# Systematic bias due to eccentricity in parameter estimation for merging
binary neutron stars

Hee-Suk Cho<EMAIL_ADDRESS>Department of Physics, Pusan National
University, Busan, 46241, Korea  Extreme Physics Institute, Pusan National
University Busan, 46241, Korea

###### Abstract

We study the impact of eccentricity on gravitational-wave parameter estimation
for binary neutron star systems. For signals with small eccentricity injected
into noise with the advanced LIGO power spectral density, we perform Bayesian
parameter estimation using the circular waveform model and show how the
recovered parameters can be biased from their true values, focusing on the
intrinsic parameters the chirp mass ($M_{c}$), the symmetric mass ratio
($\eta$), and the tidal deformability ($\tilde{\lambda}$). By comparing the
results between the Bayesian and the analytic Fisher-Cutler-Vallisneri (FCV)
methods, we obtain the valid criteria for the FCV approach. Employing the FCV
method and using the realistic population of binary neutron star sources
distributed in the $m_{1}$–$m_{2}$–$e_{0}$ space, where $e_{0}$ indicates the
eccentricity at 10Hz, we calculate the measurement errors ($\sigma_{\theta}$)
and the systematic biases ($\Delta\theta/\sigma_{\theta}$) and obtain their
generalized distributions in the range of $0\leq e_{0}\leq 0.025$. We find
that for all of the three parameters, the biases increase with increasing
$e_{0}$, and this increase is faster for larger $e_{0}$. The bias is mainly
dependent on the value of $e_{0}$ and weakly dependent on the component
masses, and thus the distribution shows a narrow band in the
$e_{0}$–$\Delta\theta/\sigma_{\theta}$ plane. We present various posterior
examples to illustrate our new findings, such as the bimodality of posteriors.
In particular, we give a specific injection-recovery example to demonstrate
the importance of including eccentricity in parameter estimation to avoid
incorrect predictions of the neutron star equation of state.

##  I Introduction

Recently, the ground-based gravitational-wave (GW) observatories, advanced
LIGO (aLIGO) [1] and advanced Virgo [2], have finished the third observing
run. The network of the two aLIGO detectors and the advanced Virgo detector
has detected $90$ GW candidates [3, 4, 5, 6], all GW sources have been
verified to be merging compact binaries. Most of the GW signals were emitted
from the binary black holes (BBHs), and a small number of signals were emitted
from two binary neutron stars (BNSs) and a few neutron star-black hole (NSBH)
binaries. In particular, the neutron star (NS) tidal parameter could be
measured from the first detected BNS signal GW170817 due to its sufficiently
high signal-to-noise ratio [7, 8].

So far, various waveform models have been developed for use in GW data
analysis. The search template bank is constructed in the three-dimensional
parameter space consisting of the component masses and the aligned-spin [9,
10]. In this pipeline, an inspiral-only post-Newtonian (PN) waveform model is
used for BNS templates [11] and a full inspiral-merger-ringdown (IMR) waveform
model is used for BBH and NSBH templates [12]. In parameter estimation, more
diverse models are used to infer source properties as accurately as possible,
taking into account the effect of waveform systematics. Various IMR models
defined in the time domain or frequency domain have been used for BBH signals,
some of which include the precession effect and/or the higher multipoles (for
example, see Table I of [13]). For BNS signals, an inspiral-only PN model and
several IMR models have been used, and these models included NS tidal effects
[8, 14].

On the other hand, the orbital eccentricity has not been taken into account in
the current LIGO-Virgo GW searches since the binary system is likely to be
circularized when they reach the detector frequency band ($\sim 10$Hz) as the
systems can radiate away their eccentricity through GWs [15, 16]. However, it
has been known that dynamically formed binaries in dense stellar environments
may still have non-negligible eccentricities ($e>0.1$) at 10Hz [17, 18].
Therefore, the presence of a dynamical formation channel is expected to be
confirmed by successful measurements of the nonsmall eccentricity. In this
context, searches for eccentric BBHs [19] or BNSs [20] have also been
performed using LIGO-Virgo data, but no new significant candidates have been
identified. Several groups have attempted to estimate the eccentricity of the
BBH sources detected by LIGO-Virgo using the inspiral-only [21], the aligned-
spin [22, 23, 24], and the numerical simulation [25] waveform models,
respectively. The authors of [26, 27] also tried to measure the eccentricity
of the BNS sources separately, and both obtained very small eccentricities.
There have also been studies on the search efficiency for eccentric BBHs [28,
29] or eccentric BNSs [30] using quasicircular templates.

Studies on the impact of eccentricity on GW parameter estimation have also
been done in several works. The authors of [31, 32, 33] found that
eccentricity can improve the measurement accuracy of mass parameters.
Recently, some injection studies have been performed in [34, 35] to
investigate the measurability of eccentricity by ground-based detectors. In
particular, a systematic parameter bias that can be produced by using a
quasicircular waveform model for eccentric BBH signals has been investigated
in the recent works [24] and [35]. They showed that the eccentric signals can
be recovered by the quasicircular binaries with higher chirp masses ($M_{c}$)
and higher symmetric mass ratios ($\eta$). The two works [36] and [31] have
also conducted research on the bias due to eccentricity in BNS parameter
estimation. The former used the analytic method introduced by Cutler and
Vallisneri [37] to calculate the systematic bias, while the latter calculated
the bias by searching the $M_{c}$ – $\eta$ overlap surface. Both considered
the equal-mass BNS system with $m_{i}=1.4M_{\odot}$ but showed different
results on the bias for $\eta$. In the former work, as the signal’s
eccentricity increases, the bias increases, exceeding the statistical error at
small eccentricities of $\sim 0.002$, while in the latter work, the bias can
be ignored compared to the statistical error even at large eccentricities of
$\sim 0.4$. As we will see in our results, such a difference is due to the
fact that since the overlap surface is restricted by the physical boundary
(i.e., $0<\eta\leq 0.25$), the recovered value of $\eta$ cannot exceed 0.25,
resulting in negligible bias. However, these boundary effects are not
considered in the analytic method, and thus the recovered $\eta$ can increase
beyond 0.25.

Unlike the case for the BBH system, the waveform of the BNS system has
additional parameters related to the NS tidal deformability. Although the
tidal parameters have a very small effect compared to other parameters such as
masses or spins, the tidal deformability is a very important source property
because this parameter is related to the internal structure of NSs. Precision
measurements of tidal deformability can directly constrain the NS equation of
state (EOS) [38, 39, 40, 41, 42]. The tidal deformability was directly
measured from GW170817 for the first time, and as a result, the soft EOS
models were preferred over stiff EOS models [8, 43]. Meanwhile, studies on the
impact of eccentricity on the measurement of tidal deformability have not yet
been conducted except for the simple analytic results presented by Favata
[36].

In this work, we study the impact of eccentricity on parameter estimation for
BNS signals. For injection signals with various small eccentricities $e_{0}$,
where $e_{0}$ indicates the eccentricity of the binary at the GW frequency of
10 Hz, we perform Bayesian parameter estimation using circular waveforms and
investigate the biases for the intrinsic parameters the chirp mass, the
symmetric mass ratio, and the tidal deformability. We also use the Fisher
matrix-based analytic method to calculate the bias and evaluate the validity
of the analytic method by comparing the results with the Bayesian parameter
estimation results. By applying the analytic method to the $10^{4}$ Monte
Carlo samples randomly generated in the $m_{1}$–$m_{2}$–$e_{0}$ space, we
obtain the general distributions of bias for the parameters. In particular, we
provide a variety of posterior examples for ease of understanding.

##  II Method

###  II.1 Waveform model: TaylorF2

So far, many waveform models have been proposed for use in GW data analysis.
In contrast to the BBH systems that require full IMR waveforms, it is a large
advantage for the BNS systems that inspiral-only waveform models can be used
efficiently in the ground-based detectors because the portion of the
postmerger is negligible as it goes out of the detector’s frequency band. This
can be directly confirmed from the data analysis results for GW170817 [8] and
GW190425 [14], which showed consistent results between the IMR and the
inspiral-only waveform models. Therefore, we adopt the TaylorF2 waveform model
in this work. Since TaylorF2 is derived from the time-domain PN waveform model
using a stationary phase approximation, this model is defined in the frequency
domain. TaylorF2 is well formulated by the PN expansion with the corresponding
higher-order corrections and has a simpler form than other models used in
current GW data analysis [12, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55,
56, 53]

The wave function of TaylorF2 is given by

$h(f)={M_{\rm c}^{5/6}\over\pi^{2/3}D_{\rm eff}}\sqrt{{5\over
24}}f^{-7/6}e^{i\Psi(f)},$ (1)

where $D_{\rm eff}$ indicates the effective distance defined by a combination
of the five extrinsic parameters; those are the true distance of the source
($d_{L}$), orbital inclination with respect to the line of sight
($\theta_{JN}$), sky position ($RA,DEC$), and polarization angle ($\psi$)
[57]. The wave phase of TaylorF2 is give by

$\displaystyle\Psi(f)$ $\displaystyle=$ $\displaystyle 2\pi
ft_{c}-2\phi_{c}-{\pi\over 4}$ $\displaystyle+$ $\displaystyle{3\over 128\eta
v^{5}}[\phi^{\rm pp,circ}(f)+\phi^{\rm tidal}(f)+\phi^{\rm ecc}(f)],$

where $v\equiv[\pi f(m_{1}+m_{2})]^{1/3}$ is the PN expansion parameter and
$t_{c}$ and $\phi_{c}$ are the coalescence time and phase, respectively. The
term $\phi^{\rm pp,circ}$ represents the point-particle, circular PN
expansion, we here use the standard 3.5PN equation given in [58] but do not
consider the spin effect. The term $\phi^{\rm tidal}$ represents the tidal
corrections. Although the PN tidal corrections first appear in the 5PN order,
its impact is comparable to the 3.5PN order point-particle correction [59].
The PN tidal corrections are given as a function of the two dimensionless
tidal deformability $\lambda_{1}$ and $\lambda_{2}$. We consider the tidal
corrections up to 6PN order, then the tidal term can be given by the two
useful parameters $\tilde{\lambda}$ and $\delta\tilde{\lambda}$ as [36, 60]

$\phi^{\rm
Tidal}=-\bigg{[}\frac{39\tilde{\lambda}}{2}v^{10}+\bigg{(}\frac{3115\tilde{\lambda}}{64}-\frac{6595\sqrt{1-4\eta}\delta\tilde{\lambda}}{364}\bigg{)}v^{12}\bigg{]}.$
(3)

In this case, the 5PN order term is only a function of $\tilde{\lambda}$, so
the contribution of $\delta\tilde{\lambda}$ can be greatly reduced in the
overall tidal effect. As we will see, $\delta\tilde{\lambda}$ has a negligible
influence on the parameter estimation results. Finally, the term $\phi^{\rm
ecc}$ represents the eccentricity corrections, we here consider the PN
eccentricity corrections derived up to 3PN order given in Eq. (6.26) of [61],
which can be expressed as

$\displaystyle\phi^{\rm ecc}$ $\displaystyle=$
$\displaystyle-\frac{2355}{1462}e_{0}^{2}\left(\frac{v_{0}}{v}\right)^{19/3}\bigg{[}1+\bigg{(}\frac{18766963}{2927736}\eta$
$\displaystyle+$
$\displaystyle\frac{299076223}{81976608}\bigg{)}v^{2}+\left(\frac{2833}{1008}-\frac{197}{36}\eta\right)v_{0}^{2}$
$\displaystyle+$ $\displaystyle O(v^{4})+...+O(v^{6})\bigg{]},$

where $e_{0}$ is the eccentricity at the reference frequency $f_{0}$ (where
the choice of $f_{0}$ is arbitrary). For the ground-based detectors, the above
eccentric PN waveform model is valid up to $e_{0}\lesssim 0.1$ for comparable
mass systems [61].

