11institutetext: 1 Instituto de Astrofísica e Ciências do Espaço, Universidade
do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal
2 DTx – Digital Transformation CoLAB, Building 1, Azurém Campus, University of
Minho, 4800-058 Guimarães, Portugal
3 INAF-Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via
Piero Gobetti 93/3, I-40129 Bologna, Italy
4 Faculdade de Ciências da Universidade do Porto, Rua do Campo de Alegre,
4150-007 Porto, Portugal
5 School of Physics, HH Wills Physics Laboratory, University of Bristol,
Tyndall Avenue, Bristol, BS8 1TL, UK
6 Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV
Groningen, The Netherlands
7 INAF-Osservatorio Astronomico di Capodimonte, Via Moiariello 16, I-80131
Napoli, Italy
8 Departamento de Física, Faculdade de Ciências, Universidade de Lisboa,
Edifício C8, Campo Grande, PT1749-016 Lisboa, Portugal
9 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências,
Universidade de Lisboa, Tapada da Ajuda, PT-1349-018 Lisboa, Portugal
10 Dipartimento di Fisica e Astronomia ”Augusto Righi” - Alma Mater Studiorum
Università di Bologna, via Piero Gobetti 93/2, I-40129 Bologna, Italy
11 Institute of Cosmology and Gravitation, University of Portsmouth,
Portsmouth PO1 3FX, UK
12 INFN-Bologna, Via Irnerio 46, I-40126 Bologna, Italy
13 Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1,
D-85748 Garching, Germany
14 Universitäts-Sternwarte München, Fakultät für Physik, Ludwig-Maximilians-
Universität München, Scheinerstrasse 1, 81679 München, Germany
15 INAF-Osservatorio Astrofisico di Torino, Via Osservatorio 20, I-10025 Pino
Torinese (TO), Italy
16 Department of Mathematics and Physics, Roma Tre University, Via della Vasca
Navale 84, I-00146 Rome, Italy
17 INFN-Sezione di Roma Tre, Via della Vasca Navale 84, I-00146, Roma, Italy
18 Dipartimento di Fisica, Universitá degli Studi di Torino, Via P. Giuria 1,
I-10125 Torino, Italy
19 INFN-Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy
20 INAF-IASF Milano, Via Alfonso Corti 12, I-20133 Milano, Italy
21 Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona), Spain
22 Port d’Informació Científica, Campus UAB, C. Albareda s/n, 08193 Bellaterra
(Barcelona), Spain
23 Institut d’Estudis Espacials de Catalunya (IEEC), Carrer Gran Capitá 2-4,
08034 Barcelona, Spain
24 Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans,
s/n, 08193 Barcelona, Spain
25 INAF-Osservatorio Astronomico di Roma, Via Frascati 33, I-00078 Monteporzio
Catone, Italy
26 Department of Physics ”E. Pancini”, University Federico II, Via Cinthia 6,
I-80126, Napoli, Italy
27 INFN section of Naples, Via Cinthia 6, I-80126, Napoli, Italy
28 Dipartimento di Fisica e Astronomia ”Augusto Righi” - Alma Mater Studiorum
Universitá di Bologna, Viale Berti Pichat 6/2, I-40127 Bologna, Italy
29 INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125,
Firenze, Italy
30 Centre National d’Etudes Spatiales, Toulouse, France
31 Institut national de physique nucléaire et de physique des particules, 3
rue Michel-Ange, 75794 Paris Cédex 16, France
32 Institute for Astronomy, University of Edinburgh, Royal Observatory,
Blackford Hill, Edinburgh EH9 3HJ, UK
33 Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy,
University of Manchester, Oxford Road, Manchester M13 9PL, UK
34 European Space Agency/ESRIN, Largo Galileo Galilei 1, 00044 Frascati, Roma,
Italy
35 ESAC/ESA, Camino Bajo del Castillo, s/n., Urb. Villafranca del Castillo,
28692 Villanueva de la Cañada, Madrid, Spain
36 Univ Lyon, Univ Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822,
F-69622, Villeurbanne, France
37 Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique
Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix,
Switzerland
38 Mullard Space Science Laboratory, University College London, Holmbury St
Mary, Dorking, Surrey RH5 6NT, UK
39 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências,
Universidade de Lisboa, Campo Grande, PT-1749-016 Lisboa, Portugal
40 Department of Astronomy, University of Geneva, ch. dÉcogia 16, CH-1290
Versoix, Switzerland
41 Université Paris-Saclay, CNRS, Institut d’astrophysique spatiale, 91405,
Orsay, France
42 Department of Physics, Oxford University, Keble Road, Oxford OX1 3RH, UK
43 INFN-Padova, Via Marzolo 8, I-35131 Padova, Italy
44 AIM, CEA, CNRS, Université Paris-Saclay, Université de Paris, F-91191 Gif-
sur-Yvette, France
45 INAF-Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, I-34143
Trieste, Italy
46 FRACTAL S.L.N.E., calle Tulipán 2, Portal 13 1A, 28231, Las Rozas de
Madrid, Spain
47 Aix-Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
48 Istituto Nazionale di Astrofisica (INAF) - Osservatorio di Astrofisica e
Scienza dello Spazio (OAS), Via Gobetti 93/3, I-40127 Bologna, Italy
49 Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46,
I-40126 Bologna, Italy
50 INAF-Osservatorio Astronomico di Padova, Via dell’Osservatorio 5, I-35122
Padova, Italy
51 Dipartimento di Fisica ”Aldo Pontremoli”, Universitá degli Studi di Milano,
Via Celoria 16, I-20133 Milano, Italy
52 INAF-Osservatorio Astronomico di Brera, Via Brera 28, I-20122 Milano, Italy
53 INFN-Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy
54 Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029
Blindern, N-0315 Oslo, Norway
55 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak
Grove Drive, Pasadena, CA, 91109, USA
56 von Hoerner & Sulger GmbH, SchloßPlatz 8, D-68723 Schwetzingen, Germany
57 Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg,
Germany
58 Université de Genève, Département de Physique Théorique and Centre for
Astroparticle Physics, 24 quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland
59 Department of Physics and Helsinki Institute of Physics, Gustaf Hällströmin
katu 2, 00014 University of Helsinki, Finland
60 NOVA optical infrared instrumentation group at ASTRON, Oude Hoogeveensedijk
4, 7991PD, Dwingeloo, The Netherlands
61 Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71,
53121 Bonn, Germany
62 INFN-Sezione di Bologna, Viale Berti Pichat 6/2, I-40127 Bologna, Italy
63 Department of Physics, Institute for Computational Cosmology, Durham
University, South Road, DH1 3LE, UK
64 Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire
Lagrange, Bd de l’Observatoire, CS 34229, 06304 Nice cedex 4, France
65 Institut d’Astrophysique de Paris, 98bis Boulevard Arago, F-75014, Paris,
France
66 Sorbonne Universités, UPMC Univ Paris 6 et CNRS, UMR 7095, Institut
d’Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France
67 University of Applied Sciences and Arts of Northwestern Switzerland, School
of Engineering, 5210 Windisch, Switzerland
68 European Space Agency/ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The
Netherlands
69 Department of Physics and Astronomy, University of Aarhus, Ny Munkegade
120, DK-8000 Aarhus C, Denmark
70 Université Paris-Saclay, Université Paris Cité, CEA, CNRS, Astrophysique,
Instrumentation et Modélisation Paris-Saclay, 91191 Gif-sur-Yvette, France
71 Institute of Space Science, Bucharest, Ro-077125, Romania
72 Dipartimento di Fisica e Astronomia ”G.Galilei”, Universitá di Padova, Via
Marzolo 8, I-35131 Padova, Italy
73 Departamento de Física, FCFM, Universidad de Chile, Blanco Encalada 2008,
Santiago, Chile
74 IFPU, Institute for Fundamental Physics of the Universe, via Beirut 2,
34151 Trieste, Italy
75 INFN-Sezione di Roma, Piazzale Aldo Moro, 2 - c/o Dipartimento di Fisica,
Edificio G. Marconi, I-00185 Roma, Italy
76 Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas
(CIEMAT), Avenida Complutense 40, 28040 Madrid, Spain
77 Universidad Politécnica de Cartagena, Departamento de Electrónica y
Tecnología de Computadoras, 30202 Cartagena, Spain
78 Infrared Processing and Analysis Center, California Institute of
Technology, Pasadena, CA 91125, USA
79 Université Paris Cité, CNRS, Astroparticule et Cosmologie, F-75013 Paris,
France
80 INAF-IASF Bologna, Via Piero Gobetti 101, I-40129 Bologna, Italy
81 INFN, Sezione di Trieste, Via Valerio 2, I-34127 Trieste TS, Italy
82 SISSA, International School for Advanced Studies, Via Bonomea 265, I-34136
Trieste TS, Italy
83 Departamento de Astrofísica, Universidad de La Laguna, E-38206, La Laguna,
Tenerife, Spain
84 Instituto de Astrofísica de Canarias (IAC); Departamento de Astrofísica,
Universidad de La Laguna (ULL), E-38200, La Laguna, Tenerife, Spain
85 Institut de Recherche en Astrophysique et Planétologie (IRAP), Université
de Toulouse, CNRS, UPS, CNES, 14 Av. Edouard Belin, F-31400 Toulouse, France
86 Dipartimento di Fisica - Sezione di Astronomia, Universitá di Trieste, Via
Tiepolo 11, I-34131 Trieste, Italy
87 INAF, Istituto di Radioastronomia, Via Piero Gobetti 101, I-40129 Bologna,
Italy
88 Dipartimento di Fisica e Scienze della Terra, Universitá degli Studi di
Ferrara, Via Giuseppe Saragat 1, I-44122 Ferrara, Italy
89 Institute for Theoretical Particle Physics and Cosmology (TTK), RWTH Aachen
University, D-52056 Aachen, Germany
90 Instituto de Astrofísica de Canarias, Calle Vía Láctea s/n, E-38204, San
Cristóbal de La Laguna, Tenerife, Spain
91 Department of Physics & Astronomy, University of California Irvine, Irvine
CA 92697, USA
92 University of Lyon, UCB Lyon 1, CNRS/IN2P3, IUF, IP2I Lyon, France
93 INFN-Sezione di Genova, Via Dodecaneso 33, I-16146, Genova, Italy
94 INAF-Istituto di Astrofisica e Planetologia Spaziali, via del Fosso del
Cavaliere, 100, I-00100 Roma, Italy
95 Instituto de Física Teórica UAM-CSIC, Campus de Cantoblanco, E-28049
Madrid, Spain
96 Department of Physics, P.O. Box 64, 00014 University of Helsinki, Finland
97 Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
98 Observatoire de Paris, PSL Research University 61, avenue de
l’Observatoire, F-75014 Paris, France
99 Université de Paris, F-75013, Paris, France, LERMA, Observatoire de Paris,
PSL Research University, CNRS, Sorbonne Université, F-75014 Paris, France
100 Department of Physics and Astronomy, University College London, Gower
Street, London WC1E 6BT, UK
101 Code 665, NASA Goddard Space Flight Center, Greenbelt, MD 20771 and SSAI,
Lanham, MD 20770, USA
102 Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of
Helsinki, Helsinki, Finland
103 Centre de Calcul de l’IN2P3, 21 avenue Pierre de Coubertin F-69627
Villeurbanne Cedex, France
104 Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2,
I-00185 Roma, Italy
105 Aix-Marseille Univ, CNRS, CNES, LAM, Marseille, France
106 Institut für Theoretische Physik, University of Heidelberg, Philosophenweg
16, 69120 Heidelberg, Germany
107 Zentrum für Astronomie, Universität Heidelberg, Philosophenweg 12, D-
69120 Heidelberg, Germany
108 Department of Mathematics and Physics E. De Giorgi, University of Salento,
Via per Arnesano, CP-I93, I-73100, Lecce, Italy
109 INFN, Sezione di Lecce, Via per Arnesano, CP-193, I-73100, Lecce, Italy
110 Institute for Computational Science, University of Zurich,
Winterthurerstrasse 190, 8057 Zurich, Switzerland
111 Junia, EPA department, F 59000 Lille, France
112 Department of Astrophysical Sciences, Peyton Hall, Princeton University,
Princeton, NJ 08544, USA
113 Department of Physics, P.O.Box 35 (YFL), 40014 University of Jyväskylä,
Finland
114 Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical
Institute (AIRUB), German Centre for Cosmological Lensing (GCCL), 44780
Bochum, Germany

