# Classification of pulsars with Dirichlet process Gaussian mixture model

Fahrettin Ay,1 Gökhan İnce,1 Mustafa E. Kamaşak1 and K. Yavuz Ekşi2
1Istanbul Technical University, Faculty of Computer and Informatics, Computer
Engineering Department, 34469, İstanbul, Turkey
2Istanbul Technical University, Faculty of Science and Letters, Physics
Engineering Department, 34469, İstanbul, Turkey E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Young isolated neutron stars (INS) most commonly manifest themselves as
rotationally powered pulsars (RPPs) which involve conventional radio pulsars
as well as gamma-ray pulsars (GRPs) and rotating radio transients (RRATs).
Some other young INS families manifest themselves as anomalous X-ray pulsars
(AXPs) and soft gamma-ray repeaters (SGRs) which are commonly accepted as
magnetars, i.e. magnetically powered neutron stars with decaying super-strong
fields. Yet some other young INS are identified as central compact objects
(CCOs) and X-ray dim isolated neutron stars (XDINSs) which are cooling objects
powered by their thermal energy. Older pulsars, as a result of a previous long
episode of accretion from a companion, manifest themselves as millisecond
pulsars and more commonly appear in binary systems. We use Dirichlet process
Gaussian mixture model (DPGMM), an unsupervised machine learning algorithm,
for analyzing the distribution of these pulsar families in the parameter space
of period and period derivative. We compare the average values of the
characteristic age, magnetic dipole field strength, surface temperature and
transverse velocity of all discovered clusters. We verify that DPGMM is robust
and provides hints for inferring relations between different classes of
pulsars. We discuss the implications of our findings for the magneto-thermal
spin evolution models and fallback discs.

###### keywords:

methods: data analysis – methods: statistical – stars: neutron – pulsars:
general

††pubyear: 2019††pagerange: Classification of pulsars with Dirichlet process
Gaussian mixture model–Classification of pulsars with Dirichlet process
Gaussian mixture model

##  1 Introduction

Rotationally powered pulsars (RPPs), the most common manifestation of neutron
stars, were discovered first as radio pulsars (Hewish et al., 1968). More than
$\sim 2600$ RPPs discovered to date are understood to be strongly magnetized
($B\sim 10^{12}$ G), rapidly rotating (with spin periods $P\sim 0.1$ s)
neutron stars spinning down by torques due to magnetic dipole radiation and
particle emission. These objects emit most of the energy, tapped from their
rotational kinetic energy, in X- and gamma-ray bands. The total radiative
luminosity of RPPs is thus below their spin-down power $L_{\rm
sd}\equiv-I\Omega\dot{\Omega}$ where $I$ is the moment of inertia of the
compact object, $\Omega=2{\mathrm{\pi}}/P$ is the spin angular frequency and
$\dot{\Omega}$ is its time derivative (see Manchester, 2017, for a review).
The bulk of the RPP population are the classical radio pulsars with periods
$P=0.2-2$ s, period derivatives $\dot{P}=10^{-16}-10^{-13}$ s s-1, magnetic
fields $B\sim 10^{10}-10^{13}$ G and characteristic ages $\tau_{\rm c}\equiv
P/2\dot{P}\sim 10^{3}-10^{7}$ years. Some of the RPPs are associated with
supernova remnants (SNRs). Young RPPs are usually isolated objects in the
sense that they have no binary companion. The pulsations are attributed to the
beaming of the emitted radiation and the “lighthouse” effect produced by the
spin of the object. Thus, apart from distant and dim objects, there could be
many RPPs whose beam does not sweep our line of sight.

A population among the RPPs are the millisecond pulsars (MSP; Backer et al.,
1982) which are old but “recycled” pulsars (Alpar et al., 1982); these objects
are understood to descent from low-mass X-ray binaries (LMXBs; see
Bhattacharya & van den Heuvel, 1991; Tauris & van den Heuvel, 2006, for
reviews) where the neutron star spins-up by accreting matter from a companion
via a disc. The accreted matter transfers angular momentum to the neutron star
thereby spinning it up to millisecond periods and the magnetic field of the
neutron star is reduced to values $B<10^{9}$ G in the process.

More than 200 $\gamma$-ray pulsars (GRPs) discovered by the Large Area
Telescope onboard Fermi (Fermi-LAT; Atwood et al., 2009) constitute a subclass
of RPPs (Abdo et al., 2010a, 2013). All known GRPs have $L_{\rm sd}>10^{33}$
erg s-1. About half of the GRPs detected by Fermi are radio-quite. This
possibly indicates that the “radio beam” which is narrower than the “gamma
beam” does not pass from our line of sight. Approximately half of the
discovered GRPs are MSP which was unexpected before the discovery (Abdo et
al., 2009).

Another recently discovered family of RPPs is the rotating radio transients
(RRATs) identified in the Parkes multibeam survey (McLaughlin et al., 2006).
Unlike ordinary pulsars detected by searches in the frequency domain, RRATs
show sporadic emission detected through their bright single pulses. These
objects show bursts of duration $2-30$ ms with an interval in the range 4
min-3 hr. Their spin periods are in the range $0.4-7$ s. The period
derivatives are measured in 3 sources implying magnetic fields $B\sim
10^{12}-10^{14}$ G and characteristic ages $\tau_{\rm c}\sim 0.1-3$ Myr.

In the last two decades many young isolated neutron star (INS) families (see
Popov, 2008; Kaspi, 2010; Harding, 2013; Safi-Harb, 2017, for reviews) other
than radio pulsars are identified. These objects are usually radio-quiet and
have X-ray luminosities exceeding the spin-down power of the compact object.
This indicates the availability of energy budgets other than rotational
kinetic energy and the possibility of evolutionary paths other than those
leading to RPPs.

Of these groups of objects soft gamma ray repeaters (SGRs) and anomalous X-ray
pulsars (AXPs) are commonly assumed to be magnetars (see Woods & Thompson
(2006); Mereghetti (2008); Rea (2014); Turolla et al. (2015); Mereghetti et
al. (2015); Kaspi & Beloborodov (2017); Gourgouliatos & Esposito (2018) for
reviews) i.e. INSs with large dipole magnetic fields $B\sim 10^{14}-10^{15}$ G
as inferred from their rapid spin down $\dot{P}\sim 10^{-13}-10^{-11}$ s s-1
(Kouveliotou et al., 1998) and slow periods clustered between $P=2-12$
seconds. According to the magnetar model (Duncan & Thompson, 1992; Thompson &
Duncan, 1996) the persistent X-ray emission of these objects with X-ray
luminosity of $L_{\rm X}\sim 10^{35}-10^{36}$ erg s-1 is powered by the decay
of this strong magnetic field (Paczynski, 1992; Thompson & Duncan, 1995) in
excess of the quantum critical limit $B_{\rm QED}=4.4\times 10^{13}$ G. These
objects occasionally show super-Eddington outbursts and, very rarely, giant
bursts which are addressed in the magnetar model with the breaking of the
neutron star crust due to magnetic stresses and reconfiguration of their
fields, respectively. The magnetars are young objects as implied by their
characteristic ages $\tau_{\rm c}\sim 10^{4}$ years and about half of them
being associated with SNRs.

X-ray dim isolated neutron stars (XDINSs), or sometimes called “magnificent
seven”, are the 7 nearby neutron stars identified through their thermal X-ray
emission (Özel, 2013; Potekhin et al., 2015; Mereghetti, 2011) with
luminosities of order $L_{\rm X}\sim 10^{30}-10^{32}$ erg s-1 (see Haberl,
2007; Kaplan, 2008; Turolla, 2009, for reviews). They have a period range
similar to the AXP/SGR family (Hambaryan et al., 2017), but are typically
older, with characteristic ages $\tau_{\rm c}\sim 10^{5}-10^{6}$ years and
kinematic ages of a few $10^{6}$ years (Tetzlaff et al., 2010, 2011, 2012).
They have inferred dipole magnetic fields of $B\sim 10^{13}$ G, an order of
magnitude lower than magnetars and an order of magnitude higher than
conventional pulsars, but their surface magnetic fields as inferred from the
narrow absorption features are 7 (Borghese et al., 2015) and 5 (Borghese et
al., 2017) times larger in the case of RX J0720.4–3125 and RX J1308.6+2127,
respectively.