Therefore, in order to produce the TaylorF2 waveforms for eccentric and
nonspinning BNS systems, we need the five extrinsic parameters
($d_{L},\theta_{JN},\psi,RA,DEC$), the five intrinsic parameters
($M_{c},\eta,\tilde{\Lambda},\delta\tilde{\Lambda},e_{0}$), and the two
arbitrary constants ($t_{c},\phi_{c}$), hence a total of 12 parameters are
required.

###  II.2 Bayesian parameter estimation

The main purpose of the GW search pipeline is to identify GW signals buried in
the detector’s data stream [57]. A more detailed analysis is conducted in the
parameter estimation pipeline to determine the source properties by exploring
the entire parameter space [62]. Using Bayesian inference statistics, the
result of parameter estimation can be given as posterior probability density
functions (PDFs) for the parameters considered. The Bayesian parameter
estimation process is based on iterative computations of the overlap between
the model waveforms ($h$) and the detector data ($x$) containing the signal
waveform ($s$). The overlap between $x$ and $h$ is given by

$\langle x|h\rangle=4{\rm Re}\int_{f_{\rm min}}^{f_{\rm
max}}\frac{\tilde{x}(f)\tilde{h}^{*}(f)}{S_{n}(f)}df,$ (5)

where the tilde denotes the Fourier transform of the time-domain waveform,
$S_{n}(f)$ is the detector’s noise power spectral density (PSD), $f_{\rm min}$
and $f_{\rm max}$ are the minimum and the maximum cutoff frequencies,
respectively. For a given prior distribution $p(\theta)$, where $\theta$ is
the set of parameters considered in the analysis, the posterior probability
that the GW signal is characterized by the parameters $\theta$ is expressed as

$p(\theta|x)\propto p(\theta)L(x|\theta),$ (6)

where $L(x|\theta)$ indicates the likelihood function. Using the overlap
defined in Eq. (5), the likelihood can be given by [63, 64]

$L(x|\theta)\propto{\rm exp}\bigg{[}-\frac{1}{2}\langle
x-h(\theta)|x-h(\theta)\rangle\bigg{]}.$ (7)

In the limit of high SNRs where the noise effect can be ignored, the above
equation can be expressed as

$L(\theta)\propto{\rm exp}\bigg{[}-\frac{1}{2}\\{\langle s|s\rangle+\langle
h(\theta)|h(\theta)\rangle-2\langle s|h(\theta)\rangle\\}\bigg{]}.$ (8)

For a given signal $s$, the SNR is given by [57]

$\rho=\sqrt{\langle s|s\rangle}.$ (9)

If we use an accurate waveform model, the likelihood distribution has a
maximum value when the parameter values of the model waveform are equal to
those of the signal waveform (i.e., $h(\theta_{0})\simeq s$, hence
$\rho^{2}=\langle s|s\rangle\simeq\langle
h(\theta_{0})|h(\theta_{0})\rangle$). In this case, around the maximum
position, the likelihood distribution in Eq. (8) can be given as [65]

$L(\theta)\propto{\rm
exp}[-\rho^{2}\\{1-\langle\hat{s}|\hat{h}(\theta)\rangle\\}],$ (10)

where $\hat{h}\equiv h/\rho$ denotes the normalized waveform. Therefore, in
Bayesian parameter estimation, the shape of the log–likelihood is determined
by the overlap between the signal and the model waveforms, and its scale of
interest depends on the SNR [65].

Since the prior information is just applied at the beginning of the process,
the construction of the parameter estimation algorithm is a matter of finding
the global maximum of the likelihood distribution over the entire parameter
space as quickly and accurately as possible. In this work, we perform Bayesian
parameter estimation for BNS injection signals using the Bilby library [66],
which is one of the parameter estimation packages, with the Dynesty [67]
nested sampling algorithm. In particular, to speed up the parameter estimation
runs, we apply the multibanded likelihood technique described in [68].
Furthermore, we select the phase marginalization option for efficiency, which
enables analytical marginalization over $\phi_{c}$.

###  II.3 Fisher matrix approach: Parameter measurement error and systematic
bias

In the high SNR limit, since the likelihood distribution obeys the Gaussian
approximately, the marginalized 1–d posterior PDF for the parameter can also
be expressed as a Gaussian function [63, 64]. The standard deviation of the
PDF corresponds to the measurement error of the corresponding parameter. On
the other hand, for a given signal with the true parameter values
$\theta_{0}$, the measurement errors can be easily obtained by using the
Fisher matrix defined by

$\Gamma_{ij}=\bigg{\langle}{\partial
h(\theta)\over\partial\theta_{i}}\bigg{|}{\partial
h(\theta)\over\partial\theta_{j}}\bigg{\rangle}\bigg{|}_{\theta=\theta_{0}}.$
(11)

The inverse of Fisher matrix gives the covariance matrix
($\Sigma_{ij}=(\Gamma^{-1})_{ij}$) of the parameter errors, and the
measurement error ($\sigma_{i}$) of each parameter and the correlation
coefficient ($C_{ij}$) between two parameters are given by

$\sigma_{i}=\sqrt{\Sigma_{ii}},\ \ \
C_{ij}={\Sigma_{ij}\over\sqrt{\Sigma_{ii}\Sigma_{jj}}}.$ (12)

Despite some well-known limitations (for more details, refer to [69]), the
Fisher matrix method has been widely used in the parameter estimation
literature since it was first used by [64, 70].

Another formula for the Fisher matrix can be given by the likelihood function
as [71, 69]

$\Gamma_{ij}=-\frac{\partial^{2}{\rm
ln}L(\theta)}{\partial\theta_{i}\partial\theta_{j}}\bigg{|}_{\theta=\theta_{0}}.$
(13)

Applying the likelihood function in Eq. (10) to the above equation gives

$\Gamma_{ij}=-\rho^{2}{\partial^{2}\langle\hat{s}|\hat{h}(\theta)\rangle\over\partial\theta_{i}\partial\theta_{j}}\bigg{|}_{\theta=\theta_{0}},$
(14)

and this represents the curvature of the overlap surface at the maximum
position. Thus, one can infer that specific isomatch contours in the overlap
surface correspond to the confidence regions of the posterior PDF for a given
SNR [72, 73, 74].

One of the limitations of the Fisher matrix method is the applicability of
prior information. Unlike Bayesian parameter estimation, only Gaussian prior
functions can be imposed analytically on the Fisher matrix. Assuming that the
variance of the Gaussian prior function for the parameter $\theta_{i}$ is
$P^{2}_{\theta_{i}}$, the covariance matrix containing the prior information
can be given by adding the component $\Gamma_{ii}^{0}=P_{\theta_{i}}^{-2}$ as
[64, 70]

$\Sigma^{P}_{ij}=(\Gamma_{ij}+\Gamma_{ii}^{0})^{-1},$ (15)

where $\Gamma_{ij}$ is the original Fisher matrix with no prior information.
The prior is independent of the signal strength, while the original Fisher
matrix strongly depends on the SNR as seen in Eq. (14). Therefore, the lower
the SNR, the stronger the prior effect on the parameter measurement.

Based on the Fisher matrix formalism, Cutler and Vallisneri [37] developed an
analytic method (henceforth denoted as “FCV” following the notation in [35])
to calculate the systematic bias. In this work, the bias can be induced by
ignoring the eccentricity in the model waveforms in the analysis of the
eccentric signals. Following Eq. (29) of [37], the systematic bias can be
given by [36]

$\Delta\theta_{i}=\theta^{\rm
rec}_{i}-\theta_{i0}=\Sigma_{ij}(\partial_{j}h_{\rm AP}|h_{\rm T}-h_{\rm
AP}),$ (16)

where $\theta^{\rm rec}$ represents the recoverd parameter value, $h_{\rm T}$
indicates the true signal waveform, and $h_{\rm AP}$ indicates the approximate
waveform in which the eccentricity corrections are ignored. Here,
$\Sigma_{ij}$ should be calculated from $h_{\rm AP}$. By representing the wave
amplitude of Eq. (1) as $A$, the above equation can be rewritten as

$\Delta\theta_{i}=4A^{2}\Sigma_{ij}\int_{f_{\rm min}}^{f_{\rm
max}}df\frac{f^{-7/3}}{S_{n}(f)}(\Psi_{\rm T}-\Psi_{\rm
AP})\partial_{j}\Psi_{\rm AP},$ (17)

where $\Psi_{\rm T}$ indicates the true wave phase given in Eq. (II.1) and
$\Psi_{\rm AP}$ indicates the approximate phase. Since $A^{2}\propto\rho^{2}$
and $\Sigma_{ij}\propto\rho^{-2}$, the bias is independent of the SNR. To
include the prior effect in the bias computation, the prior-incorporated
covariance matrix ($\Sigma^{P}_{ij}$) can be used in this equation.

##  III Result

###  III.1 Setup

The eccentric TaylorF2 waveform model is implemented in LAL (LIGO Algorithm
Library) [75] named as “TaylorF2Ecc”. Using this model, we generate eccentric
and nonspinning BNS signals and inject those into the LIGO-Hanford detector
with zero noise. We use the aLIGO sensitivity curve labeled as
“aligo$\\_$O4high” 111https://dcc.ligo.org/LIGO-T2000012/public, and the
minimum frequency cutoff is set to $f_{\rm min}=20$Hz. The maximum frequency
cutoff is set to $f_{\rm max}=f_{\rm ISCO}$, where $f_{\rm ISCO}$ is the
innermost-stable-circular-orbit frequency defined by $f_{\rm
ISCO}=1/[6^{3/2}\pi(m_{1}+m_{2})]$. We choose the fiducial BNS masses as
$(m_{1},m_{2})=(2M_{\odot},1M_{\odot})$, and then the tidal parameters can be
given as ($\lambda_{1},\lambda_{2})=(14.7,1744)$. Here, we assume the APR4
model [76], which is one of the soft EOS models preferred by the parameter
estimation results for GW170817 [8].

When running Bilby, the algorithm must explore the entire 11–d parameter
space. However, we can reduce the exploration space to 6–d space by holding
the extrinsic parameters equal to their true values. Note that, including the
extrinsic parameters in the analysis has a little impact on the results of the
intrinsic parameters (see, Appendix A). We assume the flat priors in the
ranges of $[0.5M_{\odot},\ 2.5M_{\odot}]$ for the component masses
($m_{1},m_{2}$) and [0, 5000] for the component tidal parameters
($\lambda_{1},\lambda_{2}$). The priors of $t_{c}$ and $\phi_{c}$ are given as
$[t_{c0}-1s,t_{c0}+1s]$ and [$0,2\pi$], respectively. Since our parameter
estimation is performed with circular waveforms, the eccentricity is not
included in the prior setting. When displaying the posteriors in Bilby, one
can choose the parameters of interest, and we select
($\tilde{\lambda},\delta\tilde{\lambda})$ rather than
$(\lambda_{1},\lambda_{2})$ in our results.

