# Calibrating global behaviour of equation of state by combining nuclear
and astrophysics inputs in a machine learning approach

Sk Md Adil Imam1,2<EMAIL_ADDRESS>Prafulla Saxena3
<EMAIL_ADDRESS>Tuhin Malik4 N. K. Patra 5 B. K. Agrawal1,2
<EMAIL_ADDRESS>1Saha Institute of Nuclear Physics, 1/AF Bidhannagar,
Kolkata 700064, India  2Homi Bhabha National Institute, Anushakti Nagar,
Mumbai 400094, India  3Malaviya National Institute of Technology, Jaipur,
India  4CFisUC, Department of Physics, University of Coimbra, 3004-516
Coimbra, Portugal  5Department of Physics, BITS-Pilani, K. K. Birla Goa
Campus, Goa 403726, India

(January 2024)

###### Abstract

We implemented symbolic regression techniques to identify suitable analytical
functions that map various properties of neutron stars (NSs), obtained by
solving the Tolman-Oppenheimer-Volkoff (TOV) equations, to a few key
parameters of the equation of state (EoS). These symbolic regression models
(SRMs) are then employed to perform Bayesian inference with a comprehensive
dataset from nuclear physics experiments and astrophysical observations. The
posterior distributions of EoS parameters obtained from Bayesian inference
using SRMs closely match those obtained directly from the solutions of TOV
equations. Our SRM-based approach is approximately 100 times faster, enabling
efficient Bayesian analyses across different combinations of data to explore
their sensitivity to various EoS parameters within a reasonably short time.

##  I Introduction

Understanding the behavior of matter at densities beyond those found in atomic
nuclei is crucial for advancements in modern astrophysics and nuclear physics.
The dead remnants of stellar evolution, neutron stars (NS) are perfect
laboratories for studying particles in dense environments as their central
densities go up to 5-8 times the nuclear saturation density ($\rho_{0}\sim$
0.16 fm-3). So multi-messenger observations related to NS properties such as
gravitational mass, radius, and tidal deformability, provide a way to study
the matter at extreme conditions. Observations from massive radio pulsars
Fonsecaet al. (2021); Romani  _et al._ (2022), the multimessenger event
GW170817 Abbott et al. (2017, 2019), observations of Neutron Star Interior
Composition Explorer (NICER)Riley et al. (2019); Miller et al. (2019); Riley
et al. (2021); Miller et al. (2021) of two pulsars, terrestrial endeavors such
as heavy ion collision experiments Akiba et al. (2015); Meehan (2016);
Adamczewski-Musch  _et al._ (2022); Senger and for the CBM Collaboration
(2021), and theoretical calculations based on chiral effective field theory
Hebeler  _et al._ (2015); Lynn  _et al._ (2019); Drischler and Bogner (2021);
Drischler  _et al._ (2021) have played crucial roles in deciphering the global
behavior of the Equation of State (EoS).

Future, observations of coalescing BNS events by detectors like LIGO-Virgo-
KAGRA, Einstein Telescope Punturo et al. (2010), and Cosmic Explorer Reitze
_et al._ (2019) are likely to occur more frequently, enabling a more precise
determination of the high-density part of the EoS Most  _et al._ (2018);
Traversi  _et al._ (2020); G.et al. (2021); Annala et al. (2022); Ghosh  _et
al._ (2022); Iacovelli  _et al._ (2023); Huxford  _et al._ (2023); Rose  _et
al._ (2023); Pradhan  _et al._ (2023a, b); Walker  _et al._ (2024); Huang  _et
al._ (2024); Patra  _et al._ (2023). The EoS at low densities $\rho<2\rho_{0}$
are constrained by various nuclear physics experiments measuring the bulk
properties of finite nuclei such as the nuclear masses, neutron skin thickness
in 208Pb, the dipole polarizability and isobaric analog states as well as the
heavy-ion collisions. Usually one utilizes these data from nuclear physics
experiments and astrophysical observations to constrain the EoS through a
Bayesian approachTews  _et al._ (2013); Hebeler  _et al._ (2013); Le Fèvre
_et al._ (2016); Russotto et al. (2016); Lynn  _et al._ (2016); Abbott  _et
al._ (2017); Coughlin  _et al._ (2017); Drischler  _et al._ (2019); Zhang  _et
al._ (2018); Drischler  _et al._ (2020); Dietrich et al. (2020); Raaijmakers
_et al._ (2020); Huth  _et al._ (2022); Biswas  _et al._ (2021a); Tsang  _et
al._ (2024); Imam  _et al._ (2024a); Malik  _et al._ (2024). Moreover, one
needs to perform the analysis using different combinations of these dataset to
asses their importance in determining various key parameters of a EoS. The
Bayesian approach utilizing the likelihood functions yielding the posterior
distributions for various NS observable is generally time-consuming Romero-
Shaw et al. (2020); Riley  _et al._ (2018); Brandes et al. (2023); Tiwari and
Pai (2024). This is primarily due to the computation of NS properties through
the solutions of the Tolman-Oppenheimer-Volkoff (TOV) equations. To overcome
this issue recently there have been few attempts to map the EoS to the NS
properties so that the NS properties can be calculated without recourse to the
solutions of TOV equations Soma  _et al._ (2024); Richter and Li (2023); Imam
_et al._ (2024b); Divaris  _et al._ (2024); Carvalho  _et al._ (2024);
Ofengeim  _et al._ (2024); Reed  _et al._ (2024); Kumar  _et al._ (2024).

In Ref. Imam  _et al._ (2024b), we have performed a systematic analysis and
correlation studies involving multiple parameters to identify the essential
nuclear matter parameters crucially influencing the tidal deformability of
neutron stars derived from solutions of TOV equations for neutron star masses
ranging from 1.2 - 1.8 M⊙. Here, we demonstrate the mapping of neutron star
properties to the key nuclear matter parameters. Our established empirical
relation allows for a direct application in facilitating Bayesian analysis for
diverse astrophysical observations. In Ref. Richter and Li (2023), three
cutting-edge methodologies for constructing regression models-bidirectional
stepwise feature selection, LASSO regression, and neural network regression
are utilized. They are applied to pinpoint the crucial EoS parameters
influencing the radius R1.4 of canonical NSs. This analysis is based on the
posterior EoSs derived from Bayesian analyses of observational data on NS. All
three regression techniques indicate that only a few parameters governing the
NS EoS are significant predictors of R1.4.

