SNR) between signal and templates.
The parameter maximizing SNR is the estimated value.

SNR(✓) =<latexit sha1_base64="qtS9Fili0zCUnDyzN30LuW8gAjY=">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</latexit>

Z 1h̃⇤ (f ; ✓)s̃(f ) + h̃(f ; ✓)s̃⇤ (f )df
Sn (f )
1h: template, s: GW signal, S̲n: noise power spectral density

Posterior distribution of the parameters isSNR(✓)2p(✓|s) / exp
2<latexit sha1_base64="6BY6bu/jDdh4lHB0dv9L5HuFY4M=">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</latexit>MF with IMR waveform
IMR waveforms are determined by the binary parameters(e.g. masses & spins of binary).

<latexit sha1_base64="OFRs1wGDPm84AvmVL5K2c5kB+fg=">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</latexit>

Binary masses & spins

Mass&spin of ﬁnal BH

p(m1 , m2 , a1 , a2 |s)p(Mf , af |s)<latexit sha1_base64="DbF/6GTb79hYQanF/ixQyZzfbFg=">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</latexit>

Fitting formulaobtained from NR simulations

QNM frequency<latexit sha1_base64="WM7Yxvwz8NihWvXjsxtzXRFPOoo=">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</latexit>p(fR , fI |s)

BH perturbation theory- Based on GR.
- Mainly aﬀected from inspiral part.

strain

Let us assume that・modiﬁed theories of gravity
・exotic compact objects
modify the merger-ringdown part without changing inspiral.

time

Can we estimate the QNM frequency
using only ringdown gravitational wave?

MF with damped sinusoid
When is the appropriate start time?

Amplitude

Frequency

If itʼs early, high SNR, but biased

MF with damped sinusoid
When is the appropriate start time?

Amplitude

Frequency

If itʼs late, no bias, but low SNR.

MF with damped sinusoid

the source
From the12
final blackof the leas
10tone) [61]1
at 90% co8
Testing2⇡fI
6
form a tes
dicted lea
4
3.0 msthis purpo
5.0 mssinusoid w
2
the corres
7.0 ms0
used is h(
200
220
240
260
280
300
t0 ) = 0, w
QNM frequency (Hz) fR
the unkno⌧ 2 [0.5,
Scientific Collaboration
Virgo Collaboration (2016)
FIG. 4. We show the posterior 90% LIGO
confidence
regionsandfrom
7.
0

m
s

IMR (l = 2, m = 2, n = 0)

1.0
ms<latexit sha1_base64="8x9vb1Z42D+ypTv3e3mLXJBVQz0=">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</latexit>

QNM decay time (ms)14<latexit sha1_base64="dkTsdvrw94MkHdbp57628h1Gs04=">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</latexit>

Nakano+TSY+ PRD 99, 124032

Mock Data Challenge• Investigate the optimal method for analysis of ringdown• Comparing 5 methods,- Hilbert-Huang transformation（Oohara, Sakai, Takahashi)- Auto-regressive model(Shinkai, Yamamoto)- MF with ringdown (Uchikata)- MF with merger-ringdown (Tanaka)- Deep learning (TSY)• Here, I will talk about deep learning method, comparing
with MF.

Nakano+TSY+ PRD 99, 124032

Mock Data Challenge• Mockdata is based on the GR template and modiﬁed only
ringdown part. We estimate the QNM frequency of themodiﬁed waveform.
amplitudefrequency
Mock template
GR

Mock template

GRNakano+TSY+ PRD 99, 124032

Mock Data Challenge• Prepare 2 types of modiﬁcation.
• NN (MF) use one of them as training data (templates).
4

set is not strictly rehe peak of the amplilation, we adopt the

0.5
0.4
0.4
0.2
0.3

ARD
22 (t),
4M !IGR x

0(2.14)

0

0.01

0.02

0.03

0.2

0.1

eM !I x,(2.15)

termined so that the

0

-0.2

-0.1

0

0.1

t [s]

type
30, 20)
Examplesof the(20set Adata
and B.for
The SNR=60,
inset shows• 60 test dataFIG.for1: each