When computing the Fisher matrix, we use the circular waveform model that can
be obtained by ignoring the eccentric term in Eq. (II.1). Thus, the input
parameters of the Fisher matrix are given as
($M_{c},\eta,\tilde{\lambda},\delta\tilde{\lambda},t_{c},\phi_{c}$). Although
the choice of $t_{c}$ and $\phi_{c}$ does not affect the results of the
intrinsic parameters, they should be considered with the intrinsic parameters,
as they usually have strong correlations with the intrinsic parameters. We
adopt a single-detector configuration and only focus on the intrinsic
parameters in this work. The extrinsic parameters only scale the signal
strength, and most of the information about the intrinsic parameters is
contained in the wave phase. Therefore, we do not consider the extrinsic
parameters by choosing a fixed effective distance ($D_{\rm eff}$) as in the
case of Bayesian parameter estimation. If $D_{\rm eff}$ is used as one of the
input parameters in the Fisher matrix, its correlations with the intrinsic
parameters are relatively very small [77].

###  III.2 Measurement error: Comparison between Bayesian and Fisher matrix
methods

Figure 1: Marginalized 1–d posterior PDF and 2–d confidence regions with the
SNRs $\rho=17,34,68,$ and $114$. The contours indicate 39, 86, and $99\%$
confidence regions. Injection values are
$(m_{1},m_{2},[M_{c},\eta],\lambda_{1},\lambda_{2},[\tilde{\lambda},\delta\tilde{\lambda}])=(2M_{\odot},1M_{\odot},[1.21673M_{\odot},0.22222],14.7,1744,[237,101])$
marked in orange.

First, we perform parameter estimation using circular waveforms for a circular
signal with our fiducial parameter values assuming various SNRs. In this case,
in principle, the posterior should be unbiased because we use the same
waveform model as the signal model and assume zero noise in the injections.
Figure 1 shows the results with the SNRs of (17, 34, 68, 114), where the true
values are marked in orange. The SNRs can be simply obtained by choosing
$D_{\rm eff}=(200,100,50,30)$ Mpc. For each SNR, the marginalized 1–d
posterior PDF and the 2–d confidence regions are shown for the parameters
$M_{c},\eta,$ and $\tilde{\lambda}$. It can be seen that the injection signals
can be recovered well overall, showing decreasing measurement errors with
increasing SNR. The PDFs are slightly biased from the true values for low
SNRs, but rarely biased for high SNRs. For $\rho=17$, the contours are cut by
the physical boundary for $\tilde{\lambda}$ and asymmetric around the true
value, while the contours are clearly symmetric for high SNRs. The
marginalized PDF for $\delta\tilde{\lambda}$ is given in Fig. 2 separately.
Unlike the three parameters given in Fig. 1, $\delta\tilde{\lambda}$ is poorly
recovered for all SNRs. In particular, the PDFs for $\rho=68$ and 114 are very
similar, and they show no Gaussian distribution.

Figure 2:  Marginalized PDFs for $\delta\tilde{\lambda}$. The true value is
marked in red. Figure 3: Prior distributions for $\tilde{\lambda}$ (upper) and
$\delta\tilde{\lambda}$ (lower) used in Bayesian parameter estimation. These
can be obtained by converting the flat priors for
$(m_{1},m_{2},\lambda_{1},\lambda_{2})$ in the ranges of $[0.5M_{\odot},\
2.5M_{\odot}]$ for the component masses and [0, 5000] for the component tidal
parameters.

Next, by applying the circular waveform model to the Fisher matrix method, we
calculate the measurement errors assuming no prior information. The Fisher
matrix is given as a 6–d matrix with the parameter components
$(M_{c},\eta,\tilde{\lambda},\delta\tilde{\lambda},t_{c},\phi_{c})$, and we
use the same parameter values as in the previous Bayesian parameter
estimation. By comparing the results between the Fisher matrix and the
Bayesian posterior, we find that the measurement errors from the Fisher matrix
are much larger than those from the Bayesian parameter estimation, especially
for the tidal parameters.

The discrepancy between the Fisher matrix and the parameter estimation is
mainly due to the prior effect. Figure 3 shows the prior distributions for
$\tilde{\lambda}$ and $\delta\tilde{\lambda}$ used in the Bayesian parameter
estimation, which can be obtained by converting our prior distributions for
$(m_{1},m_{2},\lambda_{1},\lambda_{2})$. It can be seen that the posterior
PDFs for $\delta\tilde{\lambda}$ in Fig. 2 are significantly reduced compared
to the prior in Fig. 3, as expected. However, we found that the measurement
error of $\delta\tilde{\lambda}$ obtained from the Fisher matrix was much
larger than the width of the Bayesian prior for $\delta\tilde{\lambda}$ in
Fig. 3. Motivated by that, we impose the Gaussian prior for the parameter
$\delta\tilde{\lambda}$, whose variance is similar to the Bayesian prior size
($P_{\delta\tilde{\lambda}}\sim 500$), to the Fisher matrix. Using the prior-
incorporated Fisher matrix, we calculate the measurement errors ($\sigma$) of
$M_{c},\eta,$ and $\tilde{\lambda}$ and their correlation coefficients. Then,
we compare the 1–d Gaussian functions determined by $\sigma$ assuming
$\rho=17,34,68$, and $114$ with the marginalized Bayesian PDFs shown in Fig.
1. In Fig. 4, the black and the blue lines correspond to the Fisher matrix and
the Bayesian results, respectively, where the contour indicates $68\%$
confidence region given in the $M_{c}$–$\eta$ plane. This result clearly shows
that the analytic Gaussian functions agree remarkably well with the Bayesian
PDFs for all SNRs. The confidence regions are also nearly identical between
the two methods, indicating that the correlations are also very similar. Note
that, the Bayesian posteriors are shifted slightly to match the Gaussian
functions, so the small biases are not shown here. In Table 1, we list the
measurement error and correlation coefficient for the parameters
$M_{c},\eta,\tilde{\lambda},$ and $\delta\tilde{\lambda}$ assuming $\rho=34$.
The result with no prior information is given in the top row, and the result
with the prior $P_{\delta\tilde{\lambda}}\sim 500$ is given in the middle row.

Figure 4: Comparison between the Fisher matrix (black) and Bayesian parameter estimation (blue). The 2–d contour indicates $68\%$ confidence region given in the $M_{c}$–$\eta$ plane. Note that, the small biases are not shown here since the Bayesian posteriors are shifted slightly to match the Gaussian functions. $P_{\delta\tilde{\lambda}}$ | $\sigma_{M_{c}}/M_{c}$ | $\sigma_{\eta}$ | $\sigma_{\tilde{\lambda}}$ | $\sigma_{\delta\tilde{\lambda}}$ | $C_{M_{c}\delta\tilde{\lambda}}$ | $C_{\eta\delta\tilde{\lambda}}$ | $C_{\tilde{\lambda}\delta\tilde{\lambda}}$
---|---|---|---|---|---|---|---
- | 4.521E-5 | 1.739E-3 | 990.1 | 19780 | 0.7271 | 0.8607 | 0.9965
500 | 3.105E-5 | 0.886E-3 | 87.0 | 499.8 | 0.0268 | 0.0426 | 0.2865
- | 3.104E-5 | 0.886E-3 | 83.4 | - | - | - | -
Table 1: Measurement errors ($\sigma_{i}$) and correlation coefficients
($C_{ij}$) calculated by the Fisher matrix method assuming $\rho=34$. The top
row is obtained from the 6–d Fisher matrix with no prior information. The
middle row is obtained from the 6–d Fisher matrix incorporating the Gaussian
prior on the parameter $\delta\tilde{\lambda}$ with the variance
($P_{\delta\tilde{\lambda}}\sim 500$) that is similar to the Bayesian prior
size shown in Fig. 3. The bottom row is obtained from the 5–d Fisher matrix
where $\delta\tilde{\lambda}$ is not considered.

Figure 5 describes how the prior information for the parameter
$\delta\tilde{\lambda}$ affects the measurement of the other intrinsic
parameters, showing the ratio between $\sigma^{\rm prior}$ and $\sigma^{\rm
priorless}$ as a function of the fractional prior size
$P_{\delta\tilde{\lambda}}/\sigma^{\rm priorless}$. Here, $\sigma^{\rm prior}$
indicates the error calculated with the prior on $\delta\tilde{\lambda}$ with
the variance $P_{\delta\tilde{\lambda}}$, while $\sigma^{\rm priorless}$
indicates the error calculated with no prior information. The result shows
that the contribution of the prior is the highest (lowest) for
$\tilde{\lambda}\ (M_{c})$. For example, if the prior size is similar to the
error size i.e., $P_{\delta\tilde{\lambda}}\sim\sigma^{\rm priorless}$,
$\sigma^{\rm prior}$ can be $\sim 70\ (85)\%$ of $\sigma^{\rm priorless}$ for
$\tilde{\lambda}\ (M_{c})$. Note that, since both the error ($\sigma^{\rm
prior}$) and the prior ($P_{\delta\tilde{\lambda}}$) are divided by
$\sigma^{\rm priorless}$, this result is independent of the SNR (for more
details, refer to Fig. 7 of [77]).

Figure 5:  Contribution of the prior on $\delta\tilde{\lambda}$ to the
measurement of the other intrinsic parameters. Here, $\sigma^{\rm prior}$
indicates the error calculated with the prior information, while $\sigma^{\rm
priorless}$ indicates the error calculated with no prior. Note that, this
result is independent of the SNR [77]. Figure 6:  Dependence of the
correlation on the prior. The curves represent the correlation coefficients
between $\delta\tilde{\lambda}$ and the other parameters.

On the other hand, we also find in Table 1 that the correlations between
$\delta\tilde{\lambda}$ and the other parameters can be reduced by
incorporating the prior information. In particular, the correlations with the
mass parameters can be reduced to almost zero, meaning that the parameter
$\delta\tilde{\lambda}$ does not affect the measurement of the mass
parameters. To confirm this, we compute the 5–d Fisher matrix, in which
$\delta\tilde{\lambda}$ is not included. The result is given in the bottom row
in Table 1. We find that the errors of the mass parameters are nearly the same
between the 5–d and the 6–d ($P_{\delta\tilde{\lambda}}=500$) Fisher matrices.
The dependence of the correlation on the prior is also described in Fig. 6,
showing the correlation coefficient as a function of
$P_{\delta\tilde{\lambda}}/\sigma^{\rm priorless}$.

###  III.3 Systematic bias due to eccentricity

Figure 7:  Marginalized PDFs for $M_{c}$ (top), $\eta$ (middle), and
$\tilde{\lambda}$ (bottom). We assume a high SNR ($\rho=114$) to remove unknow
bias sources. For reference, the unbiased PDF is also given. The red dashed
line represents the true value.