In this study, we refrain from limiting our parameter space, in contrast to
the approach outlined in Ref.Imam  _et al._ (2024b). Instead, we employ a
symbolic regression technique to construct analytical expressions of neutron
star properties in terms of a few key EoS parameters. In Ref. Imam  _et al._
(2024b), we focused solely on constructing functions for the tidal
deformability of NS at some specific masses. However, in the present
investigation, we have developed analytical expressions for NS maximum mass,
radii, and tidal deformabilities across a broad spectrum of masses ranging
from 1.1 to 2.15 M⊙. These expressions are then incorporated into a Bayesian
framework, utilizing data from nuclear physics experiments and astrophysical
observations, to probe the overall behavior of the EoS.

This paper is organized as follows. In Sec II the construction of EoSs,
symbolic regression method and Bayesian framework are briefly described. The
results are discussed in Sec.III. The conclusions are drawn in Sec.IV.
Finally, the equations obtained using symbolic regression are provided in the
Appendix for future use.

##  II Methodology

Symbolic regression, a machine learning approach is utilized to map the NS
properties to a few key EoS parameters. The EoS are constructed using
$\frac{n}{3}$ expansion as detailed in Lattimer and Prakash (2016); Gil  _et
al._ (2017); Imam  _et al._ (2022).

###  II.1 EoS for NS Matter

We constructed the EoS for the core region of NS by employing a $\frac{n}{3}$
expansion within the density range of 0.5$\rho_{0}$ to 8$\rho_{0}$. For
densities below 0.5$\rho_{0}$, the neutron star is characterized by an outer
and inner crust. The EoS for the outer crust, covering up to
0.00016$\rho_{0}$, is derived from Baym  _et al._ (1971), while a polytropic
form of the EoS, described in Carriere  _et al._ (2003), is chosen for the
inner crust. These EoSs are utilized in computing neutron star properties
through the solutions of the Tolman-Oppenheimer-Volkoff equations Oppenheimer
and Volkoff (1939); Tolman (1939); Hinderer (2008). The chosen EoS satisfy
conditions of : (i)thermodynamic stability, (ii)positive semi-definiteness of
symmetry energy, (iii)causality of the speed of sound, and (iv) maximum mass
of stable non-rotating neutron stars, M${}_{max}\geq$ 2.15 M⊙.

###  II.2 Symbolic Regression

Symbolic regression is a type of regression analysis that aims to discover the
best mathematical expression or formula out of all possible equations
representing the relationship between input and output variables Koza (1992);
Schmidt and Lipson (2009); Sun  _et al._ (2019); Makke and Chawla (2023). This
process is beneficial when the underlying relationship between variables is
unknown or complex. The process involves searching through a vast space of
mathematical expressions, typically represented as combinations of
mathematical operators (e.g., addition, subtraction, multiplication, division)
and a wide range of functions (e.g., sin, cos, exp, log), and can also include
user-defined functions, constants, or conditional statements. This search is
often performed using various optimization algorithms. While genetic
programming and evolutionary algorithms are popular choices due to their
ability to efficiently explore large search spaces, other techniques like
simulated annealing or gradient-based methods can also be employed, depending
on the specific problem and computational constraints. To perform symbolic
regression, one has to provide the features (input variables) and their
corresponding target values (output variables). A K-fold cross-validation
procedure is often used to assess model performance and avoid overfitting.
This procedure yields multiple equations in each of the ’K’ folds. To choose
the best equation, the goodness of fit and model complexity are good metrics
to filter out among various searched equations. Simpler models are often
preferred to avoid overfitting, even with slightly lower goodness-of-fit
scores. The goodness of fit evaluates how well a statistical model, such as a
regression equation, fits the observed data. Several metrics are commonly used
for this purpose, two of them are (i) R2 which is defined as

$\displaystyle R^{2}$ $\displaystyle=$ $\displaystyle
1-\frac{SS_{residual}}{SS_{total}}$ (1) $\displaystyle SS_{residual}$
$\displaystyle=$ $\displaystyle\sum(y_{i}-\hat{y}_{i})^{2}$ (2) $\displaystyle
SS_{total}$ $\displaystyle=$ $\displaystyle\sum(y_{i}-\bar{y})^{2}.$ (3)

Here yi is the observed or actual value, $\hat{y}_{i}$ is the predicted value
for the ith data point and $\bar{y}$ is the mean of the ’n’ observed values.
R2 measures the proportion of the variance in the dependent variable that is
explained by the independent variables in the model. It ranges from 0 to 1,
with higher values indicating a better fit.
(ii) RMSE which is defined as,

$\displaystyle{\rm RMSE}=\sqrt{\frac{SS_{residual}}{n}}$ (4)

This also provide information about the accuracy of a regression model, with
lower values indicating a better fit.

###  II.3 Bayesian Inference

The Bayesian likelihood is a fundamental concept in Bayesian statistics,
providing a way to update the probability estimate for a hypothesis as more
evidence or information becomes available. In the context of Bayesian
inference, the posterior distribution measures the plausibility of a set of
parameter values given the observed data.

Bayes’ Theorem:- Bayes’ theorem is used to update the probability estimate of
a hypothesis as more evidence is available. It is formulated as:

$P(\theta|{\mathcal{D}})=\frac{{\mathcal{L}}(\mathcal{D}|{\theta})P(\theta)}{P({\mathcal{D}})}$
(5)

In the above equation:

  * •

$P(\theta|{\mathcal{D}})$ is the posterior probability of the parameters
$\theta$ given the data, $\mathcal{D}$.

  * •

${\mathcal{L}}(\mathcal{D}|{\theta})$ is the likelihood of the data under the
parameters (likelihood function).

  * •

$P(\theta)$ is the prior probability of the parameters $\theta$.

  * •

$P({\mathcal{D}})$ is the marginal likelihood or evidence, often acting as a
normalizing constant.

For the details about the likelihood function see the Bayesian inference
section of Ref.Agathoset al. (2015); Wysocki  _et al._ (2020); Landry  _et
al._ (2020); Biswas  _et al._ (2021b); Imam  _et al._ (2024a).