#  Euclid preparation. XXII. Selection of quiescent galaxies from mock
photometry using machine learning

Euclid Collaboration: A. Humphrey1,2<EMAIL_ADDRESS>L. Bisigello3
P. A. C. Cunha1,4 M. Bolzonella3 S. Fotopoulou5 K. Caputi6 C. Tortora7 G.
Zamorani3 P. Papaderos0000-0002-3733-81748,9 D. Vergani3 J. Brinchmann1 M.
Moresco0000-0002-7616-713610,3 A. Amara11 N. Auricchio3 M. Baldi10,3,12 R.
Bender13,14 D. Bonino15 E. Branchini16,17 M. Brescia7 S. Camera18,15,19 V.
Capobianco15 C. Carbone20 J. Carretero0000-0002-3130-020421,22 F.J.
Castander0000-0001-7316-457323,24 M. Castellano25 S. Cavuoti26,7,27 A.
Cimatti28,29 R. Cledassou30,31 G. Congedo32 C.J. Conselice33 L. Conversi34,35
Y. Copin36 L. Corcione15 F. Courbin37 M. Cropper0000-0003-4571-946838 A. Da
Silva8,39 H. Degaudenzi40 M. Douspis41 F. Dubath40 C.A.J. Duncan42 X. Dupac35
S. Dusini43 S. Farrens44 S. Ferriol36 M. Frailis45 E. Franceschi3 M. Fumana20
P. Gómez-Alvarez35,46 S. Galeotta45 B. Garilli0000-0001-7455-875020 W.
Gillard47 B. Gillis32 C. Giocoli48,49 A. Grazian50 F. Grupp13,14 L.
Guzzo0000-0001-8264-519251,52,53 S.V.H. Haugan0000-0001-9648-726054 W.
Holmes55 F. Hormuth56 K. Jahnke57 M. Kümmel14 S. Kermiche47 A. Kiessling55 M.
Kilbinger0000-0001-9513-713844 T. Kitching38 R. Kohley35 M. Kunz58 H. Kurki-
Suonio59 S. Ligori15 P. B. Lilje54 I. Lloro60 E. Maiorano3 O.
Mansutti0000-0001-5758-465845 O. Marggraf61 K. Markovic55 F.
Marulli0000-0002-8850-030328,3,62 R. Massey63 S. Maurogordato64 H.J.
McCracken0000-0002-9489-776565,66 E. Medinaceli48 M. Melchior67 M.
Meneghetti3,12 E. Merlin25 G. Meylan37 L. Moscardini62,10,3 E. Munari45 R.
Nakajima61 S.M. Niemi68 J. Nightingale63 C. Padilla21 S. Paltani40 F. Pasian45
K. Pedersen69 V. Pettorino70 S. Pires44 M. Poncet30 L. Popa71 L. Pozzetti3 F.
Raison13 A. Renzi72,43 J. Rhodes55 G. Riccio7 E. Romelli45 M. Roncarelli10,3
E. Rossetti10 R. Saglia0000-0003-0378-703213,14 D. Sapone73 B. Sartoris74,45
R. Scaramella0000-0003-2229-193X25,75 P. Schneider61 M. Scodeggio20 A.
Secroun47 G. Seidel57 C. Sirignano72,43 G. Sirri62 L. Stanco43 P. Tallada-
Crespí76,22 D. Tavagnacco0000-0001-7475-989445 A.N. Taylor32 I. Tereno8,9 R.
Toledo-Moreo77 F. Torradeflot76,22 I. Tutusaus58 L.
Valenziano0000-0002-1170-01043,62 T. Vassallo14 Y. Wang78 J. Weller13,14 A.
Zacchei45 J. Zoubian47 S. Andreon52 S. Bardelli3 A.
Boucaud0000-0001-7387-263379 R. Farinelli80 J. Graciá-Carpio13 D.
Maino51,20,53 N. Mauri28,62 S. Mei79 N. Morisset40 F. Sureau44 M. Tenti62 A.
Tramacere0000-0002-8186-379340 E. Zucca3 C. Baccigalupi74,45,81,82 A.
Balaguera-Antolínez83,84 A. Biviano74,45 A. Blanchard85 S.
Borgani0000-0001-6151-643986,45,81,74 E. Bozzo40 C. Burigana87,12,88 R.
Cabanac85 A. Cappi64,3 C.S. Carvalho9 S. Casas0000-0002-4751-513889 G.
Castignani10,3 C. Colodro-Conde90 A.R. Cooray91 J. Coupon40 H.M. Courtois92 O.
Cucciati3 S. Davini93 G. De Lucia0000-0002-6220-910445 H.
Dole0000-0002-9767-383941 J.A. Escartin13 S. Escoffier47 M. Fabricius13 M.
Farina94 F. Finelli3,12 K. Ganga0000-0001-8159-820879 J. Garcia-Bellido95 K.
George14 F. Giacomini62 G. Gozaliasl96 I. Hook97 M. Huertas-
Company0000-0002-1416-848398,99,84,90 B. Joachimi100 V. Kansal44 A.
Kashlinsky101 E. Keihanen96 C.C. Kirkpatrick59 V. Lindholm96,102 G.
Mainetti103 R. Maoli104,25 S. Marcin67 M. Martinelli95 N. Martinet105 M.
Maturi0000-0002-3517-2422106,107 R. B. Metcalf10,3 G. Morgante3 A.A.
Nucita108,109 L. Patrizii62 A. Peel37 J.E. Pollack79 V. Popa71 C. Porciani61
D. Potter110 P. Reimberg65 A.G. Sánchez13 M. Schirmer0000-0003-2568-999457 M.
Schultheis64 V. Scottez65,111 E. Sefusatti74,45,81 J. Stadel110 R. Teyssier112
C. Valieri62 J. Valiviita113,102 M. Viel74,45,81,82 F. Calura3 H.
Hildebrandt114

(Received June 20, 2022; accepted September 15, 2022)

The Euclid Space Telescope will provide deep imaging at optical and near-
infrared wavelengths, along with slitless near-infrared spectroscopy, across
$\sim 15\,000\,{\rm deg}^{2}$ of the sky. Euclid is expected to detect $\sim
12$ billion astronomical sources, facilitating new insights into cosmology,
galaxy evolution, and various other topics. In order to optimally exploit the
expected very large dataset, appropriate methods and software tools need to be
developed. Here we present a novel machine-learning-based methodology for the
selection of quiescent galaxies using broadband Euclid
$I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$,
$J_{\scriptscriptstyle{\rm E}}$, and $H_{\scriptscriptstyle{\rm E}}$
photometry, in combination with multi-wavelength photometry from other large
surveys (e.g. the Rubin LSST). The ARIADNE pipeline uses meta-learning to fuse
decision-tree ensembles, nearest-neighbours, and deep-learning methods into a
single classifier that yields significantly higher accuracy than any of the
individual learning methods separately. The pipeline has been designed to have
‘sparsity awareness’, such that missing photometry values are informative for
the classification. In addition, our pipeline is able to derive photometric
redshifts for galaxies selected as quiescent, aided by the ‘pseudo-labelling’
semi-supervised method, and using an outlier detection algorithm to identify
and reject likely catastrophic outliers. After the application of the outlier
filter, our pipeline achieves a normalised mean absolute deviation of $\la
0.03$ and a fraction of catastrophic outliers of $\la 0.02$ when measured
against the COSMOS2015 photometric redshifts. We apply our classification
pipeline to mock galaxy photometry catalogues corresponding to three main
scenarios: (i) Euclid Deep Survey photometry with ancillary $ugriz$, WISE, and
radio data; (ii) Euclid Wide Survey photometry with ancillary $ugriz$, WISE,
and radio data; and (iii) Euclid Wide Survey photometry only, with no
foreknowledge of galaxy redshifts. In a like-for-like comparison, our
classification pipeline outperforms $UVJ$ selection, in addition to the Euclid
$I_{\scriptscriptstyle{\rm E}}-Y_{\scriptscriptstyle{\rm E}}$,
$J_{\scriptscriptstyle{\rm E}}-H_{\scriptscriptstyle{\rm E}}$ and
$u-I_{\scriptscriptstyle{\rm E}}$, $I_{\scriptscriptstyle{\rm
E}}-J_{\scriptscriptstyle{\rm E}}$ colour-colour methods, with improvements in
completeness and the F1-score (the harmonic mean of precision and recall) of
up to a factor of 2.