In this subsection, we investigate how the recovered parameters of eccentric
BNS signals can be biased from their true values by using circular waveforms
in likelihood calculations. To this end, we inject three BNS signals with
small eccentricities $e_{0}=(0.008,0.016,0.024)$ into the aLIGO PSD with zero
noise. Motivated by the results in Fig. 1, we assume a high SNR ($\rho=114$)
to remove unknown bias sources. Figure 7 shows the marginalized posterior PDFs
for the parameters $M_{c}$ (top), $\eta$ (middle), and $\tilde{\lambda}$
(bottom), respectively. As expected, as $e_{0}$ increases, the magnitude of
the bias also increases. On the other hand, all of the biases show a positive
direction, meaning that the recovered parameters are larger than the true
parameter values. This increasing behavior for the mass parameters is also
seen in the BBH systems [24, 35]. In particular, the bias on the tidal
parameter is important because the value of $\tilde{\lambda}$ can constrain
the NS EOS. For example, the soft EOS model ARP4 [76], which we employ in our
analysis, gives $\tilde{\lambda}\sim 240$ for the binary with
$(2M_{\odot},1M_{\odot})$ but the stiffer EOS model MPA1 [76] gives
$\tilde{\lambda}\sim 450$. Roughly speaking, if the astrophysical NSs are
represented by the soft EOS, parameter estimation using circular waveform
models for eccentric BNS signals with $e_{0}\gtrsim 0.016$ can yield results
in favor of the stiff EOS models. We will show a concrete example in Sec.
III.5.

| Bayesian | FCV
---|---|---
$e_{0}$ | $\Delta{M_{c}}/\sigma_{M_{c}}$ | $\Delta{\eta}/\sigma_{\eta}$ | $\Delta{\tilde{\lambda}}/\sigma_{\tilde{\lambda}}$ | $\Delta{M_{c}}/\sigma_{M_{c}}$ | $\Delta{\eta}/\sigma_{\eta}$ | $\Delta{\tilde{\lambda}}/\sigma_{\tilde{\lambda}}$
0.008 | 2.841 | 1.443 | 0.685 | 2.955 | 1.542 | 0.696
0.016 | 11.42 | 5.902 | 2.634 | 11.82 | 6.170 | 2.784
0.024 | 24.71 | 12.84 | 6.531 | 26.60 | 13.88 | 6.264
Table 2: Comparison of biases between the Bayesian and the FCV methods. We use
the errors ($\sigma$) given in Table 1 with $P_{\delta\tilde{\lambda}}=500$.
Figure 8:  $99\%$ confidence regions of the biased posteriors. The recovered
values ($\theta^{\rm rec}$) for the different signals coincide with the
coordinates (0, 0). We assume $\rho=114$.

In Table 2, we list the biases obtained from the Bayesian and the FCV methods
for the eccentric signals given in Fig. 7. For the FCV results, we use Eq.
(17) assuming the prior $P_{\delta\tilde{\lambda}}=500$. For efficiency, we
give the fractional bias defined by $\Delta\theta/\sigma_{\theta}$ where
$\sigma$ is the unbiased error given in Table 1 with
$P_{\delta\tilde{\lambda}}=500$. Overall, the results of the FCV method are
similar to those of the Bayesian method. When $e_{0}=0.024$, the differences
between the two methods are $\sim 7.6,8.1,$ and $4.1\%$ for $M_{c},\eta,$ and
$\tilde{\lambda}$, respectively.

On the other hand, one can see in Fig. 7 that the PDF for $e_{0}=0.024$ is
somewhat wider than the unbiased PDF, especially for $\tilde{\lambda}$. This
implies that a biased posterior can give a larger measurement error than the
unbiased error. This is clearly illustrated in Fig. 8, which shows the
confidence regions of the posteriors together, where the recovered values
($\theta^{\rm rec}$) for the different signals coincide with the coordinates
(0, 0).222In this plot, we used a covariance fitting function (implemented in
Mathematica) to the marginalized 2–d posteriors to obtain clear contours.

Figure 9:  Systematic bias due to eccentricity given as a function of $e_{0}$
obtained by using the Bayesian and the FCV methods. We present the fractional
bias $\Delta\theta/\sigma_{\theta}$ where the measurement error is given Table
1 with $P_{\delta\tilde{\lambda}}=500$. The horizontal red line represents the
upper boundary of $\eta$, and the vertical red line indicates the value of
$e_{0}$ when $\eta^{\rm rec}$ first touches the boundary. Figure 10:  Biased
posteriors showing the bimodality. We assume $\rho=114$. Note that, the
maximum posterior position (marked with a red star) suddenly moves from the
left peak to the right peak when $e_{0}$ is between 0.0251 and 0.0252.

One can also see in Fig. 7 that the increase in bias is faster for larger
$e_{0}$. To see a generalized tendency, we inject eccentric signals, varying
$e_{0}$ up to 0.04, and perform parameter estimation. In Fig. 9, we display
the bias as a function of $e_{0}$ for $M_{c},\eta,$ and $\tilde{\lambda}$,
where we present together the bias calculated by the FCV method. Favata et
al.[35] showed that the bias due to ignoring eccentricity can be given as
$\Delta\theta\propto e_{0}^{2}$. We also find that our analytic curves can
almost exactly match their quadratic fitting functions.

On the other hand, the Bayesian curves look similar to the analytic FCV curves
up to $e_{0}\sim 0.025$, but beyond that, they exhibit quite irregular
behavior. One can see that the Bayesian curve rises abruptly at $e_{0}\sim
0.025$ and $\sim 0.032$, which is especially noticeable for $\tilde{\lambda}$.
To figure out how these sudden jumps occur, we generate a densely distributed
signal set near $e_{0}=0.025$ and perform parameter estimation. We choose four
posterior results required to describe the sudden jump. We display the results
for $e_{0}=(0.0250,0.0251,0.0252,0.0252)$ in Fig. 10. Interestingly, these
posteriors show a bimodal distribution, and we found that such bimodality only
appears very near $e_{0}\sim 0.025$ and $\sim 0.032$. The maximum posterior
position (marked with a red star) suddenly moves from the left peak to the
right peak when $e_{0}$ is between 0.0251 and 0.0252, resulting in a sudden
jump in the bias curve.333Cho [74] also found sudden jumps in the bias curves
for BBH systems (see Fig. 16 therein), in which the bias was produced by using
an inspiral-only waveform model for the IMR signal. We also found similar
situation near $e_{0}\sim 0.032$. The posterior’s fine-scale structure becomes
clearer if the SNR increases. Thus, the bimodality will be more likely seen in
the third-generation detectors, such as Einstein Telescope [78] and Cosmic
Explorer [79].

Figure 11: Biased posteriors for the signals with $e_{0}=0.038$ (upper panel)
and $e_{0}=0.041$ (lower panel), showing the increasing (decreasing) behavior
for $\Delta M_{c}\ (\Delta\tilde{\lambda})$ in the region of $e_{0}>e_{0}^{\rm
crit}$. We assume $\rho=114$.

Although not the case for the chirp mass, the upper bound of the symmetric
mass ratio is limited to $\eta\leq 0.25$. Thus, the maximum bias for $\eta$ is
given by $\Delta\eta=0.25-\eta_{0}$, independently of increasing $e_{0}$. This
relation implies that if the true value of $\eta$ is close to 0.25, the bias
$\Delta\eta$ can be negligible compared to the measurement error. By exploring
the $M_{c}$–$\eta$ overlap surface for an equal-mass BNS source, Sun et
al.[31] also found ignorable biases even at large eccentricities of $\sim
0.4$. In Fig. 9, the horizontal red line represents the boundary $\eta=0.25$.
The vertical red line indicates the value of $e_{0}$ when $\eta^{\rm rec}$
first touches the boundary, and we denote this critical eccentricity as
$e_{0}^{\rm crit}$. On the other hand, one can see that the increasing
behavior of the bias curves changes significantly if $e_{0}$ increases beyond
$e_{0}^{\rm crit}$. The increase in $\Delta M_{c}$ slows down, $\Delta\eta$
remains at $0.25$, and $\Delta\eta$ even decreases. In Fig. 11, we present two
posterior results that illustrate such behavior well. The result for
$e_{0}=0.038$ corresponds to the case of $e_{0}\sim e_{0}^{\rm crit}$, and
$e_{0}=0.041$ corresponds to $e_{0}>e_{0}^{\rm crit}$. In both cases, the 1–d
PDFs for $\eta$ are truncated at the boundary and their peak positions
coincide with the boundary, hence $\eta^{\rm rec}=0.25$. The 2–d posteriors
show that between $e_{0}=0.038$ and $e_{0}=0.041$, $\Delta M_{c}$ increases,
but $\Delta\tilde{\lambda}$ decreases. A detailed description is given in
Appendix B

###  III.4 Monte Carlo study

Figure 12:  Distribution of the fractional biases for the selected 4582
signals that satisfy $\eta^{\rm rec}\leq 0.25$. When calculating the analytic
bias ($\Delta\theta$) and error ($\sigma_{\theta}$), we incorporate the prior
$P_{\delta\tilde{\lambda}}=500$. The red line is the 1–d bias curve for the
signal with masses of ($2M_{\odot},1M_{\odot}$).

We create $10^{4}$ Monte Carlo samples randomly generated in the parameter
space $m_{1}$–$m_{2}$–$e_{0}$. The parameter ranges are given as
$1M_{\odot}\leq m_{1,2}\leq 2M_{\odot}\ (m_{2}\leq m_{1})$ and $0\leq
e_{0}\leq 0.025$. Employing these samples as the true values, we produce
$10^{4}$ eccentric signals and calculate biases using the FCV method assuming
$P_{\delta\tilde{\lambda}}=500$. We also calculate the unbiased errors using
the same signals but assuming $e_{0}=0$ for all signals. Finally, we select a
total of 4582 signals that satisfy $\eta^{\rm rec}\leq 0.25$. Figure 12 shows
the distributions of the fractional biases for the selected signals. For
reference, the 1–d bias curves for the signal with masses of
($2M_{\odot},1M_{\odot}$) are given by the red lines. For a given value of
$e_{0}$, the width of the distribution represents the dependence on the
component masses. We find that the distribution shows a narrow band along the
red line, which means that the bias is mainly dependent on the value of
$e_{0}$ and weakly dependent on the component masses.

To see a more useful explanation of the bias distribution, we also display the
biases for the component masses ($m_{1},m_{2}$) and the total mass ($M_{t}$)
in Fig. 13. Note that, in this plot, we present only the bias $\Delta\theta$,
not the fractional bias $\Delta\theta/\sigma_{\theta}$. The massive components
have a negative bias, but the lighter components have a positive bias, and
both exhibit a wide distribution for a given value of $e_{0}$. The maximum
bias is given as $\Delta m_{1}\ (\Delta m_{2})\sim-0.4\ (0.3)M_{\odot}$ at
$e_{0}=0.025$. For $M_{t}$, the biases are negative. In particular, the bias
distribution is almost independent of the signal mass, so it even looks like a
single curve. The distribution for $M_{t}$ is quite consistent with the
quadratic fitting function $\Delta M_{t}/M_{\odot}=-160e_{0}^{2}$.

Figure 13:  Distribution of the biases for the same signals as in Fig. 12 but
given for the component masses ($m_{1},m_{2}$) and the total mass ($M_{t}$).
The distribution for $M_{t}$ looks like a single curve, which means that the
distribution is almost independent of the component masses. Note that, in this
plot, we present only the bias, not the fractional bias.

###  III.5 Injection and recovery

Figure 14:  This example shows that different eccentric signals can be
recovered as one identical binary source. The two posteriors have almost the
same distribution and the same recovered parameters, i.e.,
$(m_{1},m_{2},\tilde{\lambda})\simeq(2M_{\odot},1M_{\odot},240)$. We assume
$\rho=114$.