Yet another young NS family is the central compact objects (CCOs; see De Luca,
2017, for a review) in supernova remnants (SNRs). These $\sim 10$ objects show
no sign of RPP activity. Their X-ray spectra is dominated by the thermal
emission showing some similarities with magnetars yet they are two orders of
magnitude less luminous ($\sim 10^{33}-10^{34}$ erg s-1). The periods measured
from 3 CCOs are in the range $0.1-0.4$ s and the measured period derivatives
imply that the dipole fields of these objects are in the range $B\sim
10^{10}-10^{11}$ G (Gotthelf & Halpern, 2007, 2009; Halpern & Gotthelf, 2011),
and hence sometimes are called “anti-magnetars” (Halpern & Gotthelf, 2010;
Gotthelf et al., 2013). Yet there is evidence that these objects have much
stronger “hidden” magnetic fields (Viganò & Pons, 2012; Viganò et al., 2013;
Torres-Forné et al., 2016) as implied by the highly anisotropic emission
(Shabaltas & Lai, 2011) leading to the observed high pulsed fraction.
According to the “field burial scenario” (Muslimov & Page, 1995; Young &
Chanmugam, 1995; Geppert et al., 1999; Ho, 2011; Bernal et al., 2012; Igoshev
et al., 2016) this is due to an initial fallback accretion episode the nascent
neutron star had suffered soon after the SN explosion. An exception among the
CCO family is 1E 161348–5055 in SNR RCW 103 which has an unusually long period
of 6.7 hours (de Luca, 2008) and has recently shown magnetar-like bursts (Rea
et al., 2016).

Recent discoveries are blurring the borders of the classes: (i) The existence
of high magnetic field RPPs (Pivovaroff et al. (2000); see Ng & Kaspi (2011)
for a review) that have magnetic field strengths comparable and, in some
cases, exceeding that of some magnetars and some of which has even shown
magnetar-like X-ray bursts (Gavriil et al., 2008; Archibald et al., 2016;
Göğüş et al., 2016); (ii) radio detection from some magnetars (Camilo et al.,
2006, 2007; Levin et al., 2010); (iii) identification of low-magnetic field
magnetars (Rea et al., 2010) i.e. INS showing magnetars bursts but with
ordinary inferred dipole field strengths (see Turolla & Esposito, 2013, for a
review). These discoveries indicate that the dipole field strength is not the
single parameter leading to the different manifestations favouring the early
suggestions (Gavriil et al., 2002; Ekşi & Alpar, 2003; Ertan & Alpar, 2003;
McLaughlin et al., 2003) that what causes the magnetar activity could be in
the higher multipoles (Alpar et al., 2011; Rodríguez Castillo et al., 2016).

In fact there are other observations suggesting that AXP/SGRs have “low”
dipole fields: (i) They are not detected in the Fermi/LAT observations (Şaşmaz
Muş & Göğüş, 2010; Abdo et al., 2010b) in the GeV range though, according to
the outer gap model (Zhang & Cheng, 1997), they are expected (Cheng & Zhang,
2001) to emit high energy gamma-rays should they have super-strong magnetic
dipole fields $\gtrsim 10^{14}$ G (Tong et al., 2011). (ii) their transverse
velocities are measured (Helfand et al., 2007; Deller et al., 2012; Tendulkar
et al., 2012, 2013) to be $200\pm 100$ km s-1, similar to the velocities of
RPPs though they are expected to have exceptionally large space velocities,
$\sim 1000$ km s-1, due to getting stronger kicks (Duncan & Thompson, 1992;
Thompson & Duncan, 1993) via the rocket propulsion effect should they have
super-strong dipole fields. (iii) They do not commonly show the signature of
the expected strong energy injection in the SNRs due to hosting a neutron star
with a birth period of milliseconds (Vink & Kuiper, 2006; Martin et al., 2014;
Borkowski & Reynolds, 2017) (but see Torres (2017)) that could produce super-
strong fields by dynamo action (Duncan & Thompson, 1992; Thompson & Duncan,
1993).

Given the transitivity among the INS families the question naturally arises
whether the classification of INS above is robust. Could we expect, for
example, that more of the RPPs to show magnetar-like behaviour, or that more
magnetars to appear in the radio band? What parameters could be leading to the
different manifestations for objects with similar period and period
derivatives? The magneto-thermal evolutionary theory (e.g. Viganò & Pons,
2012; Viganò et al., 2013) addresses this diversity by the presence of
toroidal magnetic fields hidden in the crust as an extra parameter shaping the
lives of pulsars. The fallback disc model (Alpar, 2001; Chatterjee et al.,
2000) invokes the mass and specific angular momentum (Ertan et al., 2007,
2009) of a putative supernova fallback disc as two parameters leading to the
diversity.

A well-known unsupervised machine learning algorithm, Gaussian mixture model
(GMM; Press et al., 2007), had been used by Lee et al. (2012) to analyze the
distribution of pulsars on the $P-\dot{P}$ diagram and identified six Gaussian
clusters (2 for millisecond—recycled—pulsars and 4 for young pulsars). Igoshev
& Popov (2013) showed that GMM is over-sensitive to the data and does not
demonstrate robust clustering performance. There are more advanced variations
of GMM such as Dirichlet process Gaussian mixture model (DPGMM). This model
has the advantage over GMM that the number of distributions are determined
automatically though it is not used as widely as GMM especially by the
astronomy community. In addition its generalization performance is better than
GMM, as Bayesian methods are more robust against overfitting (Rasmussen, 2000;
Heller, 2008; Witten et al., 2016). We have recently showed that DPGMM is
better than GMM in both clustering and classifying pulsars in the $P-\dot{P}$
parameter space (Ay et al., 2019).

The organization of our paper is as follows. In § 2 we review DPGMM employed
in this work. In § 3 we present our results and finally, in § 4, we discuss
the implications of our findings.

##  2 Method

###  2.1 Dirichlet process Gaussian mixture model

Here we will briefly sketch the method employed in this work. The detailed
information for DPGMM can be found in (Shin et al., 2009; Del Pozzo et al.,
2018).

In finite mixture models, it is assumed that the data consists of a certain
number of clusters in which each cluster is generated by a probability
distribution. Let $\boldsymbol{X}^{\rm T}=\\{\boldsymbol{x_{\rm j}}\\}_{\rm
j=1}^{\rm N}$ be the collection of $d$-dimensional observed data with $N$
number of instances and assume further that there are $m$ distributions that
generate $\boldsymbol{X}$. Because it is not known by which distribution a
certain data point is generated, it is useful to define a latent data
$\boldsymbol{Z}$ for distribution assignments; $\boldsymbol{Z}^{\rm
T}=\\{z_{\rm j}\\}_{\rm j=1}^{\rm N}$ where $z_{\rm j}=i$ means that
$\boldsymbol{x_{\rm j}}$ is generated by the i th distribution,
$j\in\\{1,2,\ldots,N\\}$ and $i\in\\{1,2,\ldots,m\\}$.

The joint probability of a data point and i th distribution is defined as

$p(\boldsymbol{x_{\rm j}},z_{\rm j}=i)=p(\boldsymbol{x_{\rm j}}\,|\,z_{\rm
j}=i)\,p(z_{\rm j}=i)\enskip.$ (1)

The joint probability value indicates how likely a data point belongs to a
cluster. Because $\boldsymbol{x_{\rm j}}$ must be populated by one of the $m$
distributions, we can obtain marginal probability of $\boldsymbol{x_{\rm j}}$
from equation (1) as follows

$p(\boldsymbol{x_{\rm j}})=\sum_{i=1}^{m}{p(\boldsymbol{x_{\rm j}}\,|\,z_{\rm
j}=i)\,p(z_{\rm j}=i)}\enskip.$ (2)

The marginal probability value can be considered as the probability that a
data point belongs to any cluster. On the right side of the equation, the
second term is the probability of the i th distribution which is also known as
the weight of the i th cluster such that $w_{\rm i}\equiv p(z_{\rm j}=i)$
where $\sum_{i=1}^{m}{w_{\rm i}}=1$. Besides, if we assume that each cluster
is distributed normally, then the first term on the right side of the equation
is defined as Gaussian distribution;

$\mathcal{N}\left(\boldsymbol{x_{\rm j}}\,|\,\boldsymbol{\mu_{\rm
i}},\boldsymbol{\Sigma_{\rm
i}}\right)=\frac{\exp\left(-\frac{1}{2}(\boldsymbol{x_{\rm
j}}-\boldsymbol{\mu_{\rm i}})^{\rm T}\,\boldsymbol{\Sigma_{\rm
i}}^{-1}(\boldsymbol{x_{\rm j}}-\boldsymbol{\mu_{\rm
i}})\right)}{(2{\mathrm{\pi}})^{d/2}\lvert\boldsymbol{\Sigma_{\rm
i}}\rvert^{1/2}}\enskip,$ (3)

where $\boldsymbol{\mu_{\rm i}}$ is $d$-length mean vector and
$\boldsymbol{\Sigma_{\rm i}}$ is $d\times d$ covariance matrix of the i th
Gaussian distribution. In this case, the finite mixture model is called as
Gaussian mixture model (GMM).

