# Influence of a cosmic string on the rate of pairs produced by the Coulomb
potential

B. Belbaki and A. Bounames
Laboratory of Theoretical Physics, Department of Physics,
University of Jijel, Jijel 18000, Algeria. E-mail<EMAIL_ADDRESS>mail<EMAIL_ADDRESS>
###### Abstract

We study particle creation phenomenon by the Coulomb potential of an external
electric field in the presence of a gravitational field of a static cosmic
string. For that, the generalized Klein-Gordon and Dirac equations are solved,
and by using the Bogoliubov transformation we calculate the probability and
the number density of created particles. It is shown that the presence of the
cosmic string enhances the particle production. For the grand unified theory
(GUT) cosmic string, the production of spinless particles is possible if the
Coulomb potential nucleus charge $Z\geq 206$, and for spin-1/2 particles if
$Z\geq 275$.

Keywords: Particle creation, Coulomb Potential, Bogoliubov transformation,
Topological defects, Cosmic string, Deficit angle, Conical spacetime.

##  1 Introduction

The creation of particles by an electric field in curved spacetime is an
active research field and continue to attract attentions as in the
cosmological models of an expanding universe [1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19]. In particular, the creation of particles
by the Coulomb potential of an external electric field is a topic that has
received a special interest [4, 20, 21, 22, 23, 24, 25] and also in the
references cited in the three relevant reviews [26, 27, 28]. Moreover,
particle production under the effect of topological defects has been also
discussed in the literature, like in the field of a magnetic monopole and
domain walls [29, 30], and in the presence of a cosmic string [31, 32, 33, 34,
35, 36, 37, 38, 39, 40].

The cosmic strings are hypothetical objects which may have been formed during
the inflationary phase in the primordial universe [41, 42, 43]. They are one-
dimensional topological defects in the spacetime structure and resulting from
a symmetry breaking at an energy close to $10^{16}$ $GeV$ ($10^{-36}$ seconds)
[43], and with a maximum mass per unit length $\mu_{\max}=6.73\times
10^{27}g.cm^{-1}$ [44]. They have a nontrivial topology where the spacetime is
locally flat and globally conical with an azimuthal deficit angle [45]. A
cosmic string induces a repulsive force on an electric charge at rest [46] or
on a current [47], and an attractive force on a neutral particle [48]. It has
also relevant effects like gravitational lensing [43], Casimir effect and the
gravitational Aharonov-Bohm effect [49, 50]. The various observation programs
of the anisotropies in cosmic background radiation (CMB) by COBE, WMAP and the
Planck satellite have not observed any cosmic strings effects on the
primordial density perturbations [51]. Nevertheless, it is possible that they
have a role in the production of gravitational waves [52, 53, 54, 55] and
generation of high-energy cosmic rays [56]. More recently, a two-level static
atom coupled to an electromagnetic field, in a cosmic string spacetime, is
suggested as a detector to estimate the deficit angle [57].

In order to introduce the deficit angle, let us use the metric of an infinite
straight string in cylindrical coordinates $(t,\rho,\phi,z)$ [43, 58]

$ds^{2}=-dt^{2}+d\rho^{2}+\alpha^{2}\rho^{2}d\phi^{2}+dz^{2},$ (1)

where the cosmic string parameter $\alpha=1-4\bar{\mu}$ varies in the interval
$]0,1]$ with $\bar{\mu}=\frac{G}{c^{2}}\mu$, $\mu$ the linear mass density of
the string, $G$ the gravitational constant, $c$ the speed of light and the
variation of the angular variable is $0\leq\phi\leq 2\pi$. The metric (1) has
been obtained by solving Einstein’s equations [43, 58]. If we introduce the
change of variable $\varphi=(1-4\bar{\mu})\phi$, the metric (1) reduces to the
Minkowski line element

$ds^{2}=-dt^{2}+d\rho^{2}+\rho^{2}d\varphi^{2}+dz^{2},$

where the interval of variation of $\varphi$ is

$0\leq\varphi\leq 2\pi(1-4\bar{\mu}),$ (2)

and consequently the cosmic string spacetime is locally flat and globally
conical with a deficit angle $\Delta=8\pi\bar{\mu}=2\pi(1-\alpha)$, i.e., its
geometry has a conical singularity and the curvature tensor is defined using
the 2D Dirac delta function $\delta^{(2)}(\rho)$ [59]

$R\indices{{}^{12}_{12}}=2\pi\frac{\alpha-1}{\alpha}\delta^{(2)}(\rho),$ (3)

which means that the curvature is concentrated on the cosmic string axis and
zero outside. In the absence of the string $\alpha=1$, the curvature tensor
vanishes. The case $\alpha>1$ corresponds to an anti-conical spacetime with
negative curvature [60, 61].

Thereby, it is physically meaningful to analyze the role of cosmic string
deficit angle on the rate of pairs produced by an external electric field. For
this purpose, in this paper we study particle creation by a vector Coulomb
potential in the presence of a static cosmic string. Once the solutions of the
Klein-Gordon and Dirac equations obtained, the expressions of the probability
and the number density of the created particles will be computed using the
Bogoliubov transformation. Thereafter, we discuss the influence of the cosmic
string on the rate of created particles induced by the Coulomb potential for
both spin-$0$ boson and spin$-1/2$ fermion particles. On the other hand, it is
worth mentioning that the self-adjoint extensions method has recently been
used to construct rigorous mathematical formulation of wave equation solutions
for the Coulomb potential and similar singular potentials [62, 63].

In this work, we will use spherical coordinates since the problem has
spherical symmetry, for that we consider the coordinate transformation
$\rho=r\sin\theta$ and $z=r\cos\theta$, and the line element of the cosmic
string becomes

$ds^{2}=-dt^{2}+dr^{2}+r^{2}d\theta^{2}+\alpha^{2}r^{2}\sin^{2}\theta
d\phi^{2},$ (4)

where $(t,r,\theta,\phi)$: $-\infty<t<+\infty$, $r\geq 0$,
$0\leq\theta\leq\pi/2$ and $0\leq\phi\leq 2\pi$. In the absence of the string
$\alpha=1$, the metric (4) reduces to the Minkowski one.
The present paper is organized as follows, in sections 2 and 3 we present
respectively the solutions of the Klein-Gordon and Dirac equations in the
presence of a Coulomb potentiel in a cosmic string spacetime, for both cases
we calculate the probability and the number density of created particles. The
last section is devoted to the conclusion.

##  2 Klein-Gordon equation in cosmic string space-time

In curved spacetime, the generalized Klein-Gordon equation of spin$-0$
particle of mass $M$ and charge $q$ minimally coupled to an external
electromagnetic field $A_{\mu}$ is

$\left[{-\frac{1}{\sqrt{-g}}D_{\mu}{g^{\mu\nu}}\sqrt{-g}D_{\nu}+{M^{2}}}\right]\Psi=0,$
(5)

where $D_{\mu}={\partial_{\mu}}+iq{A}_{\mu}$ is the covariant derivative,
${{g^{\mu\nu}}}$ the metric tensor of the curved spacetime and
$g=\det({{g^{\mu\nu}}})$.
In this work, we consider the case where the spacial components of the
external field have a null value $({A}_{i}=0;i=1,2,3)$ and its time component
$A_{0}$ is the Coulomb potential generated by a point-like source charge
$Q_{s}$ where its expression is

${A_{0}}\equiv{V(r)}=\frac{Q_{s}}{r},$ (6)

and for an electron in a hydrogen-like atom $Q_{s}=Ze$ and $q=-e$.
Thus, the Klein-Gordon equation (5) can be written explicitely in the cosmic
string metric like

$\left[{-{{\left({\frac{\partial}{{\partial
t}}+iq{A_{0}}}\right)}^{2}}+\frac{1}{{{r^{2}}}}\frac{\partial}{{\partial
r}}\left({{r^{2}}\frac{\partial}{{\partial
r}}}\right)+\frac{1}{{{r^{2}}\sin\theta}}\left({\cos\theta\frac{\partial}{{\partial\theta}}+\sin\theta\frac{{{\partial^{2}}}}{{\partial{\theta^{2}}}}}\right)+\frac{1}{{{\alpha^{2}}{r^{2}}{{\sin}^{2}}\theta}}\frac{{{\partial^{2}}}}{{\partial{\phi^{2}}}}-{M^{2}}}\right]\Psi=0.$
(7)

Since the potential (6) is time-independent and the problem has a spherical
symmetry, the solution can be taken in the form

$\Psi(t,r,\theta,\phi)=\frac{{u(r)}}{r}f(\theta){e^{i(-Et+m\phi)}},$ (8)

where $m=0,\pm 1,\pm 2,...$ and $E$ is the energy of the spin-$0$ particle.
By Substituting Eq. (8) in Eq. (7), the angular function $f(\theta)$ satisfies
the following differential equation

$\left[{\frac{1}{{\sin\theta}}\frac{d}{{d\theta}}\left({\sin\theta\frac{d}{{d\theta}}}\right)-\frac{{{m^{2}}}}{{{\alpha^{2}}{{\sin}^{2}}\theta}}}\right]f(\theta)=-{\lambda_{\alpha}}f(\theta),$
(9)

where its solution is given in terms of the generalized Legendre functions
$P_{\nu}^{\mu}(x)$, with

${\lambda_{\alpha}}={l_{\alpha}}({l_{\alpha}}+1),$ (10)

where
${l_{\alpha}}=n+\left|{{m_{\alpha}}}\right|=l+\left|m\right|\left(\frac{1}{\alpha}-1\right)$
with $n$ a non-negative integer, $l=n+\left|m\right|$ and
$m_{\alpha}={\frac{m}{\alpha}}$. In addition $m_{\alpha}$ and $l_{\alpha}$ are
not necessarily integers, and $m_{\alpha}$ varies in the range
$-{l_{\alpha}}\leq{m_{\alpha}}\leq{l_{\alpha}}$. Finally, $l$ and $m$ are the
orbital angular momentum and the magnetic quantum numbers in the flat space,
respectively [64].

On the other hand, the radial function $u(r)$ satisfies the following
differential equation [64]

$\frac{{{d^{2}}u}}{{d{r^{2}}}}+\left[{{K^{2}}-{V_{eff}}-\frac{{{l_{\alpha}}({l_{\alpha}}+1)}}{{{r^{2}}}}}\right]u=0,$
(11)

where $K^{2}$ and $V_{eff}$ are defined as

${K^{2}}={E^{2}}-{M^{2}},\qquad{V_{eff}}=2qE{V(r)}-q^{2}{V}^{2}(r).$ (12)

By substitution of the expression (6) of the potential, Eq. (11) can be
written in the form

$\frac{{{d^{2}}u}}{{d{r^{2}}}}+\left[{{K^{2}}-\frac{2EV_{0}}{r}-\frac{{{l_{\alpha}}({l_{\alpha}}+1)}-V_{0}^{2}}{{{r^{2}}}}}\right]u=0,$
(13)

with ${V_{0}}\equiv{qQ_{s}}$. If we study only the case $V_{0}>0$, the results
for the case of $V_{0}<0$ can be obtained by the charge conjugation
transformation [62].
In order to search the solutions corresponding to the condition
$\left|E\right|>M$, we use the following notations

${\gamma_{l}}=\pm\sqrt{{{\left({{l_{\alpha}}+\frac{1}{2}}\right)}^{2}}-V_{0}^{2}},\qquad\eta=\frac{{E}V_{0}}{K},$
(14)

and the change $z=-2iKr$ transforms Eq. (13) into the following form

$\frac{{{d^{2}}u}}{{d{z^{2}}}}+\left[{-\frac{1}{4}-\frac{{i\eta}}{z}+\frac{\frac{1}{4}{-\gamma_{l}^{2}}}{z{{{}^{2}}}}}\right]u=0,$
(15)

which admits two regular linearly independent solutions in terms of the
Whittaker functions ${M_{-i\eta,{\gamma_{l}}}}(z)$ and
${W_{-i\eta,{\gamma_{l}}}}(z)$ [65]

$u_{1}(z)=C_{1}{M_{-i\eta,{\gamma_{l}}}}(z)=C_{1}{z^{({\gamma_{l}+}\frac{1}{2})}\exp(-z/2)}M({\gamma_{l}}+i\eta+\frac{1}{2},1+2{\gamma_{l}};z),$
(16)
$u_{2}(z)=C_{2}{W_{-i\eta,{\gamma_{l}}}}(z)=C_{2}{z^{({\gamma_{l}+}\frac{1}{2})}\exp(-z/2)}U({\gamma_{l}}+i\eta+\frac{1}{2},1+2{\gamma_{l}};z),$
(17)

$C_{1}$ and $C_{2}$ are the normalization constant, $M(a,b,z)$ and $U(a,b,z)$
are the Kummer’s confluent hypergeometric functions. The first solution (16)
is bounded at $z=0$ while the second (17) is bounded at
$|z|\rightarrow\infty$.

###  2.1 Creation of scalar particles

In order to obtain the expressions of the probability and the number density
of created particles, we use the Bogoliubov transformation which links the
asymptotic behavior of the obtained solutions.

Indeed, the asymptotic behavior of ${W_{-k,{\mu}}}(z)$ for
$|z|\rightarrow\infty$ is

$W_{-k,{\mu}}(z)\rightarrow{z^{-k}\exp(-z/2)},$ (18)

and the states $\phi^{+}_{out}$ and $\phi^{-}_{out}$ can be defined as

$\phi^{+}_{out}=C^{+}_{2out}W_{-k,{\mu}}(z),\quad\phi^{-}_{out}=[C^{+}_{2out}W_{-k,{\mu}}(z)]^{*}=[C^{+}_{2out}]^{*}W_{k,{\mu}}(-z).$
(19)

Similarly, the asymptotic behavior of ${M_{-k,{\mu}}}(z)$ for $z\rightarrow 0$
is

$M_{-k,{\mu}}(z)\rightarrow{z^{({\mu+}\frac{1}{2})}},$ (20)

and the states $\phi^{+}_{in}$ and $\phi^{-}_{in}$ are

$\phi^{+}_{in}=C^{+}_{1in}M_{-k,{\mu}}(z),\quad\phi^{-}_{in}=[C^{+}_{1in}M_{-k,{\mu}}(z)]^{*}=[C^{+}_{1in}]^{*}(-1)^{(-\mu+1/2)}M_{-k,{-\mu}}(z).$
(21)

Therefore, the Bogoliubov transformation can be written as

$\phi^{+}_{out}(z)={A}\phi^{+}_{in}(z)+B\phi^{-}_{in}(z),$ (22)

and with the help of the two following functional relations [65]

${W_{-k,\mu}}(z)=\frac{{\Gamma(-2\mu)}}{{\Gamma(-\mu+k+\frac{1}{2})}}{M_{-k,\mu}}(z)+\frac{{\Gamma(2\mu)}}{{\Gamma(\mu+k+\frac{1}{2})}}M_{-k,-\mu}(z),$
(23) $M_{-k,-\mu}(z)={e^{i\pi(\mu-1/2)}}[{{M_{k,\mu}}(z)}]^{*},$ (24)

the Bogoliubov coefficients $A$ and $B$ satisfy

$\left|{\frac{B}{A}}\right|=\left|{\frac{{\Gamma(-{{{\gamma}_{l}+i}\eta+{\frac{1}{2}})}}}{{\Gamma({{{\gamma}_{l}+i}\eta+{\frac{1}{2}})}}}{e^{i\pi({\gamma_{l}}-1/2)}}}\right|,$
(25)

where ${k={{i}\eta}}$ and ${\mu={{{\gamma}_{l}}}}$.