In this subsection, we present two specific cases via injection and recovery.
First, we show how different eccentric signals can be recovered as one
identical signal similar to our fiducial binary source i.e.,
$(m_{1},m_{2},\tilde{\lambda})\simeq(2M_{\odot},1M_{\odot},240)$. To this end,
we inject two signals with the true values of
$(m_{1},m_{2},\tilde{\lambda},e_{0})=(2.0182M_{\odot},0.9920M_{\odot},179,0.008)$
and $(2.0727M_{\odot},0.9686M_{\odot},14,0.016)$ and perform parameter
estimation using circular waveforms. The results are given in Fig. 14.
Although the signal in the lower panel has an $e_{0}$ value twice that of the
signal in the upper panel, their posteriors have almost the same distribution
in the parameter space $m_{1}$–$m_{2}$–$\tilde{\lambda}$ and the same
recovered parameters. On the other hand, the biases of $M_{t}$ are $\Delta
M_{t}\sim 0.010$ and $\sim 0.41$ for $e_{0}=0.008$ and 0.016, respectively.
Thus, the relation $\Delta M_{t}/M_{\odot}=-160e_{0}^{2}$ is well satisfied.
Here, we estimated the injection parameters reversely from the fiducial values
referring to the results in Fig. 9.

As discussed in Sec. III.3, parameter estimation using circular waveforms for
eccentric BNS signals can yield incorrect predictions for the NS EOS. Here, we
present a concrete example. We assume that the astrophysical NSs are
represented by one of the soft EOS models APR4 [76], and then the tidal
parameter value can be determined as $\tilde{\lambda}\sim 237$ for the given
masses of $(2M_{\odot},1M_{\odot})$. So, we inject a signal with the true
values of
$(m_{1},m_{2},\tilde{\lambda},e_{0})=(2M_{\odot},1M_{\odot},237,0.0152)$ and
perform parameter estimation using circular waveforms. The posterior result is
given in Fig. 15, where the recovered parameters are
$\sim(1.93M_{\odot},1.03M_{\odot},440)$. The tidal parameter of
$\tilde{\lambda}\sim 440$ is consistent with the prediction of one of the
stiff EOS models MPA1 [76]. Therefore, this example demonstrates the
importance of including eccentricity for the correct estimation of the NS EOS.

Figure 15:  This example shows that parameter estimation using circular
waveforms can yield inaccurate NS EOS predictions. The injection value
($\tilde{\lambda}=237$) is chosen according to the soft EOS model ARP4, but
the recovered value ($\tilde{\lambda}^{\rm rec}\sim 440$) representes the
stiffer EOS (MPA1). We assume $\rho=114$.

##  IV Summary and discussion

In this work, we investigated how the small eccentricity of BNS signals can
affect parameter estimation for the mass parameters $M_{c}$ and $\eta$ and the
tidal deformability $\tilde{\lambda}$. We showed how the recovered parameters
can be biased from their true values by presenting various posterior samples.
We also showed concrete posterior examples that display irregular behavior of
bias, such as the abrupt increase in bias.

Using the realistic population of BNS sources distributed in the
$m_{1}$–$m_{2}$–$e_{0}$ space, we calculated the measurement errors and the
systematic biases for the parameters $M_{c},\eta$, and $\tilde{\lambda}$ and
obtained their generalized distributions in the range of $0\leq e_{0}\leq
0.025$. For all of the three parameters, the fractional biases
($\Delta\theta/\sigma_{\theta}$) increase with increasing $e_{0}$, and this
increase is faster for larger $e_{0}$. For a given value of $e_{0}$, the bias
is weakly dependent on the masses and thus its distribution displays a
relatively narrow band. In particular, the bias distribution for the total
mass exhibits a very narrow band, just like a single curve, and can be fitted
by the quadratic function $\Delta M_{t}/M_{\odot}=-160e_{0}^{2}$.

We presented two specific injection-recovery examples. The first example
showed that independent parameter estimations for different signals with
different eccentricities can give almost identical recovered values for the
component masses and $\tilde{\lambda}$. In the second example, we showed that
parameter estimation for signals with small eccentricity ($\sim 0.0152$) can
yield predictions that have shifted from the “true” soft EOS to the “false”
stiff EOS, demonstrating the importance of including eccentricity in the
waveform model.

We employed both the Bayesian and the analytic Fisher matrix methods for error
calculations and demonstrated that the analytic method can predict the
parameter estimation errors well by appropriately imposing the prior
information. Unlike the Fisher matrix, the FCV method has been used very
limitedly in the literature because its general validity has not been
investigated. We provided specific comparison results between the Bayesian and
the FCV methods. For the first time, we suggested the valid criteria of the
FCV method by showing the bias as a function of the parameter that induces the
bias. In the era of the third-generation detectors, the analytic methods will
be more useful because the length of the GW signal is greatly increased and
parameter estimation requires a very long computational time. The impact of
bias will also be more pronounced because the measurement errors can be
reduced due to the increase in SNR by the third-generation detectors.
Meanwhile, the spin effect can be included in the Fisher matrix and the FCV
method by employing the spinning TaylorF2 model, and our approach will be
extended to spinning binary systems in future works.

###### Acknowledgements.

This work was supported by the National Research Foundation of Korea (NRF)
grants funded by the Korea government (No. 2018R1D1A1B07048599, No.
2019R1I1A2A01041244, and No. 2020R1C1C1003250)

##  Appendix A PARAMETER ESTIMATION FOR THE ENTIRE 11 PARAMETERS

Figure 16:  Parameter estimation result for the same BNS source as in Fig. 1
with $\rho=34$. We use the entire 11 parameters including the 5 extrinsic
parameters. The zoom-in view of the posteriors for the three main parameters
$M_{c},\eta,$ and $\tilde{\lambda}$ is given in the inset. $\phi_{c}$ was
automatically marginalized and thus is not shown here. True values are marked
in orange.

Here, we present the parameter estimation result for the entire 11 parameters
using the same BNS source as in Fig. 1. In the inset, we show the zoom-in view
for the three main parameters $M_{c},\eta,$ and $\tilde{\lambda}$. We assume
$\rho=34$, so this signal is the same as the one in Fig. 1 with $\rho=34$. It
can be seen that the posteriors in the inset are well consistent with those in
Fig. 1, including the small biases. Thus, in a single-detector analysis,
reducing the extrinsic parameters to a single parameter $D_{\rm eff}$ yields
nearly the same parameter estimation results for the intrinsic parameters
compared to using the entire parameters. Since $\phi_{c}$ was analytically
marginalized in the likelihood calculation process, it cannot be shown in the
posterior. The priors for the extrinsic parameters are given in default in
Bilby.

##  Appendix B CONFIDENCE REGION AT $e_{0}\geq e_{0}^{\rm crit}$

Here, we show a schematic view describing the increasing bias for $M_{c}$ and
the decreasing bias for $\tilde{\lambda}$ shown in Fig. 11 at $e_{0}\geq
e_{0}^{\rm crit}$. Figure 17 displays two overlap contours in the
$M_{c}$–$\eta$ (upper) and $\eta$–$\tilde{\lambda}$ planes, respectively.
These biased contours can be obtained by using the Fisher matrix and the FCV
methods for our fiducial BNS source. The maximum overlap position ($P_{\rm
max}$) is located at the center of the contour. The star indicates the maximum
overlap position only in the region below the boundary $\eta=0.25$. In the
upper panel, the difference between the red and the black stars is smaller
than the difference between $P_{\rm max}$ of the red contour and $P_{\rm max}$
of the black contour. This means that the increase in $\Delta M_{c}$ slows
down. In the lower panel, although $P_{\rm max}$ of the red contour is higher
than $P_{\rm max}$ of the black contour, the red star is lower than the black
star, and that indicates the decrease in $\Delta\tilde{\lambda}$. Note that,
these contours are comparable to the confidence regions in Fig. 11, except
that these contours are extended beyond the $\eta$–boundary.

Figure 17:  Overlap contours showing the increasing bias for $M_{c}$ (upper)
and the decreasing bias for $\tilde{\lambda}$ (lower) in the region of
$e_{0}>e_{0}^{\rm crit}$.

##  Appendix C DEPENDENCE ON THE TRUE VALUES OF $\tilde{\lambda}$ AND
$\delta\tilde{\lambda}$

Here, we show how the true values of $\tilde{\lambda}$ and
$\delta\tilde{\lambda}$ affect our results of errors and biases. We assume a
BNS signal with the true values of
$(m_{1},m_{2},e_{0})=(2M_{\odot},1M_{\odot},0.02)$. Using the Fisher matrix
and the FCV methods, we calculate the errors and the biases for the parameters
$M_{c},\eta,$ and $\tilde{\lambda}$ varying the true values of the tidal
parameters $\tilde{\lambda}$ and $\delta\tilde{\lambda}$. We find that the
results for the mass parameters are independent of the tidal parameters, so
they are not shown here. The errors and the biases for $\tilde{\lambda}$ are
given in the upper and the lower panels, respectively, in Fig. 18. It can be
seen that the dependence on $\delta\tilde{\lambda}$ is almost negligible.
Although the dependence on $\tilde{\lambda}$ is relatively large, the
variation in the results is less than $10\%$ in our parameter range. For
example, for the soft ARP4 EOS model,
$\sigma_{\tilde{\lambda}}=86.9,\Delta{\tilde{\lambda}}=544.9$, and for the
stiffer MPA1 EOS model,
$\sigma_{\tilde{\lambda}}=88.4,\Delta{\tilde{\lambda}}=569.1$. Note that, for
$M_{c}$ and $\eta$, the differences between the two EOS models are less than
$0.01\%$.

Figure 18: Errors and fractional bias for $\tilde{\lambda}$ calculated by
varying the true values of the tidal parameters. The other true values are
given as $(m_{1},m_{2},e_{0})=(2M_{\odot},1M_{\odot},0.02)$. We assume
$\rho=34$.