Deep Learning
for GW analysis• Rapid parameter estimation George&Huerta, PRD 97 044039
Gabbard+ 1909.06296• How accurately DL can estimate parameters?
Nakano+TSY+ PRD 99, 124032• Classiﬁcation of glitch noises
George+ PRD 97, 101501• Generating the waveforms
Chua+ PRLett. 122, 211101
This is incomplete list…

Deep Learning
for GW analysis• Rapid parameter estimation George&Huerta, PRD 97 044039
Gabbard+ 1909.06296• How accurately DL can estimate parameters?
Nakano+TSY+ PRD 99, 124032• Classiﬁcation of glitch noises
George+ PRD 97, 101501• Generating the waveforms
Chua+ PRLett. 122, 211101
This is incomplete list…

Neural network
in a nutshell
NN is the function having many parameters. y = NeuralNet(x; {w})<latexit sha1_base64="nVO1hObqeUTfSxNkpTGLxA6l4E8=">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</latexit>

Before applying to the real data, these parameters need to be
optimized using the training data. D = {(xn , tn )}(ex : the pairs of the simulated signal and QNM frequency.)<latexit sha1_base64="wpFuJSSIExlAUIlcIKuAjJ6NZmo=">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</latexit>

Optimization is done by iteratively using the loss function.

L [y (xn ) , tn ]<latexit sha1_base64="dh72d5zP6sGqxI8UDTf+Z1TABck=">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</latexit>

w!w<latexit sha1_base64="6EeDmiTiY6GERKs6ym+y7VlGCWk=">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</latexit>⌘rw L

NN can learn the relation between x and t from data.

Input : noisy signal
Output : predicted
QNM frequency

NeuralNet

fR
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Nakano+TSY+ PRD 99, 124032• The segment (512 points, 4096Hz) starting frompeak time is used for input data.

• Training data is generated from type B.
• # of templates = 441 ( 21 x 21 )• 4 convolutional & pooling + 2 linear layers• loss: mean relative error, optimizer: Adam
• PyTorch• GeForce1080Ti (@Nagaoka)

ErrorFor 120 data, each challenging group 3.
handedin (fRTABLE III: We show the values of log fR , (fR ), log fI andas the result of their blind analyses. In order to com
five methods, we introduce the logarithmic average(fI ) for various methods. The results limited to set A are
variance definedby
MF-MR
and AR methods
given on the first law of each method while those limited to!
N in Sec.
(estimate)explainedIII.
The coX
1Qn
set B on the second.
log Q =
log
,
(inj)
N n=1
mates isbriefly checked
belo
Qn
2
!2 31/2
N(estimate)In thecase ofme
1 X
Qn MF-MR
4
5
(Q) =
log
,
(inj)
log fR (%)(fR )(%)
log fI (%)(fI )(%) pectedisobtained.
N
NresultQn
n=1

Nakano+TSY+ PRD 99, 124032

Results

MF-R

A
B
MF-MR A
B
HHT
A
B
AR
A
B
NN
A
B

-12.88
-0.82
6.25
2.47
-13.38
-8.08
0.2
1.91
-6.64
-6.65

28.36
27.53
17.27
10.41
21.91
19.81
9.93
8.57
16.48
11.97

-71.51-46.11-12.62
7.18-44.11-28.784.88
6.2-15.239.96

in the present test (60 real pa
97.79
as indicators of the bias and the average magnitu(estimate)The
90%
confidence
interval
theparameter
estimation
error,
where
Q
i
n
75.48
estimatedability
value of thequantityon
Q for
the nthdata
regions
both
smal
(inj)
37.9Qn is the corresponding injected values. In Tabl
in
we show estimate
the values of oflogthe
fR , confidence(fR ), log fI and
27.61
for the methods
we tried. We show
the results
limit
the statisticalerror.
Thet
set A on the first law and those limited to set B on
61.58
confidence interval 27 times
second law.
recallthanthat in thethe actual
implementati
49.61 We shouldworse
expectation.
the MF-MR described in Sec. III B we adopt theinterval
need
38.75
modified dence
template thatis used tomay
generate
the setmo
Bmdata. Also,
the happened
NN method introducedin Sec. whi
III E
this
45 times,
34.64
the template bank generated in the same way as thtionto ofthebias i
B mock datatrain
the systematic
network.
33.96
For AR method, the true
23.76

better for set B, as expected.
The variance might be determined by a small numberof data with a large error. To check if it is the case or

90% confidence interval 21
the imaginary part while itreal part.