##  III Results and Discussions

Table 1: Prior distributions of the NMPs ($\theta$). Here $\rho_{0}$ = 0.16 fm-3, e0 = -16.0 MeV. All other parameters are uniformly distributed between a minimum ($\theta_{min}$) and a maximum ($\theta_{max}$). For the symbolic regression all the NMPs are reduced to the standard variable, $\hat{\theta}=\frac{(\theta-\bar{\theta})}{\sigma_{\theta}}$ with mean $\bar{\theta}$ and standard deviation $\sigma_{\theta}$. The unit of all the NMPs are in MeV.  | K0 | Q0 | Z0 | J0 | L0 | Ksym,0 | Qsym,0 | Zsym,0
---|---|---|---|---|---|---|---|---
$\theta_{min}$ | 180 | -1500 | -3000 | 25 | 20 | -300 | 300 | 3000
$\theta_{max}$ | 330 | 1500 | 3000 | 40 | 120 | 300 | 1500 | -3000
$\bar{\theta}$ | 256.54 | -0.52 | -1049.42 | 31.99 | 76.04 | -41.12 | 987.38 | 200.38
$\sigma_{\theta}$ | 48.99 | 266.00 | 1558.57 | 4.28 | 27.57 | 154.15 | 331.14 | 1694.79
Figure 1: Prediction of symbolic regression models for radius, tidal
deformability of NS of mass 1.425 M⊙ and maximum mass of NS. The root mean
squared error (RMSE) is also quoted in each panel.

We construct the Symbolic Regression Models(SRMs) to express the NS maximum
mass, radii and tidal deformabilities in terms of various nuclear matter
parameters as briefly outlined in Sec II.2. Once SRMs are obtained using the
solutions of the TOV equations, they can facilitate constraining the EoS
through Bayesian inference. We first validate SRMs for various NS properties
considered. Next, we study the sensitivity of the EoS to various nuclear
physics and astrophysical inputs considered. Finally, we compared the results
from Bayesian inference for a realistic set of data with SRMs and with those
obtained directly from the solutions of TOV equations.

###  III.1 Construction and Validation of the SRMs

A large number ($\sim$ 104) of EoSs are generated for the neutron star matter
as described in Sec II.1. The NMPs required for the EoSs are drawn randomly
from their uniform distributions (see Table 1). The average ($\bar{\theta}$)
and standard deviation ($\sigma_{\theta}$) of each NMPs($\theta$) listed in
Table 1 are obtained for the valid EoSs which satisfy the physical conditions
and lower bound on NS maximum mass as stated in Sec II.1. The NS properties
obtained for these EoSs through the solutions of the TOV equations are then
fitted to the NMPs through symbolic regression to construct the SRMs. For the
purpose of symbolic regression first all the features (NMPs), $\theta$ are
reduced to standardized form
$\hat{\theta}=\frac{(\theta-\bar{\theta})}{\sigma_{\theta}}$, so that each of
them can be treated equally. We have modelled NS radii and tidal
deformabilities with masses between 1.1 - 2.15 M⊙ and maximum mass (Mmax) of
NS in terms of the nuclear matter parameters directly through the symbolic
regression. We employed symbolic regression to explore a diverse set of
potential equations, aiming to identify the most optimal mathematical
representation. We allowed basic elementary mathematical operations along with
more complex operations like square, cube, square root, inverse, sine, cosine,
and tangent. This approach allows for a comprehensive search across various
functional forms, ensuring the best possible model fit. The expression of the
NS property,${\mathcal{T}}$ takes the form :

$\displaystyle{\mathcal{T}}_{\rm SRM}$ $\displaystyle=$ $\displaystyle
a_{0}+\sum_{i}a_{i}\hat{\theta_{i}}+\sum_{i}\sum_{j}a_{ij}\hat{\theta_{i}}\hat{\theta_{j}}$
(6)

where, $\theta_{i}\in$ {K0, Q0, Z0, J0, L0, Ksym0, Qsym0, Zsym0} with a0, ai
and aij are the coefficients of the equations obtained through the cross-
validation method in symbolic regression. The equations are selected based on
a balance between achieving a high R2 (Eq 1), low RMSE (Eq 4), and less model
complexity. These equations are provided in the Appendix. Using these
equations we compute the radius and tidal deformability of neutron stars with
masses ranging between 1.1 - 2.15 M⊙ in the interval of 0.05 M⊙ for which we
have the analytical expressions from symbolic

Figure 2: The distribution of L0 (in MeV) and Ksym0 (in MeV) for the cases for which symbolic regression model predictions yield, $\frac{\delta\Lambda}{\Lambda}>$0.3(see Fig 1) for tidal deformability of 1.425 M⊙ NSs. Table 2: Pseudo data on a few of the NS properties and key EoS quantities generated for a given set of NMPs listed as ‘ref’ in Table 3. The NS properties considered are radii (RM in km), dimensionless tidal deformabilities($\Lambda_{\rm M}$), maximum mass (Mmax in M⊙) with uncertainties 6$\%$, 30$\%$ and 2$\%$ respectively. The key EoS quantities are symmetry energy (esym($\rho$)), symmetry energy pressure (Psym($\rho$)), pressure due to snm (Psnm($\rho$)) referred as “pseudo experimental data” and energy of the pure neutron matter (epnm($\rho$)), slope of incompressibility (${\mathcal{M}}$($\rho$)) at the density $\rho$ fm-3. The uncertainties on epnm and ${\mathcal{M}}$ are 0.5 MeV and 100 MeV, respectively. The uncertainties on esym($\rho$), Psym($\rho$), Psnm($\rho$) are same as in Table-I of Refs Imam  _et al._ (2024a). The units of all the quantities related to the EoS are MeV. NS properties | EoS quantities
---|---
R1.4 |  13.24 |  epnm(0.1) |  10.43
|  |  ${\mathcal{M}}$(0.1) |  906
|  |  esym(0.035) |  12.86
R2.08 |  12.65 |  esym(0.05) |  15.77
|  |  esym(0.069) |  19.20
|  |  esym(0.101) |  24.5
$\Lambda_{1.4}$ |  597 |  esym(0.106) |  25.28
|  |  esym(0.115) |  26.66
|  |  esym(0.232) |  42.88
Mmax |  2.27 |  Psym(0.11) |  1.86
|  |  Psym(0.232) |  7.25
|  |  Psym(0.24) |  7.8
|  |  Psnm(0.32) |  17.23

regression. We use these calculated values to compute the NS radius and tidal
deformability for any mass M within the above mass range through interpolation
for which we do not have the analytical expression.