In Bayesian approach, GMM parameters
$\boldsymbol{\theta}=\\{\boldsymbol{w},\boldsymbol{\mu},\boldsymbol{\Sigma}\\}$
are also modeled with probability distributions. The choice of distribution
families is based on conjugate prior principle (Raiffa & Schlaifer, 1961) for
the computational simplicity. Accordingly, the prior distributions for the
parameters of Gaussian distributions are selected as Normal-Inverse-Wishart
distribution;

$\\{\boldsymbol{\mu},\boldsymbol{\Sigma}\\}\sim\mathcal{NIW}\left(\mathcal{\boldsymbol{\mu_{\rm
0}}},\beta,\boldsymbol{\psi},\nu\right)\enskip,$ (4)

where $\\{\mathcal{\boldsymbol{\mu_{\rm 0}}},\beta,\boldsymbol{\psi},\nu\\}$
are mean, scaling factor, scale matrix and degrees of freedom. Furthermore, if
we assume that the cluster weights are Dirichlet process (DP; Ferguson, 1973)
distributed

$\boldsymbol{w}\sim DP(\alpha,\mathcal{H})\enskip,$ (5)

then the mixture model is called as Dirichlet process Gaussian mixture model
(DPGMM). In the equation, $\alpha$ is the concentration parameter and
$\mathcal{H}$ is the base distribution for DP. Because a realization from DP
is a infinite length probability vector, DPGMM is a non-parametric, i.e.
infinite, mixture model.

In this study, we employed scikit-learn (Pedregosa et al., 2011)
implementation of DPGMM that uses variational inference (Attias, 2000; Bishop,
2006; Blei & Jordan, 2006) for learning model parameters. We also benefited
from many open source software for scientific computations, analyzes and
visualizations, such as scipy (Virtanen et al., 2019), numpy (Oliphant, 2006),
pandas (McKinney et al., 2010), matplotlib (Hunter, 2007), seaborn (Waskom et
al., 2017) and scikit-image (van der Walt et al., 2014). In addition,
Spyder111https://www.spyder-ide.org/ and IPython (Pérez & Granger, 2007) were
used as development environments.

###  2.2 Data Collection

We examine pulsar distribution on $P-\dot{P}$ parameter space in logarithmic
scale. In order to increase data samples as much as possible, various neutron
star catalogs are combined. Most of the pulsar data are obtained from
Australia Telescope National Facility (ATNF) Pulsar
Catalog222http://www.atnf.csiro.au/research/pulsar/psrcat/ that contains up to
2658 instances with numerous source types. In addition, 30 instances of SGRs
and AXPs (either confirmed or candidates) from McGill Online Magnetar
Catalog333http://www.physics.mcgill.ca/p̃ulsar/magnetar/main.html (Olausen &
Kaspi, 2014) and 40 instances of other thermally emitting neutron stars such
as CCOs and XDINSs from (Viganò et al.,
2013)444http://www.neutronstarcooling.info/ are collected. Additionally, 107
RRATs from RRATOLOG555http://astro.phys.wvu.edu/rratalog/ (McLaughlin et al.,
2006; Keane et al., 2010). and 117 GRPs from The Second Fermi-LAT
catalogue666https://heasarc.gsfc.nasa.gov/W3Browse/fermi/fermil2psr.html (Abdo
et al., 2013) are included. Finally, a recently discovered (Tan et al., 2018)
RPP with an exceptional long period is also included to the data set. As a
result, we obtained an amount of 2166 INSs with the observed spin parameters
by combining all of these data sources.

##  3 Results

###  3.1 Application of DPGMM

DPGMM has a relatively large number of hyper-parameters to be fixed. For
example, although DPGMM is an infinite mixture model, scikit-learn
implementation of DPGMM requires the hyper-parameter $m_{\rm max}$ that
corresponds to the maximum number of clusters to be set to an appropriate
value as it would not be practical to consider infinite number of clusters
during fitting. We have set the value of $m_{\rm max}$ as 10 as we did not
expect to find more than 10 clusters given the number of pulsar families and
the 6 clusters already determined by GMM (Lee et al., 2012).

Another hyper-parameter is the concentration parameter $\alpha$ that
determines how sparse distributions are allowed to be and thus may effect the
number of clusters the model can find. For a given $\alpha$, the optimum
number of clusters that best represent the data among the $m_{\rm max}$
clusters and their parameters are determined by DPGMM during the fitting
process. We have seen that the value of $\alpha$ in the literature varies
significantly: e.g. Haines & Xiang (2013) set $\alpha=0.01$ while Shin et al.
(2009) set $\alpha=1$. In another example, Chen et al. (2015) have searched
optimum value for $\alpha$ in a range between $0.001$ and $100$, and observed
that the results are relatively same independent of different $\alpha$ values.

We created DPGMMs for different $\alpha$ values within a wide range from
$10^{-10}$ to $10^{10}$. As a result, we have observed that DPGMM for this
wide range of $\alpha$ values discovered either 6 or 7 clusters depending on
whether $\alpha$ is lower or higher than $10^{4}$, respectively. We have also
observed that the estimated model parameters are remarkably independent of
$\alpha$ for all models that discovered 6 clusters ($\alpha\leq 10^{4}$). The
same situation is also true for all models that discovered 7 clusters
($\alpha>10^{4}$). In short, we have obtained two different stable models; one
having 6 clusters and the other 7 clusters. The recommended value of $\alpha$
is much less than 1 and for such values our model finds 6 clusters. We have
employed Bayesian information criteria (Schwarz, 1978) and Akaike information
criteria (Akaike, 1974) in a former study (Ay et al., 2019) to determine the
optimum values of the clusters in which we found 6 clusters. This result also
is consistent with the number of clusters found by Lee et al. (2012) who used
multidimensional Kolmogorov-Smirnov test. We thus selected the model with 6
clusters as the optimal one and our results with 7 clusters are not shown
here. In Table 1, the estimated parameters of DPGMM created with $m_{\rm
max}=10$ and $\alpha=10^{-5}$ are shown.

DPGMM Parameters
---
Cluster | $w_{\rm~{}}$ | $\boldsymbol{\mu}$ | $\boldsymbol{\Sigma}$
1st | 0.0787 | $\begin{bmatrix}-2.3782&-19.8306\end{bmatrix}$ | $\begin{bmatrix}0.0731&0.1019\\\ 0.1019&0.4127\end{bmatrix}$
2nd | 0.0384 | $\begin{bmatrix}-1.5694&-18.4996\end{bmatrix}$ | $\begin{bmatrix}0.2776&0.2627\\\ 0.2627&0.9198\end{bmatrix}$
3rd | 0.3344 | $\begin{bmatrix}-0.0148&-15.3005\end{bmatrix}$ | $\begin{bmatrix}0.1303&0.2647\\\ 0.2647&0.9584\end{bmatrix}$
4th | 0.4111 | $\begin{bmatrix}-0.2817&-14.5136\end{bmatrix}$ | $\begin{bmatrix}0.0662&0.0536\\\ 0.0536&0.4303\end{bmatrix}$
5th | 0.1129 | $\begin{bmatrix}-0.7303&-13.5903\end{bmatrix}$ | $\begin{bmatrix}0.1145&0.0096\\\ 0.0096&0.5683\end{bmatrix}$
6th | 0.0244 | $\begin{bmatrix}0.3350&-12.2949\end{bmatrix}$ | $\begin{bmatrix}0.1864&0.4824\\\ 0.4824&2.1642\end{bmatrix}$
Table 1: The estimated parameters of DPGMM ($m_{\rm max}=10$,
$\alpha=10^{-5}$). $w_{\rm~{}}$, $\boldsymbol{\mu}$ and $\boldsymbol{\Sigma}$
indicate weight, location and shape of the corresponding Gaussian
distribution, respectively.

