11institutetext: Departamento de Física e Astronomia, Faculdade de Ciências,
Universidade do Porto, Rua do Campo Alegre 687, PT4169-007 Porto, Portugal
22institutetext: Instituto de Astrofísica e Ciências do Espaço, Universidade
do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal 33institutetext:
DTx – Digital Transformation CoLab, Building 1, Azurém Campus, University of
Minho, PT4800-058 Guimarães, Portugal 44institutetext: Departamento de Física,
Faculdade de Ciências, Universidade de Lisboa, Edifício C8, Campo Grande,
PT1749-016 Lisboa, Portugal 55institutetext: Instituto de Astrofísica e
Ciências do Espaço, Universidade de Lisboa, OAL, Tapada da Ajuda, PT1349-018
Lisboa, Portugal

# Identifying type II quasars at intermediate redshift with few-shot learning
photometric classification

P. A. C. Cunha , Corresponding author, 1122<EMAIL_ADDRESS>A.
Humphrey 2233 J. Brinchmann 1122 S. G. Morais 1122 R. Carvajal 4455 J. M.
Gomes 22 I. Matute 4455 A. Paulino-Afonso 22

(Received ; accepted )

###### Abstract

Context. A sub-population of AGNs where the central engine is obscured are
known as type II quasars (QSO2s). These luminous AGNs have a thick and dusty
torus that obscures the accretion disc from our line of sight. Thus, their
special orientation allows for detailed studies of the AGN-host co-evolution.
Increasing the sample size of QSO2 sources in critical redshift ranges is
crucial for understanding the interplay of AGN feedback, the AGN-host
relationship, and the evolution of active galaxies.

Aims.  We aim to identify QSO2 candidates in the ‘redshift desert’ using
optical and infrared photometry. At this intermediate redshift range (i.e.
$1\leq z\leq 2$), most of the prominent optical emission lines in QSO2 sources
(e.g. CIV$\lambda 1549$; [OIII]$\lambda\lambda 4959,5008$) fall either outside
the wavelength range of the SDSS optical spectra or in particularly noisy
wavelength ranges, making QSO2 identification challenging. Therefore, we
adopted a semi-supervised machine learning approach to select candidates in
the SDSS galaxy sample.

Methods. Recent applications of machine learning in astronomy focus on
problems involving large data sets, with small data sets often being
overlooked. We developed a ‘few-shot’ learning approach for the identification
and classification of rare-object classes using limited training data (200
sources). The new AMELIA pipeline uses a transfer-learning based approach with
decision trees, distance-based, and deep learning methods to build a
classifier capable of identifying rare objects on the basis of an
observational training data set.

Results. We validated the performance of AMELIA by addressing the problem of
identifying QSO2s at $1\leq z\leq 2$ using SDSS and WISE photometry, obtaining
an F1-score above 0.8 in a supervised approach. We then used AMELIA to select
new QSO2 candidates in the ‘redshift desert’ and examined the nature of the
candidates using SDSS spectra, when available. In particular, we identified a
sub-population of [NeV]$\lambda 3426$ emitters at $z\sim 1.1$, which are
highly likely to contain obscured AGNs. We used X-ray and radio cross-matching
to validate our classification and investigated the performance of photometric
criteria from the literature showing that our candidates have an inherent
dusty nature. Finally, we derived physical properties for our QSO2 sample
using photoionisation models and verified the AGN classification using an SED
fitting.

Conclusions. Our results demonstrate the potential of few-shot learning
applied to small data sets of rare objects, in particular QSO2s, and confirms
that optical-IR information can be further explored to search for obscured
AGNs. We present a new sample of candidates to be further studied and
validated using multi-wavelength observations.

###### Key Words.:

obscured quasars – galaxies – photometry – spectroscopy – machine learning

##  1 Introduction

The study of the Universe around us is extremely dependent on our
understanding of galaxies since various key processes, such as star formation,
accretion, and feedback are present on galaxy scales (Silk & Rees 1998;
Ferrarese & Merritt 2000; Gebhardt et al. 2000; Fabian 2012; Hopkins et al.
2006). Active galactic nuclei (AGNs) – supermassive black holes (SMBHs)
powered by gas accretion (Lynden-Bell 1969) – are particularly relevant in
this context due to their often significant impact on the properties of their
host galaxies (e.g. Ciotti & Ostriker 1997; Kauffmann et al. 2003; Bower et
al. 2006; Constantin et al. 2008; Silverman et al. 2008; Pimbblet et al. 2013;
Porqueres et al. 2018; Soussana et al. 2020; Torrey et al. 2020; Harrison et
al. 2021; Ruffa et al. 2022; Elford et al. 2023).

From the panoply of currently known AGN classes, obscured AGNs – or type II
quasars (QSO2s) – are particularly useful for studying AGN hosts. According to
the unified AGN model (Antonucci 1993; Urry & Padovani 1995; Ramos Almeida &
Ricci 2017), the luminous central engine of QSO2s is hidden from view by what
is typically argued to be a fortuitously orientated dusty torus (e.g.
Kleinmann et al. 1988). This favourable geometry allows the host galaxy to be
studied in greater detail, compared to the case of unobscured quasars (QSO1s),
where the active nucleus often outshines the host galaxy. In some obscured
AGNs, infrared and sub-millimetre observations have hinted at a different
scenario in which the obscuration of the central engine is caused by a two-
component dusty structure with equatorial and polar components, with radiation
pressure playing an important rule (Hönig 2019; Stalevski et al. 2023, and
references therein). Thus, the selection and study of obscured AGNs are
essential for understanding the AGN population, the co-evolution of AGNs with
their host galaxies, and other open issues, such as the radiative efficiency
of SMBHs (Hickox & Alexander 2018).

Since their discovery in significant numbers in the Sloan Digital Sky Survey
(SDSS; Zakamska et al. 2004), QSO2s have been studied in various wavelengths
and with various observational techniques. They typically have an optical
emission line spectrum consisting of narrow lines from low-ionisation species
(e.g. [OII]$\lambda\lambda 3727,3730$) and high-ionisation species (e.g.
[OIII]$\lambda\lambda 4959,5008$) in addition to recombination lines such as
H$\alpha$. The luminosities of their narrow forbidden lines are comparable to
those found in QSO1s (e.g Reyes et al. 2008). Moreover, IR observations have
played a crucial role in the selection of active galaxies, including QSO2s,
across different redshifts (e.g. Stern et al. 2005; Martínez-Sansigre et al.
2006; Zakamska et al. 2008). X-ray surveys have also detected this type of
source, with L${}_{X}\geq 10^{44}$ erg s-1 (e.g. Barger et al. 2003; Szokoly
et al. 2004; Zakamska et al. 2004; Severgnini et al. 2005; Ptak et al. 2006;
Severgnini et al. 2006; Tajer et al. 2007), revealing a wide range of X-ray
luminosities and HI column densities. For example, in Ptak et al. (2006), five
of the eight studied QSO2s showed column densities consistent with Compton-
thick sources (i.e. N${}_{H}\geq 10^{22}$ cm-2). Additionally, Compton-thick
sources have very faint emission at 0.1-10 keV (restframe) due to absorbed
emission by the circumnuclear clouds of gas and dust. Therefore, QSO2s
typically have similar relations between X-ray, optical, and IR as an
extremely obscured AGN (Tajer et al. 2007).

These sources also showcase a variety of unique features, with frequent tidal
features from mergers or interactions (e.g Villar-Martín et al. 2012; Bessiere
et al. 2012; Pierce et al. 2023), intense star-formation activity (e.g. Lacy
et al. 2007; Zakamska et al. 2008; Hiner et al. 2009), ionised gas outflows
(Humphrey et al. 2010; Villar-Martín et al. 2016), and polarised emission with
an obscured non-thermal continuum (e.g. Vernet et al. 2001; Zakamska et al.
2005). Thus, multi-wavelength observations remain crucial for the validation
of the nature of this type of source (e.g. Stern et al. 2002; Della Ceca et
al. 2003; Zakamska et al. 2004; Yang et al. 2016; Matsuoka et al. 2022)

In the ‘redshift desert’ (i.e. $1\leq z\leq 2$), most of the prominent
emission lines in QSO2 sources (e.g. CIV$\lambda 1549$; [OIII]$\lambda\lambda
4959,5008$) fall either outside the wavelength range of the SDSS optical
spectra or in particularly noisy wavelength ranges (Zakamska et al. 2003,
2004; Alexandroff et al. 2013; Ross et al. 2014). This intermediate redshift
regime is still only scarcely explored. For example, in ultraluminous infrared
galaxies, QSO2s (ULIRG, L${}_{\rm{IR}}\geq 10^{12}L_{\odot}$) have been
detected at z<1 and z $\geq 2$ (e.g. Zakamska et al. 2004; Reyes et al. 2008;
Rodríguez et al. 2014). Studying QSO2s in this redshift range will provide new
insights into the properties of AGNs and their host galaxies near the peak of
star-formation rate in the universe. The potential correlation between star-
formation rate, merger history, and QSO2 sources makes this type of study
essential in order to complete our view of the AGN-galaxy co-evolution
(Magorrian et al. 1998; Di Matteo et al. 2008; Greene et al. 2011; Bessiere et
al. 2012; Humphrey et al. 2015b; Padovani et al. 2017; Villar Martín et al.
2020, 2021).

Following the constant technological innovation in astronomical surveys, the
demand to deal with data in a fast and effective manner is not only relevant
but also becoming mandatory. Traditionally, classification and physical
property estimation has been done using colour-colour methods (e.g. Haro 1956;
Daddi et al. 2004; Richards et al. 2002; Glikman et al. 2008; Stern et al.
2012; Lacy et al. 2004) and spectral fitting codes (template or model-based,
e.g. Bolzonella et al. 2000; Ilbert et al. 2006; Gomes & Papaderos 2018).
While spectral fitting codes have been shown to be extremely useful and to
provide solid results, they become difficult to use with massive data sets due
to time and computational constraints. In the last decade, the use of machine
learning (Samuel 1959) in astronomy has increased rapidly, allowing new and
more complex models to be developed and helping astronomers retrieve physical
meaning from observational data (Baron 2019). Machine learning has proved to
be particularly efficient at dealing with multidimensional data sets and often
gives similar or better results compared to traditional methods (e.g. Odewahn
et al. 1993; Collister & Lahav 2004; Baqui et al. 2021; Euclid Collaboration
et al. 2023b). The flexibility of this method allows for a wide range of
applications, including object classification using photometry data (e.g.
Cavuoti et al. 2014; Logan & Fotopoulou 2020; Clarke et al. 2020; Mechbal et
al. 2024; Hviding et al. 2024; Pérez-Díaz et al. 2024; Zeraatgari et al.
2024); morphological classification with images (e.g. Huertas-Company et al.
2015; Holincheck et al. 2016; Domínguez Sánchez et al. 2018; Bowles et al.
2021; Walmsley et al. 2022; Bickley et al. 2023); estimation of the redshift
of galaxies with tabular and imaging data (e.g. Euclid Collaboration: Desprez
et al. 2020; Razim et al. 2021; Curran et al. 2022; Lin et al. 2022; Henghes
et al. 2022; Cunha & Humphrey 2022; Cabayol et al. 2023); and physical
property estimation (e.g. Bonjean et al. 2019; Mucesh et al. 2021; Simet et
al. 2021; Euclid Collaboration et al. 2023a; Humphrey et al. 2023).