Using the bosonic condition $\left|A\right|^{2}-\left|B\right|^{2}=1$ and
$\gamma_{l}=i{\tilde{\gamma}_{l},}$ the probability for one pair production is

$P={\left|{\frac{B}{A}}\right|^{2}}={\left|\frac{{\Gamma(-i\tilde{\gamma}_{l}+{{i}\eta+{\frac{1}{2}})}}}{{\Gamma(i\tilde{\gamma}_{l}{{+i}\eta+{\frac{1}{2}})}}}{{e^{i\pi({i\tilde{\gamma}_{l}}-1/2)}}}\right|^{2}.}$
(26)

Then, using the property of gamma function

${\left|{\Gamma\left({\frac{1}{2}+ix}\right)}\right|^{2}}=\frac{\pi}{{\cosh(\pi
x)}},$ (27)

the expression of the probability is reduced to

$P=\frac{\cosh\left[\pi(\tilde{\gamma}_{l}+\eta)\right]}{\cosh\left[\pi(\tilde{\gamma}_{l}-\eta)\right]}e^{-2\pi\tilde{\gamma}_{l}},$
(28)

where

${\tilde{\gamma}_{l}}=\sqrt{V_{0}^{2}-\left[l+\left|m\right|\left(\frac{1}{\alpha}-1\right)+\frac{1}{2}\right]^{2}}>0,$
(29)

is the condition for pair production of spinless particle which depends on
${V_{0}}={qQ_{s}}$, the cosmic string paramter $\alpha$ and the two quantum
numbers $l$ and $m$.

The number density of the scalar particles created is given by

$\bar{n}=\left|B\right|^{2}=\frac{1}{P^{-1}-1}=\frac{\cosh\left[\pi(\tilde{\gamma}_{l}+\eta)\right]}{\cosh\left[\pi(\tilde{\gamma}_{l}-\eta)\right]e^{2\pi\tilde{\gamma}_{l}}-\cosh\left[\pi(\tilde{\gamma}_{l}+\eta)\right]},$
(30)

which is positive since ${{\tilde{\gamma}}_{l}}>0$.

At large frequencies $(E\gg M)$, the probability (28) reduces to

$P=\frac{\cosh\left[\pi(\tilde{\gamma}_{l}+V_{0})\right]}{\cosh\left[\pi(\tilde{\gamma}_{l}-V_{0})\right]}e^{-2\pi\tilde{\gamma}_{l}},$
(31)

and the number density takes the form

$\bar{n}=\frac{\cosh\left[\pi(\tilde{\gamma}_{l}+V_{0})\right]}{\cosh\left[\pi(\tilde{\gamma}_{l}-V_{0})\right]e^{2\pi\tilde{\gamma}_{l}}-\cosh\left[\pi(\tilde{\gamma}_{l}+V_{0})\right]},$
(32)

which is not a thermal distribution.

For the case when $\left|\tilde{\gamma_{l}}\pm V_{0}\right|\gg 1$, the number
density (32) takes the form of Bose-Einstein distribution

$\bar{n}=\frac{1}{e^{2\pi(\tilde{\gamma}_{l}-V_{0})}-1}.$ (33)

In the absence of the cosmic string ($\alpha=1)$, or for $m=0$, the
probability (31) and the number density (32) reduce to

$P(\alpha=1)=\frac{\cosh\left[\pi(\tilde{\gamma}_{1}+V_{0})\right]}{\cosh\left[\pi(\tilde{\gamma}_{1}-V_{0})\right]}\exp\left[-2\pi\tilde{\gamma}_{1}\right],$
(34)
$\bar{n}(\alpha=1)=\frac{\cosh\left[\pi(\tilde{\gamma}_{1}+V_{0})\right]}{\cosh\left[\pi(\tilde{\gamma}_{1}-V_{0})\right]e^{2\pi\tilde{\gamma}_{1}}-\cosh\left[\pi(\tilde{\gamma}_{1}+V_{0})\right]},$
(35)

where
$\tilde{\gamma}_{1}\equiv\tilde{\gamma}_{l}(\alpha=1)=\sqrt{V_{0}^{2}-{{\left({l+\frac{1}{2}}\right)}^{2}}}$.

###  2.2 Discussion of results

$\bullet$ First, let us discuss the consequences of the condition
$(\ref{cond1})$, for particle production of spin-0 boson, which can be written
as

$\left|qQ_{s}\right|>\left[l+\left|m\right|\left(\frac{1}{\alpha}-1\right)+\frac{1}{2}\right].$
(36)

In the absence of the cosmic string ($\alpha=1)$, or for $m=0$, the above
condition is reduced to

$\left|qQ_{s}\right|>\left(l+\frac{1}{2}\right),$ (37)

which is similar to that obtained in Eq. (37) for pair creation in a magnetic
monopole field [30], and also similar to the results discussed after Eq. (84)
for pair creation by charged black holes [66].

For an electron in a hydrogen-like atom, $Q_{s}=Ze$ and $q=-e$, the condition
(36) can be expressed for the nucleus charge $Z$ as

$Z>Z_{cr}\equiv\frac{1}{e^{2}}\left[l+\left|m\right|(\frac{1}{\alpha}-1)+\frac{1}{2}\right].$
(38)

For the particular values of the quantum numbers $(l=0,m=0)$, the pair
production of scalar particles is possible for :
$Z>Z_{cr}=\frac{1}{2e^{2}}=\frac{137}{2}$ (in units $\hbar=c=1$, $e^{2}\simeq
1/137$), which is a well-known result in the absence of the cosmic string
[22].

In table 1, for the first values of the quantum numbers $(l,m),$ we give the
critical values $Z_{cr}$ as function of the cosmic string parameter $\alpha$
and an estimation of the effect of the GUT cosmic string (for which [67]:
$\alpha=1-10^{-6}=0.999999$) on the values and the rate of $Z_{cr}^{GUT}$.

Table 1: $Z_{cr}$ in terms of $\alpha$, values and the rate of $Z_{cr}^{GUT}$ for boson particle $l$ |  $m$ |  $Z_{cr}$ |  $Z_{cr}^{GUT}$ |  $\frac{Z_{cr}^{GUT}-Z_{cr}(\alpha=1)}{Z_{cr}(\alpha=1)}$
---|---|---|---|---
$0$ |  $0$ |  $\frac{137}{2}=68.5$ |  |
$1$ |  $0$ |  $137\times\frac{3}{2}=205.5$ |  |
|  $\pm 1$ |  $137\times\left(\frac{1}{\alpha}+\frac{1}{2}\right)$ |  $205.500137$ |  $6.666666\times 10^{-5}\%$
|  $0$ |  $137\times\frac{5}{2}=342.5$ |  |
$2$ |  $\pm 1$ |  $137\times\left(\frac{1}{\alpha}+\frac{3}{2}\right)$ |  $342.500137$ |  $4.000004\times 10^{-5}\%$
|  $\pm 2$ |  $137\times\left(\frac{2}{\alpha}+\frac{1}{2}\right)$ |  $342.500274$ |  $8\times 10^{-5}\%$

We note that for the sub-states with $m=0$, the critical value $Z_{cr}$
increases with $l$ and is independent of the cosmic string parameter $\alpha$
(Note that these sub-states are equivalent to the case $\alpha=1$). For the
sub-states for which $m\neq 0$, $Z_{cr}$ increases with $l$ and depends
inversely on the cosmic string parameter $\alpha$. For the GUT cosmic string,
the obtained critical values $Z_{cr}^{GUT}$ do not exist for the moment (at
present, the maximum is $Z=118$ in Mendeleev table), the rate of
$Z_{cr}^{GUT}$ is about $10^{-5}\%$ as compared with the case $\alpha=1$
(without cosmic string) and the production of scalar particles is possible if
the Coulomb potential nucleus charge (or atomic number) $Z\geq 206$.

$\bullet$ Secondly, in Figure 1 the number density curves have been plotted
for fixed values of the quantum numbers $(l=1,m=1)$ and where the condition
$(\ref{cond1})$ for pair production of spinless particles is reduced to
$V_{0}>(\frac{1}{2}+\frac{1}{\alpha})$ which should be always fullfilled. Plot
(a) display, in three (3D) dimensions, the number density $(\ref{47})$ in
terms of $V_{0}=qQ_{s}$ and the cosmic string parameter $\alpha$. Plot (b)
display, in two (2D) dimensions, the number density $(\ref{47})$ in terms of
$V_{0}=qQ_{s}$ for different values the cosmic string parameter $\alpha$.

|
---|---
(a) | (b)
Figure 1: Plot (a) is the 3D number density (32) in terms of $V_{0}$ and
$\alpha$. Plot (b) is the 2D number density (32) in terms of $V_{0}$ for
different values of $\alpha=0.3,0.6,0.8$ and $1$ (without cosmic string) .

From plot (b) we note that the number density $\bar{n}$ decreases when the
cosmic string parameter $\alpha$ increases (i.e. the linear mass density $\mu$
of the string decreases) where its smallest values correspond to the case
$\alpha=1$ (in the absence of the cosmic string). Therefore, we deduce that
the presence of the cosmic string improves the number density of scalar
particles compared to the case without the cosmic string ($\alpha=1$).

##  3 Dirac equation in cosmic string space-time

In curved spacetime, the generalized Dirac equation of spin$-1/2$ particle of
mass $M$ and charge $q$ minimally coupled to an external electromagnetic field
$A_{\mu}$ is

$\left[{i{\gamma^{\mu}}(x)\left({{\nabla_{\mu}}+i{qA_{\mu}}}\right)-M}\right]\Psi(x)=0,$
(39)

where ${\nabla_{\mu}}={\partial_{\mu}}+{\Gamma_{\mu}}(x)$ is the covariant
derivative, $\Gamma_{\mu}$ is the spinor affine connection

${\Gamma_{\mu}}=\frac{1}{4}{\gamma^{(a)}}{\gamma^{(b)}}{e^{\nu}}_{(a)}\left[{{\partial_{\mu}}{e_{(b)\nu}}-\Gamma_{\mu\nu}^{a}{e_{(b)\sigma}}}\right],$
(40)

$\Gamma_{\mu\nu}^{a}$ is the Christoffel symbol of the second kind,
$\\{{e^{\mu}}_{(a)}(x)\\}$ the tetrad basis and ${{\gamma^{\mu}}(x)}$ the
generalized Dirac matrix satisfying the relation

$\left\\{{{\gamma^{\mu}}(x),{\gamma^{\nu}}(x)}\right\\}=2{g^{\mu\nu}}(x),$
(41)

which can be written as a function of the tetrad basis
$\\{{e^{\mu}}_{(a)}(x)\\}$ and the standard flat spacetime Dirac matrices
${\gamma_{(a)}}$ as

${\gamma^{\mu}}(x)={e^{\mu}}_{(a)}(x){\gamma_{(a)}}.$ (42)

The tetrad basis $\\{{e^{\mu}}_{(a)}(x)\\}$ satisfies the relations

${\eta^{(a)(b)}}{e^{\mu}}_{(a)}(x){e^{\nu}}_{(b)}(x)={g^{\mu\nu}}(x),$ (43)

where the Greek letters are for tensor indices and Latin letters for tetrad
indices.

We consider the case where the spacial components of the external field are
nuls $({A}_{i}=0,i=1,2,3)$ and its time component ${A_{0}}(r)\equiv{V}(r)$ is
the Coulomb potential defined in Eq. (6).

In order to write the Dirac equation in the cosmic string metric (4), we use
the associated tetrad $e_{(a)}^{\mu}(x)$ defined by [67, 68]

${e^{\mu}}_{(a)}=\left({\begin{array}[c]{cccc}1&0&0&0\\\
0&{\sin\theta\cos\phi}&{\sin\theta\sin\phi}&{\cos\theta}\\\
0&{\frac{{\cos\theta\cos\phi}}{r}}&{\frac{{\cos\theta\sin\phi}}{r}}&{-\frac{{\sin\theta}}{r}}\\\
0&{-\frac{{\sin\phi}}{{\alpha r\sin\theta}}}&{\frac{{\cos\phi}}{{\alpha
r\sin\theta}}}&0\end{array}}\right),$ (44)

and the expressions of the generalized Dirac matrices $\gamma^{\mu}(x)$ are

${\gamma^{0}}(x)={\gamma^{(0)}},$ (45)
${\gamma^{1}}(x)=\sin\theta\cos\phi{\gamma^{(1)}}+\sin\theta\sin\phi{\gamma^{(2)}}+\cos\theta{\gamma^{(3)}},$
(46)
${\gamma^{2}}(x)=\frac{{\cos\theta\cos\phi}}{r}{\gamma^{(1)}}+\frac{{\cos\theta\sin\phi}}{r}{\gamma^{(2)}}-\frac{{\sin\theta}}{r}{\gamma^{(3)}},$
(47) ${\gamma^{2}}(x)=-\frac{{\sin\phi}}{{\alpha
r\sin\theta}}{\gamma^{(1)}}+\frac{{\cos\phi}}{{\alpha
r\sin\theta}}{\gamma^{(2)}}.$ (48)

Since the problem is time-independent, the wavefunction can be written as

$\Psi(x)={e^{-iEt}}\chi(\vec{r}),$ (49)

and by substitution of Eq. (49) in Eq. (39), we obtain [67]

$\left[{i{\gamma^{(r)}}{\partial_{r}}+i\frac{{{\gamma^{(\theta)}}}}{r}{\partial_{\theta}}+i\frac{{{\gamma^{(\phi)}}}}{{\alpha
r\sin\theta}}{\partial_{\phi}}+{\gamma^{(0)}}E+i\frac{1}{{2r}}\left({1-\frac{1}{\alpha}}\right)\left({{\gamma^{(r)}}+\cot\theta{\gamma^{(\theta)}}}\right)-{\gamma^{(0)}}{qA_{0}}-M}\right]\chi(\vec{r})=0,$
(50)

where

$\left({\begin{array}[c]{c}{{\gamma^{(r)}}}\\\ {{\gamma^{(\theta)}}}\\\
{{\gamma^{(\phi)}}}\end{array}}\right)=\left({\begin{array}[c]{ccc}{\sin\theta\cos\phi}&{\sin\theta\sin\phi}&{\cos\phi}\\\
{\cos\theta\cos\phi}&{\cos\theta\sin\phi}&{-\sin\phi}\\\
{-\sin\phi}&{\cos\phi}&0\end{array}}\right)\left({\begin{array}[c]{c}{{\gamma^{(1)}}}\\\
{{\gamma^{(2)}}}\\\ {{\gamma^{(3)}}}\end{array}}\right).$ (51)

Then, since the problem has a spherical symmetry, the solution of Eq. (50) can
be taken as

$\chi(\vec{r})={r^{-\frac{1}{2}\left({1-\frac{1}{\alpha}}\right)}}{(\sin\theta)^{-\frac{1}{2}\left({1-\frac{1}{\alpha}}\right)}}R(r)Y_{{l_{(\alpha)}}}^{{m_{(\alpha)}}}(\theta,\phi),$
(52)

where $R(r)$ is the radial part and
$Y_{{l_{(\alpha)}}}^{{m_{(\alpha)}}}(\theta,\phi)$ the spherical harmonics.
The parameters $m_{\alpha}={\frac{m}{\alpha}}$ and
${l_{\alpha}}=n+\left|{{m_{\alpha}}}\right|=l+\left|m\right|\left({\frac{1}{\alpha}-1}\right)$
with $l=n+\left|m\right|$ and $n$ a non-negative integer [69].