## References

  * Aasi  _et al._ [2015] J. Aasi _et al._ , Classical and Quantum Gravity 32, 074001 (2015).
  * Acernese  _et al._ [2014] F. Acernese _et al._ , Classical and Quantum Gravity 32, 024001 (2014).
  * Abbott  _et al._ [2019a] B. P. Abbott _et al._ , Phys. Rev. X 9, 031040 (2019a).
  * Abbott  _et al._ [2021a] R. Abbott _et al._ , Phys. Rev. X 11, 021053 (2021a).
  * Abbott  _et al._ [2021b] R. Abbott _et al._ , arXiv:2108.01045 (2021b).
  * Abbott  _et al._ [2021c] R. Abbott _et al._ , arXiv:2111.03606 (2021c).
  * Abbott  _et al._ [2017a] B. P. Abbott _et al._ , Phys. Rev. Lett. 119, 161101 (2017a).
  * Abbott  _et al._ [2019b] B. P. Abbott _et al._ , Phys. Rev. X 9, 011001 (2019b).
  * Roy  _et al._ [2019] S. Roy, A. S. Sengupta, and P. Ajith, Phys. Rev. D 99, 024048 (2019).
  * Mukherjee  _et al._ [2021] D. Mukherjee, S. Caudill, R. Magee, C. Messick, S. Privitera, S. Sachdev, _et al._ , Phys. Rev. D 103, 084047 (2021).
  * Buonanno  _et al._ [2009] A. Buonanno, B. R. Iyer, E. Ochsner, Y. Pan, and B. S. Sathyaprakash, Phys. Rev. D 80, 084043 (2009).
  * Bohé  _et al._ [2017] A. Bohé, L. Shao, A. Taracchini, A. Buonanno, S. Babak, I. W. Harry, _et al._ , Phys. Rev. D 95, 044028 (2017).
  * Abbott  _et al._ [2020a] R. Abbott _et al._ , Phys. Rev. D 102, 043015 (2020a).
  * Abbott  _et al._ [2020b] B. P. Abbott _et al._ , The Astrophysical Journal Letters 892, L3 (2020b).
  * Peters and Mathews [1963] P. C. Peters and J. Mathews, Phys. Rev. 131, 435 (1963).
  * Peters [1964] P. C. Peters, Phys. Rev. 136, B1224 (1964).
  * Rodriguez  _et al._ [2018] C. L. Rodriguez, P. Amaro-Seoane, S. Chatterjee, and F. A. Rasio, Phys. Rev. Lett. 120, 151101 (2018).
  * Samsing [2018] J. Samsing, Phys. Rev. D 97, 103014 (2018).
  * Abbott  _et al._ [2019c] B. P. Abbott _et al._ , The Astrophysical Journal 883, 149 (2019c).
  * Nitz  _et al._ [2020] A. H. Nitz, A. Lenon, and D. A. Brown, The Astrophysical Journal 890, 1 (2020).
  * Wu  _et al._ [2020] S. Wu, Z. Cao, and Z.-H. Zhu, Monthly Notices of the Royal Astronomical Society 495, 466 (2020).
  * Romero-Shaw  _et al._ [2019] I. M. Romero-Shaw, P. D. Lasky, and E. Thrane, Monthly Notices of the Royal Astronomical Society 490, 5210 (2019).
  * Romero-Shaw  _et al._ [2021] I. Romero-Shaw, P. D. Lasky, and E. Thrane, The Astrophysical Journal Letters 921, L31 (2021).
  * O’Shea and Kumar [2021] E. O’Shea and P. Kumar, (2021), arXiv:2107.07981 [astro-ph.HE] .
  * Gayathri  _et al._ [2022] V. Gayathri, J. Healy, J. Lange, B. O’Brien, M. Szczepańczyk, I. Bartos, M. Campanelli, S. Klimenko, C. O. Lousto, and R. O’Shaughnessy, Nature Astronomy 6, 344 (2022).
  * Romero-Shaw  _et al._ [2020] I. M. Romero-Shaw, N. Farrow, S. Stevenson, E. Thrane, and X.-J. Zhu, Monthly Notices of the Royal Astronomical Society: Letters 496, L64 (2020).
  * Lenon  _et al._ [2020] A. K. Lenon, A. H. Nitz, and D. A. Brown, Monthly Notices of the Royal Astronomical Society 497, 1966 (2020).
  * Tiwari  _et al._ [2016] V. Tiwari, S. Klimenko, N. Christensen, E. A. Huerta, S. R. P. Mohapatra, A. Gopakumar, _et al._ , Phys. Rev. D 93, 043007 (2016).
  * Huerta  _et al._ [2018] E. A. Huerta, C. J. Moore, P. Kumar, D. George, A. J. K. Chua, R. Haas, _et al._ , Phys. Rev. D 97, 024031 (2018).
  * Huerta and Brown [2013] E. A. Huerta and D. A. Brown, Phys. Rev. D 87, 127501 (2013).
  * Sun  _et al._ [2015] B. Sun, Z. Cao, Y. Wang, and H.-C. Yeh, Phys. Rev. D 92, 044034 (2015).
  * Gondán  _et al._ [2018] L. Gondán, B. Kocsis, P. Raffai, and Z. Frei, The Astrophysical Journal 855, 34 (2018).
  * Gondán and Kocsis [2019] L. Gondán and B. Kocsis, The Astrophysical Journal 871, 178 (2019).
  * Lower  _et al._ [2018] M. E. Lower, E. Thrane, P. D. Lasky, and R. Smith, Phys. Rev. D 98, 083028 (2018).
  * Favata  _et al._ [2022] M. Favata, C. Kim, K. G. Arun, J. C. Kim, and H. W. Lee, Phys. Rev. D 105, 023003 (2022).
  * Favata [2014] M. Favata, Phys. Rev. Lett. 112, 101101 (2014).
  * Cutler and Vallisneri [2007] C. Cutler and M. Vallisneri, Phys. Rev. D 76, 104018 (2007).
  * Hinderer  _et al._ [2010] T. Hinderer, B. D. Lackey, R. N. Lang, and J. S. Read, Phys. Rev. D 81, 123016 (2010).
  * Damour  _et al._ [2012] T. Damour, A. Nagar, and L. Villain, Phys. Rev. D 85, 123007 (2012).
  * Del Pozzo  _et al._ [2013] W. Del Pozzo, T. G. F. Li, M. Agathos, C. VanDenBroeck, and S. Vitale, Phys. Rev. Lett. 111, 071101 (2013).
  * Agathos  _et al._ [2015] M. Agathos, J. Meidam, W. Del Pozzo, T. G. F. Li, M. Tompitak, J. Veitch, S. Vitale, and C. VanDenBroeck, Phys. Rev. D 92, 023012 (2015).
  * Harry and Hinderer [2018] I. Harry and T. Hinderer, Classical and Quantum Gravity 35, 145010 (2018).
  * Abbott  _et al._ [2018] B. P. Abbott _et al._ , Phys. Rev. Lett. 121, 161101 (2018).
  * Cotesta  _et al._ [2018] R. Cotesta, A. Buonanno, A. Bohé, A. Taracchini, I. Hinder, and S. Ossokine, Phys. Rev. D 98, 084028 (2018).
  * Cotesta  _et al._ [2020] R. Cotesta, S. Marsat, and M. Pürrer, Phys. Rev. D 101, 124040 (2020).
  * Ossokine  _et al._ [2020] S. Ossokine, A. Buonanno, S. Marsat, R. Cotesta, S. Babak, T. Dietrich, _et al._ , Phys. Rev. D 102, 044055 (2020).
  * Pan  _et al._ [2014] Y. Pan, A. Buonanno, A. Taracchini, L. E. Kidder, A. H. Mroué, H. P. Pfeiffer, M. A. Scheel, and B. Szilágyi, Phys. Rev. D 89, 084006 (2014).
  * Babak  _et al._ [2017] S. Babak, A. Taracchini, and A. Buonanno, Phys. Rev. D 95, 024010 (2017).
  * London  _et al._ [2018] L. London, S. Khan, E. Fauchon-Jones, C. García, M. Hannam, S. Husa, X. Jiménez-Forteza, C. Kalaghatgi, F. Ohme, and F. Pannarale, Phys. Rev. Lett. 120, 161102 (2018).
  * Khan  _et al._ [2019] S. Khan, K. Chatziioannou, M. Hannam, and F. Ohme, Phys. Rev. D 100, 024059 (2019).
  * Khan  _et al._ [2020] S. Khan, F. Ohme, K. Chatziioannou, and M. Hannam, Phys. Rev. D 101, 024056 (2020).
  * Varma  _et al._ [2019] V. Varma, S. E. Field, M. A. Scheel, J. Blackman, L. E. Kidder, and H. P. Pfeiffer, Phys. Rev. D 99, 064045 (2019).
  * Hannam  _et al._ [2014] M. Hannam, P. Schmidt, A. Bohé, L. Haegel, S. Husa, F. Ohme, G. Pratten, and M. Pürrer, Phys. Rev. Lett. 113, 151101 (2014).
  * Hinderer  _et al._ [2016] T. Hinderer, A. Taracchini, F. Foucart, A. Buonanno, J. Steinhoff, M. Duez, _et al._ , Phys. Rev. Lett. 116, 181101 (2016).
  * Pürrer [2014] M. Pürrer, Classical and Quantum Gravity 31, 195010 (2014).
  * Khan  _et al._ [2016] S. Khan, S. Husa, M. Hannam, F. Ohme, M. Pürrer, X. J. Forteza, and A. Bohé, Phys. Rev. D 93, 044007 (2016).
  * Allen  _et al._ [2012] B. Allen, W. G. Anderson, P. R. Brady, D. A. Brown, and J. D. E. Creighton, Phys. Rev. D 85, 122006 (2012).
  * Arun  _et al._ [2009] K. G. Arun, A. Buonanno, G. Faye, and E. Ochsner, Phys. Rev. D 79, 104023 (2009).
  * Flanagan and Hinderer [2008] E. E. Flanagan and T. Hinderer, Phys. Rev. D 77, 021502 (2008).
  * Lackey and Wade [2015] B. D. Lackey and L. Wade, Phys. Rev. D 91, 043002 (2015).
  * Moore  _et al._ [2016] B. Moore, M. Favata, K. G. Arun, and C. K. Mishra, Phys. Rev. D 93, 124061 (2016).
  * Veitch  _et al._ [2015] J. Veitch, V. Raymond, B. Farr, W. Farr, P. Graff, S. Vitale, _et al._ , Phys. Rev. D 91, 042003 (2015).
  * Finn [1992] L. S. Finn, Phys. Rev. D 46, 5236 (1992).
  * Cutler and Flanagan [1994] C. Cutler and E. E. Flanagan, Phys. Rev. D 49, 2658 (1994).
  * Cho  _et al._ [2013] H.-S. Cho, E. Ochsner, R. O’Shaughnessy, C. Kim, and C.-H. Lee, Phys. Rev. D 87, 024004 (2013).
  * Ashton  _et al._ [2019] G. Ashton _et al._ , The Astrophysical Journal Supplement Series 241, 27 (2019).
  * Speagle [2020] J. S. Speagle, Monthly Notices of the Royal Astronomical Society 493, 3132 (2020).
  * Morisaki [2021] S. Morisaki, Phys. Rev. D 104, 044062 (2021).
  * Vallisneri [2008] M. Vallisneri, Phys. Rev. D 77, 042001 (2008).
  * Poisson and Will [1995] E. Poisson and C. M. Will, Phys. Rev. D 52, 848 (1995).
  * Jaranowski and Krolak [1994] P. Jaranowski and A. Krolak, Phys. Rev. D 49, 1723 (1994).
  * Baird  _et al._ [2013] E. Baird, S. Fairhurst, M. Hannam, and P. Murphy, Phys. Rev. D 87, 024035 (2013).
  * Cho and Lee [2014] H.-S. Cho and C.-H. Lee, Classical and Quantum Gravity 31, 235009 (2014).
  * Cho [2015] H.-S. Cho, Classical and Quantum Gravity 32, 235007 (2015).
  * LIGO Scientific Collaboration [2018] LIGO Scientific Collaboration, LIGO Algorithm Library - LALSuite, free software (GPL) (2018).
  * Read  _et al._ [2009] J. S. Read, B. D. Lackey, B. J. Owen, and J. L. Friedman, Phys. Rev. D 79, 124032 (2009).
  * Cho [2022] H.-S. Cho, Classical and Quantum Gravity 39, 085006 (2022).
  * Punturo  _et al._ [2010] M. Punturo _et al._ , Classical and Quantum Gravity 27, 194002 (2010).
  * Abbott  _et al._ [2017b] B. P. Abbott _et al._ , Classical and Quantum Gravity 34, 044001 (2017b).


2021

[1]Selvakumar R

1]Department of Mathematics, Vellore Institute of Technology, Vellore, Tamil
Nadu, India, 632014

# Detection of Alzheimer’s Disease using MRI scans based on Inertia Tensor and
Machine Learning

Krishna Mahapatra<EMAIL_ADDRESS><EMAIL_ADDRESS>[

###### Abstract

Alzheimer’s Disease is a devastating neurological disorder that is
increasingly affecting the elderly population. Early and accurate detection of
Alzheimer’s is crucial for providing effective treatment and support for
patients and their families. In this study, we present a novel approach for
detecting four different stages of Alzheimer’s disease from MRI scan images
based on inertia tensor analysis and machine learning. From each available MRI
scan image for different classes of Dementia, we first compute a very simple
$2\times 2$ matrix, using the techniques of forming a moment of inertia
tensor, which is largely used in different physical problems. Using the
properties of the obtained inertia tensor and their eigenvalues, along with
some other machine learning techniques, we were able to significantly classify
the different types of Dementia. This process provides a new and unique
approach to identifying and classifying different types of images using
machine learning, with a classification accuracy of ($90\%$) achieved. Our
proposed method not only has the potential to be more cost-effective than
current methods but also provides a new physical insight into the disease by
reducing the dimension of the image matrix. The results of our study highlight
the potential of this approach for advancing the field of Alzheimer’s disease
detection and improving patient outcomes.