Few-shot learning (Li et al. 2006; Abdo et al. 2013; Wang et al. 2020) is a
machine learning method that allows for the construction of models using
relatively small data sets. It takes advantage of the size of the data set to
reduce computation time and better recognise rare examples or classes. This is
achieved by learning the distribution of rare labelled features, contrary to
the common case where the main statistical distribution of the features is
obtained from a large sample of data. Although this method was first applied
to computer vision tasks (e.g. Tian et al. 2020; Qin et al. 2020; Li et al.
2023; Jiang et al. 2023b), it can also be applied to tabular data (e.g. Chen
et al. 2019; Khajezade et al. 2021; Hegselmann et al. 2023; Nam et al. 2023;
eon Lee & Lee 2024). Domain knowledge (e.g. Shi et al. 2008; Mirchevska et al.
2014; Dash et al. 2022) plays an important role to be able to use this
technique, especially in astronomy. Prior knowledge, from previous
observations, and the physical correlations in astronomical data allows the
machine learning model to learn in an efficient manner (e.g. Gong et al.
2023). By combining both techniques, the identification of rare astronomical
sources can potentially be enhanced.

Transfer learning in machine learning involves using models pre-trained with
specific data sets, either from simulations or real data, to make predictions
in new data sets or in different domains (e.g. Smith et al. 2022; Almeida et
al. 2022; Krishnakumar 2022; Villaescusa-Navarro et al. 2022). Instead of
training and optimising a completely new model, transfer learning allows
learned patterns and mappings from the original training episode to be re-used
for similar tasks. This approach is particularly useful when labelled data for
the target task is scarce or expensive to obtain (e.g. spectroscopic data).
Therefore, transfer learning enables the application of machine learning to
new domains or tasks with smaller data sets, which is particularly relevant in
astronomy (e.g. Vilalta 2018; Kouw & Loog 2018; Tang et al. 2019).

Traditional single-colour selection methods of obscured AGNs are often
unreliable (Hickox & Alexander 2018), prompting the exploration of machine
learning techniques for more robust selections. Motivated by the difficulty to
identify QSO2 sources at the ‘redshift desert’, we have developed a few-shot
machine-learning methodology that correlates optical-infrared photometry by
linking faint optical emissions and dust features, which are characteristic of
obscured AGNs (e.g. Piconcelli et al. 2015; Assef et al. 2015; Hviding et al.
2018; Yang et al. 2023; Ishikawa et al. 2023). Our pipeline is design to
handle smaller training data sets ($\sim 200$ sources) and to be applied at
the ‘redshift desert’ via transfer learning. To test our pipeline, we
performed a blind test, using semi-supervised learning to identify QSO2
candidates. The methodology outlined here offers a photometric pre-selection
of QSO2 candidates for multi-wavelength follow-up.

This paper is organised as follows. We briefly describe the photometric data
used to distinguish QSO2s from galaxies and the pre-processing steps necessary
for the machine learning pipeline in Sect.2. The AMELIA pipeline is described
in Sect.3, and the statistical metrics to evaluate the pipeline are in Sect.4.
A proof-of-concept task and a semi-supervised approach to identify QSO2s are
presented in Sect. 5. In Sect. 6, we test our ability to recover QSO2
candidates through spectroscopic analysis, X-ray and radio cross-matching, and
comparison with AGN photometric criteria in the literature. Then, in Sect.7,
we derive the physical properties of the QSO2 candidate sample using
photoionisation models and spectral energy distribution analysis. Finally, in
Sect. 8, we discuss the nature of our sample and how it can be placed into the
AGN-galaxy co-evolution scheme. Last, we summarise our results in Sect.9.
Throughout this paper, we adopt a flat-universe cosmology with $H_{0}$ = 70 km
s-1 Mpc-1, $\Omega_{M}=0.3$ and $\Omega_{\Lambda}=0.7$ (Planck Collaboration
et al. 2020).

##  2 Sample and data

###  2.1 Photometric data

####  2.1.1 Quasar sample

Our methodology requires a sample from which the defining characteristics of
QSO2 may be learnt. For this purpose, we use the 144 sources identified and
described by Alexandroff et al. (2013) as ‘class A’ QSO2 candidates, which the
authors selected from BOSS in SDSS DR9 (Ahn et al. 2012). Alexandroff et al.
(2013) used a selection approach that combined the properties of the emission
line and continuum properties: They required a detection of Ly$\alpha\lambda
1216$ and CIV $\lambda 1550$ at 5$\sigma$ significance or higher, and an
emission line full-width half-maximum (FWHM) $\leq$ 2000 km s-1. The final
sources have narrow emission lines, no associated absorption and weak
continuum. The authors discarded narrow-line Seyfert 1 galaxies and broad-
absorption-line quasars, since they are Type 1 objects. The selected QSO2
candidates have a redshift between 2 and 4, and a median FWHM of the CIV
$\lambda 1550$ emission line equal to 1260 km s-1. (For more details, see
Alexandroff et al. (2013).)

####  2.1.2 Galaxy sample

In addition, we select a sample of sources previously classified as galaxies
in the SDSS DR16 catalogue (Ahumada et al. 2020) with spectroscopic redshifts,
to be used as a ‘test set’ for models. Our selection criteria are designed to
yield galaxies with SDSS magnitudes in a similar range to those of the QSO2
sample above, but without any specific constraints on their colours or
morphology. Specifically, our magnitude selection criteria are: (i) $19\leq
u\leq 26$; (ii) $19\leq g\leq 24$; (iii) $19\leq r\leq 24$; (iv) $19\leq i\leq
24$; (v) $19\leq z\leq 25$ (AB magnitudes). An SQL query on
CasJobs111https://skyserver.sdss.org/CasJobs/ was performed using these
criteria. We opted to use the aperture-matched modelMag magnitudes, since they
are expected to be more reliable than cModelMag, for example, due to the
higher signal-to-noise of the resulting colours (Stoughton et al. 2002). Since
we are interested in the ‘redshift desert’, we restrict the galaxy redshift to
z >1 and z <2 with z=2 being the maximum value for sources classified as a
galaxy in SDSS.

When searching for QSO2 in UV-optical photometric surveys, the primary
challenge lies in distinguishing them from non-AGN galaxies, given the optical
depth of dust and the faint emission diluted by star formation (e.g. Calzetti
et al. 1994; Draine 2003). The uncertainty in the photometric classification
boundary, caused by dust and gas obscuration, introduces the possibility of
”hidden” QSO2 sources in archival data, which can be untangled by adding
infrared information (Kleinmann 1988). Therefore, we include Wide-field
Infrared Survey Explorer (WISE; Wright et al. 2010) $W1$, $W2$, $W3$, and $W4$
(3.4, 4.6, 12, and 22 $\mu m$) photometry. These data were obtained by
including a cross-match with the AllWISE catalogue in our SQL query. The final
galaxy sample is composed of $21,943$ sources.

###  2.2 Spectroscopic data

We also make use of SDSS BOSS spectra when required for the validation of our
machine learning results. We used the SDSS Science Archive
Server222https://dr16.sdss.org/optical/ to retrieve FITS spectra. The zWARNING
flag is used as a quality filter: Only sources with zWARNING$=0$, no problems
detected, or $4$, MANY$\\_$OUTLIERS, are considered.

In the ‘redshift desert’, the [OII]$\lambda\lambda 3727,3730$ emission is used
to determine the redshift of the source. This makes the redshift
determinations more prone to errors. Therefore, we will consider the
spectroscopic redshift provided by SDSS with caution (more details are
provided in Sect.6.1). It is also important to mention that the spectroscopic
data available is close to the limit that can be achieved with a 2.5m
telescope and faint sources (r magnitude $\sim 21$). Faint emission lines
becomes more difficult to detect as they may be diluted in the noisy
continuum.

##  3 The AMELIA pipeline

AMELIA 333https://github.com/pedro-acunha/AMELIA is an interactive, modular,
and adaptive machine learning pipeline capable of performing classification
tasks using tabular data from astronomy surveys. It takes advantage of six
different classifications algorithms and an ensemble method, generalised
stacking (Wolpert 1992), to combine the outputs using soft-vote and meta-
learner correction (Zitlau et al. 2016; Cunha & Humphrey 2022; Euclid
Collaboration et al. 2023b). The following subsections describe the technical
details of AMELIA. In Sect. 3.1, we describe data pre-processing tasks
required to extract the most out of the available data. Then, in Sect. 3.2 the
machine learning methodology is described. In particular, in Sect. 3.2.1 a
brief overview of the inner works of our methodology is described.

###  3.1 Feature engineering

####  3.1.1 Feature creation

As the input features for all the models, we used both magnitudes and colours.
Due to the diversity of machine learning models, we perform feature scaling
(Wan 2019) using classes from Scikit-Learn (Pedregosa et al. 2011). For all
algorithms, the RobustScaler was applied, which ensures that potential
outliers in the data are kept. All unique permutations of broadband colours
were calculated from the following features: modelMag (i.e. $u-g$, …, $i-z$);
WISE (i.e. $W1-W2$, …, $W3-W4$); and optical-infrared colours (i.e. $u-W1$,…,
$z-W4$). When we perform binary classification tasks, a binary (0/1) target
feature is created, where 0 indicates a normal galaxy and 1 indicates a QSO2.

####  3.1.2 Imputation

In astronomy, missing values are commonly due to instrumental limitations
(i.e. non-detections) and do not necessarily represent an incomplete data set,
as they may carry a relevant physical meaning. For instance, some galaxies
have sudden spectral breaks at specific wavelengths that can manifest
themselves as ”dropouts”, revealing both the source’s nature and the redshift.
Nonetheless, missing values can cause problems on how different algorithms
interplay and digest data. A common solution is imputation, where missing
values are replaced with an alternative value with statistical or physical
meaning.

The effects of imputation have also been extensively studied in Euclid
Collaboration et al. (2023b). The authors explored the impact that different
imputation techniques, such as the mean, median, minimum and a so-called
”magic value” (e.g. -99.9), have in decision-tree and distance based, deep
learning and meta-learner ensemble models. The chosen imputation techniques
impact the model’s performance and should therefore be selected according to
the scientific problem.

For the QSO2 sources from Alexandroff et al. (2013), missing values are
present in the WISE photometry for 74 sources, which represents $\sim 51\%$ of
the sample (identified using a $2\arcsec$ cone search). To impute the missing
values, we used the KNNImputer, from Scikit-Learn, version 1.2.1, with the
following configuration: missing$\\_$values=0; n$\\_$neigbors=6; and weights=
distance. As in the k-nearest neighbours algorithm, this method predicts the
missing value by computing the mean value based on the distance between points
in the training set and the number of defined neighbours.

###  3.2 Machine learning

After the feature engineering process, described in Sect.3.1, the data are
divided into training and test set, depending on the final objective (for
further details, see Sect.5). Every time a data set is split, the procedure is
performed randomly, never considering the balance between the properties of
the features. This allows experiments in the stacking process to be applied
and compared with the each base learner. However, this variability can easily
be changed and different training-test splits tested.