By substitution of Eq. (52) into Eq. (50), we obtain [67]

$\left({{\sigma_{y}}{p_{r}}+i\frac{{{\sigma_{y}}}}{r}{\gamma^{(0)}}{k_{(\alpha)}}+{qA_{0}}+M{\gamma^{(0)}}}\right)R(r)=ER(r),$
(53)

where $p_{r}$ is the radial momentum operator,
${k_{(\alpha)}}=\pm\left[{j+\left|m\right|\left({\frac{1}{\alpha}-1}\right)+\frac{1}{2}}\right]$
with $j$ is the eigenvalue of the total angular momentum operator in flat
Minkowski spacetime [69], and $\sigma_{y}$ is Pauli’s matrix

${\sigma_{y}}=\left({\ \begin{array}[c]{cc}0&{-i}\\\ i&0\end{array}}\right).$
(54)

For the radial solution we use the following ansatz [70]

$R(r)=\frac{1}{r}\left({\ \begin{array}[c]{c}{{u_{1}}(r)}\\\
iu{{{}_{2}}(r)}\end{array}}\right),$ (55)

and by substitution of Eq.(55) in Eq. (53) and using Eq. (54), we have

$\frac{{d{u_{1}}(r)}}{{dr}}=-\frac{{{k_{(\alpha)}}}}{r}{u_{1}}(r)+\left[{E+M-{qA_{0}}}\right]u{{}_{2}}(r),$
(56)
$\frac{{d}u{{{}_{2}}(r)}}{{dr}}=\frac{{{k_{(\alpha)}}}}{r}u{{}_{2}}(r)-\left[{E-M-{qA_{0}}}\right]{u_{1}}(r).$
(57)

By taking ${V_{0}}\equiv{qQ_{s}}$ and $A_{0}$ as the Coulomb potential defined
in Eq. (6), the above equations become

$\frac{{d{u_{1}}(r)}}{{dr}}=-\frac{{{k_{(\alpha)}}}}{r}{u_{1}}(r)+\left[{E+M-}\frac{{V{{}_{0}}}}{r}\right]u{{}_{2}}(r),$
(58)
$\frac{{d}u{{{}_{2}}(r)}}{{dr}}=\frac{{{k_{(\alpha)}}}}{r}u{{}_{2}}(r)-\left[{E-M-}\frac{{V{{}_{0}}}}{r}\right]{u_{1}}(r).$
(59)

If we study only the case $V_{0}>0$, the results for the case of $V_{0}<0$ can
be obtained by the charge conjugation transformation [62]. Its convenient to
introduce the new coordinate

$z=2iKr,\qquad K=\sqrt{E^{2}-M^{2}},$ (60)

then Eqs. (58)-(59) transform into

$\frac{{d{u_{1}}(z)}}{{dz}}=-\frac{{{k_{(\alpha)}}}}{z}{u_{1}}(z)+\left[\frac{E+M}{2iK}-\frac{V_{0}}{z}\right]{u_{2}}(z),$
(61)
$\frac{{d}u{{{}_{2}}(z)}}{{dz}}=\frac{{{k_{(\alpha)}}}}{z}u{{}_{2}}(z)-\left[\frac{E-M}{2iK}-\frac{V_{0}}{z}\right]{u_{1}}(z),$
(62)

and using the ansatz

$u_{1}(z)=\sqrt{E+M}\left[F(z)+G(z)\right],\\\ \quad
u{{}_{2}}(z)=i\sqrt{E-M}\left[F(z)-G(z)\right],$ (63)

in Eqs. (61)-(62), we have

$\frac{{d{F}(z)}}{{dz}}=\left(\frac{1}{2}-i\frac{V_{0}E}{Kz}\right){F}(z)+\left(-\frac{{{k_{(\alpha)}}}}{z}-i\frac{V_{0}M}{Kz}\right){G}(z),$
(64)
$\frac{{d{G}(z)}}{{dz}}=\left(-\frac{1}{2}+i\frac{V_{0}E}{Kz}\right){G}(z)+\left(-\frac{{{k_{(\alpha)}}}}{z}+i\frac{V_{0}M}{Kz}\right){F}(z)$
(65)

and from which one easily obtains the following second order differential
equation

$\frac{d^{2}{F}(z)}{{dz^{2}}}+\frac{1}{z}\frac{{d{F}(z)}}{dz}-\left[\frac{1}{4}+\left(\frac{1}{2}-i\frac{EV_{0}}{K}\right)\frac{1}{z}+\frac{\gamma_{l}^{2}}{z^{2}}\right]F(z)=0,$
(66)

with $\gamma_{l}^{2}=k_{(\alpha)}^{2}-V_{0}^{2}$.

The following ansatz

$M(z)=\sqrt{z}F(z),$ (67)

transforms the last equation to

$\frac{d^{2}{M}(z)}{{dz^{2}}}+\left[-\frac{1}{4}-\frac{i\lambda}{z}+\frac{\frac{1}{4}-\gamma_{l}^{2}}{z^{2}}\right]M(z)=0,$
(68)

where $i\lambda=\frac{1}{2}-i\eta$ and $\eta=\frac{EV_{0}}{K}$.
This equation admits two regular linearly independent solutions in terms of
the Whittaker functions [65]

$u_{1}{(z)}={C_{3}}{M_{-i\lambda,{\gamma_{l}}}}(z)={C_{3}}{z^{({\gamma_{l}}+\frac{1}{2})}}{e^{-z/2}}M({\gamma_{l}}+i\lambda+\frac{1}{2},2{\gamma_{l}}+1;z),$
(69)
$u_{2}{(z)}={C_{4}}{W_{-i\lambda,{\gamma_{l}}}}(z)={C_{4}}{z^{({\gamma_{l}}+\frac{1}{2})}}{e^{-z/2}}U({\gamma_{l}}+i\lambda+\frac{1}{2},2{\gamma_{l}}+1;z),$
(70)

where the first solution is bounded at $z=0$ while the second is bounded at
$|z|\rightarrow\infty$, $C_{3}$ and $C_{4}$ are the normalization constants.

###  3.1 Creation of fermionic particles

In order to obtain the expressions of the probability and the number density
of created particles, we proceed as in the Klein-Gordon case using the
formulas (18-24) with ${k=i\lambda}$ and
${\mu={{{\gamma}_{l}=i}}}\tilde{\gamma_{l}}$.

Then, the Bogoliubov coefficients $A$ and $B$ verify

$\left|\frac{B}{A}\right|=\left|\frac{\Gamma(1-i(\tilde{\gamma_{l}}+\eta))}{\Gamma(1+i(\tilde{\gamma_{l}}-\eta))}e^{i\pi\left(i\tilde{\gamma_{l}}-\frac{1}{2}\right)}\right|,$
(71)

and the probability of created fermionic particles is

$P=\left|\frac{B}{A}\right|^{2}=\left|\frac{\Gamma(1-i(\tilde{\gamma_{l}}+\eta))}{\Gamma(1+i(\tilde{\gamma_{l}}-\eta))}\right|^{2}e^{-2\pi\tilde{\gamma_{l}}},$
(72)

where

$\tilde{\gamma_{l}}=\sqrt{V_{0}^{2}-\left[{j+\left|m\right|\left({\frac{1}{\alpha}-1}\right)+\frac{1}{2}}\right]^{2}}>0,$
(73)

is the condition for pair production of spin-1/2 particle which depends on
${V_{0}}={qQ_{s}}$, the cosmic string parameter $\alpha$ and the quantum
numbers $j$ and $m$.

Using the fermionic condition $\left|A\right|^{2}+\left|B\right|^{2}=1$ and
the following property of the gamma function

${\left|{\Gamma(1+ix)}\right|^{2}}=\frac{\pi x}{\sinh(\pi x)},$ (74)

the expression of the probability is

$P=\frac{(\tilde{\gamma_{l}}+\eta)}{(\tilde{\gamma_{l}}-\eta)}\frac{\sinh\left[\pi(\tilde{\gamma_{l}}-\eta)\right]}{\sinh\left[\pi(\tilde{\gamma_{l}}+\eta)\right]}e^{-2\pi\tilde{\gamma_{l}}}.$
(75)

Then, the number density of the created fermionic particles is calculated as

$\bar{n}=\frac{1}{P^{-1}+1}=\frac{(\tilde{\gamma_{l}}+\eta)\sinh\left[\pi(\tilde{\gamma_{l}}-\eta)\right]}{(\tilde{\gamma_{l}}-\eta)\sinh\left[\pi(\tilde{\gamma_{l}}+\eta)\right]e^{2\pi\tilde{\gamma_{l}}}+(\tilde{\gamma_{l}}+\eta)\sinh\left[\pi(\tilde{\gamma_{l}}-\eta)\right]}.$
(76)

At a large frequencies $(E\gg M)$, the probability (75) reduces to

$P=\frac{(\tilde{\gamma_{l}}+V_{0})}{(\tilde{\gamma_{l}}-V_{0})}\frac{\sinh\left[\pi(\tilde{\gamma_{l}}-V_{0})\right]}{\sinh\left[\pi(\tilde{\gamma_{l}}+V_{0})\right]}e^{-2\pi\tilde{\gamma_{l}}}.$
(77)

and the number of created particle (76) can be written as

$\bar{n}=\frac{(\tilde{\gamma_{l}}+V_{0})\sinh\left[\pi(\tilde{\gamma_{l}}-V_{0})\right]}{(\tilde{\gamma_{l}}-V_{0})\sinh\left[\pi(\tilde{\gamma_{l}}+V_{0})\right]e^{2\pi\tilde{\gamma_{l}}}+(\tilde{\gamma_{l}}+V_{0})\sinh\left[\pi(\tilde{\gamma_{l}}-V_{0})\right]}.$
(78)

which is not a thermal distribution.

For the case when $\left|\tilde{\gamma_{l}}\pm V_{0}\right|\gg 1$, the number
density takes the form of Fermi-Dirac distribution

$\bar{n}=\frac{1}{e^{2\pi(\tilde{\gamma}_{l}-V_{0})}+1}.$ (79)

In the absence of the cosmic string $(\alpha=1)$, or for $m=0,$ the
probability (77) and the number density (78) reduce to

$P(\alpha=1)=\frac{(\tilde{\gamma}_{1}+V_{0})}{(\tilde{\gamma}_{1}-V_{0})}\frac{\sinh\left[\pi(\tilde{\gamma}_{1}-V_{0})\right]}{\sinh\left[\pi(\tilde{\gamma}_{1}+V_{0})\right]}e^{-2\pi\tilde{\gamma}_{1}},$
(80)
$\bar{n}(\alpha=1)=\frac{(\tilde{\gamma}_{1}+V_{0})\sinh\left[\pi(\tilde{\gamma}_{1}-V_{0})\right]}{(\tilde{\gamma}_{1}-V_{0})\sinh\left[\pi(\tilde{\gamma}_{1}+V_{0})\right]e^{2\pi\tilde{\gamma}_{1}}+(\tilde{\gamma}_{1}+V_{0})\sinh\left[\pi(\tilde{\gamma}_{1}-V_{0})\right]},$
(81)

where
$\tilde{\gamma}_{1}\equiv\tilde{\gamma}_{l}(\alpha=1)=\sqrt{V_{0}^{2}-{{\left({j+\frac{1}{2}}\right)}^{2}}}$.

###  3.2 Discussion of results

$\bullet$ First, let us discuss the consequences of the condition
$(\ref{cond2})$ for pair production of spin-1/2 fermion. It can be written as

$\left|qQ_{s}\right|>\left[j+\left|m\right|\left(\frac{1}{\alpha}-1\right)+\frac{1}{2}\right].$
(82)

In the absence of the cosmic string ($\alpha=1)$, or for $m=0$, the above
condition is reduced to

$\left|qQ_{s}\right|>\left(j+\frac{1}{2}\right)=\left(l+1\right).$ (83)

For an electron in a hydrogen-like atom, $Q_{s}=Ze$ and $q=-e$, the condition
(82) can be expressed for the nucleus charge $Z$ as

$Z>Z_{cr}\equiv\frac{1}{e^{2}}\left[j+\left|m\right|(\frac{1}{\alpha}-1)+\frac{1}{2}\right].$
(84)

For the particular values of the quantum numbers $(l=0,m=0,s=1/2,j=1/2)$, the
pair production of fermionic particles is possible for
$Z>Z_{cr}=\frac{1}{e^{2}}=137$ (in units $\hbar=c=1$, $e^{2}\simeq 1/137$),
which is a well-known result in the absence of the cosmic string [22].