The values of R2 are $\sim$ 0.99, 0.93 and 0.92 for NS radii, tidal
deformabilities and maximum mass, respectively. To assess the model prediction
we calculate the radii and tidal deformabilities of NS with mass 1.425 M⊙ by
interpolating the results from the SRMs. We chose this mass because most of
the NSs clustered close to $\sim$ 1.4 M⊙. The Fig (1) displays the predictions
of the SRMs for R1.425, $\Lambda_{1.425}$ and Mmax. The RMSE scores are 0.33
km, 64 and 0.03 M⊙ for R1.425, $\Lambda_{1.425}$ and Mmax, respectively. The
$\frac{\delta{\mathcal{T}}}{{\mathcal{T}}}$ = $\frac{({\mathcal{T}}_{\rm
TOV}-{\mathcal{T}}_{\rm SRM})\times 100}{{\mathcal{T}}_{\rm TOV}}$ is the
relative percentage error for a given target variable ${\mathcal{T}}$, plotted
as a function of ${\mathcal{T}}_{\rm TOV}$. Here ${\mathcal{T}}_{\rm TOV}$ and
${\mathcal{T}}_{\rm SRM}$ denote the NS property derived using the solutions
of TOV equations and using symbolic regression models, respectively. For the
case of $\Lambda_{1.425}$, there are a few samples that show a relative error
greater than 30$\%$. To analyze this in more detail, in the Fig 2 we plot the
variation of L0 with Ksym0 for these specific cases. The large relative errors
($>$30$\%$) occur for the EoSs with very low Ksym0($<$-200 MeV) and/or very
small L0 ($<$40 MeV). These values of L0 and Ksym0 are somewhat away from the
commonly accepted ranges.

###  III.2 Sensitivity of the EoS to Various Data

We use the SRMs instead of solving TOV equations to expedite our investigation
based on Bayesian inferences. We perform Bayesian inference for various
scenarios to examine the sensitivity of the EoS to various nuclear and
astrophysics inputs. Before embarking on analysis using the realistic set of
data, we considered a set of pseudo data as listed in Table 2. These pseudo
data are generated using a suitable set of NMPs referred to as ‘ref’ in Table
3. The pseudo inputs related to NS are generated through the solutions of TOV
equations. We use the empirical data on symmetry energy (esym), symmetry
energy pressure (Psym) and pressure of symmetric nuclear matter (Psnm) from
various nuclear experiments spanning the density range 0.03 - 0.32 fm-3
referred hereafter as “pseudo experimental data”. In addition, we employ the
empirical values of energy per nucleon for pure neutron matter epnm and slope
of the incompressibility coefficients ${\mathcal{M}}$ at the crossing density
($\rho\sim$ 0.1 fm-3 ). The empirical values of epnm and ${\mathcal{M}}$ are
derived from nuclear masses and iso-scalar giant monopole resonances Brown
(2013); Khan and Margueron (2013). The significance of epnm is that it has a
contribution from both symmetric nuclear matter and symmetry energy. To
analyse the influence of these data on the EoS, we have considered different
scenarios as follows:

  * •

S1 : Only maximum mass Mmax of NS.

  * •

S2 : Radii for 1.4 M⊙ (R1.4) and 2.08 M⊙ (R2.08) and tidal deformability for
1.4 M⊙ ($\Lambda_{1.4}$) NS along with Mmax.

  * •

S3 : The empirical values of epnm and ${\mathcal{M}}$ at crossing density
($\rho\sim$ 0.1 fm-3) along with the astrophysical inputs considered in the
scenario S2.

  * •

S4 : Data from scenario S2 together with “pseudo experimental data” (see Table
2).

  * •

S5 : All the data, including those from scenario S4, as well as data on epnm
and ${\mathcal{M}}$.

Table 3: Distributions of the NMPs within 68$\%$ confidence interval obtained from Bayesian inference using symbolic regression models. The reference values (‘ref’) of the NMPs employed to generate the pseudo data of Table 2 used for Bayesian inference are also listed. We have constructed 5 scenarios for different combinations of these data (see text). Among them only the S5 scenario is also performed with the traditional approach of computing EoS and solving TOV equations which is the last row denoted as ${\mathcal{T}}_{\rm TOV}$. The unit of All the NMPs are in MeV. | Scenario | K0 | Q0 | Z0 | J0 | L0 | Ksym0 | Qsym0 | Zsym0
---|---|---|---|---|---|---|---|---|---
ref | - | 266 | -90 | -996 | 33.17 | 67 | -47 | 458 | -38
| S1 | 241${}_{-58}^{+56}$ | -118${}_{-249}^{+228}$ | -50${}_{-1782}^{+1929}$ | 32.42${}_{-4.94}^{+4.75}$ | 74${}_{-26}^{+28}$ | 6${}_{-188}^{+177}$ | 936${}_{-392}^{+355}$ | -3${}_{-1827}^{+1873}$
S2 | 242${}_{-57}^{+56}$ | -142${}_{-233}^{+251}$ | 19${}_{-1968}^{+1919}$ | 32.59${}_{-4.59}^{+4.76}$ | 66${}_{-14}^{+15}$ | -22${}_{-147}^{+158}$ | 877${}_{-348}^{+390}$ | 29${}_{-1858}^{+1869}$
${\mathcal{T}}_{\rm SRM}$ | S3 | 259${}_{-23}^{+25}$ | -194${}_{-181}^{+183}$ | 2${}_{-1881}^{+1951}$ | 33.27${}_{-1.85}^{+1.78}$ | 67${}_{-16}^{+16}$ | -40${}_{-152}^{+169}$ | 886${}_{-382}^{+397}$ | -1${}_{-1989}^{+2017}$
| S4 | 256${}_{-56}^{+46}$ | -148${}_{-266}^{+255}$ | -263${}_{-1790}^{+2079}$ | 32.92${}_{-1.33}^{+1.36}$ | 65${}_{-14}^{+14}$ | -27${}_{-79}^{+81}$ | 911${}_{-398}^{+380}$ | 197${}_{-2056}^{+1844}$
| S5 | 262${}_{-23}^{+21}$ | -167${}_{-194}^{+175}$ | -281${}_{-1805}^{+2034}$ | 32.97${}_{-1.26}^{+1.29}$ | 65${}_{-14}^{+14}$ | -26${}_{-80}^{+79}$ | 882${}_{-387}^{+402}$ | 160${}_{-2023}^{+1889}$
${\mathcal{T}}_{\rm TOV}$ | S5 | 262${}_{-20}^{+20}$ | -203${}_{-176}^{+182}$ | -40${}_{-1392}^{+1638}$ | 32.74${}_{-1.24}^{+1.24}$ | 63${}_{-13}^{+14}$ | -34${}_{-74}^{+76}$ | 845${}_{-358}^{+403}$ | 339${}_{-1691}^{+2120}$