Figure 1: Flow diagram for the AMELIA pipeline. The process begins with the
assembly of SDSS and WISE data. As seen in the orange box, feature creation is
used to generate optical, optical-IR, and IR colours. The imputation method is
applied to the training set and followed by feature scaling due to the use of
different algorithms. The blue box shows a description of the model training
procedures, but we recount them here as well: Each base learner is first
loosely optimised for generalisation. Subsequently, out-of-fold predictions
are combined with the average predictions from the base learners, utilising a
soft-vote mechanism for the training of the meta-learners. The predictions
from the meta-learners are then combined through ensembling, using a soft-vote
fusion that results in the final predictions for the test set.

In Fig.1, the flow diagram of the AMELIA pipeline is shown. Six different
algorithms were used to perform a binary classification, which we call base
learners. We combine one distance-based algorithm, four decision tree-based
algorithms, and a deep learning-based model (see more details in Sect.3.2.2).

####  3.2.1 AMELIA overview

In our pipeline, we explored two distinct approaches: supervised and semi-
supervised learning. In the supervised learning approach, we adhered to the
conventional procedure, where the data set is divided into training and
testing sets, utilising k-fold cross-validation, with the class labels of all
sources known.

In contrast, with the semi-supervised learning approach, we adopted a unique
strategy where the test set consists exclusively of a single class, that is,
galaxy class, (see Jiang et al. 2023a, for an application to galaxy morphology
classification). Our approach is based on the foundational one-class
classification problem (Moya & Hush 1996; Schölkopf et al. 1999), taking
advantage of the capabilities of one-class classifiers to discern distinct
patterns within a multidimensional space. These classifiers are trained using
a set of positive examples (e.g. Tax & Duin 2001; Yu et al. 2002). To augment
the generalisation capability of our models, we embraced a semi-supervised
one-class approach (e.g Bauman & Bauman 2017; Akcay et al. 2019; Cai et al.
2023), introducing a limited number of contrasting examples during training
and applying to a single class test set. Therefore, while the model is trained
with two distinct classes, when the model is applied to the one-class test
set, the main target is to identify the galaxy class. Due to the contrasting
examples, we then consider sources that are not classified as galaxy to be
QSO2 candidates.

In our approach, the model is trained with both QSO2 sources and a subset of
SDSS galaxies, ensuring a balanced training set, that is, same number of
examples for both classes. Subsequently, the model is applied to the one-class
test set (i.e. galaxies) to compute the probabilities for the predictive class
(i.e. QSO2). Then, a classification threshold is set to refine the
predictions. Specifically, this semi-supervised one-class approach identifies
sources labelled as galaxies in the SDSS spectroscopic data but exhibiting
photometric characteristics consistent with a QSO2. Our proof of concept uses
data sets with diverse redshift ranges, detailed in Sect.2, resulting from
specific selection criteria in the chosen data set. Given the absence of a
catalogue of spectroscopically confirmed QSO2 in the ‘redshift desert’, a
redshift transfer-learning method is employed by leveraging prior domain
knowledge concerning the correlation between optical and infrared photometric
properties from QSO2 at $z\geq 2$.

####  3.2.2 Algorithms

AMELIA takes advantage of the predictive capabilities of three decision tree-
based algorithms, all falling under the category of gradient-boosting decision
tree algorithms. This choice is intentional, as these algorithms demonstrate
superior performance and exhibit inherent distinctions in their underlying
mechanisms444A benchmark study between the three algorithms was performed by
Anghel et al. (2018). Additionally, an interesting analysis of why tree
ensemble models can outperform deep learning models with tabular data can be
seen in Shwartz-Ziv & Armon (2021).. The nuances between the three gradient-
boosting decision tree algorithms, namely XGBoost (version
1.7.3)555https://XGBoost.ai/ (Chen & Guestrin 2016), CatBoost (version
1.1)666https://catboost.ai// (Prokhorenkova et al. 2018), and LightGBM
(version 3.3.3)777https://lightgbm.readthedocs.io/ (Ke et al. 2017) are
described in Cunha & Humphrey (2022) and Euclid Collaboration et al. (2023b).

For completeness and diversity, we incorporate a k-nearest neighbours
algorithm, a randomised decision trees algorithm, and a multi-layer Perceptron
classifier, MLPClassifier, from Scikit-Learn (version 1.1.3)888https://scikit-
learn.org/. The inclusion of a diverse set of algorithms is crucial as it
enables a wider range of predictions. This diversity will become particularly
pertinent later within the context of the generalised stacking approach with
meta-learning correction (see Sect.3.2.4).

In Table 1, the hyper-parameters of the algorithms used as base learners and
meta-learners are described. The number of estimators were defined as being
the minimum number to obtain reliable metrics, as increasing the estimators
can lead to overfitting. This was done using a randomised grid search for
XGBoostClassifier and replicated for the remaining algorithms. For the
MLPClassifier, we set the initial hidden layer with the same number of layers
as the n_estimators. Additional layers were added and a manual optimisation
was performed. For the KNeighborsClassifier, we set a range of n_neighbours
from 0 to 20, and selected the one that had the best metrics.

Table 1: Description of the algorithms and hyper-parameters used as base learners and meta-learners. Algorithms | Hyper-parameters
---|---
KNN | n_neighbours = 12
Random Forest |  |  n_estimators = 50
---
max_depth = None
XGBoost |  |  n_estimators = 50
---
max_depth = 6
CatBoost |  |  n_estimators = 50
---
max_depth = 6
LightGBM |  |  n_estimators = 50
---
max_depth = -1
MLP |  |  Hidden layer sizes = (50,100,50)
---
activation = relu

####  3.2.3 K-fold cross-validation

To assess the performance of each base learner, we employed a data
partitioning strategy known as k-fold cross-validation (refer to, for
instance, Hastie et al. 2009). In this methodology, the data set is randomly
divided into five folds, ensuring the preservation of both the number of
sources and the target class distribution. Following this division, the models
are trained on the partitioned data set, utilising k-1 folds for training,
while the remaining fold is designated for validating the predictions, known
as out-of-fold (OOF) predictions. This iterative process repeats through the
defined k iterations and an average is calculated on the predictions for the
final metric analysis.

K-fold cross-validation proves to be an effective methodology to prevent
overfitting, especially in smaller data sets where the conventional training-
test-validation split may not be feasible (Lachenbruch & Mickey 1968; Stone
1974; LUNTZ 1969; Wang & Zou 2021). An additional advantage is to obtain OOF
predictions, even for the training set. In this study, the k-fold cross-
validation method plays a crucial role in validating the performance of each
individual learner before implementing the stacking approach.

####  3.2.4 Approach to generalised stacking

Our approach to handling the output predictions of the base learners closely
follows the generalised stacking methodology presented by Euclid Collaboration
et al. (2023b) and Cunha & Humphrey (2022). The primary difference is how the
final combination of the predictions is performed. In this study, we opt for a
soft vote, that is, average vote, coupled with a meta-learner approach to
combine the probabilistic predictions.

Upon receiving predictions from the six base learners for the test set, a new
data set is constructed, comprising individual predictions from each base
learner. Additionally, a new feature is created by averaging these
predictions. To integrate the predictions of the individual learners and the
average feature, we employ a combination of five meta-learners:
XGBoostClassifier; CatBoostClassifier; LGBMClassifier; MLPClassifier (as
described in Table 1); together with an additional MLPClassifier with
activation=logistic. The inclusion of the MLPClassifier model with a distinct
activation model aims to increase the diversity of predictions by adding an
additional deep learning based model.

The final predictions result from an ensemble of the individual outputs of the
meta-learners, utilising a soft vote to generate the final classifications.
This methodology explores how various algorithms analyse the distribution of
predictions and combine them into a robust classifier. Furthermore, the
exclusion of the Random Forest and KNN algorithms is attributed to the
influence of outliers on the number of outputs, which can impact the soft vote
combination. This decision was validated through a supervised approach, as
detailed in Sect.5.1.

In summary, one can interpret our approach as an analogous of the ensemble
random forest, where instead of randomly decision-trees, we have individual
base models. Then, instead of combining the predictions for each model, we use
a meta-learner to analyse the variance between the base learners’ predictions,
adding the average predictions as a feature. The main advantage of this method
is that the output model will be robust to outliers by studying the variance
between predictions and preventing systemic biases underlying particular
models. Finally, to avoid selection bias from a single meta-learner, we
average the predictions from four meta-learners. With this approach, one can
create more robust predictions by leveraging the statistical validation
provided by the ensemble models.

##  4 Statistical metrics

Statistical metrics quantify the ability of a particular model to retrieve the
desired information (e.g. Powers 2008). Herein, we describe the statistical
metrics used to validate the perform of our machine learning methodology.

Accuracy is the fraction of correct predictions for all classes compared to
the to the total number of predictions:

$\rm\textit{A}=\frac{TP+TN}{TP+TN+FP+FN}\,.$ (1)

Here, TP is the number of true positives, TN is the number of true negatives,
FP is the number of false positives, and FN is the number of false negatives
for a given class.

Precision, or purity, is the fraction of correct predictions for a given class
compared to the overall number of predictions for that class:

$\rm P=\frac{\rm TP}{\rm TP+\rm FP}\,.$ (2)

Unlike the accuracy metric, precision focus on the positive predictions to
asses their quality.

Recall, or completeness, is the fraction of correct predictions for a given
class compared to the overall number of positive cases for that class:

$\rm R=\frac{\rm TP}{\rm TP+\rm FN}\,,$ (3)

where FN is the number of false negatives in each class. The symbiotic
relation between precision and recall makes their use valuable to asses the
quality of the predictions.

By combining the precision and recall metrics, one can create informative
metrics such as the F1-score (e.g. Blair 1979). The F1-score is the harmonic
mean of the precision and recall,

$\rm F1=2\frac{P\cdot R}{P+R},$ (4)

where equal weight is given to precision and recall.

Specificity, also called true negative rate, computes the true negative
fraction identified by the model:

$\rm specificity=\frac{\rm TN}{\rm TN+\rm FP}\,.$ (5)

High specificity implies that a high fraction of true negatives is recognised.

##  5 Results from the application of AMELIA

###  5.1 Proof of concept for QSO2 selection

Before selecting new QSO2 candidates, we first conduct a test to ascertain the
performance of our pipeline for selection of QSO2 in a fully labelled data
set. By employing a supervised classification on SDSS-WISE photometric data,
we assess the reliability of the binary classification. Our primary goal is to
establish a control test to explore correlations in optical-infrared
photometric data.

Table 2: Classification evaluation metrics for KNNeigbors, RandomForest,
LightGBM, CatBoost, XGBoost, MLP, and generalised stacking with a threshold
$\geq 0.5$.

Algorithm | Accuracy | Precision | Recall | F1-Score | F1-Score, cv=10
---|---|---|---|---|---
KNNeigbors | 0.879 | 0.897 | 0.887 | 0.879 | 0.876 ±0.063
RandomForest | 0.879 | 0.881 | 0.882 | 0.879 | 0.867 ±0.080
XGBoost | 0.966 | 0.965 | 0.965 | 0.965 | 0.943 ±0.041
CatBoost | 0.966 | 0.966 | 0.968 | 0.965 | 0.926 ±0.041
LightGBM | 0.948 | 0.949 | 0.949 | 0.948 | 0.921 ±0.046
MLP | 0.948 | 0.950 | 0.952 | 0.948 | 0.921 ±0.035
Generalised stacking | 0.966 | 0.966 | 0.968 | 0.965 | –
999The final column shows the F1-Score metric using k-fold cross-validation,
with k=10, that allows the standard deviation for the final classifications.