In table 2, for the first values of the quantum numbers $(l,m,s=1/2,j=l+1/2),$
we give the critical values $Z_{cr}$ as function of the string parameter
$\alpha$ and an estimation of the effect of the GUT cosmic string on the
values and the rate of $Z_{cr}^{GUT}$.

Table 2: $Z_{cr}$ in terms of $\alpha$, values and the rate of $Z_{cr}^{GUT}$ for fermion particle $l$ |  $m$ |  $j=l+\frac{1}{2}$ |  $Z_{cr}$ |  $Z_{cr}^{GUT}$ |  $\frac{Z_{cr}^{GUT}-Z_{cr}(\alpha=1)}{Z_{cr}(\alpha=1)}$
---|---|---|---|---|---
$0$ |  $0$ |  $\frac{1}{2}$ |  $137$ |  |
$1$ |  $0$ |  $\frac{3}{2}$ |  $137\times 2=274$ |  |
|  $\pm 1$ |  $\frac{3}{2}$ |  $137\times\left(\frac{1}{\alpha}+1\right)$ |  $274,000137$ |  $5,000005\times 10^{-5}\%$
|  $0$ |  $\frac{5}{2}$ |  $137\times 3=411$ |  |
$2$ |  $\pm 1$ |  $\frac{5}{2}$ |  $137\times\left(\frac{1}{\alpha}+2\right)$ |  $411,000137$ |  $3,333333\times 10^{-5}\%$
|  $\pm 2$ |  $\frac{5}{2}$ |  $137\times\left(\frac{2}{\alpha}+1\right)$ |  $411,000274$ |  $6,666666\times 10^{-5}\%$

We note that for the sub-states with $m=0$, the critical value $Z_{cr}$
increases with $l$, depends on the spin value $(s=1/2)$ and is independent of
the cosmic string parameter $\alpha$ (Note that these sub-states are
equivalent to the case $\alpha=1$). For the sub-states for which $m\neq 0$,
$Z_{cr}$ increases with $l$, depends linearly on the spin value $(s=1/2)$ and
inversely on the cosmic string parameter $\alpha$. For the GUT cosmic string,
the obtained critical values $Z_{cr}^{GUT}$ do not exist for the moment, the
rate of $Z_{cr}^{GUT}$ is about $10^{-5}\%$ as compared with the case
$\alpha=1$ (without cosmic string) and the production of spin-1/2 particles is
possible if the Coulomb potential nucleus charge $Z\geq 275$.

$\bullet$ Secondly, in Figure 2 the number density curves have been plotted
for fixed values of the quantum numbers $(l=1,m=1,s=1/2,j=3/2)$ and where the
condition $(\ref{cond2})$ for pair production of spin-1/2 particles is reduced
to $V_{0}>(1+\frac{1}{\alpha})$ which should be always fullfilled. Plot (a)
display, in three (3D) dimensions, the number density $(\ref{D2})$ in terms of
$V_{0}=qQ_{s}$ and the cosmic string parameter $\alpha$. Plot (b) display, in
two (2D) dimensions, the number density $(\ref{D2})$ in terms of
$V_{0}=qQ_{s}$ for different values of the cosmic string parameter $\alpha$.

|
---|---
(a) | (b)
Figure 2: Plot (a) is the 3D number density (78) in terms of $V_{0}\in[0,+10]$
and $\alpha\in]0,1]$. Plot (b) is the 2D number density (78) in terms of
$V_{0}\in[0,5]$ for the different values of $\alpha=0.3,0.5,0.8$ and $1$
(without cosmic string) .

In order to study the shape of the curves for large values of $V_{0}$, the
number density (78) has been plotted in figure 3 in the short interval of
$V_{0}\in[50.0,50.6]$ for different values of $\alpha$. From which we note
that the number density $\bar{n}$ decreases when the cosmic string parameter
$\alpha$ increases (i.e. the linear mass density $\mu$ of the string
decreases) where its smallest values correspond to the case $\alpha=1$ (in the
absence of the cosmic string). Therefore, we deduce that the presence of the
cosmic string improves the number density of fermion particles compared to the
case without the cosmic string ($\alpha=1$).

Figure 3: Number density (78) in terms of $V_{0}\in[50.0,50.6]$ for different
values of $\alpha=0.3,0.5,0.8$ and $1$ (without cosmic string) .

##  4 Conclusion

In this paper, we have investigated the influence of a cosmic string on the
pair production rate induced by the Coulomb potential of an external electric
field. The solutions of the radial Klein-Gordon and Dirac equations are given
in terms of Whittaker functions, the probability and the number density of
created particles have been calculated.

For the spin-$0$ boson case, the creation of particles holds for
$\left|qQ_{s}\right|>{{\left[l+\left|m\right|\left({\frac{1}{\alpha}-1}\right){+\frac{1}{2}}\right]}}$,
the number density decreases when the cosmic string parameter $\alpha$
increases in the interval $]0,1]$ and its smallest values are reached for
$\alpha=1$. Thus, the presence of the cosmic string improves the number
density of created spinless particles compared to the case without the cosmic
string ($\alpha=1$). For the particular values of the quantum numbers
$(l=0,m=0)$, the pair production of scalar particles is possible if the
Coulomb potential nucleus charge $Z>\frac{137}{2}$.

For the spin-$1/2$ fermion case, the creation of particles holds for
$\left|qQ_{s}\right|>\left[{j+\left|m\right|\left({\frac{1}{\alpha}-1}\right)+\frac{1}{2}}\right]$,
the number density decreases with the increasing of the cosmic string
parameter $\alpha$ in the interval $]0,1]$ and its smallest values are reached
for $\alpha=1$. Thus, the presence of the cosmic string improves also the
number density of created spin-1/2 particles compared to the case without the
cosmic string ($\alpha=1$). For the particular values of the quantum numbers
$(l=0,m=0,s=1/2,j=1/2)$, the pair production of fermionic particles is
possible if the Coulomb potential nucleus charge $Z>137$.

As a result for the GUT cosmic string, the pair production for spinless
particles is possible if the Coulomb potential nucleus charge $Z\geq 206$, and
for spin-1/2 particles if $Z\geq 275$.

In limiting case of the Minkowski spacetime ($\alpha=1$), as expected we
retrieve the expressions of the number density of the scalar and fermionic
particles created by the Coulomb potential in the absence of the cosmic
string. Finally, we note that the results for the sub-states for which the
magnetic quantum number $m=0$ are independent of the cosmic string parameter
$\alpha$, i.e. these sub-states are insensitive to the presence of the cosmic
string.

## References

  * [1] N. D. Birrell, P. C. W. Davies, Quantum Fields in Curved Space, Cambridge University Press, Cambridge (1982).
  * [2] S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975). https://doi.org/10.1007/BF02345020.
  * [3] C. Bernard, A. Duncan, Regularization and renormalization of quantum field theory in curved space-time, Ann. Phys. 107, 201 (1977). https://doi.org/10.1016/0003-4916(77)90210-X.
  * [4] A. S. Lapedes, Euclidean quantum field theory and the Hawking effect, Phys. Rev. D 17, 2556 (1978). https://doi.org/10.1103/PhysRevD.17.2556.
  * [5] I. L. Buchbinder, S. D. Odintsov, Creation of particles and the effective lagrangian in the quasieuclidean model of the universe with an electromagnetic field, Sov. Phys. J. 25, 385 (1982). https://doi.org/10.1007/BF00891754.
  * [6] E. Mottola, Particle creation in de Sitter space, Phys. Rev. D 31, 754 (1985). https://doi.org/10.1103/PhysRevD.31.754.
  * [7] K.H. Lotze, Production of massive spin-1/2 particles in Robertson-Walker universes with external electromagnetic fields, Astrophys. Space Sci. 120, 191 (1986). https://doi.org/10.1007/BF00649935.
  * [8] J. Garriga, Pair production by an electric field in (1+1)-dimensional de Sitter space, Phys. Rev. D 49, 6343 (1994). https://doi.org/10.1103/PhysRevD.49.6343.
  * [9] S.P. Gavrilov, D.M. Gitman, S. D. Odintsov, Quantum scalar field in FRW universe with constant electromagnetic background, Int. J. Mod. Phys. A 12, 4837 (1997).https://doi.org/10.1142/S0217751X97002589.
  * [10] V. Villalba, W. Greiner, Creation of Dirac particles in the presence of a constant electric field in an anisotropic Bianchi I universe, Mod. Phys. Lett. A 17, 1883 (2002). https://doi.org/10.1142/S0217732302008289.
  * [11] J. E. B. Mendy, Scalar and spin-1/2 particle creation in gravitational and constant electric field backgrounds, J. Math. Phys.44, 662 (2003). http://dx.doi.org/10.1063/1.1500793.
  * [12] M. Falek, M. Merad, Duffin-Kemmer-Petiau equation in Robertson-Walker space-time, Open Physics 8, 408 (2010). https://doi.org/10.2478/s11534-009-0112-y.
  * [13] S. Haouat, R. Chekireb, Schwinger effect in a Robertson-Walker space-time, Int. J. Theo. Phys. 51, 1704 (2012). https://doi.org/10.1007/s10773-011-1048-8.
  * [14] S. Haouat, R. Chekireb, Effect of electromagnetic fields on the creation of scalar particles in a flat Robertson-Walker space-time, Eur. Phys. J. C 72, 2034 (2012).https://doi.org/10.1140/epjc/s10052-012-2034-x.
  * [15] S. Haouat, R. Chekireb, Effect of the electric field on the creation of fermions in de-Sitter space-time, 2015, https://doi.org/10.48550/arXiv.1504.08201.
  * [16] K. Sogut, A. Havare, On the scalar particle creation by electromagnetic fields in Robertson-Walker spacetime, Nucl. Phys. B 901, 76 (2015). https://doi.org/10.1016/j.nuclphysb.2015.10.005.
  * [17] K. Rajeev, S. Chakraborty, T. Padmanabhan, Generalized Schwinger effect and particle production in an expanding universe, Phys. Rev. D 100, 045019 (2019). https://doi.org/10.1103/PhysRevD.100.045019.
  * [18] B. Hamil, M. Merad, T. Birkandan, Pair creation in curved Snyder space, Int. J. Mod. Phys. A 35, 2050014 (2020). https://doi.org/10.1142/S0217751X20500141.
  * [19] L.O. Pimentel, F. Pineda, Particle creation in some LRS Bianchi I models, Gen Relativ Gravit 53, 62 (2021). https://doi.org/10.1007/s10714-021-02828-w.
  * [20] W. Pieper, W. Greiner, Interior electron shells in superheavy nuclei, Z. Physik 218, 327 (1969). https://doi.org/10.1007/BF01670014.
  * [21] S.S. Gershtein, Y.B. Zeldovich, Positron Production during the Mutual Approach of Heavy Nuclei and the Polarization of the Vacuum, Sov. Phys. JETP 30, 358 (1970). http://jetp.ras.ru/cgi-bin/e/index/e/30/2/p358?a=list.
  * [22] V.S. Popov, Positron production in a Coulomb Field with Z $>$ 137, Sov. Phys. JETP, 32, 526 (1971). http://www.jetp.ras.ru/cgi-bin/dn/e_032_03_0526.pdf.
  * [23] J. Rafelski, L.P. Fulcher, A. Klein, Fermions and bosons interacting with arbitrarily strong external fields, Phys. Rep. 38,227 (1978). https://doi.org/10.1016/0370-1573(78)90116-3.
  * [24] M. Soffel, B. Muller, W. Greiner, Stability and decay of the Dirac vacuum in external gauge fields, Phys. Rep. 85, 51 (1982). https://doi.org/10.1016/0370-1573(82)90129-6.
  * [25] M.A. B$\breve{a}$loi, The production of particles with well determined angular momentum in external Coulomb field on de Sitter expanding universe, Nucl. Phys. B 980, 115796 (2022). https://doi.org/10.1016/j.nuclphysb.2022.115796.
  * [26] V.S. Popov, Critical charge in quantum electrodynamics, Phys. Atom. Nuclei 64, 367 (2001). https://doi.org/10.1134/1.1358463.
  * [27] J. Rafelski, J. Kirsch, B. Müller, J. Reinhardt, W. Greiner, Probing QED Vacuum with Heavy Ions. In: S. Schramm, M. Schäfer (eds), New Horizons in Fundamental Physics. FIAS Interdisciplinary Science Series. Springer (2017), pp 211-251. https://doi.org/10.1007/978-3-319-44165-8_17.
  * [28] D.N. Voskresensky, Electron-Positron Vacuum Instability in Strong Electric Fields. Relativistic Semiclassical Approach, Universe 7, 104 (2021). https://doi.org/10.3390/universe7040104.
  * [29] J. Pullin, E. Verdaguer, Gravitational particle production in the formation of global monopoles and domain walls, Phys. Lett. B246, 371 (1990). https://doi.org/10.1016/0370-2693(90)90616-E.
  * [30] I. H. Duru, Spontaneous pair production in a magnetic monopole field, J. Phys. A: Math. Gen. 28, 5883 (1995). https://doi.org/10.1088/0305-4470/28/20/017.
  * [31] L. Parker, Gravitational particle production in the formation of cosmic strings, Phys. Rev. Lett. 59, 1369 (1987). https://doi.org/10.1103/PhysRevLett.59.1369.
  * [32] V. Sahni, Particle creation by cosmic strings, Mod. Phys. Lett. A 3, 1425 (1988). https://doi.org/10.1142/S0217732388001719.
  * [33] G. Mendell, W. A. Hiscock, Gravitational particle production during cosmic-string formation in the sudden approximation, Phys. Rev. D 40, 282 (1989). https://doi.org/10.1103/PhysRevD.40.282.
  * [34] D. D. Harari, V. D. Skarzhinsky, Pair production in the gravitational field of a cosmic string, Phys. Lett. B 240, 322 (1990). https://doi.org/10.1016/0370-2693(90)91106-L.
  * [35] J. Garriga, D. Harari, E. Verdaguer, Gravitational particle production by cosmic strings, Nucl. Phys. B 339, 560 (1990). https://doi.org/10.1016/0550-3213(90)90362-H.
  * [36] B. Jensen, Production of quantum particles on cones, Phys. Scr. 45, 548 (1992). https://doi.org/10.1088/0031-8949/45/6/002.
  * [37] I. Brevik, T. Toverud, Electromagnetic energy production in the formation of a superconducting cosmic string, Phys. Rev. D 51,691 (1995). https://doi.org/10.1103/PhysRevD.51.691.
  * [38] V. A. De Lorenci, R. De Paola, N. F. Svaiter, Gravitational Particle Production in Spinning Cosmic String Spacetimes, Nuovo Cim. B 113, 1331 (1998). https://arxiv.org/abs/gr-qc/9705049.
  * [39] V. A. De Lorenci, R. De Paola, N. F. Svaiter, From spinning to non-spinning cosmic string spacetime, Class. Quantum Grav. 16, 3047 (1999). https://doi.org/10.1088/0264-9381/16/10/302.
  * [40] V. B. Bezerra, V. M. Mostepanenko, R. M. Teixeira Filho, Particle creation in the chiral cosmic string spacetime, Int. J. Mod. Phys. D 11, 437 (2002). https://doi.org/10.1142/S0218271802001718.
  * [41] T. W. B. Kibble, Topology of cosmic domains and strings, J. Phys. A 9, 1387 (1976). https://doi.org/10.1088/0305-4470/9/8/029.
  * [42] Y. B. Zeldovich, Cosmological fluctuations produced near a singularity, Mon. Not. R. Astron. Soc. 192, 663 (1980). https://doi.org/10.1093/mnras/192.4.663.
  * [43] A. Vilenkin, E. P. Shellard, Cosmic Strings and Other Topological Defects, Cambridge University Press, Cambridge (2000).
  * [44] J. R. Gott III, Gravitational lensing effects of vacuum strings - Exact solutions, Astrophys. J. 288, 422 (1985). https://doi.org/10.1086/162808.
  * [45] B. Linet, A vortex-line model for infinite straight cosmic strings, Phys. Lett. A 124, 240 (1987). https://doi.org/10.1016/0375-9601(87)90629-3.
  * [46] B. Linet, Force on a charge in the space-time of a cosmic string, Phys. Rev. D 33, 1833 (1986).https://doi.org/10.1103/PhysRevD.33.1833.
  * [47] E. R. Bezerra de Mello, V. B. Bezerra, C. Furtado and F. Moraes, Self-forces on electric and magnetic linear sources in the space-time of a cosmic string, Phys. Rev. D 51, 7140 (1995). https://doi.org/10.1103/PhysRevD.51.7140.
  * [48] A. G. Smith, Gravitational effects of an infinite straight cosmic string on classical and quantum fields: Self-forces and vacuum fluctuations, in G. W. Gibbons, S. W. Hawking and T. Vachaspati (eds.): The Formation and Evolution of Cosmic Strings, Cambridge University Press, Cambridge (1990), pp. 263-292.
  * [49] J. S. Dowker, Casimir effect around a cone, Phys. Rev. D 36, 3095 (1987).https://doi.org/10.1103/PhysRevD.36.3095.
  * [50] J. S. Dowker, A gravitational Aharonov-Bohm effect, Nuovo Cimento B 52, 129 (1967). https://doi.org/10.1007/BF02710657.
  * [51] Planck Collaboration, P. Ade et al., Planck 2013 results. XXV. Searches for cosmic strings and other topological defects, (2013). Astronomy & Astrophysics 571, A25 (2014).https://doi.org/10.1051/0004-6361/201321621.
  * [52] M. Sakellariadou, Gravitational waves emitted from infinite strings, Phys. Rev. D 42, 354 (1990), https://doi.org/10.1103/PhysRevD.42.354 . Erratum Phys. Rev. D 43, 4150 (1991), https://doi.org/10.1103/PhysRevD.43.4150.2.
  * [53] T. Damour, A. Vilenkin, Gravitational Wave Bursts from Cosmic Strings, Phys. Rev. Lett. 85, 3761 (2000). https://doi.org/10.1103/PhysRevLett.85.3761.
  * [54] S. Blasi, V. Brdar, K. Schmitz, Has NANOGrav Found First Evidence for Cosmic Strings?, Phys. Rev. Lett. 126, 041305 (2021). https://doi.org/10.1103/PhysRevLett.126.041305.
  * [55] G. Boileau, A.C. Jenkins, M. Sakellariadou, R. Meyer, N. Christensen, Ability of LISA to detect a gravitational-wave background of cosmological origin: The cosmic string case, Phys. Rev. D 105, 023510 (2022). https://doi.org/10.1103/PhysRevD.105.023510.
  * [56] P. Battacharjee, G. Sigl, Origin and propagation of extremely high-energy cosmic rays, Phys. Rep. 327, 109 (2000). https://doi.org/10.1016/S0370-1573(99)00101-5.
  * [57] Y. Yang, J. Jing, Z. Tian, Probing cosmic string spacetime through parameter estimation, Eur. Phys. J. C 82, 688 (2022). https://doi.org/10.1140/epjc/s10052-022-10628-y.
  * [58] M. R. Anderson, F. Moraes, The mathematical theory of cosmic strings, IOP Publishing, Bristol (2003), Chapter 7.
  * [59] D. D. Sokoloff, A. A. Starobinskii, On the structure of curvature tensor on conical singularities, Dokl. Akad. Nauk SSSR, 234, 1043 (1977) [Sov. Phys. Dokl. 22, 312 (1977)]. http://mi.mathnet.ru/eng/dan/v234/i5/p1043.
  * [60] M. O. Katanaev, I. V. Volovich, Theory of defects in solids and three-dimensional gravity, Annals of Physics 216, 1 (1992).https://doi.org/10.1016/0003-4916(52)90040-7.
  * [61] C. Furtado, F. Moraes, On the binding of electrons and holes to disclinations, Physics Letters A 188, 394 (1994). https://doi.org/10.1016/0375-9601(94)90482-0.
  * [62] B.L. Voronov, D.M. Gitman, I.V. Tyutin, The Dirac Hamiltonian with a superstrong Coulomb field, Theor. Math. Phys. 150, 34 (2007). https://doi.org/10.1007/s11232-007-0004-5.
  * [63] M.H. Al-Hashimi, U.-J. Wiese, Self-adjoint extensions for confined electrons: From a particle in a spherical cavity to the hydrogen atom in a sphere and on a cone, Ann. Phys. 327, 2742 (2012). https://doi.org/10.1016/j.aop.2012.06.006.
  * [64] F.A. Cruz Neto, F.M. da Silva, L.C.N. Santos, L.B. Castro, Scalar bosons with Coulomb potentials in a cosmic string background: scattering and bound states, Eur. Phys. J. Plus, 135: 25 (2020). https://doi.org/10.1140/epjp/s13360-019-00062-7.
  * [65] F.W.J. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge (2010). pp 334-335.
  * [66] S.P. Kim, D.N. Page, Remarks on Schwinger Pair Production by Charged Black Holes, Nuovo Cim. B 120, 1193 (2005). https://doi.org/10.1393/ncb/i2005-10148-6, https://arxiv.org/abs/gr-qc/0401057.
  * [67] G. de A. Marques, V.B. Bezerra, Hydrogen atom in the gravitational fields of topological defects, Phys. Rev. D 66, 105011 (2002). https://doi.org/10.1103/PhysRevD.66.105011.
  * [68] M. Hosseinpour, H. Hassanabadi, Scattering states of Dirac equation in the presence of cosmic string for Coulomb interaction, Int. J. Mod. Phys. A 30, 1550124 (2015). https://doi.org/10.1142/S0217751X15501249.
  * [69] G.A. Marques, V.B. Bezerra, S.G. Fernandes, Exact solution of the Dirac equation for a Coulomb and scalar potentials in the gravitational field of a cosmic string, Phys. Lett. A 341, 39 (2005).https://doi.org/10.1016/j.physleta.2005.04.031.
  * [70] W. Greiner, B. Mulller, J. Rafelski, Quantum Electrodynamics of Strong Fields, Springer, Berlin (1985). pp 71-73 and 86-89.