###### Key Words.:

Galaxies: photometry – Galaxies: high-redshift – Galaxies: evolution –
Galaxies: general

##  1 Introduction

The study of galaxies plays a pivotal role in the effort to understand how the
baryonic component of the Universe evolved across cosmic time; it is on the
size scale of galaxies that key processes such as star formation, chemical
evolution, black hole growth, and feedback predominantly take place. Large,
systematic imaging and spectroscopic surveys are among the most valuable
resources for investigating galaxy evolution, allowing a wide range of
studies, from the identification and study of rare or elusive objects (e.g.
Alexandroff et al., 2013) to statistical studies of large samples of galaxies
(e.g. Kauffmann et al., 2003). A variety of ground- or space-based surveys
have already provided rich databases for investigating questions surrounding
the evolution of galaxies, for example the Sloan Digital Sky Survey (SDSS;
York et al., 2000; Gunn et al., 1998), with even more data to be generated by
ongoing and future campaigns and facilities that will map the extragalactic
sky to even fainter flux levels, and out to even higher redshifts, for example
the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST; Ivezić et
al. 2019), the Nancy Grace Roman Space Telescope (Akeson et al. 2019), the
Dark Energy Spectroscopic Instrument survey (Dey et al. 2019), the 4-metre
Multi-Object Spectroscopic Telescope (Guiglion et al. 2019), the Multi Object
Optical and Near-infrared Spectrograph for the VLT (Taylor et al. 2018;
Cirasuolo et al. 2020), the Square Kilometer Array (Dewdney et al. 2009), and
the extended Roentgen Survey with an Imaging Telescope Array (Predehl et al.
2021). In the coming years, the Euclid Space Telescope will make a substantial
contribution to our understanding of galaxy evolution. Euclid will observe
$\sim 15\,000\,{\rm deg}^{2}$ of the extragalactic sky at visible to near-
infrared (NIR) wavelengths, to a $5\,\sigma$ point-source depth of 26.2
mag111AB magnitudes are used herein. in the Euclid VISible Instrument (VIS;
Cropper et al., 2016) $I_{\scriptscriptstyle{\rm E}}$ ($R$+$I$+$Z$) filter and
24.5 mag in the Near Infrared Spectrometer and Photometer (NISP; Maciaszek et
al., 2016) $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$,
and $H_{\scriptscriptstyle{\rm E}}$ filters (Euclid Collaboration: Scaramella
et al., 2021; Euclid Collaboration: Schirmer et al., 2022). In addition, three
fields with an area totalling 53 deg2 will be the subject of a deeper survey,
to a $5\,\sigma$ depth of 28.2 mag in $I_{\scriptscriptstyle{\rm E}}$ and 26.5
mag in $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, and
$H_{\scriptscriptstyle{\rm E}}$. Euclid is expected to detect $\sim 12$
billion astronomical sources ($3\,\sigma$), providing multi-colour imaging at
$0\aas@@fstack{\prime\prime}1$–$0\aas@@fstack{\prime\prime}4$ resolution, and
is also expected to obtain spectroscopic redshifts for $\sim 35$ million
galaxies (e.g. Laureijs et al., 2011). While the primary science drivers of
the Euclid mission are baryonic acoustic oscillations, weak lensing cosmology,
and redshift-space distortions, the survey is also expected to enable a
multitude of high-impact extragalactic science projects, either in stand-alone
form or in combination with multi-wavelength data from other surveys (e.g.
LSST).

Automated classification and derivation of physical properties are crucial
steps towards the scientific exploitation of data from any large astronomical
survey, and traditionally this would be done using colour-colour methods (e.g.
Haro, 1956; Daddi et al., 2004; Leja, Tacchella, & Conroy, 2019) or by fitting
spectral models or templates (e.g. Bolzonella, Miralles, & Pelló, 2000;
Fotopoulou et al., 2012; Gomes & Papaderos, 2017). Machine-learning techniques
have proven to be particularly powerful in this context since they have the
ability to detect structure, correlations, and outliers in large, multi-
dimensional datasets, producing models that are typically stronger and more
efficient than the traditional methods (e.g. Baqui et al., 2021; Ulmer-Moll et
al., 2019; Logan & Fotopoulou, 2020; Clarke et al., 2020; Cunha & Humphrey,
2022). While machine-learning techniques have been applied to extragalactic
problems for several decades already (e.g. Odewahn et al., 1993), in recent
years there has been a rapid growth in their application to this area. Notable
examples include detailed morphological classification via application of deep
learning to galaxy images (e.g. Dieleman, Willett, & Dambre, 2015; Huertas-
Company et al., 2015; Domínguez Sánchez et al., 2018; Tuccillo et al., 2018;
Nolte et al., 2019; Bowles et al., 2021; Bretonnière et al., 2021), high-
accuracy estimation of the redshift ($z$) of galaxies from imaging or
photometric data (e.g. Collister & Lahav, 2004; Brescia et al., 2013; Cavuoti
et al., 2017; Pasquet et al., 2019; Razim et al., 2021; Guarneri et al.,
2021), selection and classification of galaxies into various phenomenological
types (e.g. Cavuoti et al., 2014; Steinhardt et al., 2020), and derivation of
their physical properties (e.g. Bonjean et al., 2019; Delli Veneri et al.,
2019; Mucesh et al., 2021; Simet et al., 2021). A few studies have taken a
hybrid or ‘cooperative’ approach, combining results from traditional methods
(e.g. template fitting) with machine-learning methods, resulting in improved
accuracy (e.g. Fotopoulou & Paltani, 2018; Cavuoti et al., 2017).

Figure 1: Redshift distribution of galaxies in the Int Wide and Int Deep
catalogues (top panel), and the quiescent galaxy fraction as a function of
redshift up to $z=3$ (lower panel).

The identification of quiescent galaxies222We adopt the specific star-
formation rate (sSFR) of $10^{-10.5}{\rm yr}^{-1}$ as the boundary between
quiescent and star-forming. is among the most challenging classification
problems in extragalactic astronomy and represents a crucial task in our quest
to understand the evolution of galaxies across cosmic time. A particular
problem is the fact that there exist substantial degeneracies between stellar
age, metallicity, and reddening by dust (e.g. Worthey, 1994), causing the
scattering of galaxies across classification boundaries when using broadband
spectral energy distributions (SEDs). These degeneracies can be significantly
exacerbated by the absence of redshift information.

A frequently applied technique for the selection of quiescent galaxy
candidates is $UVJ$ colour-colour selection (Strateva et al., 2001; Baldry et
al., 2004; Wuyts et al., 2007; Williams et al., 2009; Muzzin et al., 2013; van
der Wel et al., 2014; Leja, Tacchella, & Conroy, 2019; Shahidi et al., 2020).
This method selects objects in rest-frame $U-V$, $V-J_{\scriptscriptstyle{\rm
E}}$ colour-colour space that are red because their UV-to-NIR SED is dominated
by an old stellar population, as opposed to star-forming galaxies that are
reddened by dust. While the $UVJ$ selection method clearly works (e.g.
Fumagalli et al., 2014), there is a substantial contamination by star-forming
galaxies in the region of $\sim 10$–30 per cent (e.g. Moresco et al., 2013;
Schreiber et al., 2018; Fang et al., 2018). Various observer-frame colour
combinations have also been proposed for the selection of quiescent galaxies,
such as the $BzK$ method (Daddi et al., 2004), or the Galaxy Evolution
Explorer (GALEX) ${\rm FUV}-V$, $V-J_{\scriptscriptstyle{\rm E}}$, and ${\rm
FUV}-V$, $V-W3$ methods proposed by Leja, Tacchella, & Conroy (2019). Once
selected via colour-colour techniques, SED fitting (e.g. Wiklind et al., 2008;
Girelli, Bolzonella, & Cimatti, 2019) and spectroscopic observations (e.g.
Belli, Newman, & Ellis, 2015; Glazebrook et al., 2017; Schreiber et al., 2018)
can then be employed to confirm their passive nature. Alternative approaches
have also been successful at selecting quiescent galaxies, such as template
fitting followed by colour selection (e.g. Laigle et al., 2016; Deshmukh et
al., 2018).

