# On Statistical Properties of Jizba-Arimitsu Hybrid Entropy

Mehmet Niyazi Çankaya<EMAIL_ADDRESS>Jan Korbel
<EMAIL_ADDRESS>Uşak University, Faculty of Art and Sciences, Department
of Statistics, Uşak, Turkey  Department of Physics, Zhejiang University,
Hangzhou 310027, P. R. China  Faculty of Nuclear Sciences and Physical
Engineering, Czech Technical University in Prague, Břehová 7, 115 19 Praha 1,
Czech Republic

###### Abstract

Jizba-Arimitsu entropy (also called _hybrid entropy_) combines axiomatics of
Rényi and Tsallis entropy. It has many common properties with them, on the
other hand, some aspects as e.g., MaxEnt distributions, are completely
different from the former two entropies. In this paper, we demonstrate the
statistical properties of hybrid entropy, including the definition of hybrid
entropy for continuous distributions, its relation to discrete entropy and
calculation of hybrid entropy for some well-known distributions. Additionally,
definition of hybrid divergence and its connection to Fisher metric is also
discussed. Interestingly, the main properties of continuous hybrid entropy and
hybrid divergence are completely different from measures based on Rényi and
Tsallis entropy. This motivates us to introduce average hybrid entropy, which
can be understood as an average between Tsallis and Rényi entropy

###### keywords:

Jizba-Arimitsu hybrid entropy; non-extensive thermodynamics; MaxEnt;
continuous entropy; information divergence

###### PACS:

05.90.+m, 02.50.-r, 65.40.Gr

##  1 Introduction

Generalized entropies have played an important role in description of
thermodynamic, statistical and informational systems in past few decades. The
main reason for using these entropies is to describe systems that cannot be
successfully described by the conventional Shannon-Boltzmann-Gibbs entropy. In
information theory appeared Shannon entropy firstly in 1948 [1] in connection
with communication theory. Shortly afterwards, there began to appear various
generalizations of Shannon entropy. To the most important belong Rényi entropy
[2], Tsallis(-Havrda-Charvát) entropy (derived independently by Tsallis [3]
from thermodynamical point of view and by Havrda and Charvát [4] from
informational point of view), Sharma-Mittal entropy [5], Frank-Daffertshofer
entropy [6] or Kapur measure [7]. Recently, there have been made several
successful attempts in order to categorize the various entropy classes and
their properties. Hanel and Thurner [8, 9] classified the entropies according
to their asymptotic scaling, Tempesta [10] studied the generalized entropies
according to group properties, Biró and Barnaf [11] derived a new class of
entropies from its interaction with heat reservoir. Ilić and Stanković [12]
classified the pseudo-additive entropies by generalization of Khinchin axioms.

Among these entropies, the most prominent two classes are Rényi entropy, also
known from theory of multifractal systems [13], and Tsallis entropy,
describing the thermodynamics of non-extensive systems (e.g., systems in
contact with finite heat bath [14]). Jizba and Arimitsu [15] suggested a new
one-parametric class of entropies called _Jizba-Arimitsu hybrid entropy_ ,
which shares some properties of Rényi and Tsallis entropy. Particularly,
hybrid entropy is non-extensive and the conditional entropy is similarly to
Rényi defined in terms of generalized Kolmogorov-Nagumo mean [16, 17].
Recently, hybrid entropy was discussed by several authors. Ilić and Stanković
correctly pointed out that there is a mistake in the original derivation and
corrected the axiomatic [18]. Jizba and Korbel [19] calculated the error in
the original definition and concluded that for thermodynamical systems with
weak interactions is still possible to use hybrid entropy in its original
form. Moreover, thermodynamic properties of hybrid entropy were extensively
discussed in [20].

The main aim of the paper is to present the statistical properties of hybrid
entropy. These properties are often remarkably different from analogous
results in the case of Rényi and Tsallis entropy, which points to the fact
that hybrid entropy can describe conceptually different systems than the
former two entropies. We start with the relation to other generalized
entropies. Subsequently, we define the continuous (also called differential)
hybrid entropy and present its relation to the discrete hybrid entropy.
Finally, we introduce hybrid divergence and study their statistical and
informational properties, which motivates us into definition of average hybrid
entropy. The rest of the paper is organized as follows: the next section is
dedicated to revision of basic properties of hybrid entropy, relation to the
maximality axiom, the concavity issue and the relation to the Rényi and
Tsallis entropies. Section 3 defines the continuous hybrid entropy and shows
its properties on popular distributions, such as triangular, Beta,
exponential, Gamma, normal, Cauchy and Student-t distribution. In Section 4 is
introduced hybrid divergence and and its informational properties are studies,
especially its connection to Fisher metric. Section 5 defines average hybrid
entropy and briefly discusses its properties. The last section is devoted to
conclusions. Appendix presents the hybrid entropy for special class of bimodal
$\varepsilon$-skew exponential power distributions, which are important in the
theory of statistical inference.

##  2 Basic properties of hybrid entropy

Hybrid entropy is defined as a synthesis between Tsallis and Rényi entropy and
it combines both $q$-non-extensivity and generalized Kolmogorov-Nagumo
$q$-averaging, which is important for multifractal systems, especially. The
resulting hybrid entropy $\mathcal{D}_{q}(P)$ is defined by four axioms [15].
The continuity axiom requires the continuity in every argument, the maximality
axiom requests that the entropy is maximal for uniform distribution and
expansibility axiom ensures that an event with zero probability does not
affect the value of entropy. The most important additivity axiom defines the
joint entropy and conditional entropy as

$\mathcal{D}_{q}(A\cup
B)=\mathcal{D}_{q}(A)+\mathcal{D}_{q}(B|A)+(1-q)\mathcal{D}_{q}(A)\mathcal{D}_{q}(B|A)$
(1)

resp.

$\mathcal{D}_{q}(B|A)=f_{q}^{-1}\left(\sum_{k}\rho_{k}(q)f_{q}(\mathcal{D}_{q}(B|A=A_{k}))\right)$
(2)

where $f_{q}$ is a positive, invertible function on $\mathds{R}^{+}$. Its
exact form is discussed in Section 2.1. Distribution
$\rho_{k}(q)=p_{k}^{q}/\sum_{j}p_{j}^{q}$ is the _escort distribution_
belonging to experiment $A$. Escort distribution, or “zooming distribution”,
was originally discovered in connection with chaotic dynamic systems [21, 22]
and is also widely used in theory of multifractals [23].

It has been shown in Ref. [18] that the additivity rule holds exactly only for
independent events. In [19] has been discussed the error for events which are
not independent. This error is relatively small for weakly interacting
systems. The form of the resulting hybrid entropy has been introduced in Ref.
[15] in the form:

$\mathcal{D}_{q}(P)=\frac{1}{1-q}\left(e^{-(1-q)\sum_{k}\rho_{k}(q)\ln
p_{k}}-1\right)\,.$ (3)

###  2.1 $q$-Deformed Calculus and Generalized Means

In order to understand the properties of hybrid entropy, let us introduce two
mathematical terms which are important in theory of generalized entropies.
First, let us briefly revise so-called $q$-deformed calculus. This calculus is
connected with non-extensive entropies, especially Tsallis entropy. The aim is
to define the non-linear generalizations of ordinary operations and functions.
For example, _$q$ -deformed addition_ (also known as _Jackson sum_ [24]) is
defined as

$x\oplus_{q}y=x+y+(1-q)xy\,.$ (4)

We can recognize that the additivity rule for hybrid entropy is nothing else
than $q$-deformed sum. Additionally, we can define the $q$-deformed versions
of logarithm and exponential functions [25]:

$\displaystyle\exp_{q}(x\oplus_{q}y)=\exp_{q}(x)\exp_{q}(y)$
$\displaystyle\Rightarrow$ $\displaystyle\exp_{q}(x)=[1+(1-q)x]^{1/(1-q)}$ (5)
$\displaystyle\ln_{q}(xy)=\ln_{q}(x)\oplus_{q}\ln_{q}(y)$
$\displaystyle\Rightarrow$ $\displaystyle\ln_{q}(x)=\frac{x^{1-q}-1}{1-q}$ (6)

For appropriate values, it holds that

$\exp_{q}(\ln_{q}(x))=\ln_{q}(\exp_{q}(x))=x.$ (7)

Second, let us define two classes of generalized means. The first class is
known as _Kolmogorov-Nagumo means_ [16, 17] and is defined as

$E[X]_{f}=f^{-1}\left(\sum_{k}p_{k}f(x_{k})\right)$ (8)

Additionally, _escort means_ [26] are based on escort distributions and are
defined as

$E[X]^{q}=\left(\sum_{k}\rho_{k}(q)x_{k}\right)$ (9)

Of course, it is possible to combine both classes to obtain a generalized
_escort Kolmogorov-Nagumo mean_ , which reads

$E[X]_{f}^{q}=f^{-1}\left(\sum_{k}\rho_{k}(q)f(x_{k})\right)\,.$ (10)

These means often appear in connection with generalized entropies and one can
recognize that generalized entropy is nothing else than a generalized mean of
its elementary information (compare with Eq. (2), as shown in the next
section.