00footnotetext: Corresponding author: Cédric Heuchenne, Email:
<EMAIL_ADDRESS>
# Anomaly Detection for Compositional Data using VSI MEWMA control chart
111This paper was submitted to “10th IFAC Conference on Manufacturing
Modelling, Management and Control” on 14/02/2022.

Thi Thuy Van Nguyen  HEC Liège - Management School of the University of Liège,
Belgium  International Research Institute for Artificial Intelligence and Data
Science, Dong A University, Danang, Vietnam  Cédric Heuchenne  International
Research Institute for Artificial Intelligence and Data Science, Dong A
University, Danang, Vietnam  Kim Phuc Tran  Univ.Lille, ENSAIT, GEMTEX, France

###### Abstract

In recent years, the monitoring of compositional data using control charts has
been investigated in the Statistical Process Control field. In this study, we
will design a Phase II Multivariate Exponentially Weighted Moving Average
(MEWMA) control chart with variable sampling intervals to monitor
compositional data based on isometric log-ratio transformation. The Average
Time to Signal will be computed based on the Markov chain approach to
investigate the performance of proposed chart. We also propose an optimal
procedure to obtain the optimal control limit, smoothing constant, and out-of-
control Average Time to Signal for different shift sizes and short sampling
intervals. The performance of proposed chart in comparison with the standard
MEWMA chart for monitoring compositional data is also provided. Finally, we
end the paper with a conclusion and some recommendations for future research.

Keyword: Compositional data, Markov chain, VSI-MEWMA, control chart, Data
Science

##  1 INTRODUCTION

In the global competitive economy nowadays, an extremely important task for
manufacturing companies is to not only offer high-quality products but also
reduce waste and increase efficiency in the production processes. The
development of advanced technologies in Artificial Intelligence and Data
Science fields makes this task more possible, but also more challenging when
competing with other companies. Therefore, making a smart decision in
manufacturing becomes a crucial task in any production company. In this
context, early detection of abnormal products as well as assignable causes to
fix the production system as soon as possible is an indispensable part, and
Statistical Control Process (SPC) is one of the most effective methods to
accomplish this task. Through control charts, SPC helps manufacturing
companies monitor product qualities and discover the defects in the production
lines. In SPC literature, many studies have been done to design a variety of
control charts for monitoring different types of process data, see 1. Among
these different data, compositional data (CoDa) are vectors whose components
are strictly positive and they often present the proportions, percentages, or
frequencies of some whole. Their applications can be found in many domains
such as chemical research, econometrics, and the food industry, see 2. Due to
the constraint on the sum of components of the CoDa vector, it can not be
treated as normal data.

In SPC literature, the studies in the control charts for monitoring CoDa data
are still limited. In 3, a chi-squared type control chart for monitoring CoDa
data was proposed. Recently, the authors in 5 investigated a $\displaystyle
T^{2}$ control chart for monitoring CoDa with $\displaystyle p=3$ and then 6
extended the work in 5 for individual observations case. Two methods for
interpretations of out-of-control signal of individual $\displaystyle
T_{C}^{2}$ control chart in case $\displaystyle p>3$ was proposed in 4. In 9,
the authors proposed a MEWMA-CoDa chart for monitoring CoDa with arbitrary
components. This type of control chart was shown to be effective in detecting
small to moderate process shift sizes and outperforming its competitor
($\displaystyle T^{2}$-CoDa chart). The influence of measurement errors on the
performance of $\displaystyle T^{2}$, MEWMA chart for monitoring CoDa were
investigated in 8 and 7, respectively. In these control charts, the authors
suggested using an isometric log-ration (ilr) transformation to transform CoDa
to vector in $\displaystyle\mathbb{R}^{p-1}$ space to handle the constraint of
CoDa and the average run length (ARL) to evaluate the performance of proposed
control charts.

In the control charts mentioned above, the fixed sampling interval (FSI) was
supposed to use. Recently, the design of control charts tends to use variable
sampling intervals (VSI). In these charts, the sampling interval between two
consecutive samples is allowed to vary due to the value of the current control
statistic. Many studies on the VSI control chart have been published so far,
see, for example, 13, and 14, among many others. As we know, the VSI MEWMA
control chart for monitoring CoDa has not been used. Consequently, in this
study, we propose a VSI MEWMA type control chart for monitoring CoDa, namely
VSI MEWMA-CoDa, with arbitrary components based on ilr transformation. The
modification of the Markov chain approach proposed by 12 will be used to
compute average time to signal (ATS), criteria to access the performance of
VSI control charts.

The rest of this paper is organized as follows: In Section 2, the modeling of
CoDa and the suggested isometric log-ratio transformation are introduced; the
VSI MEWMA-CoDa control chart together with the Markov chain approach and
optimization procedure to find the optimal parameters are given in Section 3;
in Section 4, the performance of the VSI MEWMA-CoDa chart with different
scenarios are provided; conclusions and some recommendations for further
researches are given in Section 5.