###  3.2 Parameter space classification

In Figure 1, all clusters and decision boundaries on $P-\dot{P}$ parameter
space revealed by DPGMM are illustrated. Bivariate Gaussian curves are
represented by 2$\sigma$ confidence ellipses and each cluster region is
enumerated and colored uniquely in here. The mixture models are inductive
models so that after fitting a model to a data set, it is able to predict the
cluster in which a ‘novel’ instance should belong. Formally, the cluster label
of a novel instance $\boldsymbol{x_{\rm j+1}}$ is determined as

$z_{\rm j+1}=\operatorname*{arg\,max}_{i}{p(\boldsymbol{x_{\rm j+1}},z_{\rm
j+1}=i)}\enskip.$ (6)

If we classify the entire parameter space with this approach, the separating
line between cluster regions naturally arise as shown in Figure 1 and in
Figure 2. These lines are called as decision boundaries such that the cluster
assignment by the model is different for the one side of the line and the
other side.

In the case of mixture of Gaussian distributions, there is a possibility of
being more than one region for a single cluster on classified parameter space.
To illustrate, there are two regions for the 6th cluster in Figure 2; one is
on the top left, and the other is on the bottom right. It is significant that
all pulsars belonging to the 6th cluster are in the top left region while the
bottom one does not cover any pulsars at all. Thus, it can be said that
instead of the 6th cluster, the bottom region should be merged with the
neighbor region corresponding to the 2nd cluster. Therefore, the separating
line between the 2nd cluster and the bottom region of the 6th cluster seems
incorrect. For this reason, we examined the dependability of boundary lines in
this study.

As mentioned before, the joint probability defined in equation (1) and used in
equation (6) for cluster assignments shows that how likely a data point
$\boldsymbol{x_{\rm j}}$ is a member of i th cluster. Its value depends on the
cluster weight and the multivariate Gaussian distribution defined in equation
(3) where the exponent term in the numerator is actually half the negative of
the square of Mahalanobis distance (Mahalanobis, 1936; De Maesschalck et al.,
2000) that is a distance measure in terms of standard deviation. Roughly
speaking, it can be said that the cluster assignment for a data point is
determined by the weight of clusters and the Mahalanobis distance between the
data point and cluster means. The cluster assignment is predominately
determined by the weight of the clusters for nearby data points. On the other
hand, the effect of Mahalanobis distance on the decision making of the model
increases for the receding data points. As the Mahalanobis distance is not a
deterministic measure, the cluster assignments may be incoherent for the
distant regions. To illustrate, the bottom right region in Figure 2 is
separated from the 2nd cluster and assigned to the 6th cluster by the model
even though all pulsars belonging to the 6th cluster are in top left region.
Therefore, we defined the dependability of the cluster assignments as the
marginal probability values specified in equation (2) that can be considered
as how likely a data point is a member of any cluster, as stated before. If
the marginal probability of $\boldsymbol{x_{\rm j}}$ is lower, then the
dependability of the cluster assignment for this data point is not reliable as
well. In this way, we showed the degree of the dependability of border lines
in Figure 1 and Figure 2 based on the reliability of data points over border
lines, so that the border line is darker where it is more reliable, and it is
fainter where it is less reliable. Accordingly, it is obvious that the color
of the border line between the 2nd and the 6th clusters in Figure 2 is very
faint that means it is not reliable as expected. In this way, we tried to
eliminate the handicap of parameter space classification in the case of
mixture of Gaussian distributions.

Figure 1: The $P-\dot{P}$ parameter space classified by DPGMM. Each cluster
region is uniquely enumerated and colored. The decision boundaries are
illustrated with thick solid lines of varying shades representing the degree
of the dependability of the border lines (see § 3.2). Bivariate Gaussian
curves are represented with $2\sigma$ confidence ellipses.

As we have discussed before, we have introduced 7 families of pulsars namely
rotationally powered pulsars (RPPs) that involve millisecond pulsars, rotating
radio transients (RRATs), gamma-ray pulsars (GRPs) as well as conventional
radio pulsars, magnetars (AXPs, SGRs), X-ray dim INS (XDINSs) and central
compact objects (CCOs). As seen in Figure 1, DPGMM identifies six clusters for
the value of the concentration parameter $\alpha=10^{-5}$. In this section, we
analyse whether these six clusters are linked to any parameters of pulsars
other than $P$ and $\dot{P}$ (e.g. space velocity or surface temperature) that
could attribute further meaning to the different clusters.

| Cluster |
---|---|---
Pulsar Family | 1st | 2nd | 3rd | 4th | 5th | 6th | Total
Normal | 137 | 73 | 703 | 906 | 142 | 21 | 1982
GRP | 38 | 2 | 0 | 15 | 62 | 0 | 117
RRAT | 0 | 0 | 21 | 12 | 1 | 1 | 35
AXP | 0 | 0 | 4 | 0 | 0 | 10 | 14
SGR | 0 | 0 | 0 | 0 | 0 | 8 | 8
XDINS | 0 | 0 | 6 | 0 | 0 | 1 | 7
CCO | 0 | 0 | 3 | 0 | 0 | 0 | 3
Total | 175 | 75 | 737 | 933 | 205 | 41 | 2166
Table 2: The distribution of the pulsar families over clusters.

###  3.3 Pulsar distribution over the clusters

The distribution of pulsar types over DPGMM clusters is given in Table 2.
Accordingly, the most of the ‘non-recycled’ RPPs detected in the radio band
are located in the 3rd, 4th and 5th clusters. The 1st and the 2nd regions
contain mostly millisecond pulsars. It is significant that millisecond pulsars
are separated into two subclasses by DPGMM as in the case of GMM (Lee et al.,
2012). In § 4, we discuss the possible origin of this division. The 2nd region
is very large and extends beyond the pulsar death line. The 3rd region
contains most of the RRATs and XDINSs, all CCOs, low-B magnetars (J1647–4552,
J0418+5732, J2301+5852 and J1822–1604) and those RPPs close to the death line.
There is again a 4th region populated by RPPs with a narrow $\dot{P}$ range.
About $10\%$ of GRPs overflow to this region. The 5th region mostly contains
‘non-recycled’ GRPs while the recycled ones are in the 1st cluster. The 6th
region is predominantly occupied by ’High-B‘ AXPs and SGRs excluding the low-B
magnetars. Moreover, most of the pulsars are grouped in the 3rd and 4th
clusters that are located at the center of the diagram. There are fewer
pulsars in other clusters relatively.

###  3.4 Effect of binary companion for recycled pulsars

The distribution of binary pulsars over clusters is given in Table 3.
Accordingly, the binary pulsars are heavily grouped in the 1st and the 2nd
clusters where millisecond pulsars ‘live’. This, of course, is understood
through the “recycling scenario” (Alpar et al., 1982) that address the very
existence of these objects by spin-up of the neutron star through long-term
accretion of matter and angular momentum from a binary companion. It is
interesting, however, that both GMM and DPGMM found the millisecond pulsars
are distributed into two clusters. What could be the underlying astrophysical
distinction between these two clusters?

Almost all millisecond GRPs are located in the 1st cluster as seen in Figure 1
and in Table 3. This is obviously due to the higher rotational power, $L_{\rm
sd}=4{\mathrm{\pi}}^{2}I\dot{P}/P^{3}$, of this cluster as well as the
magnetic field strength at the light cylinder radius $R_{\rm
LC}=cP/2{\mathrm{\pi}}$ given by

$B_{\rm LC}=\sqrt{\frac{24{\mathrm{\pi}}^{4}I\dot{P}}{c^{3}P^{5}}}\enskip.$
(7)

This can more readily be inferred from Figure 2 where DPGMM classification of
$B_{\rm LC}-L_{\rm sd}$ parameter space is illustrated.

As $\\{L_{\rm sd},B_{\rm LC}\\}$ are the linear functions of $\\{P,\dot{P}\\}$
in logarithmic scale such that

$\displaystyle\log(B_{\rm LC})$
$\displaystyle\equiv\frac{1}{2}\log(\dot{P})-\frac{5}{2}\log(P)\enskip,$ (8)
$\displaystyle\log(L_{\rm sd})$
$\displaystyle\equiv\log(\dot{P})-3\log(P)\enskip,$ (9)

then the data matrix $\boldsymbol{Y}=[\log\boldsymbol{B}_{\rm
LC}\,\,\log\boldsymbol{L}_{\rm sd}]$ is also distributed with the mixture of
Gaussians as is the original data
$\boldsymbol{X}=[\log\boldsymbol{P}\,\,\log\boldsymbol{\dot{P}}]$. This is
true for any data matrix $\boldsymbol{Y}$ which is linearly related with
$\boldsymbol{X}$. Therefore, an analysis in the $B_{\rm LC}-L_{\rm sd}$
parameter space is equivalent to the analysis in the $P-\dot{P}$ diagram. For
this reason, we applied the same DPGMM ($m_{\rm max}=10$, $\alpha=10^{-5}$) to
the $B_{\rm LC}-L_{\rm sd}$ parameter space. The obtained pulsar
classification in this plane is almost same as the classification in
$P-\dot{P}$ parameter space as should be. As seen in Figure 2, $\gamma$-ray
emission can not be the underlying astrophysical cause for the existence of
two separate clusters for millisecond pulsars; it will rather be a
consequence.