###  2.2 Relation to Rényi and Tsallis entropy

Definition of hybrid entropy was motivated as an overlap between Rényi [2] and
Tsallis [27] axiomatic and therefore it shares many properties with both of
them. Let us first remind definitions of Rényi entropy and Tsallis entropy:

$\displaystyle\mathcal{R}_{q}(P)$ $\displaystyle=$
$\displaystyle\frac{1}{1-q}\ln\sum_{k}p_{k}^{q}$ (11)
$\displaystyle\mathcal{S}_{q}(P)$ $\displaystyle=$
$\displaystyle\frac{1}{1-q}\left(\sum_{k}p_{k}^{q}-1\right)\,$ (12)

where both are generalizations of Shannon entropy

$\displaystyle\mathcal{H}(P)$ $\displaystyle=$ $\displaystyle-\sum_{k}p_{k}\ln
p_{k}\,.$ (13)

All entropies become Shannon entropy for $q\rightarrow 1$. It is easy to see
that these two entropies are functions of each other, for instance

$\displaystyle\mathcal{R}_{q}(P)=\frac{1}{1-q}\ln[(1-q)\mathcal{S}_{q}(P)+1]\,.$
(14)

This is not the case of hybrid entropy, because besides $\sum_{k}p_{k}^{q}$,
there also appears the term $\sum_{k}p_{k}^{q}\ln p_{k}$. Connection of hybrid
entropy to Rényi entropy is given by the exponent $E[\ln P]^{q}$ which is
equal to

$E[\ln P]^{q}=\sum_{k}\rho_{k}(q)\ln
p_{k}=\frac{\mathrm{d}\left(\ln\sum_{k}p_{k}^{q}\right)}{\mathrm{d}q}=(1-q)\frac{\mathrm{d}\mathcal{R}_{q}(P)}{\mathrm{d}q}-\mathcal{R}_{q}(P)\,.$
(15)

This term is widely known in the theory of multifractals and scaling, because
it is closely related to multifractal spectrum and other scaling exponents
[28]. We can recognize that hybrid entropy is expressible in terms of Rényi
entropy and its derivative.

Let us focus on the contribution of elementary information to see the
connection of hybrid entropy to $q$-additivity and Tsallis entropy. It can be
derived that both entropies follow the $q$-additivity rule. Let us consider
$m$ i.i.d. variables $\\{A_{i}\\}_{i=1}^{m}$ with the uniform distribution
$p_{k}=\frac{1}{n}$. Plugging into additivity axiom, we obtain that

$\displaystyle\mathcal{D}_{q}\left(\frac{1}{n^{m}}\right)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{m}{m\choose
k}(1-q)^{k-1}\mathcal{D}^{k}_{q}\left(\frac{1}{n}\right)$ (16)
$\displaystyle=$
$\displaystyle\frac{1}{1-q}\left[\left(1+(1-q)\mathcal{D}_{q}\left(\frac{1}{n}\right)\right)^{m}-1\right]\,.$

Solution of this relation can be expressed as a $q$-deformed logarithm, i.e.,
$\ln_{q}(n)$. Thus, similarly to Tsallis entropy, the elementary information
contribution, also called _Hartley information_ , of an event with probability
$p_{k}$ is equal to

$\mathcal{I}_{q}(p_{k})=\ln_{q}\left(\frac{1}{p_{k}}\right).$ (17)

On the contrary, Hartley information of Rényi entropy is (similarly to Shannon
entropy) equal to

$\mathcal{I}_{1}(p_{k})=\ln\left(\frac{1}{p_{k}}\right)\,,$ (18)

which is a consequence of additivity rule. It is clear that the elementary
information can be determined from the additivity rule for independent events.
Consequently, all introduced entropies can be represented as a generalized
mean of the Hartley information [29]. In the case of Tsallis entropy is the
representation given by a simple mean of a non-extensive Hartley informations,
i.e. as

$\mathcal{S}_{q}(P)=E[\mathcal{I}_{q}(P)]$ (19)

while Rényi entropy can be represented as a Kolmogorov-Nagumo mean of
extensive Hartley information [30]. Alternatively, it can be viewed as an
escort Kolmogorov-Nagumo mean with the exponential Kolmogorov-Nagumo function.
This representation has been described in [30] as

$\mathcal{R}_{q}(P)=E[\mathcal{I}_{1}(P)]_{\ln_{q}\exp
x}=E[\mathcal{I}_{1}(P)]_{\exp[(q-1)x]}^{q}$ (20)

Finally, the hybrid entropy is expressible as a generalized Kolmogorov-Nagumo
escort mean of its elementary information $\mathcal{I}_{q}(p_{k})$

$\mathcal{D}_{q}(P)=E[\mathcal{I}_{q}(P)]^{q}_{\ln\exp_{q}x}$ (21)

Note that Kolmogorov-Nagumo function of hybrid entropy is the inverse function
Rényi Kolmogorov-Nagumo function.

###  2.3 Maximality axiom, concavity and Schur-concavity

When deriving the entropies from Khinchin axioms, it is necessary to note that
validity of maximality axiom (i.e., entropy is maximal for the uniform
distribution) is not completely determined and therefore it is necessary to
check its validity manually. This is generally quite complicated task.
However, it is possible to investigate several properties which are sufficient
to prove the validity. The most popular criterion is possibly concavity of
entropy, because many entropy functionals, including Shannon and Tsallis, are
concave functions. Moreover, if they belong to the _trace class_ [9] defined
as $s(P)=\sum_{i}g(p_{i})$, concavity of entropy is equivalent to concavity of
one-dimensional function $g(x)$. It can be shown that hybrid entropy is
concave only in the interval $q\in[\frac{1}{2},1]$ (see Ref. [20]).

On the other hand, concavity is only sufficient property. It is possible to
find weaker versions of concavity which also ensure the validity of maximality
axiom. One of these concepts is so-called _Schur-concavity_ , extensively
discussed e.g., in Ref. [31], which is based on the theory of majorization. A
discrete probability distribution $P=(p_{1},\dots,p_{n})$ is majorized by
$Q=(q_{1},\dots,q_{n})$ if for their ordered probabilities $p_{(1)}\geq
p_{(2)}\geq\dots\geq p_{(n)}$, resp. $q_{(1)}\geq q_{(2)}\geq\dots\geq
q_{(n)}$ the following inequalities hold

$\sum_{k=1}^{j}p_{(k)}\leq\sum_{k=1}^{j}q_{(k)}\,.$ (22)

We denote it as $P\prec Q$. A function $F$ is _Schur-concave_ if for every
$P\prec Q$ is $F(P)\geq F(Q)$ (Analogously, $G$ is Schur-convex if for every
$P\prec Q$ is $G(P)\leq G(Q)$). It is easy to show that the uniform
distribution is majorized by any other distribution

$\left(\frac{1}{n},\dots,\frac{1}{n}\right)\prec(p_{1},\dots,p_{n})\quad\forall
P=(p_{1},\dots,p_{n})\quad s.t.\ \sum_{i}p_{i}=1,$ (23)

so it is clear that Schur-concavity of every entropy functional is
_sufficient_ property for validity of maximality axiom. It is weaker than
concavity, which means that every symmetric concave function is Schur-concave.
In [32], it was shown (as a special case of Schur-concavity of so-called Gini
means) that hybrid entropy is Schur-concave for $q\geq\frac{1}{2}$, while for
$q\in[0,\frac{1}{2})$ is neither Schur-convex nor Schur-concave, and it can be
shown that it does not obey the maximality axiom.

###  2.4 MaxEnt distribution

MaxEnt distribution, originally proposed by Jaynes [33], represents the
distribution containing minimal amount of information under certain
constraints. We always require the normalization of the probability
distribution. Additionally, there are many other possible constraints. The
most common is to prescribe an average energy $\mathcal{E}$, which is usually
considered as an escort mean

$\mathcal{E}=E[\mathcal{E}]^{r}=\sum_{k}\rho_{k}(r)\mathcal{E}_{k}\,.$ (24)

Two most common choices are _linear averaging_ , i.e., $r=1$
($\sum_{k}p_{k}\mathcal{E}_{k}$) and $q$-averaging, i.e., $r=q$. The reason is
that only these two cases provide a unique real MaxEnt distribution [34].
Thus, maximization under constraints is equal to maximization of _Lagrange
function_ which reads:

$\mathcal{L}(P)=\mathcal{D}_{q}(P)-\alpha\left(\sum_{k}p_{k}\right)-\beta\left(\sum_{k}\rho_{k}(r)\mathcal{E}_{k}\right)\,.$
(25)

The MaxEnt distribution can be found by solving equations
$\frac{\partial\mathcal{L}(P)}{\partial p_{i}}=0$, which is equal to

$\displaystyle\frac{\partial\mathcal{L}(P)}{\partial
p_{i}}=e^{(q-1)E[lnP]^{q}}\left\\{q\left(E[\ln P]^{q}-\ln
p_{i}\right)-1\right\\}\frac{p_{i}^{q-1}}{\sum_{k}p_{k}^{q}}-$
$\displaystyle\alpha p_{i}-\beta
r(\mathcal{E}_{r}-\langle\mathcal{E}\rangle_{r})\frac{p_{i}^{r}}{\sum_{k}p_{k}^{r}}=0\,.$
(26)

Multiplying by $p_{i}$ and summing over $i$, we obtain that

$\alpha=-e^{(q-1)E[\ln P]^{q}}\,.$ (27)

Plugging back into Eq. (2.4), we obtain the equation for $p_{i}$:

$\alpha\left[\left\\{q\left(E[\ln P]^{q}-\ln
p_{i}\right)-1\right\\}\frac{p_{i}^{q-1}}{\sum_{k}p_{k}^{q}}+1\right]+r\beta(\mathcal{E}_{r}-\langle\mathcal{E}\rangle_{r})\frac{p_{i}^{r}}{\sum_{k}p_{k}^{r}}=0\,.$
(28)

The equation is intractable, except for the two aforementioned cases, i.e.,
$r=1$ and $r=q$. For these two cases, it is possible to express the solution
in terms of _Lambert $W$-function_

$p_{i}=\left\\{\begin{array}[]{ll}\left[\frac{1}{z_{q}}W\left(z_{q}e^{\frac{q-1}{q}(1-\frac{q\ln(-\alpha)}{q-1}-\frac{q\beta}{\alpha}\Delta_{q}\mathcal{E}_{i}}\right)\right]^{1/(1-q)}&\hbox{for
r=q,}\\\
\left[\frac{\alpha}{z_{q}(\alpha+\beta{\Delta_{1}\mathcal{E}_{i}})}W\left(\frac{z_{q}}{\alpha}\exp\left(\frac{q}{q-1}\right)\left\\{1+\frac{\beta}{\alpha}\Delta_{1}\mathcal{E}_{i}\right\\}\right)\right]^{1/(1-q)}&\hbox{for
r=1}\end{array}\right.$ (29)

where $z_{q}=\frac{(q-1)\sum_{k}p_{k}^{q}}{q}$ and
$\Delta_{r}\mathcal{E}_{i}=\mathcal{E}_{i}-\langle\mathcal{E}\rangle_{r}$.
Lambert $W$-function is a solution of equation

$x=W(x)e^{W(x)}\,.$ (30)

More details about Lambert function can be found e.g., in Ref. [35].
Particularly interestings are three cases: a) systems described by
multifractal scaling exponents, b) “high-temperature limit” ($\beta\ll 1$),
c)“low-temperature limit” ($\beta\gg 1$). These three cases provide the
interesting examples of complex dynamics of systems driven by the hybrid
entropy and are extensively discussed in Ref. [20].