###### keywords:

Alzheimer’s Disease, Magnetic Resonance Imaging, Inertia Tensor, Machine
Learning.

##  1 Introduction

Alzheimer’s disease (AD) is a progressive neurodegenerative disorder that is
characterized by a decline in cognitive and behavioral abilities
alroobaea2021alzheimer . It is the most common cause of dementia among older
adults and its prevalence is expected to increase as the population ages
world2021global .

Figure 1: Projected Growth of Alzheimer’s Disease Worldwide: 2005-2050.

In 2006, approximately twenty-seven million people suffered from this disease
worldwide brookmeyer2007o1 . AD is predicted to affect 1 person in 85 people
globally by 2050, and at least $43\%$ of prevalent cases need high-level care
for this disease revi2020alzheimer  Figure1. Although, the cause of AD is not
completely understood murphy2021mortality  so far.Two kinds of AD are
considered, one is sporadic and another one is familial. The sporadic AD
($~{}90-95\%)$ is genetical and environmental and late onset
piaceri2013genetics . The risk factors of sporadic AD are i) age (1% for age
60-65, 50% for age over 85) and ii) gene ($-$allele of apolipoprotein E gene,
or APOE-). The familial AD ( 5-10%) is early onset and some inherited dominant
gene speed up the progression of the disease. There is no proven and modifying
treatment for AD, a progressive, irreversible brain disorder characterized by
a decline in cognitive science al2020alzheimer . Therefore, the detection of
Alzheimer’s disease (AD) at an early stage is crucial for the management and
treatment of the disease as well as for planning for the future care of
patients and their families gaugler20222022 .

Techniques for brain imaging can be used to non-invasively see the
pharmacology, function, or structure of the brains spinelli2003new . Non-
invasive neuroimaging techniques, including Magnetic Resonance Image (MRI),
Positron Emission Tomography (PET), and MRI biomarkers, can be used to deduce
the clinical mechanisms of AD. However, the cost and complexity of these tools
make them difficult to use in low-resource environments chedid2022development
. There is a significant effort being made to develop strategies when
treatment may be most effective, critical goals remaining early detection of
Alzheimer’s Disease (AD), particularly at pre-symptomatic stages, to reduce or
halt disease progression nelson2015slowing . To accomplish this, cutting-edge
neuroimaging methods such as magnetic resonance imaging (MRI) and positron
emission tomography (PET) have been developed and utilized to identify
structural and molecular changes related to AD chevignard2020pediatric .
Additionally, computer aided machine learning methodologies, such as
convolutional neural networks (CNNs), support vector machines (SVMs), and
logistic regression (LR) (but it performed high prediction for heart diseases)
ambrish2022logistic , have been increasingly used for integrative analysis in
the early detection and classification of AD lazli2020survey . However, the
most of studies on Alzheimer’s disease that have been published so far have
classified it using intelligent classification methods, such as convolutional
neural network (CNN) architecture or graph neural network (GNN) architecture.

In recent years, healthcare has increasingly been using learning-based
computer-aided detection systems, such as supervised machine learning (as
Naïve Bayes (NB), Decision Trees (DT), K-Nearest Neighbour (KNN)) in the
diagnosis of several diseases, including breast cancer kumar2019machine ,
heart diseases barik2020heart , and others. These computer-based studies aim
to identify specific characteristics in MRI images that can be used to
classify different types of AD mohsen2018classification . Lu et al. (2018)
took a data set from 1051 subjects and obtained an accuracy value of 82% by
using a multiscale deep neural network structure to detect Alzheimer’s disease
lu2018multiscale . Zhao et al. in their study, used 15 healthy and 15 patient
data and obtained 92% accuracy by using the SVM methodzhang2015detection ;
zhao2015deep . Goo et al. (2017) applied CNN architecture in their study and
obtained accuracy values of 87.62% gao2017classification . Moradi et al.
stated that they used 10 folders for cross-validated and achieved 90.2%
accuracy using SVM and that played a major role in the detection of
ADmoradi2015machine . Salvatore et al. in their study for the diagnosis of AD
and obtained accuracy rate of 76% using a machine learning method that
optimizedsalvatore2015magnetic . Das et al. (2013) used least-square SVM, and
consisting of 66, 160 and 255 images, and their 5 × 5 CV showed high
classification accuracy (on an average $>$ 99%)das2013brain . There are
several drawbacks associated with using convolutional neural networks (CNNs)
for the classification of images, including:

(1) High computational requirements: CNNs are computationally intensive and
require large amounts of data and processing power, making them difficult to
implement in some settings.

(2) Lack of interpretability: CNNs are often considered ”black box” models, as
it is difficult to understand how the model is making its decisions. This
makes it hard to understand what features the model is using to classify the
images.

In this study, we propose a novel method for reducing the dimensionality of
brain images to improve the efficiency of the classification process.
Specifically, we map each image pixel value matrix to a 2x2 matrix by
utilizing the information that, in cases of dementia, brain size may be
compressed or expanded. By reducing the image matrix to a 2x2 matrix, the
computational cost is significantly reduced. The details of the methods and
techniques used are described in section 3.

##  2 Data

From the open-access Kaggle website
https://www.kaggle.com/datasets/tourist55/alzheimers-dataset-4-class-of-
images, the data set was used for this study. The Alzheimer’s Dataset
consisting of MRI images used has 4 categories:Very Mild Demented, Mild
Demented, Moderate Demented, and Non-Demented shown in Figure 2.

Figure 2: Four Different Class Image

##  3 Model Building:

There are two types of MRI data, one is structural MRI and other one is
functional MRI. The most used MRI for AD cases is structural MRI. The
structural MRI offers images which provides information about the brain
structures, including neurons, synapses, cells, etc. These images helps to
deal with the structure of the nervous system and the diagnosis of large-scale
intracranial disease (such as the tumor) and injury Baltes2010 . It measures
brain volume in vivo to detect brain degeneration (loss of tissue, cells,
neurons, etc.) which is an inevitable progressive component of AD. This work
is mostly based on structural MRI images.

A mathematical model of the brain shows how the brain is structurally changing
for dementia. In this study, we used two different types of CDR (Clinical
Dementia Rate) for two different classes of dementia: subjects with a CDR of 1
were classified as non-dementia, and CDR of 0 as moderate dementia
zhang2015detection .

The method of our work is as follows. The basic idea of identifying Dementia
from an MRI image comes due to the shrinkage of the brain cell in the images.
Note that in the images of MRI scan (which is nothing but a collection of
pixel values in a 2-dimensional plane), due to the shrinkage, the distribution
of the pixel values in the image will be intrinsically different for different
classes of Dementia. Understanding the distribution of the pixel values is
therefore key to the detection of Dementia. From the concepts of physics, we
see that the Moment of Inertia (MI) tensor is something that helps in
understanding the distribution of mass in an extended object. Therefore, by
drawing a correspondence of mass of a 2D extended object with the pixel values
of the MRI scan image, we can generically calculate the MI tensor of the image
itself, which in turn will provide information about the distribution of the
pixel values in the image. Moving forward, we consider the pixel values in the
MRI image $(a_{pq})$ to be equivalent to the mass $(m_{i})$ of a 2D object.
This assumption simplifies the calculation of the moment of inertia (MI)
tensor, which is a $2\times 2$ matrix for 2D systems. To compute the MI
tensor, we first need to define a coordinate system with the origin at the
center of the image and the x and y axes aligned horizontally and vertically,
respectively. We simplify calculations by assuming that the extent of the 2D
object is from $(-1,1)$ in the x and y directions, in some arbitrary unit. It
is important to note that these scaling factors and the pixel value to mass
conversion factor, which we take to be 1, are chosen for computational
convenience and that alternative scaling and conversion factors are unlikely
to significantly affect the results.

With the mass distribution, coordinate system, and dimensions of the 2D object
defined, the elements of the MI tensor can be calculated as follows:

$I=\begin{bmatrix}I_{00}&I_{01}\\\ I_{10}&I_{11}\end{bmatrix}$ (1)

where the different elements of the matrix are defined as follows:

$I_{00}=\sum_{i}m_{i}.y_{i}^{2}$

$I_{01}=\sum_{i}-m_{i}.x_{i}.y_{i}$

$I_{11}=\sum_{i}m_{i}.x_{i}^{2}$

$I_{10}=I_{01}$

where the sum is over all the masses in the 2D system, which is basically the
pixel values in the image. $m_{i}$ is the $i-th$ pixel value in the image.
$x_{i}$ and $y_{i}$ are the $x$ and $y$ components of the position vector
$\vec{r_{i}}$ of the $i-th$ point in the image.

Therefore this technique treats each image as a 2D mass distribution and
computes the MI tensor for each of them. We then move on to calculate the
eigenvalues for each of the MI tensor. We then plot the eigenvalues
$\\{(I_{1},I_{2})\\}$ ($I_{1},I_{2}$ are the two eigenvalues for each MI
tensor) in to observe that the two cases of Non-dementia and Moderate dementia
can be distinguished from such plot Figure 3. Using this simple technique of
converting an image matrix a $2\times 2$ tensor and extracting its
eigenvalues, we have carried on further analyses to detect different classes
of dementia, which are reported in section 4.

Figure 3: Scatter Plot of Eigen Values

##  4 Results

###  4.1 Eigenvalue distribution

The eigenvalues of the corresponding MI tensor for each of the images related
to the known cases of Non-dementia and Moderate dementia is plotted in figure
3 which shows a wider distribution of the points for the moderate cases, where
a narrower distribution (points mostly following in a straight line) for non-
dementia cases. This distribution of the eigenvalues hence can give rise to
important classification parameters detecting the different types of dementia.

###  4.2 Asymmetry in eigen values

Figure 4: The average value of the asymmetries in the eigen values for each
class.

The physical significance of the difference between the eigenvalues of the MI
tensor is noteworthy as it reflects the amount of mass distributed along each
axis. A significant difference between two eigenvalues indicates a greater
mass far away from the axis corresponding to the eigenvector of that
eigenvalue. As the shrinkage of brain cells in Moderate Dementia alters the
pixel value distribution in MRI images, the mass distribution also changes,
which is captured by the difference of the eigenvalues (see Figure 3). Our
results show that the average difference between the largest and smallest
eigenvalues, for all datasets, decreases with the severity of the disease
(Nondementia $>$ very mild $>$ mild $>$ moderate). This technique may aid in
detecting dementia classes for unknown data, as the higher average asymmetry
of eigenvalues may indicate non-dementia.

The reason for this trend can be explained as follows: In the case of non-
dementia, the brighter region in the image (section 2) is larger than in other
cases, representing regions of higher mass as their pixel values are larger.
Conversely, darker regions denote regions of lower mass as their pixel values
are lower. In contrast, for the moderate dementia class, a more ”dark” region
is present, representing a hollow mass distribution. It can be proven that the
asymmetry in the mass distribution (thus the asymmetry in the eigenvalues of
the MI tensor) along the principle axes would be larger for solid mass
distribution compared to hollow mass distribution, given all other parameters
are the same. Therefore, the non-dementia class (equivalent to a solid mass
distribution) is expected to exhibit a larger asymmetry or difference in the
eigenvalues compared to the moderate dementia class (hollow mass
distribution), which is supported by the trend shown in Figure 4 as the mass
distribution becomes increasingly hollow in the given order.