Figure 2:  The $B_{\rm LC}-L_{\rm sd}$ parameter space classified by DPGMM.  Cluster | Normal | GRPs | Total
---|---|---|---
1st | 103 | 32 | 135
2nd | 51 | 0 | 51
3rd | 13 | 0 | 13
4th | 6 | 0 | 6
5th | 3 | 1 | 4
6th | 0 | 0 | 0
Total | 176 | 33 | 209
Table 3:  The distribution binary pulsar families over clusters.  Cluster | MS | NS | CO | HE | UL
---|---|---|---|---|---
1st | 9 | 0 | 15 | 77 | 23
2nd | 0 | 14 | 17 | 12 | 4
3rd | 0 | 2 | 3 | 4 | 2
4th | 4 | 1 | 1 | 0 | 0
5th | 2 | 1 | 0 | 1 | 0
6th | 0 | 0 | 0 | 0 | 0
Total | 15 | 18 | 36 | 94 | 29
Table 4: The distribution of the companion types of binary pulsars over
clusters. Companion types — main-sequence (MS), neutron star (NS), carbon-
oxygen (or ONeMg) white dwarf (CO), helium white dwarf (HE), and ultra-light
companion or planet (UL) — are obtained from Australia Telescope National
Facility (ATNF) Pulsar Catalog
(http://www.atnf.csiro.au/research/pulsar/psrcat/).

The companion types of some binary pulsars have been detected and the
distribution of these companions over the clusters is shown in Table 4.
Accordingly, there is a remarkable distinction between the properties of the
companion objects in the two groups indicating to different astrophysical
origins and recycling history. The 1st cluster is populated by millisecond
pulsars with main sequence (MS), helium white dwarf (HE), carbon-oxygen white
dwarf (CO) and ultra-light (UL) companions while lacking any neutron star
companions. The 2nd cluster is populated by those with neutron star (NS) and
white dwarf companions (CO and He), while having only 4 UL and no MS
companions. Lee et al. (2012) attributes the identification of two separate
clusters to the chemical composition of the companions. This implies that
millisecond pulsars in the 2nd cluster systematically had larger mass
companions that lived shorter. Their larger periods (hence the smaller
rotational power and lack of gamma-emission) can be attributed to the shorter
recycling history they suffered. It is remarkable that DPGMM distinguishes
these two groups only through their spin parameter distributions.

###  3.5 Characteristic age and magnetic field

We calculated the characteristic age, $\tau_{\rm c}\equiv P/2\dot{P}$, and
magnetic dipole field strength as inferred under the assumption of a
orthogonal rotating magnetic dipole model

$B_{\rm
d}=\left(\frac{3c^{3}I}{8{\mathrm{\pi}}^{2}R^{6}}\right)^{1/2}\sqrt{P\dot{P}}$
(10)

of each cluster where we assumed $I=10^{45}\,{\rm g\,cm^{2}}$ and
$R=10^{6}\,{\rm cm}$ for the moment of inertia and the radius of neutron
stars, respectively. These parameters calculated based on the mean value
($\boldsymbol{\mu_{\rm i}}=\\{P_{i},\dot{P}_{i}\\}$) of corresponding Gaussian
distribution. Therefore the results given in Table 5 can be assumed as the
average values for each cluster. The table is arranged such that the age of
cluster is increased with the cluster order as well as the magnetic dipole
strength. Hence, the 1st cluster is the youngest one with the highest dipole
field where the 6th cluster is the oldest one with the lowest dipole field.
Beside that, the millisecond pulsars in the 2nd cluster have an order of
magnitude larger magnetic dipole field strengths compared to those in the 1st
cluster. According to our discussion in the previous subsection, stronger
fields can be understood as a consequence of the shorter ‘recycling’ episode
these systems passed through.

| Characteristic Age | Surface Magnetic Flux
---|---|---
Cluster | (yr) | (Gauss)
1st | $4.49\times 10^{9}$ | $2.52\times 10^{8}$
2nd | $1.35\times 10^{9}$ | $2.96\times 10^{9}$
3rd | $3.06\times 10^{7}$ | $7.04\times 10^{11}$
4th | $2.70\times 10^{6}$ | $1.28\times 10^{12}$
5th | $1.15\times 10^{5}$ | $2.21\times 10^{12}$
6th | $6.75\times 10^{4}$ | $3.35\times 10^{13}$
Table 5: The characteristic age (year) and dipole magnetic field strength
(Gauss) of clusters.

###  3.6 Transverse velocity

Figure 3:  The transverse velocity of non-recycled pulsars (the 3rd, 4th, 5th and 6th clusters). | Transverse Velocity $v_{\rm T}$ (km s-1)
---|---
Cluster | $N_{v_{\rm T}}$ | $\overline{v_{\rm T}}$ | $\overline{v_{\rm T}^{*}}$ | $\sigma_{v_{\rm T}}$ | $\sigma^{*}_{v_{\rm T}}$
3rd | 50 | 335.5 | 363.3 | 625.4 | 677.4
4th | 71 | 470.0 | 373.6 | 753.6 | 570.1
5th | 20 | 302.6 | 359.1 | 248.8 | 265.4
6th | 6 | 252.2 | 231.5 | 141.0 | 79.7
Table 6: Statistics of transverse velocity $v_{\rm T}$ (km s-1) of non-
recycled pulsars (the 3rd, 4th, 5th and 6th clusters). Here $N_{v_{\rm T}}$
stands for the number of pulsars with the measured speed in each cluster.
$\overline{v_{\rm T}}$, $\overline{v_{\rm T}^{*}}$, $\sigma_{v_{\rm T}}$ and
$\sigma^{*}_{v_{\rm T}}$ stand for mean, weighted mean, standard deviation and
weighted standard deviation of the transverse speed, respectively.

According to the field burial model (Muslimov & Page, 1995; Young & Chanmugam,
1995; Geppert et al., 1999; Ho, 2011), an initial, brief but intense, fallback
accretion episode (Chevalier, 1989) following the supernova explosion is
important in shaping the magnetic fields of nascent pulsars. The amount of
matter accreted determines how much the initial magnetic field is buried under
the crust and thus the time-scale of diffusion of the field to the surface.
Space velocities of pulsars due to the kick they receive during their birth
may thus determine how much mass the neutron star can accrete, as the
accretion rate is inversely proportional to the cube of the velocity in Bondi-
Hoyle accretion. The field diffusion time-scale as inferred from the measured
braking indices of young pulsars and the space velocities are inversely
correlated (Güneydaş & Ekşi, 2013; Ekşi, 2017) as expected from the field-
burial scenario. The question then naturally arises whether the space velocity
of pulsars have any role in the observed diversity of pulsars.

In order to check this possibility we showed the transverse velocity of non-
recycled pulsars (i.e. those in the 3rd, 4th, 5th and 6th clusters in the
$P-\dot{P}$ diagram) as seen in Figure 3. The figure is colored depending on
the magnitude of the velocity so that the color changes with the magnitude
from yellow to red sequentially. The transverse velocity statistics of same
clusters are given in Table 6. Since the probability that each pulsar is a
member of the cluster to which they belong is different, we calculated the
weighted statistics of their transverse velocities by using their joint
probability values (see equation (1)) with respect to corresponding Gaussian
distributions as weights. Accordingly, the weighted standard error is very
large due to the low number of space velocity measurements in each group and
the results may not be statistically significant. We see, however, that the
space velocities of pulsars in the 4th cluster is the largest. We also see
that the average speeds of pulsars in the 6th cluster is about half of those
in the 4th cluster. The velocity data was collected from the sources in which
the spin parameters of pulsars were taken as specified in § 2.2.

###  3.7 Blackbody temperature

In Figure 4, the blackbody temperature of non-recycled pulsars are
illustrated. The statistics of black body temperature of these clusters are
given in Table 7. The weighted statistics are computed in the same way
described in the previous section. According to these results, the temperature
of the 6th cluster where magnetars are frequently hosted in is almost twice as
large as temperatures of other clusters. In other words, AXP/SGRs are the
hottest pulsar type. However, the results may not be statistically significant
due to the small sample size. The blackbody temperatures of these pulsars are
acquired from the sources mentioned in § 2.2 and also from Gonzalez et al.
(2007); Kaspi & McLaughlin (2004); Webb et al. (2004); Hu et al. (2017); Li et
al. (2005); Chang et al. (2011); Caraveo et al. (2010); Marelli et al. (2011);
Güver et al. (2012); Posselt et al. (2012).