##  3 Continuous hybrid entropy

In this section, we focus on definition of hybrid entropy for continuous
distributions and present some connections to discrete hybrid entropy.
Finally, we calculate hybrid entropy for some popular distributions and
compare it with other entropies, mainly Tsallis entropy. Similarly to other
cases [36], continuous hybrid entropy can be defined as

$\mathcal{D}_{q}[p(x)]=\frac{1}{1-q}\bigg{[}exp\bigg{(}-\frac{\int_{\mathbb{S}}p^{q}(x)log[p(x)]\mathrm{d}x}{\int_{\mathbb{S}}p^{q}(x)dx}\bigg{)}^{1-q}-1\bigg{]}$
(31)

where $\mathbb{S}$ denotes support of probability distribution $p(x)$. First
of all, it is necessary to point out that some properties of discrete
entropies do not hold for continuous entropies. For example, the positivity of
entropy functionals is not guaranteed for continuous distributions. This is
also the case of continuous hybrid entropy.

Second, when establishing connection between the discrete entropy and its
continuous analogue, the most common way is to think about the continuous
entropy as a limit of discrete entropy for $n\rightarrow\infty$. Nevertheless,
this limit is not convergent and it is necessary to make a renormalization.
Let us consider a finite support $\mathbb{S}=[0,1]$. Then, it is possible to
define a discrete approximation of $p(x)$ as

$p_{k}^{(n)}=\int_{(k-1)/n}^{k/n}p(x)\,\mathrm{d}x\,$ (32)

for $k\in\\{1,\dots,n\\}$, which is nothing than a histogram. Naturally,
$p^{(n)}$ approximates $p(x)\mathrm{d}x$ for large values of $n$. Thus, when
$n\gg 1$ we can approximate $p(x)$ for $x\in[(k-1)/n,k/n]$ as $np_{k}^{(n)}$.
Consequently, the continuous hybrid entropy can be approximated as (here we
omit the dependence of $p_{k}^{(n)}$ on $n$)

$\displaystyle\mathcal{D}_{q}[p(x)]=\ln_{q}\exp\left(-\frac{\sum_{k=1}^{n}\int_{(k-1)/n}^{k/n}p^{q}(x)\ln[p(x)]\,\mathrm{d}x}{\sum_{k=1}^{n}\int_{(k-1)/n}^{k/n}p^{q}(x)\,\mathrm{d}x}\right)$
$\displaystyle\approx\ln_{q}\exp\left(-\frac{\sum_{k=1}^{n}\int_{(k-1)/n}^{k/n}\left(np_{k}\right)^{q}\ln\left[np_{k}\right]\,\mathrm{d}x}{\sum_{k=1}^{n}\int_{(k-1)/n}^{k/n}\left(np_{k}\right)^{q}\,\mathrm{d}x}\right)$
$\displaystyle=\ln_{q}\exp\left(-\frac{n^{q-1}\sum_{k=1}^{n}p_{k}^{q}\ln
p_{k}+n^{q-1}\ln
n\sum_{k=1}^{n}p_{k}^{q}}{n^{q-1}\sum_{k=1}^{n}p_{k}^{q}}\right)=$
$\displaystyle=\ln_{q}\exp\left(-\frac{\sum_{k=1}^{n}p_{k}^{q}\ln
p_{k}}{\sum_{k=1}^{n}p_{k}^{q}}\right)\oplus_{q}\ln_{q}\frac{1}{n}\,.$ (33)

Therefore, in order to process the limit $n\rightarrow\infty$, we have to
renormalize $\mathcal{D}_{q}(p^{(n)})$ in order to keep the whole expression
finite. Consequently, it is possible to express the relation between discrete
and continuous hybrid entropy as

$\mathcal{D}_{q}[p(x)]=\lim_{n\rightarrow\infty}\left(\mathcal{D}_{q}(p^{(n)})\oplus_{q}\ln_{q}\frac{1}{n}\right)\,$
(34)

if the limit exists. The situation is analogous if $\mathbb{S}$ is any other
interval. It is necessary to mention that the renormalization in Eq. (34) is
different from renormalization of Rényi and Tsallis entropy. These can be
found e.g., in [36].

In the rest of this section, we compare continuous entropies for several
popular distributions. Table 1 compares Tsallis and hybrid entropy for several
popular distributions. These include distributions with the finite support,
the positive real support and the real support. Rényi entropy can be easily
deduced from Tsallis by relation (14). Moreover, in Appendix, we also derive a
hybrid entropy for $\varepsilon$-skew exponential power distribution [39],
which is a special class of bimodal distributions, recently introduced by
Çankaya et al [40]. This class of distribution finds its place in the
mathematical and physical problems, especially in statistical estimation. We
can immediately recognize different dependence on parameter $q$ for Tsallis
and hybrid entropy. Naturally, the functional dependence is also different,
including more advanced classes of special functions, including Harmonic
numbers, digamma function, etc. This is caused the fact that the hybrid
entropy is conceptually different from Tsallis, which becomes even more
evident in the next section, when the hybrid divergence is defined.

Distribution | $\mathcal{S}_{q}(p)$ | $\mathcal{D}_{q}(p)$
---|---|---
Finite support: $\mathbb{S}=[0,1]$
| Triangular:
---
$4x$ for $x\leq\frac{1}{2}$
$4(1-x)$ for $x\geq\frac{1}{2}$
$\frac{\frac{2^{q}}{1+q}-1}{1-q}$ | $\frac{\left(2e^{\frac{1}{q+1}}\right)^{1-q}-1}{1-q}$
| Beta:
---
$\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}$
$\frac{\frac{\Gamma(q(\alpha-1)+1)\Gamma(q(\beta-1)+1)\left(\frac{1}{B(\alpha,\beta)}\right)^{q}}{\Gamma(q(\alpha+\beta-2)+2)}-1}{1-q}$ | $\frac{\left(B(\alpha,\beta)\exp\left[(\alpha+\beta-2)H_{q(\alpha+\beta-2)+1}+(1-\alpha)H_{q(\alpha-1)}+(1-\beta)H_{q(\beta-1)}\right]\right)^{1-q}-1}{1-q}$
Half-plane: $\mathbb{S}=[0,\infty)$
| Exponential:
---
$\sigma\,e^{-\sigma x}$
$\frac{\frac{\sigma^{q-1}}{q}-1}{1-q}$ | $\frac{\left(\frac{e^{1/q}}{\sigma}\right)^{1-q}-1}{1-q}$
| Gamma:
---
$\frac{e^{-x/\beta}x^{\alpha-1}\beta^{-\alpha}}{\Gamma{(\alpha)}}$
$\frac{\left(\frac{q}{\beta}\right)^{\alpha q+q-1}\beta^{\alpha(-q)}\Gamma(\alpha)^{-q}\Gamma(1-(\alpha+1)q)-1}{1-q}$ | $\frac{\left(\left(\frac{q}{\beta}\right)^{1-\alpha}\exp\left[{\alpha+(\alpha-1)\psi(1-(\alpha+1)q)-\frac{1}{q}+1}\left(\frac{\beta^{-\alpha}}{\Gamma(\alpha)}\right)^{\Gamma(\alpha)^{q-1}}\right]\right)^{1-q}-1}{1-q}$
Whole plane: $\mathbb{S}=(-\infty,\infty)$
| Gaussian:
---
$\frac{1}{\sqrt{2\pi\sigma^{2}}}\,\exp(-\frac{(x-\mu)^{2}}{2\sigma^{2}})$
$\frac{\frac{(\sqrt{2\pi\sigma^{2}})^{1-q}}{\sqrt{q}}-1}{1-q}$ | $\frac{\left(\sqrt{2\pi\sigma^{2}}\exp\left(\frac{1}{2q}\right)\right)^{1-q}-1}{1-q}$
| Cauchy:
---
$\frac{1}{\pi(1+x^{2})}$
$\frac{\frac{\pi^{\frac{1}{2}-q}\Gamma\left(q-\frac{1}{2}\right)}{\Gamma(q)}-1}{1-q}$ | $\frac{\exp\left[\,\frac{2^{q-1}\pi^{q-\frac{1}{2}}\Gamma\left(\frac{q+1}{2}\right)^{2}\left(H_{\frac{q}{2}}-H_{\frac{q-1}{2}}+2\log(\pi)\right)}{q\Gamma\left(q-\frac{1}{2}\right)}\right]^{1-q}-1}{1-q}$
| Student:
---
$\frac{\Gamma((\nu+1)/2)}{\sqrt{\nu\pi}\Gamma(\nu/2)\sigma}\bigg{[}1+\frac{(x-\mu)^{2}}{\nu\sigma^{2}}\bigg{]}^{\frac{-\nu-1}{2}}$
$(\frac{\Gamma((\nu+1)/2)}{\sqrt{\nu\pi}\Gamma(\nu/2)})^{q}\sigma^{1-q}\sqrt{\nu\pi}\frac{\Gamma(\frac{vq+q-1}{2})}{\Gamma(\frac{vq+q}{2})}$ | $\frac{\left(\frac{\Gamma((\nu+1)/2)}{\sqrt{\nu\pi}\Gamma(\nu/2)\sigma}\right)^{1-q}\exp\left[\frac{\sigma}{2}\frac{(\nu+1)}{\pi^{1/2}}\bigg{[}\psi(\frac{\nu q+q-1}{2})-\psi(\frac{\nu q+q}{2})\bigg{]}\right]^{1-q}-1}{1-q}$
Table 1: Comparison of continuous Tsallis entropy and hybrid entropy for
several distributions. These include distributions with finite support
(triangular, Beta), on the positive half-plane (exponential, Gamma) and on the
whole real axis (Gaussian, Cauchy, Student). The resulting entropies are given
in terms of elementary functions and certain special functions, including
gamma function $\Gamma(z)$, beta function $B(\alpha,\beta)$ digamma function
$\psi(z)$ and harmonic numbers $H_{n}$. Definitions and properties of these
functions can be found, e.g. in Refs. [37, 38].