For this proof of concept, the data set was constructed using $144$ sources
randomly selected from our SDSS galaxy sample (see Sect.2.1.2), and all $144$
spectroscopically confirmed sources in our QSO2 sample (see Sect.
2.1.1)101010An examination of how the class distribution of the training data
impacts the overall performance of the models is presented in Sect.A.. After
combining the data from the two classes, we performed a train test splif using
$70\%$ of the total data for training and $30\%$ of the total data for
testing.

The AMELIA pipeline was able to successful distinguish between galaxies and
the QSO2, validating the application of machine learning for this task.
Detailed in Table 9 are the statistical metrics for each individual model,
together with the metrics obtained after generalised stacking with a threshold
$\geq 0.5$. Individual models exhibited satisfactory performance, with a
slight advantage noted for the decision-tree-based models, XGBoost and
CatBoost, compared to the rest of the models.

The metrics for the generalised stacking, also seen in Table 9, may not be
sufficiently improved to validate the advantage of generalised stacking in
this context. Methodological intricacies arise when models yield similar
results, causing meta-learners to mimic clustered values for the final
predictions, converging around the optimal value. However, this does not
diminish the potential of the methodology to identify QSO2 sources, where
individual learners achieve an F1-score $\geq 0.88$. As the size of the data
set increases, which would be the case when our methodology is applied to
future survey data, we anticipate an improvement in performance (as seen in
Euclid Collaboration et al. 2023b; Cunha & Humphrey 2022).

###  5.2 Selection of new QSO2 candidates

Having shown in Sect. 5.1 that AMELIA is capable of successfully identifying
QSO2, we now proceed to apply the pipeline to the task of identifying new QSO2
candidates. In particular, the objective is to select QSO2 that were
previously misclassified in SDSS as galaxies, after template fitting. Thus, we
now use the AMELIA pipeline in semi-supervised mode (see Sect. 3.2.1),
utilising a balanced training set (same number of sources per class), and a
classification probability threshold $\geq 0.8$. By using this probability
threshold instead of the often used ‘default’ value 0.5, we aim to select
sources with a higher confidence of being a QSO2 (see Baqui et al. 2021;
Alegre et al. 2022; Jiang et al. 2023a, for different study cases with
classification threshold tuning and Appendix A for a more detailed analysis).

For this task, the training set contains the $144$ spectroscopically confirmed
QSO2, and a further $144$ sources selected randomly from our SDSS galaxy
sample, so make up the balanced training set. The testing set was created
using the remaining $\ 21\,799$ sources from the galaxy sample; no
spectroscopically confirmed QSO2 were included in this set.

When applying the AMELIA pipeline on the test set, 366 sources were classified
as QSO2 with a classification probability of $\geq 0.8$. In the following
sections, we will address the reliability of our new QSO2 candidates by
analysing photometry and spectra, including multi-wavelength information where
available.

##  6 Testing the recovery of QSO2

In this section, we assess the robustness of our new QSO2 candidate sample by
examining available optical photometry and spectra, and X-ray and radio
detections. This procedure ensures the reliability of the identified
candidates; while not providing a definitive classification of the target
sample as QSO2 sources, with multi-wavelength data being essential.
Nonetheless, the identification of specific fingerprints indicative of AGN
activity significantly enhances our confidence in the efficacy of our
selection method.

###  6.1 Searching for AGN signatures in the spectrum

Of the $366$ candidates, only $138$ had clean spectra in SDSS DR16 (i.e.
without any problematic flag, zwarning=0). Thus, our spectroscopic analysis
includes only these $138$ candidates.

After downloading the SDSS spectra, we considered the spectroscopic redshifts
(described in Sect. 2.2) to search for AGN associated emission lines. We found
that the redshift listed in the SDSS database occasionally did not match what
we obtained from the observed emission lines (e.g. Wu & Shen 2022, detected
1900 broad-line quasars in SDSS DR16, around 0.002$\%$ of their sample, had
catastrophically wrong redshifts). In such cases, we adopted our new value of
the redshift for the source. An example is provided in Appendix D. When no
clear emission lines were detected, we could not validate the provided
spectroscopic redshift and therefore assumed the provided redshift value.

Our main objective was to identify [NeV]$\lambda 3426$, successfully used as
an AGN identification criterion (e.g. Schmidt et al. 1998; Gilli et al. 2010;
Mignoli et al. 2013; Vergani et al. 2018), and [NeIII]$\lambda 3869$ emission
lines. [NeV]$\lambda 3426$ is a high ionisation line with a relatively high
ionisation potential (IP; Ne4+ IP=$97.16$ eV) requiring a hard ionising
source, such as an accretion disk of an AGN (Haehnelt et al. 2001). (e.g.
Schmidt et al. 1998; Gilli et al. 2010; Mignoli et al. 2013; Feltre et al.
2016). Alternative, other more ”exotic” ionisation sources might be able to
produce [NeV]$\lambda 3426$ such as supernovae, Population III stars,
Wolf–Rayet stars, stripped stars in binaries, high-mass X-ray binaries, and
hot, low-mass evolved stars (e.g. Flores-Fajardo et al. 2011; Eldridge et al.
2017; Schaerer et al. 2019; Martínez-Paredes et al. 2023).

In our spectroscopic analysis, we fitted each emission line and identified
$30$ [NeV]$\lambda 3426$ and $52$ [NeIII]$\lambda 3869$ emitters with S/N
$\geq 3\sigma$. In total, we found $65$ objects with [NeV]$\lambda 3426$ or
[NeIII]$\lambda 3869$ emission lines, around $47\%$ of the total number of
candidates.

Figure 2: SDSS spectra from BOSS spectrograph for a sample of three QSO2
candidates with a [NeV]$\lambda 3426$ detection. From top to bottom: SDSS
J141542.16+522206.9, SDSS J013056.89-022638.1, and SDSS J141542.16+522206.9.
The red vertical lines indicate where the different emission lines should be
with respect to the SDSS spectroscopic redshift.

Figure 2 shows the spectrum of three [NeV] emitters from our sample of QSO2
candidates. The final candidates were divided into three groups based on the
detected emission lines 111111Throughout this paper, the sources with the
emission lines will be name as follows: [NeV] emitters for [NeV]$\lambda 3426$
emitters; and [NeIII] emitters for [NeIII]$\lambda 3869$ emitters.: Clear AGN:
[NeV]$\lambda 3426$ detected with S/N $\geq 3\sigma$; Ambiguous AGN:
[NeIII]$\lambda 3869$ detected with S/N $\geq 3\sigma$; Unclear: No
[NeV]$\lambda 3426$ or [NeIII]$\lambda 3869$ detected.

Figure 3: Stacked spectra of the [NeV] and [NeIII] emitters from the QSO2
candidate sample. The fluxes for the [NeIII] emitters is offset for a better
visualisation.

Fig. 3 shows the stacked spectra for the [NeV] and [NeIII] emitters. For
stacking the spectra, we used the software
specstack121212https://github.com/astrom-tom/specstack (Thomas 2019), with a
limit of $3\sigma$ clipping and rest-frame limit between 1795 and 3838 Å. Due
to the low S/N of the spectra only the most prominent emission lines are
detected, in particular [NeV]$\lambda 3426$, [OII]$\lambda\lambda 3727,3730$,
and [NeIII]$\lambda 3869$.

Additionally, we compared our methodology with random selection. From the test
set, $366$ sources were randomly selected. From the selected sources, we
investigated $255$ sources, with clean spectra, and found that $13$ of them
had [NeV]$\lambda 3426$ lines. This corresponds to 5$\%$ of the total random
sample. As expected, our methodology is much more efficient at selecting QSO2
candidates than a random selection. We argue that our classification pipeline
may use specific features, that is, combination of magnitudes and colours with
specific weights, within the broadband SED that indicate the presence of an
obscured AGN, even in the cases where no clear AGN emission lines could be
detected in the SDSS spectrum, most probably due to the low S/N or
instrumentation limitations (e.g. Mignoli et al. 2013; Vergani et al. 2018,
see references herein). This is mainly due to the optically faint nature of
the sample and designed strategy of the SDSS facility. Follow-up spectroscopic
both in optical and infrared regimes will allow us to improve this analysis
and distinguish QSO2 from star-forming galaxies.

An unsupervised learning analysis was also performed to study the possible
variability within the photometric data between [NeV] and [NeIII] emitters and
non-[Ne] emitters. We applied the KMeans algorithm with PCA dimensionality
reduction on the input features to identify potential clusters and structures
within the data. The number of clusters was set to three, representing the
three different emitter labels defined previously. We found no clear
separation or intrinsic property to distinguish between the sources, implying
no significant variance exists between our sub-classes. This results supports
the hypothesis that the unclear sources might have similar features as clear
and ambiguous AGNs, requiring high spectroscopic resolution.

###  6.2 Cross-matching with radio and X-ray catalogues

Given that QSO2 might have faint X-ray emission (e.g. Barger et al. 2003; Ptak
et al. 2006; Tajer et al. 2007), the detection of X-ray emission from our
sources would corroborate our classification. Additionally, radio emission is
present in previously identified QSO2 sources at low redshift (e.g. Villar
Martín et al. 2014, 2021), allowing for better characterisation of the
sources.

In this section, we examine currently available X-ray and radio data sets to
evaluate the detection of our QSO2 candidate sample. Also, we make use of the
Millions of Optical-Radio/X-ray Associations (MORX) Catalogue,
v2131313https://cdsarc.cds.unistra.fr/viz-bin/cat/V/158 (Flesch 2024) to
cross-match our candidates using the VizieR service.

####  6.2.1 X-ray cross-identification

The coverage of our sources by deep X-ray observations from XMM-Newton and
Chandra is patchy and does not lend itself to statistical studies. However, we
highlight that XMM-Newton (Lin et al. 2012) has detected the five QSO2
candidates: SDSS J092106.72+302737.8, SDSS J014104.76-005209.2, SDSS
J020711.42+021549.5, SDSS J141542.16+522206.9, and SDSS J011219.62+023732.7.
Additionally, SDSS J160852.36+443028.0 has been detected in the XMM-Newton
slew survey (Saxton et al. 2008). Chandra (Vagnetti et al. 2016) has detected
three QSO2 candidates, two of which also have XMM-Newton detection: SDSS
J141542.16+522206.9, SDSS J092106.72+302737.8, and SDSS J014546.90-043318.8.
Additionally, SDSS J084126.29+411059.9 is included in the eROSITA Data Release
1 (Merloni et al. 2024).

While these detections confirm the AGN nature for these 10 cases, we can still
be missing Compton-thick sources, as previously observed (e.g. Ptak et al.
2006; Carroll et al. 2021, 2023).

####  6.2.2 Radio cross-identification

We cross-matched our sample with the LOw-Frequency ARray (LOFAR) Two-metre Sky
Survey (LoTSS, Shimwell et al. 2022) Data Release 2, and obtained a total of
20 detected sources from our sample. Since we get a higher number of sources,
we have performed a statistical analysis for this sub sample.