##  2 Modeling of Compositional Data

By definition, a row vector,
$\displaystyle\textbf{x}=(x_{1},x_{2},\ldots,x_{p})$, is a p-part composition
when its components are strictly positive and they carry only relative
information, see 2, 11. The relative information here refers only to the
proportions between components of the composition, regardless of their
numerical values. The sum of the components of x,
$\displaystyle\sum_{i=1}^{p}x_{i}$, is a constant $\displaystyle\kappa$. For
instance, $\displaystyle\kappa=100$ refers to measurements in percentage while
$\displaystyle\kappa=1$ means that the measurements are proportions. Each
composition can be considered as an equivalent class made of proportional
factors since the ratios between its components do not change when multiplying
it by a positive constant. In this case, if
$\displaystyle\textbf{x},\textbf{y}$ are compositions and
$\displaystyle\textbf{x}=\lambda\textbf{y}$ for some constants
$\displaystyle\lambda$, we say that $\displaystyle\textbf{x},\textbf{y}$ are
compositionally equivalent. To check the equivalency of the two compositions,
we can use the closure function $\displaystyle\mathcal{C}(\textbf{x})$,
defined as

$\mathcal{C}(\textbf{x})=\left(\frac{\kappa\cdot x_{1}}{\sum_{i=1}^{p}x_{i}},\
\frac{\kappa\cdot x_{2}}{\sum_{i=1}^{p}x_{i}},\ldots,\frac{\kappa\cdot
x_{p}}{\sum_{i=1}^{p}x_{i}}\right)$

where $\displaystyle\kappa>0$ is a fixed constant; in this definition, two
p-part compositions $\displaystyle\textbf{x},\textbf{y}$ are compositionally
equivalent if $\displaystyle\mathcal{C}(\textbf{x})=\mathcal{C}(\textbf{y})$.
The sample space of CoDa is the simplex,

$\begin{split}\mathcal{S}^{p}=\Big{\\{}\textbf{x}=(x_{1},x_{2},\ldots,x_{p})\enskip|\enskip
x_{i}>0,i=1,\ldots,D;\\\ \sum_{i=1}^{p}x_{i}=\kappa\Big{\\}}\end{split}$

In $\displaystyle\mathbb{R}^{p}$ space, we can use Euclidean geometry to add
vectors or multiply vectors by scalar to obtain their properties or compute
their distance. But, due to special structure of CoDa vectors in
$\displaystyle\mathcal{S}^{p}$, this geometry can not be applied directly. The
authors of 2 introduced the Aitchison geometry, with two operations required
for a vector space structure on $\displaystyle\mathcal{S}^{p}$: Perturbation
and powering operators. The perturbation $\displaystyle\oplus$ of
$\displaystyle\textbf{x}\in\mathcal{S}^{p}$ by
$\displaystyle\textbf{y}\in\mathcal{S}^{p}$ (equivalent to the addition in
$\displaystyle\mathbb{R}^{p}$) is defined by

$\displaystyle\textbf{x}\oplus\textbf{y}=\mathcal{C}(x_{1}y_{1},\ldots,x_{p}y_{p})\in\mathcal{S}^{p}$

and the powering $\displaystyle\odot$ of
$\displaystyle\textbf{x}\in\mathcal{S}^{p}$ by a constant
$\displaystyle\alpha\in\mathbb{R}$ (equivalent to the multiplication by a
scalar operation in the $\displaystyle\mathbb{R}^{p}$) is defined by

$\displaystyle\alpha\odot\textbf{x}=\mathcal{C}(x_{1}^{\alpha},\ldots,x_{p}^{\alpha})\in\mathcal{S}^{p}$

In practice, CoDa are often transformed to vectors in the Euclidean space to
remove its constraints. The center log-ratio (clr) transformation of vector
$\displaystyle\textbf{x}\in\mathcal{S}^{p}$,
$\displaystyle\operatorname{clr}(\textbf{x})$, is an isometry from
$\displaystyle\mathcal{S}^{p}$ to a subspace $\displaystyle
U\subset\mathbb{R}^{p}$, defined by

$\displaystyle\displaystyle\operatorname{clr}(\textbf{x})$
$\displaystyle\displaystyle=\left(\ln\frac{x_{1}}{g_{m}(\textbf{x})},\ln\frac{x_{2}}{g_{m}(\textbf{x})},\ldots,\ln\frac{x_{p}}{g_{m}(\textbf{x})}\right)$
$\displaystyle\displaystyle=(\xi_{1},\xi_{2},\ldots,\xi_{p})$

where

$g_{m}(\textbf{x})=\left(\prod_{i=1}^{p}x_{i}\right)^{\frac{1}{p}}=\exp\left(\frac{1}{p}\sum_{i=1}^{p}x_{i}\right)$

is the geometric mean of the composition and
$\displaystyle\sum_{i=1}^{p}\xi_{i}=0$. The inverse center log-ratio
$\displaystyle\operatorname{clr}^{-1}(\bm{\xi})$ recovering x from
$\displaystyle\bm{\xi}=(\xi_{1},\ldots,\xi_{p})$ is

$\operatorname{clr}^{-1}(\bm{\xi})=\mathcal{C}(\exp(\bm{\xi}))=\mathcal{C}(\exp(\xi_{1}),\exp(\xi_{2}),\ldots,\exp(\xi_{p})).$

Egozcue et al. 15 showed that the constraint in the component of
$\displaystyle\operatorname{clr}(\textbf{x})$ makes singular the
$\displaystyle\operatorname{clr}(\textbf{x})$ variance-covariance matrix for
random composition. To overcome this drawback, the authors of 15 proposed a
new transformation which is associated with an orthogonal basis in
$\displaystyle\mathcal{S}^{p}$, named isometric log-ratio (irl)
transformation. Let
$\displaystyle{\textbf{e}_{1},\textbf{e}_{2},\ldots,\textbf{e}_{p-1}}$ be an
orthonormal basis of $\displaystyle\mathcal{S}^{p}$. Any composition
$\displaystyle\textbf{x}\in\mathcal{S}^{p}$ can be expressed as

$\textbf{x}=\bigoplus_{i=1}^{p-1}x_{i}^{*}\odot\textbf{e}_{i},\enskip
x_{i}^{*}=\langle\textbf{x},\textbf{e}_{i}\rangle_{a}=\langle\operatorname{clr}(\textbf{x}),\operatorname{clr}(\textbf{e}_{i})\rangle$

where $\displaystyle\langle,\rangle_{a}$ denotes the Aitchison inner product.
Thus, the ilr transformation of $\displaystyle\textbf{x}\in\mathcal{S}^{p}$ is
$\displaystyle\operatorname{ilr}(\textbf{x})=\textbf{x}^{*}=(x_{1}^{*},x_{2}^{*},\ldots,x_{p-1}^{*})$.
Let B be a $\displaystyle(p-1,p)$ matrix whose $\displaystyle i^{th}$ row is
$\displaystyle\operatorname{clr}(\textbf{e}_{i}),i=1,\ldots,p-1$. This matrix
is known as a contrast matrix associated with the orthonormal basis
$\displaystyle{\textbf{e}_{1},\textbf{e}_{2},\ldots,\textbf{e}_{p-1}}$. The
ilr transformation $\displaystyle\textbf{x}^{*}$ of composition x can be
computed by

$\displaystyle\textbf{x}^{*}=\operatorname{ilr}(\textbf{x})=(x_{1}^{*},\ldots,x_{p-1}^{*})=\operatorname{clr}(\textbf{x})\cdot\textbf{B}^{\intercal}$

There are many candidates for an orthonormal basis in
$\displaystyle\mathcal{S}^{p}$. The authors of 16 proposed a sequential binary
partition to define an orthonormal basis. In this basis,
$\displaystyle\textbf{e}_{i}$ is defined to be
$\displaystyle\mathcal{C}(e_{i,1},\ldots,e_{i,j},\ldots,e_{i,p})$ where

$\displaystyle
e_{i,j}=\left\\{\begin{array}[]{ll}\exp{\left(\sqrt{\frac{1}{i(i+1)}}\right)}&\text{\quad
if \quad}j\leq i\\\ \exp{\left(-\sqrt{\frac{i}{i+1}}\right)}&\text{\quad if
\quad}j=i+1\\\ 1&\text{\quad otherwise \quad}\end{array}\right.$

The elements $\displaystyle e_{i,j}$ are called the _balancing elements_ of
this basis. Thus, if this type of orthonormal basis is chosen to transform x,
i.e.,
$\displaystyle\operatorname{ilr}(\textbf{x})=\operatorname{clr}(\textbf{x})\cdot\textbf{B}^{\intercal}$,
the coordinates $\displaystyle x^{*}_{i}$ are called _balances_ and can be
obtained by

$\displaystyle
x^{*}_{i}=\sqrt{\frac{i}{i+1}}\ln\left(\frac{\left(\prod_{j=1}^{i}x_{j}\right)^{\frac{1}{i}}}{x_{i+1}}\right).$

From its ilr coordinate $\displaystyle\textbf{x}^{*}$, x can be recovered by
using the inverse of ilr transformation:

$\displaystyle\operatorname{ilr}^{-1}(\mathbf{x}^{*})=\operatorname{clr}^{-1}(\mathbf{x}^{*}\mathbf{B})=\mathcal{C}(\exp(\mathbf{x}^{*}\mathbf{B})).$

Table 1 illustrates the application of ilr transformation in practice for the
case $\displaystyle p=4$. The first 4 columns present the components of 6
compositions in $\displaystyle\mathcal{S}^{4}$ and the remaining 3 columns
present their corresponding ilr coordinates in $\displaystyle\mathbb{R}^{3}$.
As can be seen, these $\displaystyle\operatorname{ilr}$ coordinates
$\displaystyle x_{i}^{*}$ are not constrained any longer. For more detail on
CoDa and its properties, see 11.

Table 1: An example of ilr transformation in $\displaystyle\mathcal{S}^{4}$ $\displaystyle x_{1}$ | $\displaystyle x_{2}$ | $\displaystyle x_{3}$ | $\displaystyle x_{4}$ | $\displaystyle x^{*}_{1}$ | $\displaystyle x^{*}_{2}$ | $\displaystyle x^{*}_{3}$
---|---|---|---|---|---|---
0.10 | 0.30 | 0.50 | 0.10 | -0.78 | -0.87 | 0.78
0.20 | 0.25 | 0.20 | 0.35 | -0.16 | 0.09 | -0.42
0.50 | 0.10 | 0.20 | 0.20 | 1.14 | 0.09 | 0.06
0.60 | 0.05 | 0.05 | 0.30 | 1.76 | 1.01 | -0.83
0.35 | 0.15 | 0.10 | 0.40 | 0.60 | 0.68 | -0.72
0.20 | 0.45 | 0.05 | 0.30 | -0.57 | 1.46 | -0.52

##  3 VSI Multivariate EWMA control chart for Compositional Data

###  3.1 VSI MEWMA-CoDa control chart

Let us suppose that, at each sampling period $\displaystyle i=1,2,\ldots$, a
sample of size $\displaystyle n$ independent $\displaystyle p$-part
composition observations
$\displaystyle\\{\mathbf{X}_{i,1},\ldots,\mathbf{X}_{i,n}\\}$,
$\displaystyle\mathbf{X}_{i,j}\in\mathcal{S}^{p}$, $\displaystyle
j=1,\ldots,n$ is collected, and suppose also that each
$\displaystyle\mathbf{X}_{i,j}$, $\displaystyle j=1,\ldots,n$, follows a
multivariate normal distribution $\displaystyle
N_{\mathcal{S}^{p}}(\bm{\mu},\bm{\Sigma})$ on the simplex
$\displaystyle\mathcal{S}^{p}$, where
$\displaystyle\bm{\mu}\in\mathcal{S}^{p}$ is the center of compositions and
$\displaystyle\bm{\Sigma}$ is their variance-covariance matrix. Assume that,
when the process is in-control, the composition center is
$\displaystyle\bm{\mu}_{0}$ and when the process is out-of-control, the
composition center is $\displaystyle\bm{\mu}_{1}$. The aim of this paper is to
design a variable sampling interval MEWMA control chart (denoted by VSI-MEWMA-
CoDa) to monitor the _center_ $\displaystyle\bm{\mu}$ of a $\displaystyle
p$-part compositional process. Since CoDa data has a constant constraint on
its components, the traditional VSI-MEWMA control chart may not perform well
on monitoring this type of data. In 9, instead of directly monitoring the
composition center $\displaystyle\bm{\mu}$, the authors proposed to monitor
the mean vector $\displaystyle\bm{\mu}^{*}=\operatorname{ilr}(\bm{\mu})$ using
a FSI MEWMA control chart for the sample mean coordinates vector
$\displaystyle\bar{\mathbf{X}}^{*}_{i}$. In this study, we will apply the idea
in 9 to investigate a VSI-MEWMA control chart for monitoring a compositional
process.

Let $\displaystyle\\{\mathbf{X}^{*}_{i,1},\ldots,\mathbf{X}^{*}_{i,n}\\}$ be
the corresponding $\displaystyle\operatorname{ilr}$ coordinates of
$\displaystyle\\{\mathbf{X}_{i,1},\ldots,\mathbf{X}_{i,n}\\}$, i.e.
$\displaystyle\mathbf{X}^{*}_{i,j}=\operatorname{ilr}(\mathbf{X}_{i,j})\in\mathbb{R}^{p-1}$.
Since $\displaystyle\mathbf{X}_{i,j}$ follows a multivariate normal
distribution $\displaystyle N_{\mathcal{S}^{p}}(\bm{\mu},\bm{\Sigma})$ on
$\displaystyle\mathcal{S}^{p}$, its corresponding
$\displaystyle\operatorname{ilr}$ coordinate
$\displaystyle\mathbf{X}^{*}_{i,j}$ follows a multivariate normal distribution
$\displaystyle N_{\mathbb{R}^{p-1}}(\bm{\mu}^{*},\bm{\Sigma}^{*})$ on
$\displaystyle\mathbb{R}^{p-1}$, where
$\displaystyle\bm{\mu}^{*}=\operatorname{ilr}(\bm{\mu})\in\mathbb{R}^{p-1}$ is
the mean vector, $\displaystyle\bm{\Sigma}^{*}$ is the
$\displaystyle(p-1,p-1)$ variance-covariance matrix of the ilr transformed
data. The values of parameters $\displaystyle\bm{\mu}^{*}$ and
$\displaystyle\bm{\Sigma}^{*}$ depend on the particular choice of matrix
$\displaystyle\mathbf{B}$ chosen in ilr transformation (see 11 and section 2).
Denote the ilr coordinates of in-control composition center
$\displaystyle\bm{\mu}_{0}$ and out-of-control composition center
$\displaystyle\bm{\mu}_{1}$ are $\displaystyle\bm{\mu}_{0}^{*}$ and
$\displaystyle\bm{\mu}_{1}^{*}$, respectively. The average of $\displaystyle
n$ independent $\displaystyle p$-part compositional observations is defined by

$\displaystyle\bar{\mathbf{X}}_{i}=\frac{1}{n}\odot(\mathbf{X}_{i,1}\oplus\cdots\oplus\mathbf{X}_{i,n})$

then its ilr coordinate $\displaystyle\bar{\mathbf{X}}^{*}_{i}$ is
$\displaystyle\bar{\mathbf{X}}^{*}_{i}=\operatorname{ilr}(\bar{\mathbf{X}}_{i})=\frac{1}{n}(\operatorname{ilr}(\mathbf{X}_{i,1})+\cdots+\operatorname{ilr}(\mathbf{X}_{i,n}))=\frac{1}{n}(\mathbf{X}^{*}_{i,1}+\cdots+\mathbf{X}^{*}_{i,n})\in\mathbb{R}^{p-1}$.
We first recall the FSI MEWMA-CoDa control chart proposed by the authors in 9
as follows. Let the MEWMA vector $\displaystyle\mathbf{W}_{i}$ be

$\displaystyle\mathbf{W}_{i}=r(\bar{\mathbf{X}}^{*}_{i}-\bm{\mu}_{0}^{*})+(1-r)\mathbf{W}_{i-1},\,i=1,2,\ldots$

where $\displaystyle\mathbf{Y}_{0}=\mathbf{0}$, $\displaystyle r\in(0,1]$ is a
fixed smoothing parameter. In FSI MEWMA-CoDa control chart, Tran et al. 9
suggested to monitor the statistic

$Q_{i}=\mathbf{W}_{i}^{\intercal}\bm{\Sigma}^{-1}_{W_{i}}\mathbf{W}_{i},\,i=1,2,\ldots$
(1)

where $\displaystyle\bm{\Sigma}_{W_{i}}$ is the variance-covariance matrix of
$\displaystyle\mathbf{W}_{i}$. In this work, the asymptotic form of the
variance-covariance matrix $\displaystyle\bm{\Sigma}_{W_{i}}$

$\displaystyle\bm{\Sigma}_{W_{i}}=\frac{r}{n(2-r)}\bm{\Sigma}^{*}$

was used to compute the plotted statistic (and it is also used in our work).
An out-of-control signal is issued when $\displaystyle Q_{i}>UCL=H$, where
$\displaystyle H>0$ is chosen to achieve a specific value of in-control
$\displaystyle ATS$.