##  4 Hybrid divergence and Fisher metric

Divergence, or relative entropy defines a relative distance of distribution
$P$ from the underlying distribution $P_{0}$ [41, 42]. While entropy is
expressed as a functional of the probability density function
$p=\frac{\mathrm{d}P}{\mathrm{d}\mu}$ on a measurable space with measure
$\mu$, divergence is a functional of _Radon-Nikodym derivative_
$\frac{\mathrm{d}P}{\mathrm{d}P_{0}}$ on a space with measure $P_{0}$.
Therefore, _hybrid divergence_ can be defined straightforwardly as

$\mathcal{D}_{q}(P||P_{0})=\mathcal{D}_{q}\left(\frac{\mathrm{d}P}{\mathrm{d}P_{0}}\right)_{P_{0}}=\ln_{q}\exp\left(-\int\rho_{q}\\!\left(\frac{\mathrm{d}P}{\mathrm{d}P_{0}}\right)\,\ln\\!\left(\frac{\mathrm{d}P}{\mathrm{d}P_{0}}\right)\mathrm{d}P_{0}\right)$
(35)

where $\rho_{q}\\!\left(\frac{\mathrm{d}P}{\mathrm{d}P_{0}}\right)$ is the
_relative escort distribution_

$\rho_{q}\\!\left(\frac{\mathrm{d}P}{\mathrm{d}P_{0}}\right)=\frac{\left(\frac{\mathrm{d}P}{\mathrm{d}P_{0}}\right)^{q}}{\int\left(\frac{\mathrm{d}P}{\mathrm{d}P_{0}}\right)^{q}\mathrm{d}P_{0}}\,.$
(36)

If there exist probability density functions $p$, resp. $p_{0}$, it is
possible to rewrite the hybrid divergence as

$\mathcal{D}_{q}(p||p_{0})=\ln_{q}\exp\left(-\frac{\int
p^{q}p_{0}^{1-q}\ln\left(\frac{p}{p_{0}}\right)\mathrm{d}\mu}{\int
p^{q}p_{0}^{1-q}\mathrm{d}\mu}\right)$ (37)

Naturally, the hybrid divergence is well defined for $q\geq\frac{1}{2}$, while
for $q<\frac{1}{2}$ it is not always positive for $p\neq p_{0}$.

Let us focus on a well-known connection between the entropies and the Riemann
geometry on space of parametric probability distributions given by Fisher
metric [43]. Let us consider a parametric family of distributions defined by
parametric manifold $\mathcal{I}_{\theta}\subset\mathds{R}^{n}$:

$\mathcal{F}_{\theta}=\\{p({\bf x};\theta)|\,{\bf x}\in
M,\theta\in\mathcal{I}_{\theta}\\}\,.$ (38)

We define a pseudo-distance measure by symmetrization of hybrid divergence

$d_{q}(p_{1},p_{2})=\frac{\mathcal{D}_{q}(p||p_{0})+\mathcal{D}_{q}(p_{0}||p)}{2}\,.$
(39)

It is necessary to mention that $d_{q}(p_{1},p_{2})$ is not a proper distance
measure, because it does not obey the triangle inequality. Nevertheless, for
very close distributions it defines a metric tensor, because for $p_{1}({\bf
x})=p({\bf x},\theta)$ and $p_{2}({\bf x})=p({\bf x},\theta+\mathrm{d}\theta)$
we get that $d_{q}(p(\theta),p(\phi))\equiv d_{q}(\theta,\phi)$ is equal to

$\displaystyle d_{q}(\theta,\theta+\mathrm{d}\theta)$ $\displaystyle=$
$\displaystyle\frac{1}{2!}\sum_{i,j}\left[\frac{\partial^{2}d_{q}(\theta,\phi)}{\partial\theta_{i}\partial\theta_{j}}\right]_{\theta=\phi}\mathrm{d}\theta_{i}\mathrm{d}\theta_{j}+\mathcal{O}(\theta^{3})\,$
(40) $\displaystyle=$
$\displaystyle\frac{1}{2!}\sum_{i,j}g_{ij}^{q}(\theta)\mathrm{d}\theta_{i}\mathrm{d}\theta_{j}+\mathcal{O}(\theta^{3})\,.$

Note that the absolute term vanishes, because $d_{q}(\theta,\theta)=0$ and
linear term vanishes, because $d_{q}(\theta,\phi)$ is minimal for
$\theta=\phi$. The metric tensor is, for $q=1$, i.e., for Kullback-Leibler
divergence, equal to $g_{ij}^{1}=-F_{ij}(\theta)$, where $F_{ij}$ is the
Fisher information matrix

$F_{ij}(\theta)=E\left[\frac{\partial\ln
p(x,\theta)}{\partial\theta_{i}}\frac{\partial\ln
p(x,\theta)}{\partial\theta_{j}}\right]=\int\frac{1}{p(x,\theta)}\frac{\partial
p(x,\theta)}{\partial\theta_{i}}\frac{\partial
p(x,\theta)}{\partial\theta_{j}}\,\mathrm{d}x.$ (41)

After a straightforward calculation, we get that for hybrid divergence is the
metric tensor equal to

$g_{ij}^{q}(\theta)=(1-2q)\,F_{ij}(\theta)=(2q-1)\,g_{ij}^{1}(\theta)\,.$ (42)

It is necessary to note a few comments here. First, for $q=\frac{1}{2}$ is the
metric identically equal to zero. This is caused by the fact the second
derivative of $\mathcal{D}_{\frac{1}{2}}$ is equal to zero for
$p=\\{\frac{1}{n},\dots,\frac{1}{n}\\}$. Thus, it is impossible to recognize
two very close distributions from each other when $\mathcal{D}_{\frac{1}{2}}$
is used. Second, the metric is different from Rényi, Tsallis and Sharma-Mittal
metric of order $q$, because all are equal to $qg_{ij}^{1}(\theta)$ [44, 45].
The factor $(2q-1)$ is characteristic quantity for the whole hybrid entropy,
obtained by combination of $q$-non-extensivity and $q$-escort averaging.
Consequently, in the regime of close distributions, it is possible to relate
the hybrid entropy parameter $q_{\mathcal{D}}$ with Rényi entropy parameter
$q_{\mathcal{R}}$, resp. Tsallis entropy parameter $q_{\mathcal{S}}$ as

$q_{\mathcal{R}}=q_{\mathcal{S}}=2q_{\mathcal{D}}-1\qquad\Rightarrow\qquad
q_{\mathcal{D}}=\frac{q_{\mathcal{R}}+1}{2}=\frac{q_{\mathcal{S}}+1}{2}\,.$
(43)

These conclusions motivate us to slightly modify the class of hybrid in order
to obtain an average between Tsallis and Rényi entropy.

##  5 Average hybrid entropy

Results of the last section motivate us to introduce a slightly different
class of entropies, which would correspond to an average between Rényi and
Tsallis entropy, whereas hybrid entropy itself rather corresponds to
combination of non-extensivity and escort averaging. Following Eq. (43), we
introduce _average hybrid entropy_ as

$\mathcal{A}_{q}(p)=\mathcal{D}_{\frac{q+1}{2}}(p)\,.$ (44)

This entropy has many interesting properties. First, it is properly defined
for $q>0$, similarly to Tsallis and Rényi. Moreover its Fisher metric is now
the same as in the case of Rényi and Tsallis. Naturally, for $q\rightarrow 1$,
it boils down to Shannon entropy. Average hybrid entropy is concave for $q\leq
1$, while for $q>1$ is only Schur-concave (similarly to Rényi entropy).