We took into account the following equation from de Jong et al. (2024) to
calculate the total flux density ($S_{\nu}$):

$S_{\nu}=\frac{L_{\nu}(1+z)^{1-\alpha}}{4\pi d_{L}^{2}},$ (6)

where $L_{\nu}$ is the source luminosity, $d_{L}$ is the luminosity distance,
and $(1+z)^{1-\alpha}$ considers the $K$-correction with spectral index
$\alpha$. We could rearrange this equation to estimate $L_{\nu}$, with
$S_{\nu}$ as the total flux density estimated in the LoTSS catalogue. We used
this estimation to determine if the origin of the radio emission is star-
formation driven or AGN driven. Following Hardcastle et al. (2016), we used
the following criterion to set a confidence level on the origin of radio
emission: log10L${}_{\rm{150MHz}}\geq 23.5$ (see also Best & Heckman 2012, in
particular, Fig.3 and Table 4 therein). We estimated the following range for
the L150MHz: log10L${}_{\rm{150MHz}}=24.32-27.05$ W Hz-1. Thus, the measured
L150MHz shows a high probability that radio emission has an AGN origin since
it is within the interval mainly populated by detected LOFAR AGNs (Hardcastle
et al. 2016, see Fig. 17).

Finally, by cross-matching with Faint Images of the Radio Sky at Twenty-cm
(FIRST, Becker et al. 1994) catalogue, we identified five QSO2 candidates:
SDSS J083317.63+242113.8, SDSS J020711.42+021549.5, SDSS J115621.12+552723.7,
SDSS J000844.13-015212.0, and SDSS J160852.36+443028.0 (also detected with
LoTSS and XMM-Newton). From the Rapid ASKAP Continuum Survey (RACS, Hale et
al. 2021), three QSO2 candidates are detected: SDSS J020711.42+021549.5 (also
detected with FIRST and XMM-Newton), SDSS J084035.02+225159.2, and SDSS
J224340.32+255931.8 (also detected in LoTSS). Future studies in the radio and
X-ray wavelength, will allow us to better characterise the nature of our
candidates.

###  6.3 Comparison with AGN colour criteria from the literature

Since only 38$\%$ of our QSO candidates have clean SDSS spectra, it is
interesting to apply colour-colour criteria from the literature to further
extend our analysis. Our QSO2 candidate sample was compared with the colour
criteria for the selection of AGN sources, including: Stern et al. (2012);
Mateos et al. (2012); Mingo et al. (2016); Blecha et al. (2018); Carvajal et
al. (2023).

Figure 4: Bar plot illustrating the performance of the colour criteria in
selecting AGN for the QSO2 candidates, as described in the studies by Stern et
al. (2012); Mateos et al. (2012); Mingo et al. (2016); Blecha et al. (2018);
Carvajal et al. (2023). Each bar represents a specific subset: (a) [NeV]
emitters (30 sources), (b) sources with clean spectra (138 sources), and (c)
all candidate sources (365 sources).

Most of the criteria are based on a combination of MIR colours, and the
results are summarised in Fig. 4. Three groups were used in this analysis: (a)
[NeV] emitters (or clear AGN; see Sect.6.1), (b) candidates with clean
spectra, and (c) all QSO2 candidates.

For group (a), the results are as follows: seven sources ($23\%$) met all
criteria, 15 sources ($50\%$) met at least three criteria, 11 sources ($\sim
37\%$) did not meet any criteria. On group (b): 22 sources ($\sim 16\%$) met
all criteria, 65 sources ($\sim 47\%$) met at least three criteria, and 56
sources ($41\%$) did not met any criteria. Finally, for group (c): 49 sources
($\sim 13\%$) met all criteria, 126 sources ($\sim 35\%$) met at least three
criteria, and 179 sources ($\sim 49\%$) did not meet any criteria.

Additionally, we performed the same analysis using the colour selection
criteria proposed by Hickox et al. (2017), where a combination of a simple MIR
colour ($W1-W2\geq 0.7$; Vega) and an optical-infrared colour
[($u$-W3[AB])$\geq$ 1.4(W1-W2[Vega])+3.2], is used to select obscured quasars.
For group (a), three candidates ($\sim 10\%$) met this criterion. In group
(b), 12 candidates ($\sim 9\%$) satisfied the criterion. Finally, in group
(c), 30 candidates ($\sim 8\%$) met the criteria.

Colour criteria serve as an effective but conservative pre-selection tools,
which fall short in capturing all relationships across the SED. Relying on two
or three highly correlated features only partially taps into the available
data potential. For example, the detection percentage of [NeV] emitters
surpasses the number obtain using the condition where at least three colour
criteria are required. While computationally efficient, these methods tend to
have conservative classification boundaries, missing potentially interesting
objects, that generally lay close to the boundary conditions. This result
aligns with Vergani et al. (2018), where a sample of [NeV] emitters was
selected, and traditional AGN selection techniques showed low efficiency in
their selection.

##  7 Physical property estimation

Additional efforts can be made to analyse and derive physical properties for
our QSO2 candidate sample, using both spectroscopic and photometric SDSS data.
In this section, we explore the use of photoionisation models and SED fitting
to further characterise our sample.

###  7.1 Physical properties derived from [NeIII]

Considering that our sample has an active black hole, we can estimate relevant
physical properties, such as the black hole mass. Assuming the source is
radiating at the Eddington limit, the black hole mass can then be estimated
using the following equation:

$M_{E}=8\times
10^{5}\,\bigg{(}\frac{L_{\rm{bol}}}{10^{44}\rm{ergs}^{-1}}\bigg{)}\,M_{\odot},$
(7)

where $L_{\rm{bol}}$ is the bolometric luminosity. A common approach to
estimate $L_{\rm{bol}}$ is to use the [OIII]$\lambda 5007$ emission line as a
proxy (e.g. Kong & Ho 2018, see also references therein).

However, our new QSO2 candidates with spectra lie in the so-called redshift
desert, a redshift range in which the bright optical and UV lines are outside
or near the noisy edges and do not allow for reliable analysis. In the absence
of the bright high-ionisation line [OIII]$\lambda 5007$, we will instead take
advantage of the detected [NeIII]$\lambda 3869$ emission line. We used the
MAPPINGS 1e (Binette 1985; Ferruit et al. 1997; Binette et al. 2012)
photoionisation model from Morais et al. (2021) to compute the [OIII]$\lambda
5007/$[NeIII]$\lambda 3869$ ratio, and then estimate the [OIII]$\lambda 5007$
luminosity from the observed [NeIII]$\lambda 3869$ flux.

The photoionisation model has parameters that are expected to be in the
correct range for luminous QSO2s since the Alexandroff et al. (2013) sample
was used. The photoionisation model parameters from Morais et al. (2021),
which are also used in this work, are the following:

  * •

ionisation parameter141414$U=\frac{Q}{4\pi r^{2}n_{H}c}$, where $Q$ is the
luminosity of ionising photons emitted by the ionising source assuming it is
isotropic, $r$ is the distance between the cloud and the ionising source, and
$n_{H}$ is the hydrogen density of the cloud. = 0.01

  * •

spectral index of the ionising continuum151515We assume a power-law spectrum
of ionising radiation, $S_{\nu}\propto\nu^{+\alpha}$.: $\alpha$ = -1.5

  * •

high energy cut-off for the ionising continuum of $5\times 10^{4}$eV

  * •

metallicity161616The gas metallicity is expressed with respect to the solar
abundance set of Grevesse et al. (2010).: $Z=Z_{\odot}$

  * •

hydrogen density171717Column density of neutral hydrogen, following the
results from McCarthy et al. (1990); Villar-Martín et al. (1999).: $n_{H}$ =
100 cm-2

We obtained an [OIII]$\lambda 5007$/[NeIII]$\lambda 3869$ flux ratio equal to
$11.9$. The flux-luminosity relation to estimate the [OIII]$\lambda 5007$
luminosity can thus be defined as

$L_{\rm{[OIII]}}=11.9\,F_{\rm{[NeIII]}}\,4\pi D_{L}^{2},$ (8)

where $D_{L}$ is the luminosity distance and $F_{\rm{[NeIII]}}$ is the
[NeIII]$\lambda 3869$ flux.

Next, we estimate the bolometric luminosity, $L_{bol}$, using [OIII]$\lambda
5007$. This conversion has been extensively performed in the literature (e.g.
Heckman et al. 2004; Lamastra et al. 2009; Singh et al. 2011). Here, we
estimate $L_{\rm{bol}}$ using the relation given by Pennell et al. (2017):

$\log(L_{\rm{bol}})=(0.5617\pm 0.0978)\log(L_{\rm{[OIII]}})+(22.186\pm
4.164).$ (9)

Another important physical property of AGNs is the mass accretion rate, which
can be calculated using

$\dot{M}=\frac{L_{\rm{bol}}}{\eta c^{2}},$ (10)

where $\eta$ is the accretion efficiency, $L_{\rm{bol}}$ is the bolometric
luminosity, and $c$ is the speed of light. We considered an accretion
efficiency of $\eta=0.1$ (Trakhtenbrot et al. 2017).

Table 3: Summary of the characterisation of the QSO2 candidates.

|  $L_{[OIII]}$ (erg s-1) |  $L_{bol}$ (erg s-1) |  $\dot{M}$ ($M_{\odot}$ yr-1) |  $M_{BH}$ ($M_{\odot}$)
---|---|---|---|---
Median | 2.913$\times 10^{42}$ | 1.092$\times 10^{46}$ | 2.431 | 8.737$\times 10^{7}$
Min | 6.599$\times 10^{41}$ | 4.742$\times 10^{45}$ | 1.055 | 3.794$\times 10^{7}$
Max | 2.240$\times 10^{43}$ | 3.434$\times 10^{46}$ | 7.642 | 2.747$\times 10^{8}$
181818From left to right, [OIII] luminosity, bolometric luminosity, accretion
rate and black hole mass at the Eddington limit.

A summary of the results of the characterisation of the QSO2 candidates is
shown in Table 18 (see Appendix G for the full table). For $L_{\rm{bol}}$,
typical values for AGN sources range between $10^{42}-10^{46}$ erg s-1 (Hönig
& Beckert 2007). Additionally, the bolometric luminosities are similar to the
ones estimated for the QSO2 sample with $10^{46.3}-10^{46.8}$ erg s-1
(Alexandroff et al. 2013). The AGN black hole mass range is roughly between
$10^{7}-10^{8}M_{\odot}$ (Laor 2000). Finally, following the criterion in
Zakamska et al. (2003) for the selection of QSO2, LOIII/L⊙ >3 $\times 10^{8}$
L⊙, we obtained 47 sources that fulfil the criterion. The remaining five
sources have LOIII/L⊙ between $1.72-2.37\times 10^{8}$ L⊙. Thus, the overall
physical properties derived for the QSO2 candidates are consistent with being
a luminous AGN.

###  7.2 SED fitting with CIGALE

Due to the low S/N or the lack of an SDSS spectrum, conducting a comprehensive
spectroscopic analysis for the entire sample was not possible. Thus, for our
entire sample of QSO2 candidates we additionally performed an SED fitting
analysis using Code Investigating GALaxy Emission (CIGALE Burgarella et al.
2005; Noll et al. 2009; Boquien et al. 2019; Yang et al. 2022, version
2022.1). CIGALE performs the SED fit in a self-consistent framework
considering the energy balance between UV/optical absorption and IR emission.
From the best-fit model, various parameters can be estimated, including
parameters related to the AGN or stellar components of the SED. The models
used here are described in Yang et al. (2020) and the parameters are presented
in Sect. F.