In the FSI MEWMA-CoDa control chart, the sampling interval is a fixed constant
$\displaystyle h_{F}$. As for the VSI MEWMA-CoDa control chart, based on the
current value of $\displaystyle Q_{i}$, the time between two successive
samples $\displaystyle\bar{\mathbf{X}}_{i},\bar{\mathbf{X}}_{i+1}$ is allowed
to varied. In this chart, the control limit UCL is held the same as in the FSI
chart, and an additional warning limit $\displaystyle w=UWL$
$\displaystyle(0<UWL<UCL)$ is introduced to determine the switch between the
long and short sampling intervals: The long sampling intervals $\displaystyle
h_{L}$ is used when the control statistic $\displaystyle Q_{i}^{2}\leq
UWL^{2}$ (safe region) and the short sampling intervals $\displaystyle h_{S}$
is used when $\displaystyle UWL<Q_{i}^{2}\leq UCL^{2}$ (warning region). An
out-of-control signal is issued when $\displaystyle Q_{i}^{2}>UCL^{2}$.

###  3.2 Markov chain model

Suppose that the occurrence of an assignable cause makes the in-control
composition center $\displaystyle\bm{\mu}_{0}$ is shifted to
$\displaystyle\bm{\mu}_{1}$, or equivalently $\displaystyle\bm{\mu}_{0}^{*}$
is shifted to $\displaystyle\bm{\mu}_{1}^{*}$. In this subsection, we will
discuss a method based on the Markov chain model to compute the average of the
zero-state time to signal (ATS) for the VSI MEWMA-CoDa control chart. Let
$\displaystyle ATS_{0},ATS_{1}$ denote the ATS when the process runs in-
control, and out-of-control, respectively. In comparison with other control
charts, it is desirable to design a chart with smaller $\displaystyle ATS_{1}$
while their $\displaystyle ATS_{0}$ are the same. In the FSI chart, since the
sampling interval $\displaystyle h_{F}$ is fixed, we have

$ATS^{\mathrm{FSI}}=h_{F}\times ARL^{\mathrm{FSI}}.$

In the VSI chart, since the sampling interval is allowed to vary, the relation
between $\displaystyle ATS$ and $\displaystyle ARL$ would be:

$ATS^{\mathrm{VSI}}=E(h)\times ARL^{\mathrm{VSI}}.$

where $\displaystyle E(h)$ denote the average sampling interval.

The authors of 10 showed that the performance of a
MEWMA-$\displaystyle\bar{X}$ chart is a function of the $\displaystyle n$,
$\displaystyle\bm{\mu}_{0}^{*}$, $\displaystyle\bm{\mu}_{1}^{*}$ and
$\displaystyle\bm{\Sigma}^{*}$ only through the non-centrality parameter
$\displaystyle\delta$ where

$\displaystyle\delta=\sqrt{n(\bm{\mu}_{1}^{*}-\bm{\mu}_{0}^{*})^{\intercal}(\bm{\Sigma}^{*})^{-1}(\bm{\mu}_{1}^{*}-\bm{\mu}_{0}^{*})}.$

Without loss of generality, we can assume $\displaystyle n=1$,
$\displaystyle\bm{\mu}_{0}^{*}=\mathbf{0}$ (i.e. the in-control composition
center is
$\displaystyle\bm{\mu}_{0}=(\frac{1}{p},\frac{1}{p},\ldots,\frac{1}{p})$) and
$\displaystyle\bm{\Sigma}^{*}=\mathbf{I}_{p-1}$ (the identity matrix in
$\displaystyle\mathbb{R}^{p-1}$). In this case, the statistic $\displaystyle
Q_{i}$ in (1) is modified to $\displaystyle Q_{i}=b\left\lVert
W_{i}\right\rVert_{2}^{2}$ with $\displaystyle b=\frac{2-r}{r}$. Consequently,
the control limits UCL and UWL of VSI MEWMA-CoDa are modified to be

$\displaystyle UCL=\sqrt{H/b},\quad UWL=\sqrt{w/b}$

To calculate the in- and out-of-control ATS of the VSI
MEWMA-$\displaystyle\bar{X}$ chart, the author of 12 modified the Markov chain
approach proposed in 17 to approximate its calculation based on the statistic
$\displaystyle q_{i}=\left\lVert W_{i}\right\rVert_{2}$.

Concerning the _in-control_ case, the one dimensional Markov chain can be used
to approximate ATS. In this case, the interval
$\displaystyle[0,UCL^{\prime}]$, where $\displaystyle
UCL^{\prime}=\sqrt{H/b}$, is divided into $\displaystyle m+1$ sub-intervals
(states): the first sub-interval has length $\displaystyle\frac{g}{2}$ and the
others have length $\displaystyle g$, where $\displaystyle
g=\frac{2UCL^{\prime}}{2m+1}$. The probability of transition from state
$\displaystyle i$ to state $\displaystyle j$, denoted by $\displaystyle
p(i,j)$, is given by

  * •

for $\displaystyle i=0,1,\ldots,m$ and $\displaystyle j=1,2,\ldots,m$,

$\begin{split}p(i,j)=P\bigg{(}\Big{(}\frac{(j-0.5)g}{r}\Big{)}^{2}<\chi^{2}(p-1,c)\\\
<\Big{(}\frac{(j+0.5)g}{r}\Big{)}^{2}\bigg{)}\end{split}$

where $\displaystyle\chi^{2}(p-1,c)$ denotes a non central chi-square random
variable with $\displaystyle p-1$ degrees of freedom and non-centrality
parameter $\displaystyle c=\left(\frac{(1-r)ig}{r}\right)^{2}$,

  * •

for $\displaystyle j=0$,

$\displaystyle
p(i,0)=P\left(\chi^{2}(p-1,c)<\left(\frac{g}{2r}\right)^{2}\right).$

Let $\displaystyle\textbf{P}_{1}$ denote the $\displaystyle(m+1,m+1)$
transition probability matrix corresponding to the transient states with the
elements $\displaystyle p(i,j)$ then the zero-state in-control $\displaystyle
ATS$ of the VSI MEWMA-CoDa control chart is obtained by

$\displaystyle
ATS=\mathbf{s}^{\intercal}(\mathbf{I}_{m+1}-\mathbf{P}_{1})^{-1}\mathbf{h},$

where $\displaystyle\mathbf{s}$ is the $\displaystyle(m+1)$-starting
probability vector, i.e.
$\displaystyle\mathbf{s}=(1,0,0,\ldots,0)^{\intercal}$,
$\displaystyle\mathbf{h}$ is the $\displaystyle(m+1)$-vector of sampling
interval with the $\displaystyle i^{th}$ component $\displaystyle h_{i}$ is
defined by

$\displaystyle\displaystyle
h_{i}=\begin{cases}h_{L}~{}~{}\text{if}~{}~{}ig\leq UWL\\\
h_{S}~{}~{}\text{if}~{}~{}ig>UWL\end{cases}.$

The expected sampling interval $\displaystyle E(h)$ is calculated by

$E(h)=\frac{\mathbf{s}^{\intercal}(\mathbf{I}_{m+1}-\mathbf{P}_{1})^{-1}\mathbf{h}}{\mathbf{s}^{\intercal}(\mathbf{I}_{m+1}-\mathbf{P}_{1})^{-1}\mathbf{1}_{m+1}},$

where $\displaystyle\mathbf{1}_{m+1}=(1,1,\ldots,1)^{\intercal}$ is the
$\displaystyle m+1$ column vector of 1’s.

To calculate the zero-state ATS of VSI MEWMA-CoDa chart in the _out-of-
control_ case, The author of 12 modified the two dimensional Markov chain
approach which is originally proposed in 17. In this approach,
$\displaystyle\mathbf{W}_{i}\in\mathbb{R}^{p-1}$ is partitioned into
$\displaystyle W_{i1}\in\mathbb{R}$ with mean $\displaystyle\delta\neq 0$ and
$\displaystyle\mathbf{W}_{i2}\in\mathbb{R}^{p-2}$ with zero mean. Then,
$\displaystyle
q_{i}=\left\lVert\mathbf{W}_{i}\right\rVert_{2}=\sqrt{W^{2}_{i1}+\mathbf{W}^{\intercal}_{i2}\mathbf{W}_{i2}}$.

The transition probability $\displaystyle h(i,j)$ of $\displaystyle W_{i1}$
from state $\displaystyle i$ to state $\displaystyle j$ is used to analyze the
out-of-control component. Applying the Markov chain-based approach with the
number of states of the Markov chain is $\displaystyle 2m_{1}+1$, for
$\displaystyle i,j=1,2,\ldots,2m_{1}+1$, we have

$\displaystyle\displaystyle h(i,j)=$
$\displaystyle\displaystyle\Phi\left(\frac{-UCL^{\prime}+jg_{1}-(1-r)c_{i}}{r}-\delta\right)$
$\displaystyle\displaystyle-\Phi\left(\frac{-UCL^{\prime}+(j-1)g_{1}-(1-r)c_{i}}{r}-\delta\right)$

where $\displaystyle\Phi$ denotes the cumulative standard normal distribution
function, $\displaystyle c_{i}=-UCL^{\prime}+(i-0.5)g_{1}$ is the center point
of state $\displaystyle i$ with the width of each state $\displaystyle
g_{1}=\frac{2UCL^{\prime}}{2m_{1}+1}$.

Concerning $\displaystyle\mathbf{W}_{i2}$ component, the transition
probability $\displaystyle v(i,j)$ from state $\displaystyle i$ to state
$\displaystyle j$ is used to analyze the in-control component. In this case,
the Markov chain approach as in in-control case will be applied with
$\displaystyle p-2$ replacing $\displaystyle p-1$. The control region is
partitioned into $\displaystyle m_{2}+1$ sub-intervals (states) with the width
of each states is $\displaystyle g_{2}=\frac{2UCL^{\prime}}{2m_{2}+1}$. The
transition probability $\displaystyle v(i,j)$ is given as follows

  * •

for $\displaystyle i=0,1,2,\ldots,m_{2}$ and $\displaystyle
j=1,2,\ldots,m_{2}$

$\begin{split}v(i,j)=P\bigg{(}\Big{(}\frac{(j-0.5)g_{2}}{r}\Big{)}^{2}<\chi^{2}(p-2,c)\\\
<\Big{(}\frac{(j+0.5)g_{2}}{r}\Big{)}^{2}\bigg{)},\end{split}$

where $\displaystyle c=\left(\frac{(1-r)ig_{2}}{r}\right)^{2}$,

  * •

for $\displaystyle j=0$,

$\displaystyle
v(i,0)=P\left(\chi^{2}(p-2,c)<\left(\frac{g_{2}}{2r}\right)^{2}\right)$

Let $\displaystyle\mathbf{H}$ denote the $\displaystyle(2m_{1}+1,2m_{1}+1)$
transition probability matrix of $\displaystyle W_{i1}$ with elements
$\displaystyle h(i,j)$, $\displaystyle\mathbf{V}$ denote the
$\displaystyle(m_{2}+1,m_{2}+1)$ transition probability matrix of
$\displaystyle\left\lVert\mathbf{Y}_{i2}\right\rVert_{2}$ with elements
$\displaystyle v(i,j)$, and $\displaystyle\mathbf{P}_{2}$ denote the
transition probability matrix of two dimensional Markov chain. Since
$\displaystyle W_{i1}$ and $\displaystyle\mathbf{Y}_{i2}$ are independent, we
have $\displaystyle\mathbf{P}_{2}=\mathbf{H}\otimes\mathbf{V}$, where
$\displaystyle\otimes$ is the Kronecker’s matrices product. Matrix
$\displaystyle\mathbf{P}_{2}$ will consist of the transition probabilities of
all transient and some absorbing states of the Markov chain.

Let $\displaystyle\mathbf{T}$ be the $\displaystyle(2m_{1}+1,m_{2}+1)$\-
matrix with element $\displaystyle T(\alpha,\beta)$ given by

$\displaystyle\mathbf{T}(\alpha,\beta)=\left\\{\begin{array}[]{ll}1&\mbox{if
state ($\displaystyle\alpha,\beta$) is transient}\\\
0&\mbox{otherwise}\end{array}\right.$

and $\displaystyle\mathbf{P}$ be the transition probability matrix containing
only transient states of the Markov chain. Then, we have
$\displaystyle\mathbf{P}=\mathbf{T}(\alpha,\beta)\circledast\mathbf{P}_{2}$
where symbol $\displaystyle\circledast$ indicates the element-wise
multiplication of matrices.

Let $\displaystyle\mathbf{h}$ be the $\displaystyle(2m_{1}+1)\cdot(m_{2}+1)$
vector of sampling intervals for the bivariate chain. M.H. Lee 12 defined
$\displaystyle\mathbf{h}$ to be

$\begin{split}\mathbf{h}^{\intercal}=\big{(}(1,0),\ldots,(1,m_{2}),(2,0),\ldots,(2,m_{2}),\ldots,\\\
\ldots,(2m_{1}+1,0),\ldots,(2m_{1}+1,m_{2})\big{)}\end{split}$

with the element $\displaystyle\mathbf{h}(i,j)$ defined by

$\displaystyle\mathbf{h}(i,j)=\left\\{\begin{array}[]{ll}h_{L}&\mbox{if}\enskip
a_{i,j}\leq UWL^{2}\\\ h_{S}&\mbox{if}\enskip UWL^{2}<a_{i,j}\leq UCL^{2}\\\
0&\mbox{otherwise}\end{array}\right.$

where $\displaystyle a_{i,j}=(i-(m_{1}+1))^{2}g_{1}^{2}+j^{2}g_{2}^{2}.$

Thus, the zero-sate out-of-control $\displaystyle ATS$ of VSI MEWMA-CoDa
control chart is defined by $\displaystyle
ATS=\mathbf{s}^{\intercal}(\mathbf{I}-\mathbf{P})^{-1}\mathbf{h}$ where
$\displaystyle\mathbf{s}$ is the initial probability vector with the component
corresponding to state $\displaystyle(\alpha,\beta)=(m_{1}+1,0)$ is equal to
one and all other components are equal to zero. In case $\displaystyle
m_{1}=m_{2}=m$, M.H. Lee 18 showed that the entry corresponding to the
component with value equal to 1 of $\displaystyle\mathbf{s}$ is the
$\displaystyle(m(m+1)+1)$th entry. Concerning the performance of the program
used for the computation of the $\displaystyle ATS$, we follow the
recommendation in 9 and decide to use $\displaystyle m_{1}=m_{2}=30$.