Actually, the factor $\frac{q+1}{2}$ plays a special role in $q$-deformed
calculus. One can easily see that

$x\oplus_{\frac{q+1}{2}}y=x+y+\left(1-\frac{q+1}{2}\right)xy=x+y+\frac{1-q}{2}xy=2\left(\frac{x}{2}\oplus_{q}\frac{y}{2}\right)\,.$
(45)

This means that the non-extensivity of average entropy can be expressed
through non-extensivity of order $q$, but for rescaled quantities. Let us note
that the non-extensivity parameter $\frac{q+1}{2}$ is actually an average
between Tsallis non-extensivity parameter $q$ and Rényi (non)-extensivity
parameter $1$. Additionally, deformed logarithm appearing in the Kolmogorov-
Nagumo function can be for average hybrid entropy expressed as

$\ln_{\frac{q+1}{2}}(x)=\frac{x^{1-(q+1)/2}-1}{1-\frac{q+1}{2}}=2\frac{x^{\frac{1-q}{2}}-1}{1-q}=2\ln_{q}(\sqrt{x})\,.$
(46)

The deformed logarithm plays also role in the expression for maximal entropy.
It is possible to show that

$\ln_{q}n\geq\ln_{\frac{q+1}{2}}n\geq\ln n\quad\mathrm{for}\ q\leq
1,\qquad\qquad\ln_{q}n\leq\ln_{\frac{q+1}{2}}n\leq\ln n\quad\mathrm{for}\
q\geq 1\,.$ (47)

As a consequence, maximum of average hybrid is always between maxima of
Tsallis and Rényi entropy and can be considered as their average.

##  6 Conclusions

Hybrid entropy represents an conceptual overlap between Tsallis entropy and
Rényi entropy. Naturally, it shares many properties of both entropies: non-
extensivity of Tsallis entropy and related upper bound, Shur-concavity, which
appears at Rényi entropy and much more. On the other hand, some properties are
conceptually different, starting from accessible values of $q$ ($q\geq 1/2$),
going through functional form of MaxEnt distribution, which can be expressed
in terms of Lambert function (for Tsallis and Rényi we obtain well-known
$q$-deformed Gaussian distributions), and finally completely different
properties of continuous hybrid entropy and hybrid divergence. The latter
enables to establish connection to Fisher metric, which plays an important
role in description of parametric probability distributions. Fisher metric
obtained from hybrid divergence is conceptually different from Rényi, Tsallis
and Sharma-Mittal Fisher metric. As a result, we can identify the relation
between hybrid entropy parameter $q_{\mathcal{D}}$ and Tsallis/Rényi entropy
parameter $q_{\mathcal{S}}$, resp. $q_{\mathcal{R}}$. As a result, it is
possible to define the average hybrid entropy, which main properties can be
interpreted as an average between Tsallis entropy and Rényi entropy. All these
results point to potential usefulness of (average) hybrid entropy in
mathematical, physical or statistical applications. For a given entropy, it is
possible to find a so-called Cramér-Rao bound [45], which connects the score
function of a parametric distribution with inverse Fisher information obtained
by the prescribed entropy functional. By utilization of hybrid entropy one can
obtain a more precise bound for the estimations of model parameters. These
possible applications are a subject of ongoing research.

##  7 Acknowledgements

We acknowledge helpful conversations with Petr Jizba. J.K. was supported by
the GAČR, grant No. GA14-07983S.

##  Appendix A Continuous hybrid entropy for $\varepsilon$-skew exponential
power distribution

In this appendix, we calculate the hybrid entropy for recently introduced
$\varepsilon$-skew exponential power distribution (ESEP). The distribution was
originally introduced in [39, 40] as

$p(x;\mu,\sigma,\alpha,\beta,\eta,\varepsilon)=\frac{\alpha\beta}{2\sigma\eta^{1/\alpha}\Gamma(\frac{1}{\alpha\beta})}exp\\{-\frac{|x-\mu|^{\alpha\beta}}{\eta^{\beta}(\sigma(1-sign(x-\mu)\varepsilon))^{\alpha\beta}}\\},\
\ x\in\mathds{R}$ (48)

where $\alpha,\beta,\eta,\sigma>0$, $\mu\in\mathbb{R}$ and
$\varepsilon\in[-1,1]$. Parameters $\mu$ and $\sigma$ are location and scale
parameters, respectively. Parameters $\alpha$ and $\beta$ are shape parameters
that control the shape of peakedness. Parameter $\eta$ is a parameter
describing the underlying kernel of normal or Laplace distribution. It can
also be considered as a scale variant parameter. For $\alpha=2,\beta=1$ and
$\eta=2$, we get the $\varepsilon$-skew form for the normal distribution
proposed by [46]. ESEP is considered not only as a generalized version of
normal distribution and its $\varepsilon$-skew form but also it is a class of
the exponential power distributions. Since there are parameters that control
the shape of density, it is likely to find more applications in statistics.

Let us have $\varepsilon$-skew exponential power distribution. In order to
calculate hybrid entropy of ESEP distribution, we need to express following
integrals:

$\displaystyle
I_{ESEP_{N}}=\int_{-\infty}^{\infty}p(x;\alpha,\beta,\eta,\varepsilon)^{q}\ln[p(x;\alpha,\beta,\eta,\varepsilon)]\mathrm{d}x$
$\displaystyle=\frac{2\eta^{(1/\alpha)(1-q)}\sigma^{1-q}(\alpha\beta)^{q-1}\Gamma(\frac{1}{\alpha\beta
q})log(c)}{q[2\Gamma(\frac{1}{\alpha\beta})]^{q}}$
$\displaystyle-\bigg{[}\frac{(\alpha\beta)^{q-1}\eta^{\beta+1/\alpha(1-q)}\sigma^{\alpha\beta+1-q}}{q[2\Gamma(\frac{1}{\alpha\beta})]^{q}}\Gamma(\frac{1}{q}+\frac{1}{\alpha\beta
q})\bigg{]}\bigg{[}(1+\varepsilon)^{\alpha\beta+1}+(1-\varepsilon)^{\alpha\beta+1}\bigg{]}$

where $c$ is the normalizing constant,
$c=\frac{\alpha\beta}{2\sigma\eta^{1/\alpha}\Gamma(\frac{1}{\alpha\beta})}$.
Similarly,

$I_{ESEP_{D}}=\int_{-\infty}^{\infty}p(x;\alpha,\beta,\eta,\varepsilon)^{q}\mathrm{d}x=\frac{\bigg{[}2\sigma\eta^{1/\alpha}(\alpha\beta)^{-1}\bigg{]}^{1-q}}{\Gamma^{q}(\frac{1}{\alpha\beta})q}\Gamma(\frac{1}{\alpha\beta
q}).$

Finally, the hybrid entropy of ESEP distribution can be expressed with help of
$I_{ESEP_{N}}$ and $I_{ESEP_{D}}$ as

$\mathcal{D}_{q}=\ln_{q}exp\left(-\frac{I_{ESEP_{N}}}{I_{ESEP_{D}}}\right)=\ln_{q}\left(c\exp\left[\frac{(\eta\sigma)^{\beta}\sigma^{\alpha}\Gamma(\frac{1}{q}(1+\frac{1}{\alpha\beta}))[(1+\varepsilon)^{\alpha\beta+1}+(1-\varepsilon)^{\alpha\beta+1}]}{2\Gamma(\frac{1}{\alpha\beta
q})}\right]\right)$ (49)

Hybrid entropy depends on all parameters in the $\varepsilon-$skew exponential
power distribution, except the location parameter, which is similar to other
distributions, see e.g. [47].

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# Multilingual Pre-Trained Transformers and Convolutional NN Classification
Models for Technical Domain Identification

Suman Dowlagar
LTRC
IIIT-Hyderabad
suman.dowlagar@
research.iiit.ac.in
&Radhika Mamidi
LTRC
IIIT-Hyderabad
radhika.mamidi@
iiit.ac.in

###### Abstract

In this paper, we present a transfer learning system to perform technical
domain identification on multilingual text data. We have submitted two runs,
one uses the transformer model BERT, and the other uses XLM-ROBERTa with the
CNN model for text classification. These models allowed us to identify the
domain of the given sentences for the ICON 2020 shared Task, TechDOfication:
Technical Domain Identification. Our system ranked the best for the subtasks
1d, 1g for the given TechDOfication dataset.

##  1 Introduction

Automated technical domain identification is a categorization/classification
task where the given text is categorized into a set of predefined domains. It
is employed in tasks like Machine Translation, Information Retrieval, Question
Answering, Summarization, and so on.

In Machine Translation, Summarization, Question Answering, and Information
Retrieval, the domain classification model will help leverage the contents of
technical documents, select the appropriate domain-dependent resources, and
provide personalized processing of the given text.

Technical domain identification comes under text classification or
categorization. Text classification is one of the fundamental tasks in the
field of NLP. Text classification is the process of identifying the category
where the given text belongs. Automated text classification helps to organize
unstructured data, which can help us gather insightful information to make
future decisions on downstream tasks.

Traditional text classification approaches mainly focus on feature engineering
techniques such as bag-of-words and classification algorithms Yang (1999).
Nowadays, the sate-of-the-art results on text classification are achieved by
various NNs such as CNN Kim (2014), LSTM Hochreiter and Schmidhuber (1997),
BERT Adhikari et al. (2019), and Text GCN Adhikari et al. (2019). Attention
mechanisms Vaswani et al. (2017) have been introduced in these models, which
increased the representativeness of the text for better classification.
Transformer models such as BERT Devlin et al. (2018) uses the attention
mechanism that learns contextual relations between words or sub-words in a
text. Text GCN Yao et al. (2019) uses a graph-convolutional network to learn a
heterogeneous word document graph on the whole corpus, which helped classify
the text. However, of all the deep learning approaches, transformer models
provided SOTA results in text classification.

In this paper, We present two approaches for technical domain identification.
One approach uses the pre-trained Multilingual BERT model, and the other uses
XLM-ROBERTa with CNN model.

The rest of the paper is structured as follows. Section 2 describes our
approach in detail. In Section 3, we provide the analysis and evaluation of
results for our system, and Section 4 concludes our work.