In Table 3, we present the median and 68$\%$ confidence intervals for the NMPs
for various scenarios as discussed above. It is apparent that only scenarios
involving Mmax, and R1.4, R2.08 and $\Lambda_{1.4}$ are not enough to
reproduce the reference values. We consider the scenario S3 where the data on
epnm and ${\mathcal{M}}$ at crossing density ($\rho\sim$ 0.1 fm-3) are
included in addition to those for Mmax, R1.4, R2.08 and $\Lambda_{1.4}$. The
uncertainties for most of the NMPs are reduced. The median values of the NMPs
such as K0, J0, L0 and Ksym0 become closer to their reference values. It is
clear that epnm and ${\mathcal{M}}$ associated with low density play a
significant role in constraining the overall behaviour of the EoS.

Figure 3: Distribution of $\beta$ equilibrium matter pressure Pbeta as a
function baryon density. S2 represents the scenario where only data from NS
observations are considered while S5 represents the scenario in which all the
data from low-density nuclear experiments are also considered along with data
of S2 scenario.

The scenario S4 which includes all the pseudo data of Table 2 except for epnm
and ${\mathcal{M}}$, yields smaller uncertainties concerning NMPs associated
with symmetry energy compared to the scenario S2, where only the data on NS
radii, tidal deformabilities and maximum mass are taken into account. The
uncertainty of the K0 is also reduced. When comparing scenarios S3 and S4, it
becomes apparent that constraints from epnm and ${\mathcal{M}}$ control the
symmetric nuclear matter sector of the EoS, while the pseudo experimental data
constrains the density-dependent symmetry energy sector of the EoS. Therefore,
by combining both sets of data with information on NS radius, tidal
deformability and maximum mass in the scenario S5, the distributions of NMPs
are found to be consistent with those of S3 and S4. To assess the robustness
of our results using SRMs in Bayesian inference, we performed the calculation
again only for the scenario S5, by replacing SRMs with the solutions from the
TOV equations to compute NS properties. From the last two rows of the table,
it is evident that the results are similar in both approaches. Nevertheless,
it should be underscored that the SRMs method proves significantly more
efficient, performing computations more than 100 times quicker, often within a
few minutes. Furthermore, as displayed in Fig 3 the experimental data and
empirical values of epnm and ${\mathcal{M}}$ have constrained the pressure of
$\beta$-equilibrium matter at high density, as evidenced by the results
obtained for the scenarios S2 and S5 presented in Table 3.

###  III.3 Constraining the EoS using Realistic Data

We repeat the analysis using realistic data for the scenario S5, as the
reference values for the NMPs are successfully recovered in this case.
Bayesian inference is employed with both SRMs and solutions from TOV equations
to calculate various properties of neutron stars. The realistic data except
for epnm and ${\mathcal{M}}$ are exactly the same as in Ref Imam  _et al._
(2024a)(see Table-I of Imam  _et al._ (2024a)). The epnm and ${\mathcal{M}}$
as introduced in the present paper, there values at the crossing density are
taken from Refs Brown (2013); Khan and Margueron (2013). The SRMs provide the
results within 20 minutes being 100 times faster than the approach of
computing the EoS and solving TOV equations to calculate the NS properties.

The marginalized posterior distributions (PDs) and the confidence ellipses for
the NMPs derived from both SRMs and TOV solutions are displayed as corner
plots in Fig 4. The corner plots exhibit the one-dimensional marginalized PDs
for the NMPs on their diagonals. Concurrently, the confidence ellipses are
depicted alongside the off-diagonal elements of the corner plots, representing
1$\sigma$, 2$\sigma$, and 3$\sigma$ confidence intervals. The width and
inclination of the confidence ellipses for a pair of NMPs determines the
nature of the linear correlations among them. It shows L0 is strongly
correlated with J0 and Ksym0 with Pearson’s correlation coefficient r $\sim$
0.9. There is also a stronger correlation between J0 and Ksym0 (r $\sim$ 0.7).
Any other pairs of NMPs do not show significant correlations. It is also clear
that the results of Bayesian inference using SRMs and solutions of TOV
equations are very similar.

We also present the posterior distributions of pressure of $\beta$ equilibrium
matter (Pbeta($\rho$)), pressure of symmetric nuclear matter (Psnm($\rho$))
and symmetry energy pressure (Psym($\rho$)) in Fig 5. We observe that the low-
density parts of pressure, both for symmetric nuclear matter and symmetry
energy, are tightly constrained, and hence the $\beta$-equilibrium matter
pressure is also constrained very well at low density ($\rho\leq$
2$\rho_{0}$). So the experimental data at low densities effectively limit the
symmetry energy within this regime. However, uncertainties grow

Figure 4:  The marginalized posterior distributions of the NMPs obtained in
the Bayesian inference with the realistic data. We present the results
obtained using SRMs (blue) and solutions of TOV equations (red) to compute NS
properties in the inference. Figure 5: 95$\%$ confidence interval for the
pressure of $\beta$-equilibrate matter Pbeta($\rho$) (top), pressure of
symmetric nuclear matter Psnm($\rho$)(middle) and symmetry energy pressure
Psym($\rho$)(bottom) as a function of baryon density. These are calculated
using distributions of the NMPs as shown in Fig 4. The unit of All the NMPs
are in MeV.

significantly as we move beyond 2$\rho_{0}$, especially in the case of the
pressure from symmetry energy. This indicates that existing astrophysical data
on neutron star properties do not sufficiently constrain symmetry energy at
the high-density.

##  IV Summary

We generate a large ensemble of EoSs characterized by the nuclear matter
parameters using $\frac{n}{3}$ expansion. The neutron star properties such as
radius, tidal defromability and maximum mass corresponding to these EoSs are
obtained through the solutions of TOV equations. We have constructed
analytical expressions using symbolic regression which map these NS properties
to the nuclear matter parameters. For these expressions of radii and tidal
deformabilities of NS, we have considered the mass range from 1.1 to 2.15 M⊙,
with the interval of 0.05 M⊙. These analytical expressions (SRMs) are employed
in Bayesian inference for the faster estimation of NS properties directly in
terms of the NMPs. We employed the SRMs in the Bayesian inference to constrain
the EoS.