###  3.3 Optimization procedure

Assume that the fixed sampling interval in FSI control charts is to be a time
unit, i.e. $\displaystyle h_{F}=1$. Hence, $\displaystyle
ATS_{0}^{FSI}=ARL_{0}$. In order to evaluate the performances of VSI MEWMA-
CoDa with its FSI version, we can compare their out-of-control $\displaystyle
ATS_{1}$ while constraining the same in-control values of both $\displaystyle
ATS_{0}$ and $\displaystyle E_{0}(h)$ (average sampling interval). Thus, the
VSI MEWMA-CoDa control chart can be designed by finding the optimal
combination of parameters that minimize the out-of-control $\displaystyle
ATS_{1}$ subject to the predefined constraint of $\displaystyle ATS_{0}$ and
$\displaystyle E_{0}(h)$.

In general, a fixed couple $\displaystyle(h_{S},h_{L})$ is typically used,
which can be chosen from the suggested list as in 13. However, as discussed in
the study 14, while $\displaystyle h_{S}$ is quite reasonable to fix, it seems
not practical to fix $\displaystyle h_{L}$ due to the fact that when the
control statistic falls into the central region, the process is still in safe
and the next sampling interval can be flexible to choose if it does not
influence the performance of the chart. Based on this reason, we follow the
suggestion in 14 to fix the proportion between the UCL and UWL values. Let
$\displaystyle R$ be the number such that $\displaystyle UWL=R\cdot UCL$. When
the control limit UCL is determined, the warning limit UWL can be computed
based on the value of $\displaystyle R$.

Thus, the optimal design of the VSI MEWMA-CoDa control chart will consist of
searching the optimal parameters $\displaystyle(r,H,h_{L})$ which minimize the
out-of-control $\displaystyle ATS_{1}$ for given shift $\displaystyle\delta$
subject to constraints in the in-control $\displaystyle ATS_{0}$ and
$\displaystyle E_{0}(h)=1$, i.e,

$(r^{*},H^{*},h_{L}^{*})=\underset{(r,H,h_{L})}{\operatornamewithlimits{argmin}}\enskip\operatorname{ATS}(n,r,H,R,p-1,\delta,h_{L},h_{S})$

subject to the constraint

$\displaystyle\displaystyle\begin{cases}\operatorname{ATS}(n,r^{*},H^{*},R,p-1,\delta=0,h_{L}^{*},h_{S})=\operatorname{ATS}_{0}\\\
\operatorname{E}_{0}(h)=1\end{cases}$

By fixing the in-control predefined $\displaystyle ATS_{0}$ value, these
optimal parameters can be obtained by using the two-steps optimization
procedure as follows

  1. 1.

Find the set of triples $\displaystyle(r,H,h_{L})$ such that the in-control
$\displaystyle ATS=ATS_{0}$ and $\displaystyle E_{0}(h)=1$.

  2. 2.

Among these feasible triples $\displaystyle(r,H,h_{L})$, choose
$\displaystyle(r^{*},H^{*})$ which provides the smallest out-of-control
$\displaystyle ATS$ value for a particular shift $\displaystyle\delta$ in
vector $\displaystyle\bm{\mu}_{0}^{*}$.

As noted in 9, the value of $\displaystyle r$ must not be too small to avoid
unreliable results and the diverging ability in the Markov Chain approach. In
this paper, we fix the minimal bound to search for the smoothing parameter
$\displaystyle r$ to be $\displaystyle 0.05$, as recommended in many studies,
including 9.

##  4 PERFORMANCE OF THE VSI MEWMA-CoDa CONTROL CHART

In this section, we will compare the performance of the VSI MEWMA-CoDa chart
with the FSI MEWMA-CoDa chart proposed by 9. The comparison will be based on
the values of out-of-control $\displaystyle\operatorname{ATS}_{1}$ while
constraining on the same in-control values of both
$\displaystyle\operatorname{ATS}_{0}$ and
$\displaystyle\operatorname{E}_{0}(h)$. To take advantage of the results from
the study of 9, save the calculation costs, and simplify the application in
practice, we propose to find the near-optimal values to the VSI MEWMA-CoDa
control chart as follows:

  * •

For each optimal couple $\displaystyle(r^{*},H^{*})$ in Table 2 in study of 9,
the value of UWL and $\displaystyle h_{L}$ are chosen to achieve predefined
$\displaystyle\operatorname{ATS}_{0}$ and
$\displaystyle\operatorname{E}_{0}(h)$,

  * •

After obtaining UWL and $\displaystyle h_{L}$, together with the corresponding
$\displaystyle(r^{*},H^{*})$, we compute the
$\displaystyle\operatorname{ATS}_{1}$ of VSI MEWMA-CoDa for specific shift
sizes $\displaystyle\delta$ and compare them with
$\displaystyle\operatorname{ARL}_{1}$ of FSI MEWMA-CoDa chart (Table 3 in 9).

The procedure to find the near-optimal values is implemented based on
following scenarios:

  * •

$\displaystyle n=1,p=3$, $\displaystyle\operatorname{ATS}_{0}=200$, and
$\displaystyle\operatorname{E}_{0}(h)=1$;

  * •

$\displaystyle\delta\in\\{0.25,0.50,0.75,1.00,1.25,1.50,1.75,2.00\\}$;

  * •

$\displaystyle h_{S}\in\\{0.1,0.5\\}$.

The values $\displaystyle\operatorname{ARL}_{1}$ (FSI column) of MEWMA-CoDa
and $\displaystyle\operatorname{ATS}_{1}$ (VSI columns) for some different
scenarios are shown in Table 2. The values of $\displaystyle w$ such that
$\displaystyle UWL=\sqrt{w/b}$ and $\displaystyle h_{L}$ to obtain the near-
optimal value are also provided for each scenario. Some remarks can be drawn
from this results as follows

  * •

The VSI MEWMA-CoDa control chart always outperforms the FSI MEWMA-CoDa control
chart in detecting the process shifts. For example, when
$\displaystyle\delta=0.25,h_{S}=0.1$, we have
$\displaystyle\operatorname{ARL}_{1}=64.6$ for FSI MEWMA-CoDa chart and
$\displaystyle\operatorname{ATS}_{1}=56.8$ for VSI MEWMA-CoDa chart,

  * •

The VSI MEWMA-CoDa charts with smaller $\displaystyle h_{S}$
$\displaystyle(h_{S}=0.1)$ perform better than the ones with larger
$\displaystyle h_{S}$ $\displaystyle(h_{S}=0.5)$. For example, when
$\displaystyle\delta=0.5$, we have $\displaystyle\operatorname{ATS}_{1}=19.9$
in case $\displaystyle h_{S}=0.1$ and
$\displaystyle\operatorname{ATS}_{1}=23.5$ in case $\displaystyle h_{S}=0.5$,

  * •

When the shift sizes $\displaystyle\delta$ are large $\displaystyle(\delta\geq
1.75)$, the performance of VSI MEWMA-CoDa chart are still better than FSI
MEWMA-CoDa chart, but not much.

Table 2: Comparison between VSI MEWMA-CoDa and FSI MEWMA-CoDa charts $\displaystyle\delta$ | FSI | $\displaystyle h_{S}=0.1$ | $\displaystyle h_{S}=0.5$
---|---|---|---
$\displaystyle(w,h_{L})$ | VSI | $\displaystyle(w,h_{L})$ | VSI
0.25 | 64.6 | $\displaystyle(1.7,1.6)$ | 56.8 | $\displaystyle(0.7,2.1)$ | 63.5
0.50 | 26.4 | $\displaystyle(1.7,1.6)$ | 19.9 | $\displaystyle(0.9,1.8)$ | 23.5
0.75 | 15.1 | $\displaystyle(1.6,1.7)$ | 10.4 | $\displaystyle(1.0,1.8)$ | 12.9
1.00 | 9.9 | $\displaystyle(2.9,1.3)$ | 6.9 | $\displaystyle(0.9,1.8)$ | 8.4
1.25 | 7.1 | $\displaystyle(1.6,1.8)$ | 4.9 | $\displaystyle(0.9,2.0)$ | 6.3
1.50 | 5.4 | $\displaystyle(3.5,1.2)$ | 3.7 | $\displaystyle(0.9,1.9)$ | 4.8
1.75 | 4.3 | $\displaystyle(3.7,1.2)$ | 3.0 | $\displaystyle(0.8,2.1)$ | 4.2
2.00 | 3.5 | $\displaystyle(3.6,1.2)$ | 2.4 | $\displaystyle(1.1,1.8)$ | 3.3

##  5 Conclusion

In this paper, we proposed a VSI MEWMA-CoDa control chart to monitor a normal
multivariate random vector defined as the inverse isometric log-ratio of a
$\displaystyle p$-part composition. The optimal procedure to compute the
optimal triple $\displaystyle(r^{*},H^{*},h_{L}^{*})$ and the
$\displaystyle\operatorname{ATS}$ values of the proposed chart for different
shift sizes were presented. We also proposed a method to find the near-optimal
values for the VSI MEWMA-CoDa chart to utilize the results in the study of 9
and reduce the computation costs. The numerical performance comparison between
the VSI MEWMA-CoDa chart and standard (FSI) MEWMA-CoDa control chart in terms
of $\displaystyle\operatorname{ATS}_{1}$ (based on the near-optimal values
method) showed that the VSI MEWMA-CoDa chart always outperforms the standard
chart. Future research on monitoring CoDa could be concentrated on the
extension of the VSI MEWMA-CoDa chart to the VSI MCUSUM-CoDa chart, or
investigating the effect of measurement error on these charts. The methods to
transform CoDa into normal data before designing these controls charts are
also worthy to focus. Due to the wide applications of CoDa in the real-life,
the online monitoring of CoDa should be worthy of consideration by researchers
in the SPC field.

## References

  * 1 D.C. Montgomery (2013) _Statistical Quality Control: a Modern Introduction, 7th Edn_ , Wiley, New York.
  * 2 J. Aitchison (1986) _The Statistical Analysis of Compositional Data (Monographs on Statistics and Applied Probability)_ , Chapman & Hall Ltd., London, (Reprinted in 2003 with additional material by The Blackburn Press).
  * 3 R. A. Boyles (1997) _Using the chi-square statistic to monitor compositional process data_ , Journal of Applied Statistics, Vol 24(5), 589-602.
  * 4 M. Vives-Mestres and J. Daunis-I-Estadella and J.A. Martin-Fernandez (2016) _Signal Interpretation in Hotelling’s $\displaystyle T^{2}$ Control Chart for Compositional Data_, IIE Transactions, Vol 48(7), 661-672.
  * 5 M. Vives-Mestres and J. Daunis-I-Estadella and J.A. Martin-Fernandez (2014a) _Out-of-Control Signals in Three-Part Compositional $\displaystyle{T}^{2}$ Control Chart_, Quality and Reliability Engineering International, Vol 30(3), 337–346.
  * 6 M. Vives-Mestres and J. Daunis-I-Estadella and J.A. Martin-Fernandez (2014b) _Individual $\displaystyle{T}^{2}$ Control Chart for Compositional Data_, Journal of Quality Technology, Vol 46(2), 127-139.
  * 7 Zaidi, F. S. and Castagliola, P. and Tran, K. P. and Khoo, M. B. C. (2020) _Performance of the MEWMA-CoDa control chart in the presence of measurement errors_ , Quality and Reliability Engineering International, Vol 36(7), 2411-2440.
  * 8 F. S. Zaidi and P. Castagliola and K. P. Tran and M. B. C. Khoo (2019) _Performance of the hotelling T2 control chart for compositional data in the presence of measurement errors_ , Journal of Applied Statistics, Vol 46(14), 2583-2602.
  * 9 Tran, K. P. and Castagliola, P. and Celano, G. and Khoo, Michael B.C. (2017) _Monitoring compositional data using multivariate exponentially weighted moving average scheme_ , Quality and Reliability Engineering International, Vol 34(3), 391-402.
  * 10 Lowry, C. A. and Woodall, W.H. and Champ, C. W. and Rigdon, S. E. (1992) _A Multivariate Exponentially Weighted Moving Average control chart_ , Technometrics, Vol 34(1), 46–53.
  * 11 V. Pawlowsky-Glahn and J. J. Egozcue and R. Tolosana-Delgado (2015) _Modeling and Analysis of Compositional Data_ , John Wiley & Sons.
  * 12 M.H. Lee (2009) _Multivariate EWMA Charts with Variable Sampling Intervals_ , Economic Quality Control, Vol 24, 231-241.
  * 13 Castagliola, P. and Achouri, A. and Taleb, H. and Celano, G. and Psarakis, S. (2013) _Monitoring the Coefficient of Variation Using a Variable Sampling Interval Control Chart_ , Quality and Reliability Engineering International, Vol 29(8), 1135-1149.
  * 14 Nguyen, H. D. and Tran, K. P. and Heuchenne, C. (2018) _Monitoring the Ratio of two Normal Variables using Variable Sampling Interval EWMA control charts_ , Quality and Reliability Engineering.
  * 15 Egozcue, J. J. and Pawlowsky-Glahn, V. and Mateu-Figueras, G. and Barceló-Vidal, C. (2003) _Isometric Logratio Transformations for Compositional Data Analysis_ , Mathematical Geology, Vol 35(3), 279-300.
  * 16 J.J. Egozcue and V. Pawlowsky-Glahn (2005) _Groups of Parts and Their Balances in Compositional Data Analysis_ , Mathematical Geology, Vol 37(7), 795-828.
  * 17 G. C. Runger and S. S. Prabhu (1996) _A Markov Chain Model for the Multivariate Exponentially Weighted Moving Averages Control Chart_ , Journal of the American Statistical Association, Vol 91(436), 1701-1706.
  * 18 M.H. Lee and M.B.C. Khoo (2006) _Optimal statistical design of a multivariate EWMA chart based on ARL and MRL_ , Communications in Statistics-Simulation and Computation, Vol 35(3), 831–847.