To estimate the contribution of the AGN to the SED, we used the SKIRTOR model,
which conceptualises the torus as comprised of dusty clumps (Stalevski et al.
2012, 2016). This model considers that $97\%$ of the mass fraction within the
AGN is in the form of high-density clumps, with the remaining $3\%$ evenly
distributed, while incorporating an anisotropic disk emission.

For our study, the AGN fraction, fracAGN, is among the most interesting
parameters, as it allows one to understand whether an obscured AGN is likely
to be present, validating our AGN classification. The fracAGN is defined as
the AGN contribution to the total IR dust luminosity:

$\rm{frac_{AGN}=\frac{L_{dust,AGN}}{L_{dust,AGN}+L_{dust,galaxy}}},$ (11)

where $\rm{L_{dust,AGN}}$ and $\rm{L_{dust,galaxy}}$ are the AGN and galactic
dust luminosity, integrated over all wavelengths. We allowed $\rm{frac_{AGN}}$
to vary between zero (no AGN) to 0.99 (AGN dominated).

Figure 5: Histogram (blue) and empirical cumulative distribution function
(orange) for the AGN fraction, fracAGN, derived using CIGALE using the SKIRTOR
model.

Figure 5 displays the distribution of fracAGN obtained using CIGALE. A
significant number of QSO2 candidates require high fracAGN values, implying
that dust re-emission due to a high ionisation source is necessary to explain
the observed enhancement in the IR part of the SED. Therefore, including an
AGN component to account for the optical-NIR-MIR SED with enhanced dust
emission is mandatory to analyse these sources.

A more thorough analysis of the SED fitting results, including a comparison
with existing literature, lies beyond the scope of this paper and will be
presented in an upcoming publication (Cunha et al., in preparation).

##  8 Placing the QSO2 candidates into the galaxy evolution history

Currently, mergers and interactions between galaxies are believed to play
crucial roles in the evolutionary history of galaxies and AGNs (e.g. Sanders
et al. 1988; Hopkins et al. 2008; Treister et al. 2012; Villar-Martín et al.
2012; Goulding et al. 2018). While there is no consensus on the impact of the
merger process on the physical properties of obscured AGNs (e.g. Mechtley et
al. 2016; Marian et al. 2019), morphological studies have demonstrated its
potential as a driving mechanism (e.g. Treister et al. 2012; Urbano-Mayorgas
et al. 2019; Zhao et al. 2019; Marian et al. 2020), in particular for QSO2
(e.g. Greene et al. 2009; Villar-Martín et al. 2011; Bessiere et al. 2012;
Wylezalek et al. 2016; Araujo et al. 2023; Pierce et al. 2023).

When two galaxies merge, the first by-product is thought to be a dusty star-
forming galaxy which evolves into a dusty AGN. During this transitional phase,
the galaxy is heavily obscured by dust, also known as a dust-obscured galaxy
(DOG; Dey et al. 2008; Fiore et al. 2008; Bussmann et al. 2009; Desai et al.
2009; Bussmann et al. 2011; Yutani et al. 2022). Therefore, DOGs are believed
to be a transition phase from a gas-rich major merger to an optically thin
quasar in a gas-rich major merger scenario Dey et al. (2008). Due to the dusty
environment, DOGs have a very red optical-MIR colour with $i-[W4]_{AB}\geq
7.0$ where the i-band and W4 band are in AB magnitude (Toba et al. 2015, 2017,
2018). Applying this criterion to our new QSO2 candidates, we find that 208
($\sim 57\%$) of them met the DOG selection criterion. We note that 127 ($\sim
88\%$) of the spectroscopically confirmed QSO2 used for model training (see
Sect. 2.1.1) also meet the criterion. This result allowed us to confirm that
dust related features are also seen in our candidate sample, validating the
presence of high amount of dust, excepted in obscured AGNs.

Extremely red quasars (ERQs; e.g. Ross et al. 2015; Hamann et al. 2017; Villar
Martín et al. 2020) are of significant relevance in this context. ERQs
represent a distinct transition phase population that exhibits both AGN and
star-forming activity. Ross et al. (2015) identified ERQs using SDSS and WISE
photometry, which included various AGN types, including obscured and
unobscured QSO, starburst-dominated quasars, and quasars with broad absorption
lines. Finally, Hamann et al. (2017) refined the selection criteria based on
broadband photometry and emission line measurements: $i-[W3]_{AB}\geq 4.6$.
Interestingly, 38 ($\sim 10\%$) of our new QSO2 candidates meet these criteria
(and also meet the DOG criterion). Similarly to the previous analysis, this
analysis allowed us to better understand the reliability of this selection,
always keeping in mind the different nature of these type of sources from
QSO2s.

In summary, we find that most of our QSO candidates are consistent with being
dust-obscured host galaxies, as also supported by the results in Sect. 7.2. As
such, we argue that the sources are likely to be a transitional evolutionary
stage, similar to the findings of Ishikawa et al. (2023) (see discusion in
Sect. 4 therein). High-resolution imaging will be crucial to asses this
question.

It is tantalising to speculate further on the nature of our QSO2 candidates.
The fact that many do not show optical emission lines in their spectra,
despite showing infrared colours consistent with the presence of an AGN, might
be due to the AGN being so heavily obscured by dust that the ionising UV
continuum from the accretion disk is fully absorbed in the nucleus, and does
not escape to produce a classical narrow line region. This scenario would
represent a relatively early stage in the life of the AGN.

Furthermore, the dearth of emission lines could mean that our sample contains
some analogues of the ‘line-dark’ radio galaxies studied by Humphrey et al.
(2015a, 2016), where no significant UV continuum or lines were detected (see
also Willott et al. 2001). In a detailed case-study of one such source,
Humphrey et al. (2015a) concluded that the host galaxy is in an advanced stage
of being cleansed of its cold ISM by feedback driven by the AGN, thus
representing a relatively late stage in the life of the AGN. One particular
source, SDSS J121805.63+583904.6, falls within the definition of a ‘line-dark’
radio galaxy (see Sect. E). High spatial resolution radio and optical imaging
as well as deep optical and IR spectroscopy will be crucial to clarifying the
nature and evolutionary stage of our QSO2 candidates.

##  9 Conclusions

We have presented AMELIA, an application of the few-shot machine-transfer-
learning based method to identify QSO2s using optical and infrared photometric
data. AMELIA uses a generalised stacking ensemble technique, which acts as an
intelligent system of optimisation and model selection, to improve
classification performance. We validated the performance of AMELIA for the
selection of QSO2s by performing a binary classification of spectroscopically
confirmed QSO2s and ‘normal’ galaxies, with an F1-Score $\geq 0.8$ for QSO2
selection.

We applied the AMELIA pipeline to the problem of identifying QSO2s that were
previously misclassified as (non-active) galaxies by the SDSS classification
pipeline. We selected 366 new QSO2 candidates, using a class probability of
$\geq 0.8$.

From the $138$ candidates that have reliable spectra, we obtained a mean
redshift of $z\sim 1.1$. Of these sources, $30$ were detected in [NeV]$\lambda
3426$ and thus can be considered ‘spectroscopically confirmed’ QSO2s. For an
additional $35$ candidates, the [NeIII]$\lambda 3869$ line was instead
detected, making them ‘likely’ QSO2s. In total, $65$ of our QSO2 candidates
show [NeV]$\lambda 3426$ or [NeIII]$\lambda 3869$, so $\sim 47\%$ of the
candidates have reliable spectra.

We cross-matched our candidates with X-ray and radio catalogues and found that
ten candidates have X-ray detection and 24 candidates have radio detection. We
derived the radio luminosity of these sources and concluded that they show
L150MHz, which is consistent with an AGN source.

To complement the spectroscopic analysis, we also performed an SED fitting
using the CIGALE code. We found that all the QSO2 candidates require an AGN
component to account for their SED, corroborating their AGN classification and
dusty nature. In fact, more than half of our QSO2 candidates (208) meet the
DOG selection criterion, and a significant number (38) also meet the ERQ
criterion.

The physical properties of the AGNs were also derived by extrapolation from
the [NeIII] $\lambda 3869$ flux. We obtained the following median values:
L${}_{\rm{OIII}}=2.9\times 10^{42}$ erg $s^{-1}$; $L_{\rm{bol}}=1.1\times
10^{46}$ erg $s^{-1}$; $\dot{M}=2.4$ $M_{\odot}$ yr-1; and M${}_{\rm
BH}=8.8\times 10^{7}M_{\odot}$.

To further characterise our QSO2 candidate sample and confirm our
classification, more observations are necessary, which would also allow for
more reliable estimation of the host galaxy and AGN properties (e.g. star-
formation status, stellar mass, outflow activity status). Multi-wavelength
observations (e.g. X-ray, IR, sub-millimetre, and radio) along with high-
resolution imaging will play a pivotal role in obtaining a better
understanding of the nature of our candidates. Finally, we remark that despite
the substantial observational challenges involved with the study of sources in
the ‘redshift desert’, QSO2s in this redshift regime provide a crucial bridge
between obscured quasar activity at ‘cosmic noon’ ($z\sim 2$) and in the low-
redshift Universe ($z\leq 1$), potentially allowing us to gain insights into
the evolution of AGN and galaxy co-evolution across cosmic time.