The symbolic regression model approach is computationally faster by more than
100 times compared to those performed with the solutions of TOV equations.
This enabled us to perform Bayesian inference for different scenarios to study
the sensitivity of various data on the EoS. We have first considered a set of
pseudo data for a known value of NMPs so that the accuracy of the posterior
distributions of the NMPs can be assessed. We have considered selected data on
key EoS quantities and NS properties. The key EoS quantities considered are
the pressure for the symmetric nuclear matter, symmetry energy pressure and
the symmetry energy, empirically constrained over a range of
densities($\sim$0.03 - 0.32 fm-3) by the experimental data on bulk properties
of finite nuclei and the heavy-ion collision. We have also included the
constraints on energy per pure neutron matter and slope of incompressibility
coefficient at the crossing density ($\sim$ 0.1 fm-3). The astrophysics data
considered are the radii for NS with mass $\sim$ 1.4 M⊙ and 2.08 M⊙ and the
tidal deformability for 1.4 M⊙ NS and the NS maximum mass. Our analysis
reveals that the key EoS quantities associated with low densities in addition
to the NS observations play a significant role in confining the overall
behaviour of the EoS (see Fig 3).
Finally, we have performed Bayesian inference using the aforementioned data
obtained from nuclear experiments and astrophysical observations. The symbolic
regression models are employed in the Bayesian inference to compute the NS
properties. To verify the reliability of our findings with symbolic regression
models, we repeated the analysis using the solutions of TOV equations to
compute the NS properties. Remarkably, both approaches yielded comparable
results, while the SRMs deliver outcomes in just 20 minutes, making them over
100 times faster than the approach of using the solutions of TOV equations
within the same computational setup.

##  V Acknowledgements

BKA acknowledges partial support from the SERB, Department of Science and
Technology, Government of India with CRG/2021/000101. The authors sincerely
acknowledge the usage of the analysis software BILBY Gregory et al. (2019);
Romero-Shaw et al. (2020). T.M has received support through national funds
from FCT (Fundação para a Ciência e a Tecnologia, I.P, Portugal) for projects
UIDB/04564/2020 and UIDP/04564/2020, identified by DOIs
10.54499/UIDB/04564/2020 and 10.54499/UIDP/04564/2020, respectively, and for
project 2022.06460.PTDC with the DOI identifier 10.54499/2022.06460.PTDC. T.M
would additionally like to express gratitude for the RNCA (Rede Nacional de
Computação Avançada) advanced computing projects 2024.14108.CPCA.A3, which
received funding from the FCT.

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## Appendix

### Equations for Tidal Deformability of NS

To find the tidal deformability at any mass M between 1.1 M⊙ to 2.15 M⊙ first
we compute the tidal deformabilities using the expressions $\Lambda_{\rm M}$
given below for all the masses, then we do interpolate between this mass
range. Here each of the NMPs $\theta$ are expressed as a reduced variable
${\hat{\theta}}=\frac{\theta-\bar{\theta}}{\sigma_{\theta}}$ with
${\bar{\theta}}$ and $\sigma_{\theta}$ are the average and standard deviation
of each NMPs as described earlier.