Figure 4:  The blackbody temperature (eV) of non-recycled pulsars (the 3rd, 4th, 5th and 6th clusters). | Blackbody temperature $T_{\rm bb}$ (eV)
---|---
Cluster | $N_{T_{\rm bb}}$ | $\overline{T_{\rm bb}}$ | $\overline{T_{\rm bb}^{*}}$ | $\sigma_{T_{\rm bb}}$ | $\sigma^{*}_{T_{\rm bb}}$
3rd | 14 | 219.2 | 138.2 | 175.0 | 83.7
4th | 3 | 128.7 | 126.3 | 67.7 | 55.4
5th | 14 | 172.0 | 157.5 | 84.4 | 51.0
6th | 19 | 395.5 | 334.1 | 166.5 | 171.3
Table 7: Statistics of blackbody temperature (eV) of non-recycled pulsars (the
3rd, 4th, 5th and 6th clusters). Here $N_{T_{\rm bb}}$ stands for number of
pulsars with the measured temperatures. $\overline{T_{\rm bb}}$,
$\overline{T_{\rm bb}^{*}}$, $\sigma_{T_{\rm bb}}$ and $\sigma^{*}_{T_{\rm
bb}}$ stand for mean, weighted mean, standard deviation and weighted standard
deviation of temperature, respectively.

##  4 Discussion

We have examined the pulsar population with an unsupervised machine learning
algorithm, Dirichlet process Gaussian mixture model (DPGMM). We confirmed the
earlier result obtained with Gaussian mixture model (GMM) by Lee et al. (2012)
that the millisecond pulsar population has two clusters and that the normal
pulsar population consists of four clusters.

We have considered possible hidden parameters (space velocity and surface
temperature) as possible underlying cause of the distinct clusters. We have
first tried DPGMM on higher dimensional planes, e.g. 3-D planes
($P-\dot{P}-v_{\rm T}$ and $P-\dot{P}-T_{\rm bb}$) and 4-D plane
($P-\dot{P}-v_{\rm T}-T_{\rm bb}$). We have found that the results for the
higher dimensions were unreliable due to the lack of the data instances
available for the time being. We have then examined the transverse velocity
and the blackbody temperature statistics for each cluster. We have found that
the 4th cluster has about twice the average space velocity compared to the
slowest cluster, the 6th one, (see Figure 3 and Table 6) providing a marginal
evidence that space velocities of young neutron stars could play a role in
their astrophysical manifestations by controlling the total mass accreted
during post-supernova fallback and the depth of field burial (Güneydaş & Ekşi,
2013; Ekşi, 2017). This indicates the pulsars in the 4th cluster suffered less
supernova fallback accretion and less field burial. Indeed the average
magnetic field of pulsars in the 4th cluster are similar to those in the 5th
cluster and are twice of those in the 3rd cluster. If the magnetic fields of
pulsars decay as they age within a timescale less than a million years, the
difference in the magnetic fields of the 4th and 3rd clusters could be also
attributed to the former systems being younger and subject to less field
decay. Yet it is unlikely that the magnetic fields of typical pulsars decay in
such short timescales favouring the idea that the space velocities could be
the underlying reason for the difference in the magnetic fields of the 4th and
3rd clusters.

We note that the very high space velocities of CCOs imply that they are not
likely to suffer very strong post-supernova fallback accretion. This does not
neatly fit into the ‘field burial scenario’ (Muslimov & Page, 1995; Geppert et
al., 1999) as the small dipole fields of these objects, in this model, are
attributed to very intense fallback accretion. This may indicate the role of
the initial magnetic field and spin period as other parameters controlling the
amount of the accreted fallback mass: The low magnetic fields and relatively
slow initial periods of CCOs could allow them to accrete large amount of mass
to bury their fields in spite of their large space velocities. Another
solution high-speed-low-field puzzle could be that their dipole fields are in
fact very strong but their rotation and spin axis are aligned to a very high
precision such that the cluster of their dipole moment perpendicular to the
rotation axis inferred from spin-down is so small. This idea, however, would
work only in the vacuum dipole model i.e. if a corotating plasma does not
exist around CCOs; Magnetohydrodynamic (MHD) dipole model (Spitkovsky, 2006)
require non-vanishing spin-down even in the aligned case. This idea is worth
considering given that only three of the CCOs show pulsations while there is
evidence for strong magnetic fields (Shakura et al., 2012).

It is remarkable that the speed of objects in the 6th group is ‘typical’. This
group involves AXP/SGRs which according to the magnetar model should have
large space velocities, $\sim 1000$ km s-1 due to the “rocket effect” (Duncan
& Thompson, 1992; Thompson & Duncan, 1993). This may be because it is their
super-strong quadrupole fields, rather than the dipole fields, that
distinguishes these objects from the rest of the isolated pulsars. The higher
spin-down rates of these objects could then be attributed to additional
torques, contributing to the magnetic dipole torque, due to winds (Harding et
al., 1999; Thompson et al., 2000) or supernova fallback discs (Wang et al.,
2006; Ertan et al., 2007).

The surface temperatures of pulsars in the 6th cluster are found to be
relatively large (see Figure 4 and Table 7). This is an expected result as
this cluster involves AXP/SGRs which, according to magnetar model, have
decaying fields. This also is expected according to the fallback disc model as
a result of accretion (Alpar, 2001; Chatterjee et al., 2000). A critical
discrimination between the two models would be to compare the surface
temperatures of HBPSR’s which presumably are not accreting objects, with those
of AXP/SGRs. The upper limits on the temperatures of high magnetic field
pulsars PSR B0154+61 (Gonzalez et al., 2004), J1814-1744 and J1847-0130 (Keane
et al., 2013) imply that they are not significantly hotter than pulsars with
lower magnetic fields thus raising the question why the fields of these
objects do not decay and contribute to the X-ray luminosity. Yet the
temperature of B1509-58 (Hu et al., 2017) lies between the temperatures of
AXP/SGRs and typical RPPs, and the temperature of PSR J1119–6127 (Gonzalez et
al., 2005) is among the highest measured from typical RPPs. The observed
pulsed fraction of this source is also higher than those of typical RPPs
(Gonzalez et al., 2005) possibly indicating the role of strong fields and
unlikely to be related to the presence of a putative fallback disc. Also, the
recent analysis (Çerri-Serim et al., 2019) shows that the noise strength and
X-ray luminosity magnetars lacks the correlation known to exist in accreting
X-ray pulsars thus suggesting that they are not accreting.

If cluster regions do indeed have anything to do with the evolutionary
connections among pulsar families, the 5th and the 6th regions may be
considered as two different origins for young neutron stars on the $P-\dot{P}$
diagram. A nascent neutron star may be born as a Crab-like GRP ($P_{0}\sim 10$
ms, $\dot{P}_{0}\sim 10^{-12}$ s s-1) or as a magnetar ($P_{0}\sim 0.5$ s,
$\dot{P}_{0}\sim 10^{-8}$ s s-1) assuming the field is not produced by the
initial dynamo effect, but flux conservation (Ferrario & Wickramasinghe, 2006)
depending on its spin parameters, respectively. The classification presented
here clearly relates low-B magnetars with XDINSs and RRATs and implies they
could descend from magnetars by field decay. Yet statistics suggests magnetars
are much less common objects compared to XDINSs and RRATs. This evolutionary
scenario could, however, be validated only if magnetar stage is very short or
there are many undiscovered transient magnetars. The recent identification of
the high energy component in XDINS (Yoneyama et al., 2019) indeed suggests
that magnetars and XDINSs are evolutionary linked.

There are, of course, strong selection effects on the representation of
pulsars on the $P-\dot{P}$ diagram. CCOs, RRATs and XDINSs should be much more
common in the galaxy than they are detected. Yet it is not possible to do the
similar analysis with a ‘flux limited sample’ at present as such filtering of
data would result with much less instances of data than the model we employed,
DPGMM, requires. Given that many types of pulsars are lower-represented in our
sample, it may be argued that some of the clusters discovered by DPGMM are
artefacts of describing an intrinsically non-Gaussian distribution. Yet, DPGMM
classification we present here broadly coincides with the existing
astrophysical classification despite it is an unsupervised model.