V.

CONCLUD

We implemented five meth

ErrorFor 120 data, each challenging group 3.
handedin (fRTABLE III: We show the values of log fR , (fR ), log fI andas the result of their blind analyses. In order to com
five methods, we introduce the logarithmic average(fI ) for various methods. The results limited to set A are
variance definedby
MF-MR
and AR methods
given on the first law of each method while those limited to!
N in Sec.
(estimate)explainedIII.
The coX
1Qn
set B on the second.
log Q =
log
,
(inj)
N n=1
mates isbriefly checked
belo
Qn
2
!2 31/2
N(estimate)In thecase ofme
1 X
Qn MF-MR
4
5
(Q) =
log
,
(inj)
log fR (%)(fR )(%)
log fI (%)(fI )(%) pectedisobtained.
N
NresultQn
n=1

Nakano+TSY+ PRD 99, 124032

Results
Real part

MF-R

A
B
MF-MR A
B
HHT
A
B
AR
A
B
NN
A
B

-12.88
-0.82
6.25
2.47
-13.38
-8.08
0.2
1.91
-6.64
-6.65

28.36
27.53
17.27
10.41
21.91
19.81
9.93
8.57
16.48
11.97

-71.51-46.11-12.62
7.18-44.11-28.784.88
6.2-15.239.96

in the present test (60 real pa
97.79
as indicators of the bias and the average magnitu(estimate)The
90%
confidence
interval
theparameter
estimation
error,
where
Q
i
n
75.48
estimatedability
value of thequantityon
Q for
the nthdata
regions
both
smal
(inj)
37.9Qn is the corresponding injected values. In Tabl
in
we show estimate
the values of oflogthe
fR , confidence(fR ), log fI and
27.61
for the methods
we tried. We show
the results
limit
the statisticalerror.
Thet
set A on the first law and those limited to set B on
61.58
confidence interval 27 times
second law.
recallthanthat in thethe actual
implementati
49.61 We shouldworse
expectation.
the MF-MR described in Sec. III B we adopt theinterval
need
38.75
modified dence
template thatis used tomay
generate
the setmo
Bmdata. Also,
the happened
NN method introducedin Sec. whi
III E
this
45 times,
34.64
the template bank generated in the same way as thtionto ofthebias i
B mock datatrain
the systematic
network.
33.96
For AR method, the true
23.76

90% confidence interval 21
the imaginary part while itreal part.

For the real part,AR method gets the best result.
better for set B, as expected.
CONCLUD
Matched ﬁlter and neural net are comparable with eachV.other.
The variance might be determined by a small numberof data with a large error. To check if it is the case or

We implemented five meth

ErrorFor 120 data, each challenging group 3.
handedin (fRTABLE III: We show the values of log fR , (fR ), log fI andas the result of their blind analyses. In order to com
five methods, we introduce the logarithmic average(fI ) for various methods. The results limited to set A are
variance definedby
MF-MR
and AR methods
given on the first law of each method while those limited to!
N in Sec.
(estimate)explainedIII.
The coX
1Qn
set B on the second.
log Q =
log
,
(inj)
N n=1
mates isbriefly checked
belo
Qn
2
!2 31/2
N(estimate)In thecase ofme
1 X
Qn MF-MR
4
5
(Q) =
log
,
(inj)
log fR (%)(fR )(%)
log fI (%)(fI )(%) pectedisobtained.
N
NresultQn
n=1