$\displaystyle\Lambda_{1.1}$ $\displaystyle=$
$\displaystyle{\hat{L}}_{0}(-118.62{\hat{J}}_{0}-368.19(\frac{{\hat{J}}_{0}}{{\hat{L}}_{0}}+{\hat{K}}_{\rm
sym0}-{\hat{L}}_{0})+1194.33)+2696.03$ (7) $\displaystyle\Lambda_{1.15}$
$\displaystyle=$
$\displaystyle{\hat{L}}_{0}((201.40-{\hat{L}}_{0})(-{\hat{J}}_{0}-\frac{{\hat{J}}_{0}}{{\hat{L}}_{0}}-{\hat{K}}_{\rm
sym0}+{\hat{L}}_{0})+845.04)+2197.35$ (8) $\displaystyle\Lambda_{1.2}$
$\displaystyle=$ $\displaystyle-96.86({\hat{J}}_{0}-{\hat{K}}_{\rm
sym0}-{\hat{Q}}_{\rm
sym0}+{\hat{Z}}_{0})+{\hat{L}}_{0}({\hat{L}}_{0}+582.78)+1835.62$ (9)
$\displaystyle\Lambda_{1.25}$ $\displaystyle=$
$\displaystyle-100.89({\hat{J}}_{0}-{\hat{K}}_{\rm sym0}-{\hat{Q}}_{\rm
sym0})+{\hat{L}}_{0}(410.36-{\hat{K}}_{\rm sym0})-54.39{\hat{Z}}_{0}+1445.51$
(10) $\displaystyle\Lambda_{1.3}$ $\displaystyle=$
$\displaystyle-51.27({\hat{J}}_{0}-{\hat{K}}_{0}-{\hat{Q}}_{0}-{\hat{Q}}_{\rm
sym0})+102.54{\hat{K}}_{\rm sym0}+296.27{\hat{L}}_{0}+1160.47$ (11)
$\displaystyle\Lambda_{1.35}$ $\displaystyle=$
$\displaystyle-53({\hat{J}}_{0}-{\hat{K}}_{0}-{\hat{Q}}_{0}-{\hat{Q}}_{\rm
sym0})+108.45({\hat{K}}_{\rm sym0}+2{\hat{L}}_{0})+919.09$ (12)
$\displaystyle\Lambda_{1.4}$ $\displaystyle=$
$\displaystyle-54.02({\hat{J}}_{0}-{\hat{K}}_{0}-{\hat{Q}}_{0}-{\hat{Q}}_{\rm
sym0})+76.09{\hat{K}}_{\rm sym0}+178.12{\hat{L}}_{0}+737.93$ (13)
$\displaystyle\Lambda_{1.45}$ $\displaystyle=$
$\displaystyle-40.17({\hat{J}}_{0}-{\hat{Q}}_{0}-{\hat{Q}}_{\rm
sym0})+48.08({\hat{K}}_{0}+{\hat{K}}_{\rm sym0})+136.60{\hat{L}}_{0}+595.79$
(14) $\displaystyle\Lambda_{1.5}$ $\displaystyle=$
$\displaystyle-37.67({\hat{J}}_{0}-{\hat{K}}_{0})+82.07({\hat{K}}_{\rm
sym0}+{\hat{L}}_{0})+26.79({\hat{Q}}_{0}+{\hat{Q}}_{\rm sym0})+484.31$ (15)
$\displaystyle\Lambda_{1.55}$ $\displaystyle=$
$\displaystyle-25.89({\hat{J}}_{0}-{\hat{Q}}_{\rm
sym0}+{\hat{Z}}_{0})+27.54{\hat{K}}_{0}+63.88({\hat{K}}_{\rm
sym0}+{\hat{L}}_{0})+391.95$ (16) $\displaystyle\Lambda_{1.6}$
$\displaystyle=$
$\displaystyle-21.54({\hat{J}}_{0}-{\hat{Q}}_{0}-{\hat{Q}}_{\rm
sym0})+43.08{\hat{K}}_{0}+53.01({\hat{K}}_{\rm sym0}+{\hat{L}}_{0})+319.95$
(17) $\displaystyle\Lambda_{1.65}$ $\displaystyle=$
$\displaystyle(-{\hat{K}}_{0}-20.51)({\hat{J}}_{0}-{\hat{K}}_{0}-2.14{\hat{K}}_{\rm
sym0}-2.14{\hat{L}}_{0}-{\hat{Q}}_{0}-{\hat{Q}}_{\rm sym0}-12.75)$ (18)
$\displaystyle\Lambda_{1.7}$ $\displaystyle=$
$\displaystyle-12.47({\hat{J}}_{0}-{\hat{Q}}_{0}-{\hat{Q}}_{\rm
sym0})+24.93{\hat{K}}_{0}+34.31({\hat{K}}_{\rm sym0}+{\hat{L}}_{0})+213.74$
(19) $\displaystyle\Lambda_{1.75}$ $\displaystyle=$
$\displaystyle-11.12{\hat{J}}_{0}+24.89({\hat{K}}_{0}+{\hat{K}}_{\rm
sym0}+{\hat{L}}_{0})+14.95({\hat{Q}}_{0}+{\hat{Q}}_{\rm sym0})+174.25$ (20)
$\displaystyle\Lambda_{1.8}$ $\displaystyle=$
$\displaystyle-10.24({\hat{J}}_{0}-{\hat{Q}}_{0}-{\hat{Q}}_{\rm
sym0})+18.04({\hat{K}}_{0}+{\hat{L}}_{0})+25.83{\hat{K}}_{\rm sym0}+142.42$
(21) $\displaystyle\Lambda_{1.85}$ $\displaystyle=$
$\displaystyle-8.05{\hat{J}}_{0}+19.96({\hat{K}}_{0}+{\hat{K}}_{\rm
sym0})+12({\hat{L}}_{0}+{\hat{Q}}_{0}+{\hat{Q}}_{\rm sym0})+116.10$ (22)
$\displaystyle\Lambda_{1.9}$ $\displaystyle=$
$\displaystyle-8.64({\hat{J}}_{0}-{\hat{L}}_{0}-{\hat{Q}}_{0}-{\hat{Q}}_{\rm
sym0})+15.10{\hat{K}}_{0}+17.28{\hat{K}}_{\rm sym0}+94.47$ (23)
$\displaystyle\Lambda_{1.95}$ $\displaystyle=$
$\displaystyle-6.42({\hat{J}}_{0}-{\hat{L}}_{0}-{\hat{Q}}_{\rm
sym0})+14.69({\hat{K}}_{0}+{\hat{K}}_{\rm sym0})+9.20{\hat{Q}}_{0}+76.57$ (24)
$\displaystyle\Lambda_{2.0}$ $\displaystyle=$
$\displaystyle-4.78({\hat{J}}_{0}-{\hat{L}}_{0}-{\hat{Q}}_{\rm
sym0}-{\hat{Z}}_{0})+12.84({\hat{K}}_{0}+{\hat{K}}_{\rm
sym0}+{\hat{Q}}_{0})+61.66$ (25) $\displaystyle\Lambda_{2.05}$
$\displaystyle=$ $\displaystyle-4.32({\hat{J}}_{0}-{\hat{Q}}_{\rm
sym0})+11.34{\hat{K}}_{0}+8.38({\hat{K}}_{\rm
sym0}-{\hat{Q}}_{0})+{\hat{L}}_{0}+49.33$ (26) $\displaystyle\Lambda_{2.1}$
$\displaystyle=$ $\displaystyle-3.57({\hat{J}}_{0}-{\hat{Q}}_{\rm
sym0})+11.23{\hat{K}}_{0}+7.66({\hat{K}}_{\rm
sym0}+{\hat{Q}}_{0})+{\hat{Z}}_{\rm sym0}+38.89$ (27)
$\displaystyle\Lambda_{2.15}$ $\displaystyle=$
$\displaystyle{\hat{K}}_{0}+(-{\hat{K}}_{0}-{\hat{Q}}_{0}-3.88)(0.67{\hat{J}}_{0}-1.32{\hat{K}}_{\rm
sym0}-{\hat{Q}}_{\rm sym0}-7.76)$ (28)

### Equations for Radius of NS

To calculate the radius of a neutron star at any mass M we have to follow the
same procedure as discussed about tidal deformability in the previous section.