In summary, we have shown that DPGMM, an unsupervised machine learning model,
identifies 6 pulsar classes: two for recycled millisecond pulsars
corresponding to different accretion histories; one roughly coinciding XDINSs,
low-B magnetars and RRATs, but involves many RPPs as well; one involving the
bulk of RPPs and some RRATs; one involving the very young Crab-like RPPs
including the bulk of the GRPs; and finally one that involves the AXP/SGRs and
HBPSRs.

## Acknowledgements

KYE acknowledges support from TUBITAK, the national foundation of science in
Turkey, with the project number 118F028. We thank M. Ali Alpar and Erbil
Gügercinoğlu for useful discussion.

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# A NOTE ON THE CHEBYSHEV SET PROBLEM IN NORMED LINEAR SPACES

Samson Owiti, Benard Okelo∗, Julia Owino  Department of Pure and Applied
Mathematics
School of Biological, Physical, Mathematics and Actuarial Sciences
Jaramogi Oginga Odinga University of Science and Technology
Box 210-40601, Bondo-Kenya<EMAIL_ADDRESS>
(Date: Received: 13 January, 2023.
∗ Corresponding author)

###### Abstract.

Best approximation (BA) is an interesting field in functional analysis that
has attracted a lot of attention from many researchers for a very long period
of time up-to-date. Of greatest consideration is the characterization of the
Chebyshev set (CS) which is a subset of a normed linear space (NLS) which
contains unique BAs. However, a fundamental question remains unsolved to-date
regarding the convexity of the CS in infinite NLS known as the CS problem. The
question which has not been answered is: Is every CS in a NLS convex?. This
question has not got any solution including the simplest form of a real
Hilbert space (HS). In this note, we characterize CSs and convexity in NLSs.
In particular, we consider the space of all real-valued norm-attainable
functions. We show that CSs of the space of all real-valued norm-attainable
functions are convex when they are closed, rotund and admits both Gateaux and
Fréchet differentiability conditions.

###### Key words and phrases:

Nearest point, Convex set, Chebyshev set, Best approximation, Distance
function.

###### 2010 Mathematics Subject Classification:

Primary 41A65; secondary 41A50, 46B20, 47N10, 47H05.

##  1\. Introduction

Studies in approximation theory have been carried out by many mathematicians
over decades (see [1], [2], [10] and [21] and the references therein). The
most important basic question in the field of BA is the concern about the
existence of BAs [7]. This is because BA theory has several applications
involving finding solution to systems of equations [10]. This work is useful
in contributing knowledge in functional analysis by providing at least a
partial solution to the CS problem [18]. It will also be useful in solving
convex optimization problems and finding solutions to differential equations
[16]. Best approximation (BA) is an interesting field in functional analysis
that has attracted a lot of attention from many researchers for a very long
period of time up-to-date (see [3] and [4], [9] and the references there in).
Of greatest consideration is the characterization of the CS which is a subset
of a NLS which contains unique BAs [24]. Approximation theory involves
obtaining best approximation of functions using simple functions whether they
are linear or nonlinear [4], [6] and [15]. Martin [11] characterized
remotality of sets with regard to normed linear spaces and in particular for
convex sets in Banach spaces. However, the convexity of the Banach spaces was
not done in general due to the complex nature of spaces. Mazaheri [14] also
considered weakly-Chebyshev subspaces for NLSs of Banach spaces and did their
characterization in terms of nearest and farthest points via distance
functions. However, the authors could not give a particular best approximation
for the CSs and their convexity even for the simplest case [13] of a HS of
$\emph{l}^{2}.$ Zalinescu [26] on the study of convex sets and their
characterizations in general spaces determined optimization criteria for
vector spaces and left open a question regarding convexity of these spaces.
More recently, Mazaheri and Salehi [14] studied CSs and considered conditions
under which they are convex. However, they could not determine convexity of
the CSs in NLSs even for the simplest set up of HSs. It is worth noting that
various techniques have been used in trying to get a solution to the CS
problem [4]. The first technique we discuss in this work is the Bunt-Motzkin
Theorem. This theorem is a result on the converse of the CS problem. It
asserts that if a set $C$ is Chebyshev then it implies that it is convex in
Bergman spaces [5]. However, this assertion still remains unknown if it holds
for infinite dimensional Hilbert spaces. The other technique is the
Fr$\grave{e}$chet differentiability [8]. This is a derivative defined mostly
in normed spaces. Fr$\grave{e}$chet differentiability occurs on real-valued
functions or vector valued functions of multiple variables. It is applicable
mostly and particularly on directional derivative where the continuity of the
map is essential [19]. Gateaux differentiability conditions are also very
instrumental conditions in BAs. A Gateaux derivative is a fundamental
principle in differential calculus which is a generalization on functions
which are continuous on Banach spaces. It is useful in carrying out
approximations in locally convex spaces [20]. It is useful in formalization of
functional derivatives which are important in best approximations in calculus
of variations. Moreover, its useful since it takes care of nonlinear functions
also. Best approximation techniques have also been employed in CS problem
[24]. These are techniques in approximation theory which useful in obtaining
best approximation results for functions in various spaces. They include:
Polynomial approximations; Chebyshev approximations; Remez techniques and
algorithms; and Pade’ approximation techniques for optimal polynomials.
Despite the fact that these techniques have been used in trying to solve CS
problem, the answer to this problem is still elusive [23]. It is very
important to unveil a detailed account of research work which have been done
in approximation theory. In particular, we consider literature on conditions
under which subsets of NLSs are Chebyshev. We also discuss the
characterizations on distance functions of CSs in NLSs and finally give a
review on investigations on convexity of CSs in various NLSs [6]. We begin
with some brief account on the various studies on CSs and their subsets. CSs
are important sets in approximation theory due to their properties. A lot of
characterizations have been done on these sets with interesting results
obtained. Their subsets are also interesting as they carry hereditary
properties in them. Fletcher and Moors [7] characterized CSs and showed that a
CS is particularly a subset of a NLS which has properties that helps in the
establishment of best approximations results which are unique. The authors
investigated characteristics of the metric projection and obtained necessary
and sufficient conditions under which a a subset of a NLS becomes a CS and
also conditions for a CS to satisfies convexity properties. Moreover, the
authors gave an example where they constructed a nonconvex CS as shown in the
next result.

######  Proposition 1.1.

([7], Proposition 2.12) Let $W$ be a NLS. A subset of $W$ which is closed is
also convex if and only if it is midpoint convex.

Proposition 1.1 characterizes NLSs in terms of midpoint convexity and shows
that a subset of a NLS can be convex if its closed and satisfies midpoint
convexity. However, this result does not indicate whether the space $W$ is
convex in general even if it is a CS. It is noted that closedness is useful
for midpoint convexity but not for convexity in general.
Vlasov [25] in the earlier years characterized normed spaces in terms of
approximate properties. Historically, it can be shown in brief, the main
contributions of researchers in this field of study. In 1934 Bunt proved the
convexity of a CS in the real plane. Also 1938 Kritikos followed Bunt’s
results and extended the theorem of Bunt to an $n$-dimensional real space.
This was followed by the work of Efimov and Stechkin in 1961 which showed that
a CS of a general HS which is approximately compact satisfies convexity
condition. To conclude the history, Klee considered weak closedness and proved
that CS which is weakly closed satisfies the convexity condition. This history
and more details can be found in Mantegazza [9]. To consider particular cases,
the author in [24] considered proximal sets which are related to Chebyshev
sets and gave the result below.

######  Theorem 1.2.

Let $W$ be a NLS. Consider $J$ as a proximinal set of $W$. Then $J$ is
nonempty and closed.

Theorem 1.2 considers proximal sets and characterizes closedness and the
content of the sets. However, it becomes very difficult to come up with a
structure of the metric projection function in terms of its geometry.
Nonetheless, an exception on this assertion can be considered when $J$ is a
subspace. Next we consider distance functions of Chebyshev sets in NLSs.
Distance functions are also important in characterizing CSs. By the result of
Asplund [1], the CS problem was given a different dimension to consider metric
projections as shown in the next proposition.

######  Proposition 1.3.

([7], Proposition 2.2) Let $W$ be a NLS. Consider $J$ as a CS of $W$. Then the
distance function for $J$ satisfies nonexpansivity and continuity conditions.

Proposition 1.3 describes CSs and their distance functions in terms of
continuity and nonexpansivity in a general set up. Lastly we consider CSs and
convexity. We consider literature on the key question of this study, that is,
Is every CS in HS convex?. We begin with the following analogy of the result
of Borwein [5] on closedness, reflexivity and rotundity of Hilbert spaces.