Figure 2: Distribution of galaxy magnitudes in the Int catalogues (left) and
the SED catalogues (right) for the various optical and NIR bands used herein.
Vertical dashed lines indicate the expected $10\,\sigma$ sensitivity of the
Euclid Wide and Deep Surveys in the $I_{\scriptscriptstyle{\rm E}}$,
$Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, and
$H_{\scriptscriptstyle{\rm E}}$ bands. In all bands, only photometry with a
signal-to-noise ratio $\geq 3$ is included.

In the context of preparations for the Euclid survey, Bisigello et al. (2020,
B20 hereinafter) recently developed $I_{\scriptscriptstyle{\rm
E}}-Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm
E}}-H_{\scriptscriptstyle{\rm E}}$ and $u-I_{\scriptscriptstyle{\rm E}}$,
$I_{\scriptscriptstyle{\rm E}}-J_{\scriptscriptstyle{\rm E}}$ colour-colour
criteria to separate quiescent galaxies from star-forming galaxies up to
$z=2.5$, for use in the case where spectroscopic or photometric redshifts are
available. The proposed colour-colour criteria significantly outperform the
traditional $UVJ$ technique that, when derived using only the four Euclid
filters, provides a completeness of only $\sim 0.2$ at $z<3$. For example,
using their $u-I_{\scriptscriptstyle{\rm E}}$, $I_{\scriptscriptstyle{\rm
E}}-J_{\scriptscriptstyle{\rm E}}$ criteria, B20 were able to select a sample
of quiescent galaxies at $0.75<$ $z<1$ with a completeness of $\sim 0.7$ and a
precision $>0.85$. Their $I_{\scriptscriptstyle{\rm
E}}-Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm
E}}-H_{\scriptscriptstyle{\rm E}}$ criteria also allowed the authors to select
quiescent galaxies at $1<z<2$ with a completeness $>0.65$ and a precision
$>0.8$.

An important limitation of colour-colour selection techniques is that they
are, in effect, lossy dimensionality-reduction methods. As such, they are
likely to discard a significant quantity of otherwise useful information that
is present in a broadband SED. In this context, machine-learning methods offer
a promising alternative since they are able to perform selection from within
highly multi-dimensional datasets, and can be tuned to make a trade-off
between purity and completeness that is appropriate for a desired science
application. Indeed, Steinhardt et al. (2020) recently explored the selection
of quiescent galaxies using the unsupervised $t$-distributed stochastic
neighbour embedding method (van der Maaten & Hinton, 2008; van der Maaten,
2014), reporting a significant improvement over the $UVJ$ and template fitting
methods.

In this paper we present a new supervised machine-learning method for the
separation of quiescent and star-forming galaxies using Euclid and ancillary
photometry, which is designed to handle sparse data and can provide
photometric redshift estimates where necessary. The paper is organised as
follows. In Sect. 2 we describe the mock photometry catalogues used in this
study. The metrics of model quality we use are defined in Sect. 3. Full
details of the ARIADNE pipeline are given in Sect. 4. We describe in Sect. 5
the results of applying our separation methods to the mock photometric data.
Next, in Sect. 6 we compare our method with colour-colour methods previously
proposed by B20 and others. In Sect. 7 we summarise results from a number of
further analyses and tests, with full details given in Appendix B. Finally, in
Sect. 8 we summarise our results and conclusions.

##  2 Mock galaxy catalogues

In this work we make use of an updated version of the mock catalogues
presented in B20. All catalogues were derived from magnitudes in the
COSMOS2015 multi-wavelength catalogues (Laigle et al., 2016). Objects labelled
as stars or X-ray sources, and objects with inadequate333Sources that are
‘masked in optical broad-bands’ in the COSMOS2015 catalogue. optical
photometry, being removed. After this selection, the COSMOS2015 catalogue
contains 518404 objects up to $z=6$. We now briefly introduce the two methods
used to derive Euclid-like mock catalogues.

In the first method, we interpolated over the observed COSMOS2015 photometry
to produce a broken-line template running from ultraviolet to infrared
wavelengths, which is then convolved with the filters of interest to derive
mock photometry. As this method is affected directly by the COSMOS2015
photometric errors, which are similar or larger than those expected in the
Euclid Wide Survey, we did not include any additional photometric scatter. The
20 cm Very Large Array (VLA) radio continuum flux, where measured, is also
included without modification. Using this approach, we derived two separate
catalogues, resembling the Euclid Wide Survey (Euclid Collaboration:
Scaramella et al., 2021) and Euclid Deep Survey. We refer to these mock
catalogues as ‘Int Wide’ and ‘Int Deep’, respectively.

The question of the potential impact of emission lines on the Int catalogues
was discussed previously by B20. In short, the broadband magnitudes in the Int
and SED catalogues include contributions from emission lines. Because the Int
catalogues were constructed using real observed photometry, this means that
for some bands and some galaxies, a contribution from nebular emission is
present. In the case of the SED catalogues, the LePhare code was used with
emission lines included (see Ilbert et al., 2006), allowing nebular emission
to contribute to the mock magnitudes.

Nevertheless, we do not expect the inclusion (or exclusion) of emission lines
to have a significant effect on our results. Given the widths of the filters
considered herein (Euclid Collaboration: Schirmer et al., 2022), the effect of
emission lines is marginal: observed equivalent widths larger than 350 Å, 260
Å, 390 Å, and 480 Å would be required to produce a boost of $\sim 0.1$ mag in
the VIS, Y, J, and H bands, respectively; such high equivalent widths are rare
in the sample of galaxies used for our mock catalogues (e.g., Amorín et al.,
2015)

Table 1: $10\,\sigma$ depth in AB magnitudes of the Wide Survey for the
filters included in the mock catalogues.

$I_{\scriptscriptstyle{\rm E}}$ | $Y_{\scriptscriptstyle{\rm E}}$ | $J_{\scriptscriptstyle{\rm E}}$ | $H_{\scriptscriptstyle{\rm E}}$ | CFIS/$u$ | SDSS/$g$ | SDSS/$r$ | SDSS/$i$ | SDSS/$z$ |  $W$1 |  $W$2
---|---|---|---|---|---|---|---|---|---|---
24.5 | 23.25 | 23.25 | 23.25 | 24.20 | 24.50 | 23.90 | 23.60 | 23.40 | 18.39 | 18.04

444The Deep Survey is expected to be two magnitudes deeper than the Wide
survey in the Euclid, CFIS, and SDSS bands.