$\displaystyle R_{1.1}$ $\displaystyle=$
$\displaystyle-0.26({\hat{J}}_{0}-{\hat{Q}}_{\rm
sym0}-{\hat{Z}}_{0})-1.26{\hat{K}}_{\rm
sym0}+2.42{\hat{L}}_{0}+{\hat{Q}}_{0}+14.76$ (29) $\displaystyle R_{1.15}$
$\displaystyle=$
$\displaystyle-0.38({\hat{J}}_{0}+{\hat{K}}_{0}-{\hat{Q}}_{\rm
sym0}+{\hat{Z}}_{0})-1.02{\hat{K}}_{\rm sym0}+2.30{\hat{L}}_{0}+14.66$ (30)
$\displaystyle R_{1.2}$ $\displaystyle=$
$\displaystyle-0.36({\hat{J}}_{0}+{\hat{K}}_{\rm
sym0}{\hat{L}}_{0}-{\hat{Q}}_{0}-{\hat{Q}}_{\rm sym0})-{\hat{K}}_{\rm
sym0}+2{\hat{L}}_{0}+14.55$ (31) $\displaystyle R_{1.25}$ $\displaystyle=$
$\displaystyle-0.33({\hat{J}}_{0}-{\hat{Q}}_{0}-{\hat{Q}}_{\rm
sym0})-0.18{\hat{K}}_{0}-{\hat{K}}_{\rm sym0}+2{\hat{L}}_{0}+14.43$ (32)
$\displaystyle R_{1.3}$ $\displaystyle=$
$\displaystyle-0.21({\hat{J}}_{0}+{\hat{K}}_{0}-{Q}_{0}-{\hat{Q}}_{\rm
sym0})-0.79K_{\rm sym0}+2{\hat{L}}_{0}+14.36$ (33) $\displaystyle R_{1.35}$
$\displaystyle=$ $\displaystyle-0.30(\hat{J}_{0}-\hat{Q}_{0}-\hat{Q}_{\rm
sym0})-0.70\hat{K}_{\rm sym0}+1.70\hat{L}_{0}+14.28$ (34) $\displaystyle
R_{1.4}$ $\displaystyle=$
$\displaystyle-0.31({\hat{J}}_{0}-{\hat{Q}}_{0}-{\hat{Q}}_{\rm
sym0})-0.62{\hat{K}}_{\rm sym0}+1.75{\hat{L}}_{0}+14.22$ (35) $\displaystyle
R_{1.45}$ $\displaystyle=$
$\displaystyle{\hat{L}}_{0}+14.15-(0.51({\hat{J}}_{0}-{\hat{L}}_{0}-{\hat{Q}}_{0}-{\hat{Q}}_{\rm
sym0})+{\hat{K}}_{\rm sym0})/(2.37-{\hat{L}}_{0})$ (36) $\displaystyle
R_{1.5}$ $\displaystyle=$ $\displaystyle-0.25(\hat{J}_{0}-\hat{Q}_{\rm
sym0})-0.43(\hat{K}_{\rm sym0}-\hat{Q}_{0})+1.43\hat{L}_{0}+14.08$ (37)
$\displaystyle R_{1.55}$ $\displaystyle=$
$\displaystyle-0.22(\hat{J}_{0}-\hat{Q}_{0}-\hat{Q}_{\rm
sym0})-0.45\hat{K}_{\rm sym0}+1.38\hat{L}_{0}+14.03$ (38) $\displaystyle
R_{1.6}$ $\displaystyle=$ $\displaystyle-0.25(\hat{J}_{0}+\hat{K}_{\rm
sym0}\hat{L}_{0}+\hat{K}_{\rm sym0}-\hat{Q}_{0}-\hat{Q}_{\rm
sym0})+1.25\hat{L}_{0}+13.97$ (39) $\displaystyle R_{1.65}$ $\displaystyle=$
$\displaystyle\hat{L}_{0}+13.92-(0.63\hat{J}_{0}+\hat{K}_{\rm
sym0}-\hat{L}_{0}-\hat{Q}_{0}-\hat{Q}_{\rm sym0})/(3.95-\hat{L}_{0})$ (40)
$\displaystyle R_{1.7}$ $\displaystyle=$
$\displaystyle-0.23(\hat{J}_{0}-\hat{Q}_{\rm sym0})-0.27(\hat{K}_{\rm
sym0}-\hat{Q}_{0})+1.17\hat{L}_{0}+13.85$ (41) $\displaystyle R_{1.75}$
$\displaystyle=$ $\displaystyle-0.23(\hat{J}_{0}+\hat{K}_{\rm
sym0}-\hat{Q}_{0}-\hat{Q}_{\rm sym0})-0.17\hat{K}_{\rm
sym0}\hat{L}_{0}+\hat{L}_{0}+13.81$ (42) $\displaystyle R_{1.8}$
$\displaystyle=$ $\displaystyle-0.21(\hat{J}_{0}+\hat{K}_{\rm
sym0}-\hat{Q}_{\rm sym0}+\hat{Z}_{0})-0.16\hat{K}_{\rm
sym0}\hat{L}_{0}+\hat{L}_{0}+13.74$ (43) $\displaystyle R_{1.85}$
$\displaystyle=$ $\displaystyle-0.20(\hat{J}_{0}+\hat{K}_{\rm
sym0}-\hat{Q}_{\rm sym0})+0.13\hat{K}_{0}+\hat{L}_{0}+0.32\hat{Q}_{0}+13.67$
(44) $\displaystyle R_{1.9}$ $\displaystyle=$
$\displaystyle-0.16(\hat{J}_{0}-\hat{K}_{0}+\hat{K}_{\rm
sym0})+\hat{L}_{0}+0.32\hat{Q}_{0}+0.24\hat{Q}_{\rm sym0}+13.58$ (45)
$\displaystyle R_{1.95}$ $\displaystyle=$
$\displaystyle-0.18(\hat{J}_{0}-\hat{K}_{0}+\hat{K}_{\rm
sym0}-2\hat{Q}_{0}-\hat{Q}_{\rm sym0})+0.82\hat{L}_{0}+13.52$ (46)
$\displaystyle R_{2.0}$ $\displaystyle=$
$\displaystyle-0.18\hat{J}_{0}+0.24\hat{K}_{0}+0.76\hat{L}_{0}+0.34\hat{Q}_{0}+0.24\hat{Q}_{\rm
sym0}+13.43$ (47) $\displaystyle R_{2.05}$ $\displaystyle=$
$\displaystyle-0.15\hat{J}_{0}+0.24\hat{K}_{0}+0.69\hat{L}_{0}+0.31\hat{Q}_{0}+0.24\hat{Q}_{\rm
sym0}+13.37$ (48) $\displaystyle R_{2.1}$ $\displaystyle=$
$\displaystyle-0.16\hat{J}_{0}+0.31\hat{K}_{0}+0.61\hat{L}_{0}+0.36\hat{Q}_{0}+0.37\hat{Q}_{\rm
sym0}+13.21$ (49) $\displaystyle R_{2.15}$ $\displaystyle=$
$\displaystyle-0.19\hat{J}_{0}+0.28\hat{K}_{0}+0.57\hat{L}_{0}+0.38\hat{Q}_{0}+0.28\hat{Q}_{\rm
sym0}+13.06$ (50)

### Equation for Maximum Mass of Neutron Star

The equation for the maximum mass of a neutron star is given as,

$\displaystyle M_{\rm max}$ $\displaystyle=$ $\displaystyle
0.14{\hat{K}}_{0}-0.02{\hat{L}}_{0}+0.27{\hat{Q}}_{0}+0.16{\hat{Z}}_{0}+2.33$
(51)