###### Acknowledgements.

PACC also thanks Daniel Vaz, Catarina Marques, Christina Thörne and José
Fernández for the very useful comments and suggestions. PACC acknowledges
financial support by Fundação para a Ciência e Tecnologia (FCT) through the
grant 2022.11477.BD, and through the research grants UIDB/04434/2020,
UIDP/04434/2020 and EXPL/FIS-AST/1085/2021. AH acknowledges supported by
Fundação para a Ciência e Tecnologia (FCT) through grants UID/FIS/04434/2019,
UIDB/04434/2020, UIDP/04434/2020 and PTDC/FIS-AST/29245/2017, and an FCT-CAPES
Transnational Cooperation Project. SGM acknowledges support from Fundação para
a Cência e Tecnologia (FCT) through the Fellowships PD/BD/135228/2017
(PhD::SPACE Doctoral Network PD/00040/2012), POCH/FSE (EC) and
COVID/BD/152181/2021. SGM was also supported by Fundação para a Ciência e
Tecnologia (FCT) through the research grants UIDB/04434/2020 and
UIDP/04434/2020, and an FCT-CAPES Transnational Cooperation Project ”Parceria
Estratégica em Astrofísica Portugal-Brasil”. R.C. acknowledges support from
the Fundação para a Ciência e a Tecnologia (FCT) through the Fellowship
PD/BD/150455/2019 (PhD:SPACE Doctoral Network PD/00040/2012) and POCH/FSE (EC)
and through research grants PTDC/FIS-AST/29245/2017, EXPL/FIS-AST/1085/2021,
UIDB/04434/2020, and UIDP/04434/2020. JB acknowledges financial support from
the Fundação para a Ciência e a Tecnologia (FCT) through national funds
PTDC/FIS-AST/4862/2020 and work contract 2020.03379.CEECIND. A.P.A.
acknowledges support from the Fundação para a Ciência e a Tecnologia (FCT)
through the work Contract No. 2020.03946.CEECIND. AH acknowledges support from
NVIDIA through an NVIDIA Academic Hardware Grant Award. This publication makes
use of data products from the Wide-field Infrared Survey Explorer, which is a
joint project of the University of California, Los Angeles, and the Jet
Propulsion Laboratory/California Institute of Technology, funded by the
National Aeronautics and Space Administration. Funding for the Sloan Digital
Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S.
Department of Energy Office of Science, and the Participating Institutions.
SDSS-IV acknowledges support and resources from the Center for High
Performance Computing at the University of Utah. The SDSS website is
www.sdss.org. SDSS-IV is managed by the Astrophysical Research Consortium for
the Participating Institutions of the SDSS Collaboration including the
Brazilian Participation Group, the Carnegie Institution for Science, Carnegie
Mellon University, Center for Astrophysics — Harvard & Smithsonian, the
Chilean Participation Group, the French Participation Group, Instituto de
Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the
Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the
Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz
Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie
(MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-
Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical
Observatories of China, New Mexico State University, New York University,
University of Notre Dame, Observatário Nacional / MCTI, The Ohio State
University, Pennsylvania State University, Shanghai Astronomical Observatory,
United Kingdom Participation Group, Universidad Nacional Autónoma de México,
University of Arizona, University of Colorado Boulder, University of Oxford,
University of Portsmouth, University of Utah, University of Virginia,
University of Washington, University of Wisconsin, Vanderbilt University, and
Yale University. LOFAR is the Low Frequency Array designed and constructed by
ASTRON. It has observing, data processing, and data storage facilities in
several countries, which are owned by various parties (each with their own
funding sources), and which are collectively operated by the ILT foundation
under a joint scientific policy. The ILT resources have benefited from the
following recent major funding sources: CNRS-INSU, Observatoire de Paris and
Université d’Orléans, France; BMBF, MIWF-NRW, MPG, Germany; Science Foundation
Ireland (SFI), Department of Business, Enterprise and Innovation (DBEI),
Ireland; NWO, The Netherlands; The Science and Technology Facilities Council,
UK; Ministry of Science and Higher Education, Poland; The Istituto Nazionale
di Astrofisica (INAF), Italy. This research made use of the Dutch national
e-infrastructure with support of the SURF Cooperative (e-infra 180169) and the
LOFAR e-infra group. The Jülich LOFAR Long Term Archive and the German LOFAR
network are both coordinated and operated by the Jülich Supercomputing Centre
(JSC), and computing resources on the supercomputer JUWELS at JSC were
provided by the Gauss Centre for Supercomputing e.V. (grant CHTB00) through
the John von Neumann Institute for Computing (NIC). This research made use of
the University of Hertfordshire high-performance computing facility and the
LOFAR-UK computing facility located at the University of Hertfordshire and
supported by STFC [ST/P000096/1], and of the Italian LOFAR IT computing
infrastructure supported and operated by INAF, and by the Physics Department
of Turin university (under an agreement with Consorzio Interuniversitario per
la Fisica Spaziale) at the C3S Supercomputing Centre, Italy. This research has
made use of the VizieR catalogue access tool, CDS, Strasbourg, France
(Ochsenbein 1996). The original description of the VizieR service was
published in Ochsenbein et al. (2000). In preparation for this work, we used
the following codes for Python: Numpy Harris et al. (2020), Scipy (Virtanen et
al. 2020), Astropy (Astropy Collaboration et al. 2013, 2018, 2022), Scikit-
Learn (Pedregosa et al. 2011), Pandas (Wes McKinney 2010), XGBoost (Chen &
Guestrin 2016), CatBoost (Prokhorenkova et al. 2018), LightGBM (Ke et al.
2017), specstack (Thomas 2019), matplotlib (Hunter 2007), seaborn (Waskom
2021).

## Data availability

All data used within this work are publicly available in the SDSS archive. The
code is publicly available on GitHub (https://github.com/pedro-acunha/AMELIA).
The complete candidates data set are available from the corresponding author
upon reasonable request.

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##  Appendix A The impact of the training data distribution

Figure 6: Visual representation of the three different training scenarios.
From left to right: Balanced training set, unbalanced training set, and equal
ratio training set. The yellow colour represents the QSO2 sources, and the
blue colour indicates SDSS galaxies.

We explored two distinct methods for the training-test split and assessed
their implications within the semi-supervised one-class approach. We
established three different training scenarios for the training set, as
illustrated in Fig. 6 for visual reference:

  * •

balanced training set: The balanced training set comprises 100 QSO2 sources
and 100 galaxy sources. The QSO2 data is distributed in a $7:3$ ratio, with
70$\%$ allocated to the training set and 30$\%$ to the test set. Conversely,
the galaxy data follows a $10:90$ ratio, where 10$\%$ is included in the
training set, and 90$\%$ is reserved for the testing set.

  * •

unbalanced training set: In the unbalanced training set, there are 100 QSO2
sources and 500 galaxy sources. The QSO2 data maintains a $7:3$ ratio, while
the galaxy data adopts a $1:1$ ratio, signifying an equal $50\%$ distribution
between the training and testing sets.

  * •

equal ratio training set: The equal ratio training set ensures the equality of
class ratios across both training and testing sets, maintaining a constant
$7:3$ distribution for both QSO2 and galaxy classes.

Table 4: Percentage of False Positives and False Negatives for the different
strategies of train-test splits and threshold values.

| Equal Ratio ($\%$) | Balance with $0.5$ ($\%$) | Balance with $0.8$ ($\%$) | Unbalance ($\%$)
---|---|---|---|---
False Positive | 2 | 14 | 9 | 3
False Negative | 17 | 5 | 9 | 18

191919False positive (galaxy) and false negative (QSO2) comparison for the
XGBoost algorithm: Equal ratio training set, balanced training set with
threshold $50\%$ and $80\%$, and unbalanced training set. The results are
presented in percentage based on the total number of objects in each
classification class.

To explore the implications of various configurations in one-class semi-
supervised tasks, as detailed in Sect.5.2, our investigation focusses on the
evaluation of two key metrics. Specifically, we concentrate on evaluating the
total number of false positives (FP) for the galaxy class (sources labelled as
‘galaxy’ but classified as QSO2) as well as the false negatives (FN) for the
QSO2 class (sources labelled as ‘QSO2’ but classified as galaxy). Table 19
displays the analysis of four distinct training data distributions utilising
the XGBoost algorithm.

Figure 7: Histogram with the average probability predictions from the
generalised stacking method with meta-learner correction. The black dashed
line shows the ‘default value’, 0.5, and the yellow dashed line shows the
chosen classification threshold in this study, 0.8.

The balanced training set exhibits a lower FN percentage, indicating greater
efficiency in identifying QSO2 sources. These results hold significant value
as they aid in selecting the most suitable training distribution for a given
task. For instance, the equal ratio training set can be used for selecting
galaxies, while the balanced training set demonstrates more promising results
for selecting QSO2 sources. This result is not surprising, as it is well-
established in the machine learning community (e.g. LemaÃŽtre et al. 2017, and
references herein).

The primary goal of our machine learning pipeline is to identify QSO2 sources
within a data set that includes known and unknown classification labels. In
the context of dealing with misclassified data in binary classification, false
positive (FP) is the most relevant metric for analysis. As the volume of test
data increases, the number of FPs also increases. Therefore, we opted for the
balanced training set with a threshold $\geq 0.8$, as it achieves a balance
between FPs and false negatives (FNs), as seen in Table 19. In Fig. 7, we show
the distribution of the average classification probability from our meta-
learners. Since this is a semi-supervised approach, it is not possible to
conduct any threshold optimisation analysis, as the ground-truth is not known.
Choosing a lower threshold, between 0.5 and 0.8, would substantially increase
the number of targets to validate. Our choice, threshold $\geq$ 0.8, enables
us to mitigate the bias of our model by balancing the number of
misclassifications, due to the increase of the classification threshold, while
reducing the number of potential candidates for further validation.

##  Appendix B Feature importance analysis

One of the primary advantages of employing a decision-tree-based algorithm
lies in its capability to identify crucial features. These models can quantify
the utility of individual features (or observables) in selecting QSO2 sources.

Table 5: Top five features with the highest feature importance for
theRandomForest, XGBoost, CatBoost, and LightGBM models.

Model | Features
---|---
RandomForest | g-z, g, r-W1, g-W1, g-W2
XGBoost | W4, u-g, u-r, g, r-z
CatBoost | g, u-g, W4, g-i, r
LightGBM | u-g, g, W4, r-z, r-W4
202020The features’ order is from the highest feature importance metric to the
lowest. To improve the readability, the feature names were simplified.

We provide a quick analysis of the feature importance provided by
RandomForest, LightGBM, CatBoost, and XGBoost models. The majority of the
methodologies use different techniques to compute the feature importances.
While their examination can inform about the relevant features, their
interpretation should be taken with a grain of salt. In the RandomForest
algorithm, the feature importance is impurity-based, meaning that it is
calculated based on the mean and standard deviation of the impurity
measurement from top to bottom of the tree. LightGBM can compute feature
importance based on split and gain. In this work, we used the split method
which is based on the number of times the feature is used within the model to
compute the final prediction. For the CatBoost algorithm, we used the
PredictionValuesChange method, which computes the average change in the
predicted values if the feature value changes. Finally, for XGBoost the weight
importance was used, which computes the number of times a feature appears in a
tree, similar to the method used in LightGBM.

The top five features with the highest importance of the features are
summarised in Table 20, ordered by decreasing importance. Some features, such
as u-g, g, and W4, appear as some of the most important features in different
models. Even though some features appear as less important, it does not mean
they do not make a difference in the final classification. We inferred that it
is the balance between optical-infrared information that allows the machine
learning models to distinguish between the two classes successfully. Some
studies explore the relationship between feature importance and final
predictions in more depth (e.g. Ansari et al. 2022; He et al. 2022). We did
not perform such an analysis due to the limited size of our training set.

In Euclid Collaboration et al. (2023b), the authors conducted A/B tests
(randomised experiments where one or more feature is removed and the
performance model re-evaluated) to understand the impact of features in the
classification models, while doing cross-validation and randomly changing
random seeds. They have shown that removing multiple features (e.g. broad-band
colours, magnitudes) impacts negatively the F1-scores metrics when classifying
quiescent galaxies, in the context of Euclid. We performed similar A/B tests
for the selection of QSO2 and found that removing features independently of
their importance in individual learners has a negative impact.

##  Appendix C Performance prediction functions to estimate statistical
metrics

Since we do not have access to ground-truth labels for our test set, it is
interesting to quantify the performance of the model based on the model
output. In Humphrey et al. (2022), the authors used a Confidence-Based
Performance Estimation
(CBPE)212121https://nannyml.readthedocs.io/en/stable/how_it_works/performance_estimation.html
method from the NANNYML222222https://github.com/NannyML/nannyml package to
predict the impact of population shifts in unlabelled data for the selection
of quasars using decision-tree ensemble models.

Therein, performance prediction functions allow the estimation of the
following metrics for our semi-supervised approach: F1-score, precision,
accuracy, recall and specificity. For the performance prediction metrics, we
obtain A$=0.90$, P$=0.92$, R$=0.26$, F1-Score$=0.40$ and specificity of
$0.997$. Low recall can be explained with the high-threshold setup.

##  Appendix D QSO2 candidate with wrong spec-z estimation: SDSS
J013056.89-022638.1

Figure 8: Spectrum of SDSSJ013154.44 + 224437.9 from the BOSS spectrograph. In
the top plot, the red vertical lines indicate the emission lines for z$\approx
1.633$, computed by idlspec2d from SDSS. In the middle plot, the red vertical
lines indicate the emission lines for z$\approx 0.960$, computed by the
authors. In the bottom plot, the spectrum without any smoothing is shown.