###  4.3 Total pixel value

Figure 5: The average values of total mass for each image in various classes.

As previously mentioned, we have observed that the MRI images corresponding to
the Moderate dementia class exhibit a hollower mass distribution in a physical
sense, and that the hollowness increases with the severity of the disease
($H_{non}<H_{very-mild}<H_{mild}<H_{moderate}$, where $H$ denotes the
hollowness and the subscripts denote the different disease classes).
Consequently, a more fundamental classification scheme can be based on the
total mass of the system. We therefore computed the total mass (i.e., the sum
of the pixel values) for each image in the various classes and found that the
average values of the total mass are consistent with our expectations.
Mathematically, since $H\propto 1/M$, it follows that
$M^{a}_{non}>M^{a}_{very-mild}>M^{a}_{mild}>M^{a}_{moderate}$ (see Figure 5),
where $M$ and $M^{a}$ respectively denote the total mass and the average of
the total mass for the images, and the subscripts denote the different disease
classes. This trend is also illustrated in Figure 5. Based on this simple
analysis, we can gain insights into the underlying disease type of unknown
image data by comparing the total pixel values with the average value. This
analysis will be explore briefly using machine learning methods. There we can
use different model to train and test of different classes of asymmetry in
eigen values and get the result of classification.

##  5 Machine learning methods

Prior research has used machine learning techniques to differentiate between
Alzheimer’s disease patients’ images and healthy MRI images. These techniques
include decision trees, support vector machines, convolutional neural
networks, linear discriminant analysis, multi-regression linear analysis, and
others. However, the pre-processed image; features are extracted and given to
the classifier as input. The numerous features of class data are explained,
and the use of computationally intensive methods that obstruct explainability,
such as Inertia Tensor and: black box; ML models used in this work. Image
classification can be accomplished by using feature extraction techniques to
reduce a large input data set into important key features.

###  5.1 Feature Extraction:

In this study, simplification of the original input data space in order to
reduce the original dimensionality into 2D- inertia matrix and obtain a new
simpler representation that includes the largest part of the original data
variability.Every feature vector of MRI brain images characterizing the images
has to be extracted to get exact features. The inertia Tensor Matrix is used
as a feature in the design of the mathematical model Figure 6.

Figure 6: Feature Extraction Process

###  5.2 ML-Based Classification Techniques:

We now give a brief overview of the machine learning (ML) classifiers that
enable us to differentiate between the four different classes of Alzheimer’s
disease. We are classifying every dementia class eigenvalue. These ML
techniques are known as supervised classifiers. In this paper, we have used
one supervised classifiers,i.e., SVM (Support Vector Machine). Support vector
machines (SVM), a mathematical model extension of neural networks, can
classify both linear and non-linear data. By transforming the input training
data into higher dimensional space and constructing hyperplanes (nothing but
decision planes). Different kernel functions, such as linear, polynomial, and
gaussian radial basis functions, can be used to specify the decision boundary
function. It is a supervised learning technique that is most employed for
regression, classification. One of the best classification techniques in
machine learning, SVM, has been arguably acknowledged zhang2012classification
. SVMs are linear classifiers, but for non-linear data, we introduced the
kernel SVMs (KSVMs), RBF kernel. So, Predictions can be made using machine
learning classifiers that the algorithm can predict the class the data belongs
to.

###  5.3 Inertia tensor method

As has been shown in the results of section 4, the novel method of creating an
inertia tensor from an image by taking the pixel values as masses of a 2D body
and scaling the figure dimension to the length and width of the body, which
makes it possible to identify an MRI scan image with a physical 2D mass
distribution of a body, can also help in differentiating or classifying the
different classes of dementia. The technique is summarised in Figure 6. The
figure depicts the initial distribution of pixel values in the image. Then by
appropriate choice of origin and axes system, we create a $2\times 2$ inertia
tensor. Then we calculate the eigenvalues, which essentially are the same as
the diagonal elements of the inertia tensor after diagonalization. Then from
the eigenvalues themselves, and their asymmetries, we have shown that
classification can be done for the different classes of dementia fig 7. The
calculation of total mass also is an effective tool for classification.

Figure 7: (a)SVM classification of moderate and non dementia images (b) SVM
classification of moderate, mild, and non dementia images (c) SVM
classification of moderate, mild, very mild, and nondementia images Figure 8:
All Asymmetry Class

##  6 Conclusion

This study presents a new method for detecting Alzheimer’s disease using MRI
scans and machine learning. The approach draws a correspondence between a 2D
image and a 2D mass distribution of a physical body and then uses inertia
tensor analysis from that mass distribution, calculates the eigenvalue of the
inertia tensor matrix, and classify different types of images. With a
classification accuracy of ($90\%$), this method has the potential to be more
cost-effective and provide physical insight into the disease. The results
demonstrate the potential of this approach for advancing Alzheimer’s disease
detection and improving patient outcomes.

##  7 Conflict of Interest

There is no conflict of interest.

##  8 Data Availability

https://www.kaggle.com/datasets/tourist55/alzheimers-dataset-4-class-of-
images.

## References

  *   * (1) Alroobaea, R., Mechti, S., Haoues, M., Rubaiee, S., Ahmed, A., Andejany, M., Bragazzi, N.L., Sharma, D.K., Kolla, B.P., Sengan, S.: Alzheimer’s disease early detection using machine learning techniques (2021)
  * (2) Organization, W.H., et al.: Global status report on the public health response to dementia (2021)
  * (3) Brookmeyer, R., Johnson, E., Ziegler-Grahamm, K., Arrighi, H.M.: O1–02–01: Forecasting the global prevalence and burden of alzheimer’s disease. Alzheimer’s & Dementia 3(3S_Part_3), 168–168 (2007)
  * (4) Revi, M.: Alzheimer’s disease therapeutic approaches. GeNeDis 2018: Genetics and Neurodegeneration, 105–116 (2020)
  * (5) Murphy, S.L., Kochanek, K.D., Xu, J., Arias, E.: Mortality in the united states, 2020 (2021)
  * (6) Piaceri, I., Nacmias, B., Sorbi, S.: Genetics of familial and sporadic alzheimer’s disease. Frontiers in Bioscience-Elite 5(1), 167–177 (2013)
  * (7) Al-Shoukry, S., Rassem, T.H., Makbol, N.M.: Alzheimer’s diseases detection by using deep learning algorithms: a mini-review. IEEE Access 8, 77131–77141 (2020)
  * (8) Gaugler, J., James, B., Johnson, T., Reimer, J., Solis, M., Weuve, J., Buckley, R.F., Hohman, T.J.: 2022 alzheimer’s disease facts and figures. ALZHEIMERS & DEMENTIA 18(4), 700–789 (2022)
  * (9) Spinelli, L., Lazeyras, F., Willi, J.-P., Slosman, D.O., Delavelle, J., Seeck, M., Michel, C.M.: New brain imaging techniques. Revue médicale de la Suisse romande 123(1), 53–61 (2003)
  * (10) Chedid, N., Tabbal, J., Kabbara, A., Allouch, S., Hassan, M.: The development of an automated machine learning pipeline for the detection of alzheimer’s disease. Scientific Reports 12(1), 18137 (2022)
  * (11) Nelson, L., Tabet, N.: Slowing the progression of alzheimer’s disease; what works? Ageing research reviews 23, 193–209 (2015)
  * (12) Chevignard, M., Câmara-Costa, H., Dellatolas, G.: Pediatric traumatic brain injury and abusive head trauma. In: Handbook of Clinical Neurology vol. 173, pp. 451–484. Elsevier, ??? (2020)
  * (13) Ambrish, G., Ganesh, B., Ganesh, A., Srinivas, C., Mensinkal, K., et al.: Logistic regression technique for prediction of cardiovascular disease. Global Transitions Proceedings 3(1), 127–130 (2022)
  * (14) Lazli, L., Boukadoum, M., Mohamed, O.A.: A survey on computer-aided diagnosis of brain disorders through mri based on machine learning and data mining methodologies with an emphasis on alzheimer disease diagnosis and the contribution of the multimodal fusion. Applied Sciences 10(5), 1894 (2020)
  * (15) Kumar, A., Sushil, R., Tiwari, A.K.: Machine learning based approaches for cancer prediction: a survey. In: Proceedings of 2nd International Conference on Advanced Computing and Software Engineering (ICACSE) (2019)
  * (16) Barik, S., Mohanty, S., Rout, D., Mohanty, S., Patra, A.K., Mishra, A.K.: Heart disease prediction using machine learning techniques. In: Advances in Electrical Control and Signal Systems: Select Proceedings of AECSS 2019, pp. 879–888 (2020). Springer
  * (17) Mohsen, H., El-Dahshan, E.-S.A., El-Horbaty, E.-S.M., Salem, A.-B.M.: Classification using deep learning neural networks for brain tumors. Future Computing and Informatics Journal 3(1), 68–71 (2018)
  * (18) Lu, D., Popuri, K., Ding, G.W., Balachandar, R., Beg, M.F., Initiative, A.D.N., et al.: Multiscale deep neural network based analysis of fdg-pet images for the early diagnosis of alzheimer’s disease. Medical image analysis 46, 26–34 (2018)
  * (19) Zhang, Y., Dong, Z., Phillips, P., Wang, S., Ji, G., Yang, J., Yuan, T.-F.: Detection of subjects and brain regions related to alzheimer’s disease using 3d mri scans based on eigenbrain and machine learning. Frontiers in computational neuroscience 9, 66 (2015)
  * (20) Zhao, Y., He, L.: Deep learning in the eeg diagnosis of alzheimer’s disease. In: Computer Vision-ACCV 2014 Workshops: Singapore, Singapore, November 1-2, 2014, Revised Selected Papers, Part I 12, pp. 340–353 (2015). Springer
  * (21) Gao, X.W., Hui, R., Tian, Z.: Classification of ct brain images based on deep learning networks. Computer methods and programs in biomedicine 138, 49–56 (2017)
  * (22) Moradi, E., Pepe, A., Gaser, C., Huttunen, H., Tohka, J., Initiative, A.D.N., et al.: Machine learning framework for early mri-based alzheimer’s conversion prediction in mci subjects. Neuroimage 104, 398–412 (2015)
  * (23) Salvatore, C., Cerasa, A., Battista, P., Gilardi, M.C., Quattrone, A., Castiglioni, I., Initiative, A.D.N.: Magnetic resonance imaging biomarkers for the early diagnosis of alzheimer’s disease: a machine learning approach. Frontiers in neuroscience 9, 307 (2015)
  * (24) Das, S., Chowdhury, M., Kundu, M.K.: Brain mr image classification using multiscale geometric analysis of ripplet. Progress In Electromagnetics Research 137, 1–17 (2013)
  * (25) Baltes, C., Mueggler, T., Rudin, M.: In: Stolerman, I.P., Price, L.H. (eds.) Structural and Functional Magnetic Resonance Imaging, pp. 1–12. Springer, Berlin, Heidelberg (2010). https://doi.org/10.1007/$978-3-642-27772-6_298-2$. https://doi.org/10.1007/$978-3-642-27772-6_298-2$
  * (26) Zhang, Y., Wu, L.: Classification of fruits using computer vision and a multiclass support vector machine. sensors 12(9), 12489–12505 (2012)