##  2 Our Approach

Here we present two approaches for the TechDOfication task.

###  2.1 BERT for TechDOfication

Figure 1: The architecture of the BERT model for sentence classification.

In the first approach, we use the pre-trained multilingual BERT model for
domain identification of the given text. Bidirectional Encoder Representations
from Transformers (BERT) is a transformer encoder stack trained on the large
corpora. Like the vanilla transformer model Vaswani et al. (2017), BERT takes
a sequence of words as input. Each layer applies self-attention, passes its
results through a feed-forward network, and then hands it off to the next
encoder. The BERT configuration model takes a sequence of words/tokens at a
maximum length of 512 and produces an encoded representation of dimensionality
768.

The pre-trained multilingual BERT models have a better word representation as
they are trained on a large multilingual Wikipedia and book corpus. As the
pre-trained BERT model is trained on generic corpora, we need to finetune the
model for the given domain identification tasks. During finetuning, the pre-
trained BERT model parameters are updated.

In this architecture, only the [CLS] (classification) token output provided by
BERT is used. The [CLS] output is the output of the 12th transformer encoder
with a dimensionality of 768. It is given as input to a fully connected neural
network, and the softmax activation function is applied to the neural network
to classify the given sentence.

###  2.2 XLM-ROBERTa with CNN for TechDOfication

Figure 2: The architecture of the XLM-ROBERTa with CNN for sentence
classification.

XLM-ROBERTa Conneau et al. (2019) is a transformer-based multilingual masked
language model pre-trained on the text in 100 languages, which obtains state-
of-the-art performance on cross-lingual classification, sequence labeling, and
question answering. XLM-ROBERTa improves upon BERT by adding a few changes to
the BERT model such as training on a larger dataset, dynamically masking out
tokens compared to the original static masking, and uses a known pre-
processing technique (Byte-Pair-Encoding) and a dual-language training
mechanism with BERT in order to learn better relations between words in
different languages. The given model is trained for the language modeling
task, and the output is of dimensionality 768. It is given as input to a CNN
Kim (2014) because convolution layers can extract better data representations
than Feed Forward layers, which indirectly helps in better domain
identification.

##  3 Experiment

This section presents the datasets used, the task description, and two models’
performance on technical domain identification. We also include our
implementation details and error analysis in the subsequent sections.

###  3.1 Dataset

We used the dataset provided by the organizers of TechDOfication ICON-2020.
There are two subtasks, one is coarse-grained, and the other is fine-grained.
The coarse-grained TechDOfication dataset contains sentences about Chemistry,
Communication Technology, Computer Science, Law, Math, and Physics domains in
different languages such as English, Bengali, Gujarati, Hindi, Malayalam,
Marathi, Tamil, and Telugu. Whereas the fine-grained English dataset focuses
on the Computer-Science domain with sub-domain labels as Artificial
Intelligence, Algorithm, Computer Architecture, Computer Networks, Database
Management system, Programming, and Software Engineering.

###  3.2 Implementation

For the implementation, we used the transformers library provided by
HuggingFace111https://huggingface.co/. The HuggingFace contains the pre-
trained multilingual BERT, XLM-ROBERTa, and other models suitable for
downstream tasks. The pre-trained multilingual BERT model used is _“bert-base-
multilingual-cased”_ and pre-trained XLM-R model used is _“xlm-roberta-base”_.
We programmed the CNN architecture as given in the paper Kim (2014). We used
the PyTorch library that supports GPU processing for implementing deep neural
nets. The BERT models were run on the Google Colab and Kaggle GPU notebooks.
We trained our classifier with a batch size of 128 for 10 to 30 epochs based
on our experiments. The dropout is set to 0.1, and the Adam optimizer is used
with a learning rate of 2e-5. We used the hugging face transformers pre-
trained BERT tokenizer for tokenization. We used the
BertForSequenceClassification module provided by the HuggingFace library
during finetuning and sequence classification for the multilingual-BERT based
approach.

###  3.3 Baseline models

Here, we compared the BERT model with other machine learning algorithms.

#### SVM with TF_IDF text representation

We chose Support Vector Machines (SVM) with TF_IDF text representation for
technical domain identification. SVM classifier and TF_IDF vector
representation is obtained from the scikit-learn library Pedregosa et al.
(2011).

#### CNN:

Convolutional Neural Network Kim (2014). We explored CNN-non-static, which
uses pre-trained word embeddings.

###  3.4 Results

The results are tabulated in Table 1. We evaluated the performance of the
method using macro F1. The multilingual-BERT model performed well when
compared to the other SVM with TF-IDF and CNN models. Given all the languages,
we have observed an increase of 7 to 25% in classification metrics for BERT
compared to the baseline SVM classifier, it showed a 2 to 5% increase in
classification metrics compared to the CNN classifier on the validation data.
On the test data, multilingual BERT showed better performance in subtasks 1a,
1b, 1c, 1h and 2a whereas XLM-ROBERTa with CNN showed better performance in
the subtasks 1d, 1e, 1f, 1g. This increase in classification metrics is due to
the transformer model’s and convolutional NN’s capability, which learned
better text representations from the generic data than other models.

| Classifier Models
---|---
Dataset | Validation | Test
| SVM | CNN | M-Bert | XLM-R+CNN | M-Bert | XLM-R+CNN
English subtask-1a | 81.48 | 83.05 | 88.87 | 87.09 | 79.84 | 73.57
Bengali subtask-1b | 66.35 | 85.78 | 86.81 | 85.71 | 80.35 | 78.17
Gujarati subtask-1c | 69.63 | 86.27 | 87.21 | 86.89 | 68.67 | 66.73
Hindi subtask-1d | 58.21 | 81.03 | 83.40 | 82.13 | 59.89 | 60.44
Malayalam subtask-1e | 80.60 | 92.51 | 94.72 | 93.40 | 34.47 | 34.86
Marathi subtask-1f | 73.32 | 86.89 | 87.42 | 86.37 | 59.52 | 59.89
Tamil subtask-1g | 65.95 | 85.75 | 87.50 | 86.54 | 49.24 | 51.34
Telugu subtask-1h | 71.98 | 88.07 | 90.28 | 89.43 | 67.17 | 62.26
English subtask-2a | 70.24 | 72.53 | 77.36 | 76.77 | 78.98 | 78.07
Table 1: macro F1 on validation and test data for all the subtasks

##  4 Error Analysis

The multilingual-BERT model’s confusion matrix is compared with the poorly
performed model for languages, Hindi, and Tamil languages are shown in Figure
3. We chose Hindi and Tamil languages because, here, the difference in
performance is more significant. For the Hindi subtask, the SVM classifier
confused between “cse”, “com_tech”, and “mgmt” labels, whereas the BERT model
performed better. For the Tamil subtask, the SVM classifier confused between
“com_tech” and “mgmt” labels, whereas the BERT model performed better than the
other models. This is because both the approaches (pre-trained multilingual-
BERT and pre-trained XLM-ROBERTa with CNN) learned better representation of
the above data than the other models that helped in technical document
identification.

(a)

(b)

(c)

(d)

Figure 3: Confusion matrix on the given validation data for the Hindi and
Tamil languages

##  5 Conclusion and Future work

We used pre-trained bi-directional encoder representations using multilingual-
BERT and XLM-ROBERTa with CNN technical domain identification for English,
Bengali, Gujarati, Hindi, Malayalam, Marathi, Tamil, and Telugu languages. We
compared the approaches with the baseline methods. Our analysis showed that
pre-trained multilingual BERT and XLM-ROBERTa with CNN models and finetuning
it for text classification tasks showed an increase in macro F1 score and
accuracy metrics compared to baseline approaches.

Some datasets are large, like for the Hindi, Tamil, and Telugu, we can train
the BERT and XLM-ROBERTa models from scratch and consider its hidden layer
representation, and concatenate this with the representation of the pre-
trained model. It might help to classify the datasets even better.

## References

  * Adhikari et al. (2019) Ashutosh Adhikari, Achyudh Ram, Raphael Tang, and Jimmy Lin. 2019.  Docbert: Bert for document classification.  _arXiv preprint arXiv:1904.08398_.
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  * Devlin et al. (2018) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018.  Bert: Pre-training of deep bidirectional transformers for language understanding.  _arXiv preprint arXiv:1810.04805_.
  * Hochreiter and Schmidhuber (1997) Sepp Hochreiter and Jürgen Schmidhuber. 1997.  Long short-term memory.  _Neural computation_ , 9(8):1735–1780.
  * Kim (2014) Yoon Kim. 2014.  Convolutional neural networks for sentence classification.  _arXiv preprint arXiv:1408.5882_.
  * Pedregosa et al. (2011) F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. 2011.  Scikit-learn: Machine learning in Python.  _Journal of Machine Learning Research_ , 12:2825–2830.
  * Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017.  Attention is all you need.  In _Advances in neural information processing systems_ , pages 5998–6008.
  * Yang (1999) Yiming Yang. 1999.  An evaluation of statistical approaches to text categorization.  _Information retrieval_ , 1(1-2):69–90.
  * Yao et al. (2019) Liang Yao, Chengsheng Mao, and Yuan Luo. 2019.  Graph convolutional networks for text classification.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 33, pages 7370–7377.