In our QSO2 candidate sample, we noticed that the spectrum for the source SDSS
J013056.89-022638.1 showed emission lines that did not match the estimated
SDSS spectroscopic redshift. In particular, [OII]$\lambda\lambda 3727,3730$
and [OIII]$\lambda\lambda 4959,5008$ were misidentified. By computing the
redshift using the [OIII]$\lambda\lambda 4959,5008$ doublet, we obtain a
spectroscopic redshift z$\approx 0.960$, quite different from z$\approx 1.633$
(computed with idlspec2d). In Fig.8, the top plot shows where the emission
lines would be placed with z$\approx 1.633$, and the middle plot shows the
wavelength for the emission lines with the estimated redshift z$\approx
0.960$. One can easily identify the following emission lines: MgII$\lambda
2799$, [NeV]$\lambda\lambda 3347,3426$, [OII]$\lambda\lambda 3727,3730$,
[NeIII]$\lambda 3869$, HeI$\lambda 3889$, and [OIII]$\lambda\lambda 4959,5008$
doublets.

##  Appendix E Notes on individual object: SDSS J121805.63+583904.6

Previously identified in the Millions of Optical Radio/X-ray Associations
(MORX) v2 catalogue Flesch (2024), this faint optical source (u: 22.594; g:
23.394; r: 22.368; i: 20.782; z: 19.634) exhibit double radio lobes in the
image mosaics from LoTSS DR2. The SDSS spectrum shows no emission lines, being
an unclear source, assuming our spectroscopic classification (see Sect.6.1).
Also, this source is consistent with a featureless spectrum, making it a good
candidate for a ‘line-dark’ radio galaxy (Humphrey et al. 2015a, 2016). Even
though, it is not possible to confirm the redshift, from the lack of strong
AGN emission lines one can extrapolate that this source could be within the
‘redshift desert’.

(a) SDSS optical image

(b) LoTSS radio image (c) SDSS optical spectrum

Figure 9: Optical and radio imaging and spectrum for the SDSS
J121805.63+583904.6. From left to right, top to bottom: (a) grz cutout image
from SDSS with 1$\arcmin$ size; (b) LoTSS DR2 cutout image with 1$\arcmin$
size; (c) SDSS optical spectrum from BOSS spectrograph, assuming z$=1.122$ .

##  Appendix F CIGALE SED fitting parameters

Table 6: Summary of the CIGALE SED fitting parameters used in this work. Module | Parameter | Values
---|---|---
Star formation history: |  $e$-folding time $\tau_{\rm{main}}$ (Myr) | 5-20 (step 5), 30, 50, 100-300 (step 100),
|  | 500.0, 1000.0, 3000.0, 5000.0
sfhdelayed | Stellar Age, tmain (Myr) | 31, 35, 40, 50, 75, 100-500 (step 100),
|  | 1000, 3000, 4000, 5000, 5500
|  $e$-folding time $\tau_{\rm{burst}}$ (Myr) | 5, 10, 15, 20, 25, 50
| Stellar Age, tburst (Myr) | 5, 10, 15, 20, 25, 30
| Age of the last burst (Myr) | 0.0, 0.1, 0.2
Galactic dust emission: |  $\alpha$ slope | 0.5, 1, 1.5, 2.0, 2.5, 3, 3.5
Dale et al. (2014) |  |
AGN: SKIRTOR | Average edge-on optical depth at 9.7 $\mu$m | 3, 7, 11
| AGN contribution to IR luminosity, fracAGN | 0.0-0.9 (step 0.1), 0.99
Stalevski et al. (2016) | inclination (i.e. viewing angle) | 40, 60, 80
| Intrinsic disk type | Schartmann (2005) spectrum
| Polar extinction E(B-V) | 0, 0.05, 0.1, 0.2, 0.3

##  Appendix G Estimation of physical properties

Table 7: Physical properties derived for the [NeIII] emitters using Morais et al. (2021) photoionisation models. Plate-MJD-Fiber | z |  L[OIII] ( erg $s^{-1}$) | L[OIII]/L⊙ |  Lbol ( erg $s^{-1}$) |  $\dot{M}$ ( M⊙ yr-1) |  MBH ( M⊙)
---|---|---|---|---|---|---
10447-58143-90 | 1.133 | 1.676E+42 | 4.378E+08 | 8.005E+45 | 1.781 | 6.404E+07
10454-58135-808 | 1.109 | 9.084E+41 | 2.373E+08 | 5.675E+45 | 1.263 | 4.540E+07
10456-58136-680 | 1.019 | 7.033E+41 | 1.837E+08 | 4.915E+45 | 1.094 | 3.932E+07
10468-58131-630 | 1.215 | 2.936E+42 | 7.670E+08 | 1.097E+46 | 2.441 | 8.775E+07
10469-58133-915 | 1.102 | 2.378E+42 | 6.213E+08 | 9.744E+45 | 2.168 | 7.795E+07
10739-58255-860 | 1.562 | 4.064E+42 | 1.062E+09 | 1.317E+46 | 2.93 | 1.053E+08
11650-58508-299 | 1.141 | 5.245E+42 | 1.370E+09 | 1.519E+46 | 3.381 | 1.216E+08
6932-56397-443 | 1.061 | 1.067E+42 | 2.787E+08 | 6.211E+45 | 1.382 | 4.969E+07
7239-56693-902 | 1.236 | 1.437E+42 | 3.756E+08 | 7.344E+45 | 1.634 | 5.875E+07
7242-56628-268 | 1.183 | 6.326E+42 | 1.653E+09 | 1.688E+46 | 3.757 | 1.350E+08
7375-56981-607 | 1.057 | 5.606E+42 | 1.465E+09 | 1.577E+46 | 3.51 | 1.262E+08
7574-56945-776 | 1.027 | 3.551E+42 | 9.277E+08 | 1.221E+46 | 2.716 | 9.764E+07
7839-56900-431 | 1.006 | 1.936E+42 | 5.057E+08 | 8.681E+45 | 1.932 | 6.945E+07
8184-57426-269 | 1.115 | 1.959E+42 | 5.117E+08 | 8.738E+45 | 1.944 | 6.990E+07
8227-57427-716 | 1.364 | 1.572E+43 | 4.107E+09 | 2.815E+46 | 6.264 | 2.252E+08
8409-57867-708 | 1.084 | 3.721E+42 | 9.720E+08 | 1.253E+46 | 2.788 | 1.002E+08
8543-57542-675 | 1.012 | 2.891E+42 | 7.552E+08 | 1.087E+46 | 2.42 | 8.698E+07
8769-57338-24 | 1.208 | 2.879E+42 | 7.521E+08 | 1.085E+46 | 2.414 | 8.679E+07
8769-57338-810 | 1.073 | 1.919E+42 | 5.012E+08 | 8.637E+45 | 1.922 | 6.909E+07
8792-57364-92 | 1.021 | 2.478E+42 | 6.473E+08 | 9.971E+45 | 2.219 | 7.977E+07
8954-57453-648 | 1.328 | 3.463E+42 | 9.047E+08 | 1.203E+46 | 2.678 | 9.627E+07
8958-57514-103 | 1.157 | 6.697E+42 | 1.750E+09 | 1.743E+46 | 3.879 | 1.394E+08
9198-57713-103 | 1.166 | 4.802E+42 | 1.254E+09 | 1.446E+46 | 3.218 | 1.157E+08
9199-57684-394 | 1.002 | 1.463E+42 | 3.823E+08 | 7.418E+45 | 1.651 | 5.935E+07
9204-57712-580 | 1.055 | 1.495E+42 | 3.906E+08 | 7.508E+45 | 1.671 | 6.006E+07
9233-58035-981 | 1.146 | 3.766E+42 | 9.839E+08 | 1.262E+46 | 2.807 | 1.009E+08
9312-57784-764 | 1.157 | 2.843E+42 | 7.426E+08 | 1.077E+46 | 2.397 | 8.617E+07
7338-57874-237 | 1.002 | 1.608E+43 | 4.200E+09 | 2.850E+46 | 6.343 | 2.280E+08
7635-56979-874 | 0.960 | 2.240E+43 | 5.851E+09 | 3.434E+46 | 7.642 | 2.747E+08
10474-58143-153 | 1.044 | 6.824E+41 | 1.783E+08 | 4.833E+45 | 1.075 | 3.866E+07
9315-57713-481 | 1.091 | 1.377E+42 | 3.598E+08 | 7.170E+45 | 1.596 | 5.736E+07
9320-58069-26 | 1.102 | 3.823E+42 | 9.988E+08 | 1.272E+46 | 2.831 | 1.018E+08
9321-58069-779 | 1.004 | 3.671E+42 | 9.590E+08 | 1.243E+46 | 2.767 | 9.948E+07
9323-58047-417 | 1.095 | 2.151E+42 | 5.618E+08 | 9.209E+45 | 2.049 | 7.367E+07
9329-57688-155 | 1.053 | 7.342E+42 | 1.918E+09 | 1.835E+46 | 4.084 | 1.468E+08
9331-57711-141 | 1.183 | 1.164E+42 | 3.040E+08 | 6.522E+45 | 1.451 | 5.218E+07
9335-57713-514 | 1.01 | 2.670E+42 | 6.975E+08 | 1.040E+46 | 2.314 | 8.319E+07
9335-57713-675 | 1.023 | 3.339E+42 | 8.721E+08 | 1.179E+46 | 2.623 | 9.431E+07
9343-57715-575 | 1.02 | 3.469E+42 | 9.061E+08 | 1.205E+46 | 2.68 | 9.636E+07
9345-57713-444 | 1.128 | 2.225E+42 | 5.812E+08 | 9.386E+45 | 2.089 | 7.509E+07
9373-58096-257 | 1.25 | 4.309E+42 | 1.126E+09 | 1.361E+46 | 3.028 | 1.089E+08
9379-58081-222 | 1.41 | 7.954E+42 | 2.078E+09 | 1.920E+46 | 4.272 | 1.536E+08
9384-58080-706 | 1.011 | 8.429E+42 | 2.202E+09 | 1.983E+46 | 4.413 | 1.587E+08
9407-58041-946 | 1.101 | 6.599E+41 | 1.724E+08 | 4.742E+45 | 1.055 | 3.794E+07
9416-58098-471 | 1.021 | 2.033E+42 | 5.311E+08 | 8.923E+45 | 1.986 | 7.138E+07
9553-57801-634 | 1.237 | 2.242E+42 | 5.858E+08 | 9.428E+45 | 2.098 | 7.542E+07
9555-57814-860 | 1.145 | 2.382E+42 | 6.222E+08 | 9.752E+45 | 2.17 | 7.802E+07
9557-58098-437 | 1.229 | 1.442E+43 | 3.768E+09 | 2.682E+46 | 5.968 | 2.146E+08
9594-58135-741 | 1.351 | 4.930E+42 | 1.288E+09 | 1.467E+46 | 3.266 | 1.174E+08
9616-58135-297 | 1.037 | 4.382E+42 | 1.145E+09 | 1.373E+46 | 3.056 | 1.099E+08
11078-58437-179 | 1.055 | 1.958E+42 | 5.115E+08 | 8.736E+45 | 1.944 | 6.989E+07