######  Lemma 1.4.

([5], Fact 3) All CSs are closed and all closed sets satisfying convexity
condition are Chebyshev in a rotund reflexive space. Particularly, all
nonempty closed sets satisfying convexity condition in HS are Chebyshev.

Lemma 1.4 gives an elaborate characterization of Chebyshev sets and convexity
in terms of reflexivity and rotundity that requires uniqueness property
however this does not answer Cs problem in general.
In summary, with all these considerations, a fundamental question remains
unsolved to-date regarding the convexity of CSs in infinite NLSs known as the
CS problem. The question which has not been answered is: Is every CS in a NLS
convex?. This question has not been answered even in the simplest case of a
real HS. In this regard, it is worth characterizing CS and convexity in NLSs.
We also attempt to answer this question partially in a particular case of the
NLS space of all norm-attainable real-valued functions. This work is organized
as follows: For the first section, we begin with a mathematical background as
given in this introduction followed with the preliminary basic concepts that
helps us to understand this work. We then provide the main results and finally
the conclusion.

##  2\. Preliminaries

For a better understanding of this work, we outline the basic definitions that
are key to this note on CS and convexity in NLSs.

######  Definition 2.1.

([8]) Let $W$ be a NLS and $G$ be a nonempty subset of $W$. Consider the
particular point $\zeta\in W$. We define the distance from the point $\zeta$
to $G$ by $d(\zeta,G)=\inf_{\eta\in G}\|\zeta-\eta\|$, and the map
$\zeta\mapsto d(\zeta,G)$ is called the distance function for $G$. We call
$\zeta$ the nearest point in $G.$

######  Definition 2.2.

([15]) A set $G$ in a NLS $W$ is called a CS if every point in $W$ has a
unique nearest point in $G.$ That is, CS is a subset of a NLS that admits
unique best approximations.

######  Definition 2.3.

([8]) Let $D$ be a nonempty set. A subset $E$ of $D$ is said to be convex if
for all $\zeta,\eta\in E$ the line segment connecting $\zeta$ and $\eta$ is in
$E$, that is, $(1-\alpha)\zeta+\alpha\eta$ is in $E$ for $\zeta,\eta\in E$,
and $\alpha\in[0,1].$

At this point, we proceed to give the main results of this paper. These
results are restricted to the space of all norm-attainable real-valued
functions. A function $\phi$ is said to be norm-attainable if there exists a
unit vector $\xi$ in the domain of $\phi$ such that $\|\phi(\xi)\|=\|\phi\|.$
The space of all norm-attainable real-valued functions is a NLS. For details
on norm-attainability, see [16]-[20] and the references therein.

##  3\. Main results

We provide the main results of this note in this section. We characterize CSs
and their subsets and tackle the CS problem. We begin with the following
proposition which considers distance functions and Gateaux differentiability.
We note that all the spaces and their subspaces are all nontrivial and are
strictly NLS spaces of all functions that are norm-attainable unless otherwise
stated.

######  Proposition 3.1.

Let $\mathfrak{Q}$ be a NLS space of all norm-attainable real-valued functions
and $\mathfrak{J}$ be a closed and smooth subset of $\mathfrak{Q}$. Let
$\zeta\in\mathfrak{Q}\backslash\mathfrak{J}$ and $\eta$ the nearest point for
$\zeta$ in $\mathfrak{J},$ then Gateaux differentiability condition of
$\mathfrak{Q}$ holds for $(\zeta-\eta).$

###### Proof.

Since the norm of $\mathfrak{Q}$ is Gateaux differentiable from the statement
of the proposition, it suffices to prove the exixtence of the unique limit of
the the Gateaux derivative $\displaystyle{\lim_{l\rightarrow
0}\frac{d_{\mathfrak{J}}(\zeta+l(x-\eta))-d_{\mathfrak{J}}(\zeta)}{l}}.$ From
[16], we deduce that if $l>0$ then the limit exists. For the uniqueness, we
see from the result of [7] that the limit of the derivative is unique. It
follows then that $\langle d^{\prime}_{\mathfrak{J}}(\zeta),x-\eta\rangle$
holds for $d_{\mathfrak{J}}(\zeta)$. This completes the proof. ∎

Proposition 3.1 leads to the interesting question as to what happens when
$\mathfrak{Q}$ is rotund. We see this in the next lemma.

######  Lemma 3.2.

Let $\mathfrak{Q}$ be a NLS space of all norm-attainable real-valued functions
and $\mathfrak{J}$ be a closed and smooth CS of $\mathfrak{Q}$. Let
$\zeta\in\mathfrak{Q}\backslash\mathfrak{J}$ and $\partial
d_{\mathfrak{J}}(\zeta)$ be a singleton set. Then the following conditions
hold if the first dual of $\mathfrak{Q}$ is rotund:
(i). $\phi$ on $\mathfrak{J}$ is uniformly continuous.
(ii). $\phi$ on $\mathfrak{J}$ is totally bounded.
(iii). $\mathfrak{J}$ satisfies convexity condition.
(iv). $d_{\mathfrak{J}}$ satisfies convexity condition.
(v). $d_{\mathfrak{J}}$ satisfies Gateaux differentiability at $\zeta$.

###### Proof.

We proceed with the proof as follows:
Case $(i).$ $\phi$ on $\mathfrak{J}$ is uniformly continuous since every space
of norm-attainable functions contains continuous functions.
Case $(ii).$ $\phi$ on $\mathfrak{J}$ is totally bounded follows immediately
from case $(i).$
Case $(iii).$ $\mathfrak{J}$ satisfying convexity condition follows
immediately from the conditions of the statement of the lemma.
Case $(iv).$ $d_{\mathfrak{J}}$ satisfying convexity condition follows from
the fact that $\mathfrak{J}$ satisfies convexity condition.
Case $(v).$ Since $d_{\mathfrak{J}}$ satisfies convexity condition and is
uniformly continuous at $\zeta$ and from Proposition 3.1 $\partial
d_{K}(\zeta)$ is a singleton set, $d_{\mathfrak{J}}$ satisfies Gateaux
differentiability at point $\zeta$ and we attain equality of
$d^{\prime}_{\mathfrak{J}}(\zeta)$ and $\partial d_{\mathfrak{J}}(\zeta)$.
This completes the proof. ∎

At this point, we state the main theorem of our work that characterizes
convexity of $\mathfrak{Q}$ in terms of Fréchet differentiability condition.

######  Theorem 3.3.

Let $\mathfrak{Q}$ be a NLS space of all norm-attainable real-valued functions
and $\mathfrak{J}$ be a closed and smooth CS of $\mathfrak{Q}$. Let
$\zeta\in\mathfrak{Q}\backslash\mathfrak{J}$ and $\partial
d_{\mathfrak{J}}(\zeta)$ be a singleton set. Then $d_{\mathfrak{J}}$ satisfies
Fréchet differentiability condition at $\zeta$.

###### Proof.

It is known from [25] that the norm of $\mathfrak{Q}$ and hence the dual norm
of $\mathfrak{Q}^{*}$ satisfies Fréchet differentiability condition. Also from
Lemma 3.2 it implies that $\mathfrak{Q}$ is strictly reflexive. Moreover,
$\mathfrak{Q}\backslash\mathfrak{J}$ has the nearest point $\zeta$ and so
$\mathfrak{J}$ of $\mathfrak{Q}$ satisfies Fréchet differentiability condition
and so is $\mathfrak{Q}$. ∎

As consequences of Theorem 3.3, we state the following corollaries.

######  Corollary 3.4.

Every distance function of a CS of the NLS space of all norm-attainable real-
valued functions is Fréchet differentiable.

###### Proof.

Follows from the conditions of Lemma 3.2 and Theorem 3.3. The rest is clear
from the fact that every rotund CS is convex. ∎

######  Corollary 3.5.

Every distance function of a CS of the NLS space of all norm-attainable real-
valued functions is Gateaux differentiable.

###### Proof.

Follows immediately from the conditions of Theorem 3.3 and Corollary 3.5 the
proof is complete. ∎

##  4\. Conclusion

In conclusion, a fundamental question that remains unsolved to-date regarding
the convexity of the CS in infinite NLS known as the CS problem has been
studied in this work. This CS problem which has not been solved in totality
(even in this note) states that: Is every CS in a NLS convex? This question
has not got any solution even in the simplest form of a real Hilbert space
(HS). In this note, we have characterized Chebyshev sets and their convexity
in NLSs. We considered the NLS space of all real-valued norm-attainable
functions. We have shown that Chebyshev subsets of the NLS space of all real-
valued norm-attainable functions are convex when they are closed, rotund and
admits both Gateaux and Fréchet differentiability conditions.

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