In the second approach, we used the public code LePhare (Arnouts et al., 2007;
Ilbert et al., 2006) to fit the COSMOS2015 photometry with a large set of
Bruzual & Charlot (2003) templates, with the redshift fixed at its COSMOS2015
value from Laigle et al. (2016). In particular, we considered templates with
two different metallicities ($Z_{\odot}$ and 0.4 $Z_{\odot}$), exponentially
declining star-formation histories with an e-folding timescale $\tau$ varying
from 0.1 to 10 Gyr, and ages between 0.1 and 12 Gyr. For the dust extinction,
we considered the reddening law of Calzetti et al. (2000) with 12 values of
colour excess from 0 to 1. For each galaxy, we obtained the best SED template
applying a $\chi^{2}$ minimisation procedure and we convolved the resulting
template with the filters of interest to calculate the desired mock
photometry. In effect, the resulting photometric SED is a synthetic
representation of the observed one. For each galaxy, we derived ten mock
galaxies by randomising the mock photometry within the expected photometric
errors. As with the first approach, here we derived two different catalogues,
one for the Euclid Wide Survey (SED Wide) and one for the Euclid Deep Survey
(SED Deep). Because the templates do not extend into the radio regime, the SED
Wide and SED Deep mock catalogues do not include the 20 cm radio band.

We included the VIS $I_{\scriptscriptstyle{\rm E}}$ filter, the NISP
$Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, and
$H_{\scriptscriptstyle{\rm E}}$ filters (Euclid Collaboration: Schirmer et
al., 2022), and the Canada-France Imaging Survey $u$ filter (CFIS/$u$), as
previously presented in B20. In addition, we derived mock fluxes for the SDSS
(Gunn et al., 1998) $griz$ filters and the Wide-field Infrared Survey Explorer
(WISE; Wright et al., 2010) filters at 3.4 and $4.6\,\mu$m ($W$1 and $W$2).
For these two WISE filters, we considered the $5\,\sigma$ observational depths
of the WISE All-Sky survey: 0.08 and 0.11 mJy, respectively (Wright et al.,
2010). For the other filters we instead assumed the observational depths
reported in the Euclid Red Book (Laureijs et al., 2011), which are expected to
be reached with a variety of ground-based telescopes, for which we use the
SDSS filters as proxies. The complete list of observational depths are listed
in Table 1. Our choice of depths for the Euclid and ground-based photometry is
motivated primarily by the need to make a direct comparison with the colour-
colour results of B20. Thus, we adopted the depths used in B20555The Euclid
photometric depths adopted herein differ slightly from the most recent
forecasts: the $I_{\scriptscriptstyle{\rm E}}$ photometry is now forecast to
be 0.5 mag deeper, and the NISP photometry 0.25 mag deeper (Euclid
Collaboration: Scaramella et al., 2021), compared to the values adopted
herein. In the case of ground-based optical photometry overlapping the Euclid
survey areas there is no single forecast, since the photometry is expected to
come from several different surveys. For instance, at the time the Euclid DR3
release, the UNIONS survey (Chambers et al., 2020) is forecast to be 0.6 mag
shallower in $u$ and 0.2 mag deeper in $r$ compared to the Wide Survey depths
adopted herein, with similar depths in the $g$, $i$ and $z$ bands. In the
Southern Hemisphere, LSST is expected to provide deeper optical data..

For all bands, only photometry measurements with a signal-to-noise ratio of
$\geq 3$ are considered. This threshold has been chosen to ensure that only
reliable measurements are used. The use of low signal-to-noise data deserves a
detailed and in-depth study, and is under investigation for a future paper.
For all bands except $I_{\scriptscriptstyle{\rm E}}$, non-detections are
flagged as missing; objects for which $I_{\scriptscriptstyle{\rm E}}$ has a
signal-to-noise below 3 are excluded from the catalogues. We refer to B20 for
further details on the creation of the mock catalogues.

The specific star-formation rate (sSFR) was derived for each galaxy at $z\leq
3$ by fitting the 30-band photometric SED of Laigle et al. (2016) using
LePhare. Further details of this process are given in B20. The dividing line
between ‘quiescent’ and ‘star-forming’ in terms of sSFR is somewhat arbitrary,
and various different thresholds are in use in the literature, although these
are usually in the range $<10^{-10}$ yr-1 (e.g. Wu et al., 2018) to
$<10^{-11}$ yr-1 (e.g. Ilbert et al., 2013). For consistency with B20, we
define quiescent to mean that a galaxy has sSFR $<10^{-10.5}$ yr-1, and star-
forming to mean that a galaxy has sSFR $\geq 10^{-10.5}$ yr-1. In any case,
the classification metrics we obtain herein are not significantly dependent on
which value of sSFR we adopt for the threshold between quiescent and star-
forming galaxies, for values of the threshold between $10^{-10}$ yr-1 and
$10^{-11}$ yr-1.

For the redshift labels we adopt the 30-band COSMOS photometric redshifts
estimated by Laigle et al. (2016). For the SED catalogues, the redshift labels
represent the true redshifts (i.e. with no uncertainty), because the mock
photometry is derived directly from templates with known redshifts. On the
other hand, in the case of the Int catalogues, the redshift labels are merely
photometric redshift estimates, and thus have an uncertainty with systematic
and random components.

We characterise the properties of the mock catalogues in Figs. 1–4. In Fig. 1
we show the redshift distribution of galaxies. Also shown is the distribution
and fraction of quiescent galaxies as a function of redshift, up to $z=3$. It
can be seen that the quiescent fraction falls rapidly with increasing
redshift, starting at $\sim 0.16$ at $z\sim 0$ and declining to $\leq 0.05$ by
$z\sim 2.5$, illustrating the highly challenging nature of the search for
quiescent (or passive) galaxies at high redshift.

The distribution of magnitudes is shown in Fig. 2, with the expected
$10\sigma$ detection limits in the Euclid filters marked. This figure shows
that the Wide mock catalogues are complete down to the $3\sigma$ detection
limit in all optical and NIR bands. However, the Deep catalogues are not
complete, due to the limits of the COSMOS2015 photometry catalogue.

Figure 3 shows the distribution of $I_{\scriptscriptstyle{\rm
E}}-H_{\scriptscriptstyle{\rm E}}$, $u-I_{\scriptscriptstyle{\rm E}}$, and
$Y_{\scriptscriptstyle{\rm E}}-H_{\scriptscriptstyle{\rm E}}$ colours in the
four mock catalogues. There are slight colour differences between the Int and
SED catalogues that arise due to their different construction methodologies.
For instance, the SED catalogues are slightly bluer in terms of their average
observer-frame $u-I_{\scriptscriptstyle{\rm E}}$ colours, compared to the Int
catalogues.