<EMAIL_ADDRESS>
# Automated scheduler for the SOXS instrument: design and performance

Laura Asquini  INAF – Osservatorio Astronomico di Brera-Merate, via E. Bianchi
46, I-23807 Merate (LC), Italy;  Dipartimento di Scienza e Alta Tecnologia,
Università dell’Insubria, via Valleggio 11, I-22100 Como, Italy  Marco Landoni
INAF – Osservatorio Astronomico di Brera-Merate, via E. Bianchi 46, I-23807
Merate (LC), Italy;  Dave Young  Astrophysics Research Centre, School of
Mathematics and Physics, Queen’s University Belfast, Belfast BT7 1NN, UK
Laurent Marty  INAF - Osservatorio Astronomico di Capodimonte, Salita
Moiariello 16, Naples- Italy  Stephen J. Smartt  University of Oxford, Keble
Road, Oxford OX1 3RH  Sergio Campana  INAF – Osservatorio Astronomico di
Brera-Merate, via E. Bianchi 46, I-23807 Merate (LC), Italy;  Riccardo Claudi
INAF – Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5,
I-35122, Padua, Italy  Pietro Schipani  INAF - Osservatorio Astronomico di
Capodimonte, Salita Moiariello 16, Naples- Italy  Jani Achrén  Incident Angle
Oy, Capsiankatu 4 A 29, FI-20320 Turku, Finland  Matteo Aliverti  INAF –
Osservatorio Astronomico di Brera-Merate, via E. Bianchi 46, I-23807 Merate
(LC), Italy;  José A. Araiza-Durán  INAF - Osservatorio Astrofisico di
Arcetri.Largo Enrico Fermi 5, 50125 Florence - ITALY  Iair Arcavi  Tel Aviv
University, Department of Astrophysics, 69978 Tel Aviv, Israel  Federico
Battaini  INAF – Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio
5, I-35122, Padua, Italy  Andrea Baruffolo  INAF – Osservatorio Astronomico di
Padova, Vicolo dell’Osservatorio 5, I-35122, Padua, Italy  Sagi Ben-Ami
Weizmann Institute of Science, Herzl St 234, Rehovot, 7610001, Israel  Andrea
Bianco  INAF – Osservatorio Astronomico di Brera-Merate, via E. Bianchi 46,
I-23807 Merate (LC), Italy;  Alex Bichkovsky  Weizmann Institute of Science,
Herzl St 234, Rehovot, 7610001, Israel  Anna Brucalassi  INAF - Osservatorio
Astrofisico di Arcetri.Largo Enrico Fermi 5, 50125 Florence - ITALY  Rachel
Bruch  Weizmann Institute of Science, Herzl St 234, Rehovot, 7610001, Israel
Giulio Capasso  INAF - Osservatorio Astronomico di Capodimonte, Salita
Moiariello 16, Naples- Italy  Enrico Cappellaro  INAF – Osservatorio
Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122, Padua, Italy
Mirko Colapietro  INAF - Osservatorio Astronomico di Capodimonte, Salita
Moiariello 16, Naples- Italy  Rosario Cosentino  INAF – Osservatorio
Astrofisico di Catania, Via S. Sofia 78 30, I-95123 Catania, Italy  Francesco
D’Alessio  INAF – Osservatorio Astronomico di Roma, Via Frascati 33, I-00078
M. Porzio Catone, Italy  Paolo D’Avanzo  INAF – Osservatorio Astronomico di
Brera-Merate, via E. Bianchi 46, I-23807 Merate (LC), Italy;  Massimo Della
Valle  INAF - Osservatorio Astronomico di Capodimonte, Salita Moiariello 16,
Naples- Italy  Sergio D’Orsi  INAF - Osservatorio Astronomico di Capodimonte,
Salita Moiariello 16, Naples- Italy  Rosario Di Benedetto  INAF – Osservatorio
Astrofisico di Catania, Via S. Sofia 78 30, I-95123 Catania, Italy  Simone Di
Filippo  INAF – Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio
5, I-35122, Padua, Italy  Avishay Gal-Yam  Weizmann Institute of Science,
Herzl St 234, Rehovot, 7610001, Israel  Matteo Genoni  INAF – Osservatorio
Astronomico di Brera-Merate, via E. Bianchi 46, I-23807 Merate (LC), Italy;
Marcos Hernandez  FGG-INAF, TNG, Rambla J.A. Fernández Pérez 7, E-38712 Breña
Baja (TF), Spain  Ofir Hershko  Weizmann Institute of Science, Herzl St 234,
Rehovot, 7610001, Israel  Jari Kotilainen  Finnish Centre for Astronomy with
ESO (FINCA), FI-20014 University of Turku, Finland  Hanindyo Kuncarayakti
Finnish Centre for Astronomy with ESO (FINCA), FI-20014 University of Turku,
Finland  Gianluca Li Causi  INAF – Osservatorio Astronomico di Roma, Via
Frascati 33, I-00078 M. Porzio Catone, Italy  Seppo Mattila  Finnish Centre
for Astronomy with ESO (FINCA), FI-20014 University of Turku, Finland  Matteo
Munari  INAF – Osservatorio Astrofisico di Catania, Via S. Sofia 78 30,
I-95123 Catania, Italy  Giorgio Pariani  INAF – Osservatorio Astronomico di
Brera-Merate, via E. Bianchi 46, I-23807 Merate (LC), Italy;  Hector Pérez
Ventura  FGG-INAF, TNG, Rambla J.A. Fernández Pérez 7, E-38712 Breña Baja
(TF), Spain  Giuliano Pignata  Instituto de Alta Investigación, Universidad de
Tarapacá, Arica, Casilla 7D, Chile  Kalyan Radhakrishnan  INAF – Osservatorio
Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122, Padua, Italy
Michael Rappaport  Weizmann Institute of Science, Herzl St 234, Rehovot,
7610001, Israel  Davide Ricci  INAF – Osservatorio Astronomico di Padova,
Vicolo dell’Osservatorio 5, I-35122, Padua, Italy  Marco Riva  INAF –
Osservatorio Astronomico di Brera-Merate, via E. Bianchi 46, I-23807 Merate
(LC), Italy;  Adam Rubin  Weizmann Institute of Science, Herzl St 234,
Rehovot, 7610001, Israel  Bernardo Salasnich  INAF – Osservatorio Astronomico
di Padova, Vicolo dell’Osservatorio 5, I-35122, Padua, Italy  Salvatore
Savarese  INAF - Osservatorio Astronomico di Capodimonte, Salita Moiariello
16, Naples- Italy  Maximilian Stritzinger  Aarhus University, Ny Munkegade
120, D-8000 Aarhus, Denmark  Salvatore Scuderi  INAF – Osservatorio
Astrofisico di Catania, Via S. Sofia 78 30, I-95123 Catania, Italy  Fabrizio
Vitali  INAF – Osservatorio Astronomico di Roma, Via Frascati 33, I-00078 M.
Porzio Catone, Italy  Ricardo Zanmar Sanchez  INAF – Osservatorio Astronomico
di Padova, Vicolo dell’Osservatorio 5, I-35122, Padua, Italy

###### Abstract

We present the advancements in the development of the scheduler for the Son Of
X-shooter (SOXS, 1, 2) instrument at the ESO-NTT 3.58-m telescope in La Silla,
Chile. SOXS is designed as a single-object spectroscopic facility and features
a high-efficiency spectrograph with two arms covering the spectral range of
350-2000 nm and a mean resolving power of approximately R=4500. Its primary
purpose is to conduct UV-visible and near-infrared follow-up observations of
astrophysical transients, drawing from a broad pool of targets accessible
through the streaming services of wide-field telescopes, both current and
future, as well as high-energy satellites. The instrument is set to cater to
various scientific objectives within the astrophysical community, each
entailing specific requirements for observation planning, a challenge that the
observing scheduler must address. A notable feature of SOXS is that it will
operate at the European Southern Observatory (ESO) in La Silla, without the
presence of astronomers on the mountain. This poses a unique challenge for the
scheduling process, demanding a fully automated algorithm that is autonomously
interacting with the appropriate databases and the La Silla Weather API, and
is capable of presenting the operator not only with an ordered list of optimal
targets (in terms of observing constraints) but also with optimal backups in
the event of changing weather conditions. This requirement imposes the
necessity for a scheduler with rapid-response capabilities without
compromising the optimization process, ensuring the high quality of
observations and best use of the time at the telescope. We thus developed a
new highly available and scalable architecture, implementing API Restful
applications like Docker Containers, API Gateway, and Python-based Flask
frameworks. We provide an overview of the current state of the scheduler,
which is now ready for the approaching on-site testing during Commissioning
phase, along with insights into its web interface and preliminary performance
tests.

###### keywords:

ESO-NTT telescope – SOXS – Scheduling

##  1 INTRODUCTION

The Son Of X-Shooter (SOXS, 2) instrument is a single-object spectrograph with
a spectral resolution of approximately R=4,500, designed to observe the
southern sky from the New Technology Telescope (NTT) in La Silla, Chile. It
will operate within the ESO La Silla-Paranal Observatory environment without
an astronomer physically present on the mountain. Consequently, it
necessitates the development of a web-based software capable of autonomously
organizing and managing nightly observations, both in advance (for scientists’
approval and scrutiny) and in real-time. These processes must adhere to and
optimize observational constraints, while also periodically checking current
weather conditions on the mountain. Additionally, the scheduler must integrate
targets from both the SOXS Consortium and regular ESO proposals, respecting
the 50/50 Guaranteed Time of Observation agreement. The Targets of Opportunity
(ToOs) from the SOXS Consortium will be managed through the Marshall
application[3], while other ESO users will use the Phase 2 (P2) web page.
Throughout and at the end of each night, the scheduler needs to track the
instrument’s operations, including the percentage of completed observations,
hit/miss rate of transient events, and general observing conditions for each
target. It will automatically communicate with the official ESO Phase 2 tools,
creating and verifying Observing Blocks (OBs) from the targets in the standard
ESO format, and sending them to fill the Visitor Execution Sequence (VES)
based on target visibility, priority, and current weather conditions accessed
through the Astronomical Site Monitor (ASM) Weather API. We refer the reader
to previous works [4, 5] for the first presentation of the software and a
thorough description of its components. In this paper, we will focus on the
performance of the real-time night managing algorithm, which is the crucial
part of the nightly operations. The paper is organized as follows: in Section
2 we will briefly summarize the main features of the software. In Section 3 we
will describe the night-managing algorithm, while in Section 4 we will focus
on its performance. We draw our conclusions in Section 5.