Nakano+TSY+ PRD 99, 124032

Results

MF-R

A
B
MF-MR A
B
HHT
A
B
AR
A
B
NN
A
B

-12.88
-0.82
6.25
2.47
-13.38
-8.08
0.2
1.91
-6.64
-6.65

28.36
27.53
17.27
10.41
21.91
19.81
9.93
8.57
16.48
11.97

-71.51-46.11-12.62
7.18-44.11-28.784.88
6.2-15.239.96

in the present test
(60 real pa
Imaginary
part
97.79

as indicators of the bias and the average magnitu(estimate)The
90%
confidence
interval
theparameter
estimation
error,
where
Q
i
n
75.48
estimatedability
value of thequantityon
Q for
the nthdata
regions
both
smal
(inj)
37.9Qn is the corresponding injected values. In Tabl
in
we show estimate
the values of oflogthe
fR , confidence(fR ), log fI and
27.61
for the methods
we tried. We show
the results
limit
the statisticalerror.
Thet
set A on the first law and those limited to set B on
61.58
confidence interval 27 times
second law.
recallthanthat in thethe actual
implementati
49.61 We shouldworse
expectation.
the MF-MR described in Sec. III B we adopt theinterval
need
38.75
modified dence
template thatis used tomay
generate
the setmo
Bmdata. Also,
the happened
NN method introducedin Sec. whi
III E
this
45 times,
34.64
the template bank generated in the same way as thtionto ofthebias i
B mock datatrain
the systematic
network.
33.96

23.76

For AR method, the true
90% confidence interval 21
the imaginary part while itreal part.

For the imaginary part, AR, MF, NN are comparable.
better for set B, as expected.
The variance might be determined by a small numberof data with a large error. To check if it is the case or

V.

CONCLUD

We implemented five meth

ErrorFor 120 data, each challenging group 3.
handedin (fRTABLE III: We show the values of log fR , (fR ), log fI andas the result of their blind analyses. In order to com
five methods, we introduce the logarithmic average(fI ) for various methods. The results limited to set A are
variance definedby
MF-MR
and AR methods
given on the first law of each method while those limited to!
N in Sec.
(estimate)explainedIII.
The coX
1Qn
set B on the second.
log Q =
log
,
(inj)
N n=1
mates isbriefly checked
belo
Qn
2
!2 31/2
N(estimate)In thecase ofme
1 X
Qn MF-MR
4
5
(Q) =
log
,
(inj)
log fR (%)(fR )(%)
log fI (%)(fI )(%) pectedisobtained.
N
NresultQn
n=1

Nakano+TSY+ PRD 99, 124032

Results

MF-R

A
B
MF-MR A
B
HHT
A
B
AR
A
B
NN
A
B

-12.88
-0.82
6.25
2.47
-13.38
-8.08
0.2
1.91
-6.64
-6.65

28.36
27.53
17.27
10.41
21.91
19.81
9.93
8.57
16.48
11.97

-71.51-46.11-12.62
7.18-44.11-28.784.88
6.2-15.239.96

in the present test (60 real pa
97.79
as indicators of the bias and the average magnitu(estimate)The
90%
confidence
interval
theparameter
estimation
error,
where
Q
i
n
75.48
estimatedability
value of thequantityon
Q for
the nthdata
regions
both
smal
(inj)
37.9Qn is the corresponding injected values. In Tabl
in
we show estimate
the values of oflogthe
fR , confidence(fR ), log fI and
27.61
for the methods
we tried. We show
the results
limit
the statisticalerror.
Thet
set A on the first law and those limited to set B on
61.58
confidence interval 27 times
second law.
recallthanthat in thethe actual
implementati
49.61 We shouldworse
expectation.
the MF-MR described in Sec. III B we adopt theinterval
need
38.75
modified dence
template thatis used tomay
generate
the setmo
Bmdata. Also,
the happened
NN method introducedin Sec. whi
III E
this
45 times,
34.64
the template bank generated in the same way as thtionto ofthebias i
B mock datatrain
the systematic
network.
33.96
For AR method, the true
23.76

90% confidence interval 21
the imaginary part while itreal part.

Only the type B is used for training.
As expected, the result for type A is worse than that for type B.
better for set B, as expected.

The variance might be determined by a small numberof data with a large error. To check if it is the case or

V.

CONCLUD

We implemented five methConclusion• Neural network is competitive with matched ﬁlter.
• For estimation of the imaginary part, NN is one ofthe powerful estimator.

• (As expected, ) the accuracy for the type A (notused for training) becomes worse than that of thetype B.

Future work
• More training data (# of data, modiﬁcation)• Bayesian inferenceend