In Fig. 4 we show the fraction of missing photometry measurements as a
function of redshift for the optical and infrared bands. In the case of our
mock catalogues, photometry is flagged as missing when (i) a galaxy or area is
unobserved in that band, (ii) a galaxy or area was masked, (iii) a photometry
measurement falls below the detection threshold of the catalogue, or (iv) a
photometry measurement has a signal-to-noise ratio lower than 3. In general
terms, the missing fraction is higher for higher redshifts, although in some
cases (notably the NIR bands) there is a turnover and decrease in the missing
fraction at very high redshifts ($z\ga 4$). The WISE $W$1 and $W$2 bands and
the 20 cm radio band data are highly sparse, with missing fractions of 0.947,
0.976, and 0.992, respectively. Conversely, the $I_{\scriptscriptstyle{\rm
E}}$ band has a missing fraction of exactly 0, since detection in this band is
a requirement for inclusion in our mock catalogues.

Finally, Fig. 5 shows the impact on the number and redshift distribution of
galaxies from application of the main photometric pre-selection criteria used
herein: (i) $3\,\sigma$ detections in $I_{\scriptscriptstyle{\rm E}}$,
$Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$,
$H_{\scriptscriptstyle{\rm E}}$; (ii) $3\,\sigma$ detections in $u$,
$I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$,
$J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$; and (iii)
$3\,\sigma$ detections in $ugriz$, $I_{\scriptscriptstyle{\rm E}}$,
$Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$,
$H_{\scriptscriptstyle{\rm E}}$. As expected, the impact of requiring
detection in the $u$ band is to induce a step in the distribution at $z\sim
3$, as the Lyman break is redshifted out of the $u$-band filter; this step is
much stronger in the Wide catalogues than in the SED catalogues. In the Int
Deep catalogue, the reduction in the number of sources at $z\ga 3$ when
detection in $u$ or $ugriz$ is required is surprisingly small; this is likely
to be at least partly caused by the presence of sources with incorrect
photometric redshifts in the COSMOS2015 catalogue. Table 7 lists the number of
galaxies that are detected in each band for each of the mock catalogues.

Figure 3: Comparison between the distribution of the
$I_{\scriptscriptstyle{\rm E}}-H_{\scriptscriptstyle{\rm E}}$,
$u-I_{\scriptscriptstyle{\rm E}}$, and $Y_{\scriptscriptstyle{\rm
E}}-H_{\scriptscriptstyle{\rm E}}$ colours in the Int and SED mock catalogues.
Top row: Euclid Wide catalogues. Bottom row: Euclid Deep catalogues.
Significant differences between the Int and SED methods are apparent, most
notably with the SED method giving rise to significantly bluer
$u-I_{\scriptscriptstyle{\rm E}}$ colours compared to the Int method. These
differences arise when the galaxy templates are unable to closely match the
observed broadband SED; this may be due to the photometric redshift being
incorrect and/or due to the absence of a template that sufficiently represents
the properties of the galaxy.

##  3 Metrics of model quality

To evaluate our classification models, we used several metrics that are useful
to quantify model quality. Precision, $P$, is the fraction of assignments to a
particular class that are correct, calculated as

$P=\frac{\rm TP}{\rm TP+\rm FP}\,,$ (1)

where TP is the number of true positives and FP is the number of false
positives. Precision is also sometimes known as ‘purity’ in astronomy, or the
‘positive predictive value’.

Figure 4: Fraction of missing photometry measurements vs. redshift for each of
the filters used for model training. By construction, the
$I_{\scriptscriptstyle{\rm E}}$ band has no missing values. Very few of the
galaxies are detected in the $W1$, $W2$, or VLA 20 cm bands.

Figure 5: Effect of various photometric pre-selection criteria on the number
and redshift distribution of galaxies in the four mock catalogues. We show
four cases: (i) only a detection in $I_{\scriptscriptstyle{\rm E}}$ is
required; (ii) detection is required in all four Euclid bands
($I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$,
$J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$); (iii)
detection is required in $u$ and all four of the Euclid bands; and (iv)
detection is required in $ugriz$ and all of the Euclid bands. As described in
the main text, for all bands we adopt a signal-to-noise detection threshold
$\geq 3$.

Recall, $R$, is the fraction of galaxies of a particular class within the
dataset that are correctly classified as such. It is calculated as

$R=\frac{\rm TP}{\rm TP+\rm FN}\,,$ (2)

where FN is the number of false negatives. Recall is also sometimes known as
‘completeness’ or ‘sensitivity’.

The F1-score is the harmonic mean of the precision and the recall and, as
such, provides a more general metric for model quality (Dice, 1945; Sørensen,
1948). The F1-score is calculated as

${\rm F1{\text{-}}score}=2\;{P\,R\over P+R}\,,$ (3)

where we have opted to give equal weights to $P$ and $R$. The metrics $P$,
$R$, and F1-score have values between 0 and 1. Since there is a large
imbalance between classes in the datasets used here, we compute the metrics
separately for each class. Unless otherwise stated, the values of $P$, $R$,
and the F1-score we quote are computed for the quiescent galaxy class only.

To assess the quality of photometric redshift estimates produced by our
pipeline, we used the following measures. As a measure of accuracy, we used
the normalised median absolute deviation (NMAD), which we calculated as

$\mbox{NMAD}=1.48\,\mbox{median}\,\left(\frac{\,|z_{\rm phot}-z_{\rm
ref}|\,}{1+z_{\rm ref}}\right)\,,$ (4)

where $z_{\rm phot}$ is our photometric redshift and $z_{\rm ref}$ is the
reference redshift used as the ‘ground truth’. In this study, we adopt the
30-band photometric redshifts from COSMOS2015 (Laigle et al., 2016) for
$z_{\rm ref}$. The NMAD can be loosely interpreted as the standard deviation.

As a further measure of quality, we also calculated the fraction of
catastrophic outliers ($f_{\rm out}$). A photometric redshift estimate is
considered to be a catastrophic outlier when

$\frac{|z_{\rm phot}-z_{\rm ref}|}{1+z_{\rm ref}}>0.15.$ (5)

To test whether (and to which extent) our pipeline systematically over- or
underestimates galaxy redshifts, we defined the bias of the photometric
redshifts as

$\mbox{bias}=\mbox{median}\left(\frac{z_{\rm phot}-z_{\rm ref}}{1+z_{\rm
ref}}\right).$ (6)

It is also useful to have a metric that quantifies the degree to which a false
positive classification is in error. We define the ‘incorrectness’ of an
individual false positive classification as

$I_{\rm FP}=\operatorname{log_{10}}({\rm sSFR\leavevmode\nobreak\ yr})+10.5.$
(7)

We considered a false positive to be ‘marginal’ when $I_{\rm FP}\leq 0.5$, and
‘catastrophic’ when $I_{\rm FP}\geq 1.0$. We also defined an average
incorrectness parameter as

$\bar{I}_{\rm FP}=\sum\limits_{i=0}^{n}\frac{{\operatorname{log_{10}}({\rm
sSFR\leavevmode\nobreak\ yr})+10.5}}{{\rm FP}}\,.$ (8)

We have not used the commonly used ‘accuracy’ metric, which gives the fraction
of predictions that are correct, because it can be misleading when the test
data are significantly imbalanced, as is the case here.