##  2 Main Features

In this Section we summarize the building blocks of the software, that have
detailed description in previous works [4, 5].

The scheduler is implemented as a RESTful API using the Flask micro-framework,
serving as an intermediary between the Marshall and the ESO P2 system. OBs are
created via the SOXS scheduler and stored in a MySQL database. The core code
is adaptable for any ground-based instrument, requiring only two additional
files for instrument and telescope specifics. The scheduling algorithm is
written in Python 3, utilizing Astropy[6] libraries, particularly Astroplan,
to compute observational constraints and employ a priority scheduler. The
algorithm’s workflow is divided into two steps: the first processes the
computational workload and selects observable targets for the night, using
scripts for new and existing transients. The second step is the actual
scheduling process. The data stream starts with targets fed into the SOXS API
by the Marshall or ESO P2, which are then processed daily to ensure up-to-date
information. The scheduler creates an optimized schedule, which is sent to the
P2 API for validation and then to the VES. The system continuously requests
feedback on observation status from P2 and checks weather conditions via the
ASM API. If issues arise, the automatic night manager adjusts the schedule or
selects alternative OBs. Figure 1 shows the overall structure of the scheduler
and its data flow.

Figure 1: The overall workflow of the SOXS scheduler.

###  2.1 Filtering

Target filtering is essential to reduce the workload on the scheduler. Two
scripts were developed for this purpose: one processes new OBs, and the other
updates existing ones. These scripts operate continuously, day and night,
periodically checking the database for their respective OBs.

The algorithm, detailed in a previous paper, represents the night using 1D
arrays with five-minute time slots. Each target has an associated
observability array, where each slot is marked as zero (not observable) or one
(observable) based on the convolution of observing constraints. Additionally,
each target is graded on its likelihood of a good observation for the night,
using a system adapted from ESO’s observation planning.

The processing script evaluates each target for five consecutive nights, while
the updating script discards the past night and adds the fifth night from the
current day. This ensures a wide availability of targets for both the
scheduler and the scientific team at all times.

###  2.2 Scheduling

The scheduler works on three types of OB: ESO OBs (high-priority, coming from
the general ESO public), follow-up OBs (high-priority, coming from the SOXS
Consortium), and classification OBs (low-priority, targets that are
unclassified and are public, unless a scientist from the Consortium claims
them as follow-up). To manage the scientific prioritization of targets,
particularly for follow-up and ESO proposals, the built-in Astroplan[7]
priority scheduler is used as the first step to handle top-priority targets.
This scheduler optimizes observing times for each object, evaluating
constraints with a fine one-minute time resolution. While efficient, this
process is computationally intensive, so it is reserved for high-priority
targets.

The remaining time is managed with an algorithm called the Graded Gap Filler
(GGF[4]) using 1D boolean arrays representing the night in five-minute slots.
The scheduler fills gaps by finding the first free slot and matching it with
suitable observable targets from a pre-sorted list, based on their grades and
required slots. The algorithm selects the best candidate for each slot and
updates the night array accordingly, continuing until the night is full or no
suitable targets remain.

During an observing night, the scientific team can tweak the schedule through
a web interface but does not need direct access to the algorithms. At the
beginning of the night, or when the team approves a schedule, a new night is
opened via the interface. The scheduler proposes an optimized schedule,
respecting scientific priorities and observation-quality constraints. The team
can swap OBs with alternatives provided by the scheduler or manually add
targets from the “observable obs” page. Once the schedule is finalized and
approved, it synchronizes with P2 for verification, and the first OB is sent
to the Visitor Execution Sequence (VES), which is the list of ordered OB that
the observatory system checks for the night schedule.

Figure 2: The interface for the Science Team. It features various tools to aid
the SOXS scientists to tweak and perfect the schedule.

##  3 Night Managment

In this project, a key challenge for the scheduler is to autonomously monitor
and respond to real-time events, such as weather changes during an observing
night. This task is managed by the “Next OB” endpoint, operated by the
telescope instrument operator (TIO). The TIO requests a new OB from the
scheduler after each one is completed or fails. Upon receiving a request, the
scheduler checks current weather conditions via a weather API and verifies
time constraints before approving the next OB for observation and sending it
to the VES.

Three main event categories that might trigger schedule adjustments have been
identified:

  * •

Delays in executing observations;

  * •

Previously failed OBs;

  * •

Weather changes

To address these scenarios, three distinct “recovery loops” have been
implemented. Each recovery loop is an algorithm designed to handle specific
issues during the observing process. These loops are central to the “Next OB”
endpoint’s functionality. The approach aims to adhere closely to the initially
approved schedule, minimizing conflicts with the time share between users and
the Science Team’s decisions.

The process for handling OBs in the VES involves several checks and updates,
which are shown in Figure 3:

  * •

OB Status Check: The first check determines if the current OB in the VES is
marked as completed (status X) or failed (status A) in the P2 database. If the
OB is still in progress (status “+” for “Accepted” or “S” for “in
observation”), it is returned without further checks to avoid multiple-call
errors.

  * •

Status Update: The OB status is updated in the local database.

  * •

Time Check: A check is performed to ensure there are no major delays or
advances. If there are, the Delay Loop is triggered. It will reschedule the
remaining OBs, prioritizing the observation of high-priority targets, and
filling any gaps left.

  * •

Failed OB Check: If the schedule is on time, the presence of failed OBs is
checked. If there are failed OBs, the Failed OB Recovery Loop is triggered. It
will try and rearrange the remaining time, inserting the previously failed OBs
while still maintaining the high-priority ones.

  * •

Subsequent OB Check: If no OBs have failed, the next OB is identified, and its
status is checked to ensure it is the correct choice.

  * •

Weather Check: Current weather conditions are retrieved via the API and
checked against the OB requirements. If requirements are not met, the Weather
Check Loop is triggered. It will find a suitable OB for the current weather
conditions, to fill the gap left by the current candidate OB.

  * •

Failed OB Management: if a priority target OB fails, it is added to the
“failed obs” table, so that it will be checked again later for recovery.
Classification targets, which are interchangeable, are not added to this list.
If the schedule has undergone any recovery loops, the new schedule is saved,
and the previous one is archived in the “history schedule” table.

  * •

VES Update: If everything is in order, the VES is cleaned, and the current OB
is sent to the VES for observation.

Figure 3: The workflow of the “Next OB” algorithm.

##  4 Performance

To evaluate the overall quality of the resulting schedules, a three-month
simulation was conducted. This simulation incorporated artificial time
progression, simulated weather conditions, and introduced a thousand new
targets each night using recorded transient alerts from the ePESSTO
collaboration[3] in 2018. Targets were generated with randomized magnitudes
(not exceeding the maximum allowed magnitude of 21, valid for SOXS) to handle
1000 potentially observable new targets per night. Follow-up requests from the
SOXS Consortium and ESO users shared 50% of telescope time, averaging 10
requests per day.

Randomized weather conditions were based on ESO data for Cerro Paranal in the
Atacama desert. Unforeseen events, such as telescope malfunctions or
unregistered bad weather, were simulated as “delays” occurring approximately
every 20 OBs (1-2 days), ranging from 30 minutes to two hours. The simulation
automated scheduler functionalities, including target creation, database
cleanup, filtering processes, schedule creation, and night progression. The
stress-test was conducted under worst-case conditions, such as during Chilean
summer with shorter nights, starting January 7, 2018.

The simulation showed that, even under undesirable conditions (overbooking,
weather changes, frequent delays), the scheduler managed to optimize 54.4% of
requested targets, observing them within a day of creation. An additional
18.2% of OBs were recovered during night rearrangements and observed within 30
days, while 27.4% could not be recovered. These results highlight the
importance of scientific prioritization by the Science Team to ensure high-
priority OBs are observed on the scheduled night.

Regarding observation quality, the most significant metric was ”delta
airmass.” Approximately 90-95% of targets were observed within a 0.2 airmass
of their optimal position. The first schedule had a mean delta airmass of
0.1$\pm$0.2, which stabilized at 0.1$\pm$0.1 after night management,
maintaining consistency even with a heavier load on the GGF compared to the
Priority scheduler. The average response time for the process, from clicking
the next OB button to viewing it on the operator interface, was less than 3
seconds with a typical fiber optic internet connection.

##  5 Conclusions

The development and implementation of the scheduler for the SOXS system has
demonstrated significant robustness and efficiency, even under challenging
conditions. The scheduler successfully handles real-time events such as
weather changes through the “Next OB” endpoint, ensuring smooth operation
throughout the observing night. This capability is critical for maintaining
the integrity of the observation schedule despite unforeseen events.
Specificallt, the recovery loops (Rescheduling, Failed OB Recovery, and
Weather Check) effectively address delays and failed observations, maintaining
a high success rate for target observation. The three-month simulation
incorporated real-time weather data and a high volume of targets, and the
scheduler demonstrated its capability to manage heavy loads and adverse
conditions. Despite overbooking and frequent delays, the scheduler optimized
54.4% of the targets within a day and recovered an additional 18.2% within 30
days. The resulting observations consistently proved to be of high-quality,
with 90-95% of targets observed within a 0.2 airmass of their optimal
position. Finally, the average response time of the system was less than 3
seconds under normal internet conditions. This quick rectivity is crucial for
real-time observation adjustments and maintaining a seamless workflow.

## References

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