##  4 The ARIADNE pipeline

ARIADNE is a flexible, modular machine-learning pipeline designed for the
purpose of classification and derivation of physical properties of
astronomical sources on the basis of their photometric SEDs. In a nutshell,
ARIADNE takes a table of photometry as input, and uses algorithms to learn how
the input data maps to labels corresponding to galaxy properties. Here we
describe the functionality of ARIADNE in its classification mode (a flowchart
is also shown in Fig. 6); its application to estimation of physical
properties, such as stellar mass, star-formation rate, and extinction, will be
presented in a future publication (Humphrey et al., in prep.).

###  4.1 The learning algorithm

After completion of the feature engineering and preprocessing steps described
in Sect. 4.2, the data are split into a training set and test set with a 2:1
ratio between the two. This ratio was chosen to give a good balance between
having a large number of examples on which to train the learners, while still
having a test set that is representative of the whole dataset666When applied
to the actual Euclid survey data it is expected that the ratio used for the
train test split may be of order $\sim 1:10\,000$. The split is done randomly,
without attempting to balance target labels, redshift, or any other
observational properties, and each execution of the pipeline produces a
different train-test split.

The pipeline then trains binary classification models on the training dataset,
using five different ‘base-learners’ (see Fig. 6). Three of the base-learners
are tree-ensemble methods with somewhat different implementations
(CatBoostClassifier777https://catboost.ai version 0.23.2;
LGBMClassifier888https://lightgbm.readthedocs.io version 0.90;
RandomForestClassifier), one is nearest-neighbours-based
(KNeighborsClassifier), and one is deep-learning-based (MLPClassifier). All
are open source and are briefly summarised below.

The Python Scikit-Learn999https://scikit-learn.org version 0.22.2 package
(Pedregosa et al., 2011) offers various machine-learning methods with
excellent functionality and inter-operability with a multitude of in-package
preprocessing tools. From Scikit-Learn we make use of the nearest-neighbours-
based, non-parametric KNeighborsClassifier method, the Multi-layer Perceptron-
based MLPClassifier, and the randomised decision-tree classifier
RandomForestClassifier (Breiman, 2001).

Several other, advanced machine-learning methods are used in our pipeline.
CatBoost (Prokhorenkova et al., 2018) is a gradient-boosting, tree-ensemble
method that offers high performance classification or regression, using
‘ordered-boosting’ in place of the classic boosting algorithm to significantly
reduce the ‘prediction shift’ commonly associated with the latter.

Arguably the most advanced tree-based learning algorithm publicly available at
the time of writing is LightGBM, a gradient-boosting, decision-tree method
that deploys a number of key innovations that are especially relevant for the
classification problems we approach in this work (Ke et al., 2017). For
instance, LightGBM uses leaf-wise (best-first), rather than level-wise tree-
growth, for improved classification accuracy. Also of relevance is the use of
histogram-based algorithms that place continuous feature values into discrete
bins, significantly reducing training time and memory usage.

In addition to the five default base-learners described above, our pipeline
also includes the option to use the tree-based method
XGBoost101010https://xgboost.readthedocs.io version 0.25.3 (Chen & Guestrin,
2016), for use in situations where this learning algorithm is able to train
stronger models than those described above. The XGBoost algorithm uses
gradient boosting and has several key innovations, including sparsity-aware
split-finding, which is of particular relevance for datasets containing a
significant fraction of missing values (e.g. photometric non-detections,
masked or unobserved areas).

Our application of these learning algorithms to the data at hand reveals that
no single algorithm consistently outperforms the others over the full range of
classification problems posed herein. Thus, our pipeline employs model
ensembling and interactive algorithm selection.

The base-learners are trained within a stratified $k$-fold procedure: the
training data are shuffled and split, with replacement, into five similarly
sized folds, ensuring that each fold contains the same proportion of each
target class. Each base-learner is trained five times, each time leaving out a
different fold, for which a set of out-of-fold (OOF) class predictions is
produced. This results in an array of OOF predictions for each base-learner.
For each iteration of the k-fold procedure, class predictions are also
produced for the test set, averaging the five sets of class predictions for
each base-learner. Unless otherwise stated, we adopt a class probability
threshold of 0.5: predictions of $<0.5$ correspond to class 0 (star-forming
galaxies), and predictions of $\geq 0.5$ correspond to class 1 (quiescent
galaxies).

The pipeline then performs several iterations of non-linear combination of
class predictions from the individual learners. First, a hard-voting ensemble
is produced for the OOF and test predictions. Each base learner contributes
one vote towards the class of a galaxy, and the class with the highest number
of votes is chosen. For the data and classification problem considered in this
work, the hard-voting ensemble almost always results in a significantly higher
F1-score than any of the individual base-learners, or a simple average of the
class probabilities.

To further improve model quality, the pipeline contains our implementation of
the ‘generalised stacking’ method proposed by Wolpert (1992). A linear
discriminant analysis or MLPClassifier meta-learner is trained to classify
galaxies using the OOF predictions from the five base-learners as features.
This is performed within a stratified k-fold procedure as described above, and
produces a new set of OOF predictions for the training dataset, in addition to
a new set of predictions for the test dataset. A second iteration of meta-
learning is then performed, this time training on just two features: (i) the
results of the hard-voting ensemble and (ii) the OOF predictions produced in
the previous iteration of meta-learning. Finally, the resulting model is used
to predict classes for the test data. An alternative implementation of
generalised stacking, applied to redshift estimation, is presented in Zitlau
et al. (2016).

One of the benefits of generalised stacking is that the optimisation of base-
learner hyperparameters, while still necessary to some extent, is not as
crucial for the final performance of the pipeline as it would be when a single
learning algorithm is used. This is partly because when the meta-learner
distills the base-classifiers into a single classifier, it performs a process
broadly analogous to optimisation and model-selection. Arguably, this process
can be more efficient than traditional optimisation and model selection
methods, since it is performed in a single step and is not restricted to
selecting a single model or a single set of hyperparameters, and instead can
combine the strengths of several different classifiers that are best able to
model different subsets of the data. In the case where a single base-learner
is used within our pipeline, the generalised stacking procedure instead serves
as an ‘error-correction’ algorithm.

We have not attempted to perform an exhaustive optimisation of the base-
learners prior to applying our generalised stacking method. Instead, for each
learner we manually identified a set of default hyperparameters that gives
what we judged to be near to the global maximum of the F1-score for selecting
quiescent galaxies from the mock catalogues. This is done to avoid biasing the
individual base-learners towards producing classifiers that all succeed (or
fail) in modelling the same subset of the data, and to expand the diversity of
classification models available to the meta-learner.

In the default configuration of our pipeline, all five default base-learners
are used. However, the user can instead use a subset of the base learners, or
a single base-learner, when appropriate. For example, we opt to use
XGBoostClassifier for the selection of quiescent galaxies at $2.5<z<3.0$,
where its sparsity awareness confers a significant advantage over the other
classifiers. In addition, the pipeline has a ‘fast mode’ that uses LightGBM
for all classification or regression tasks, at the cost of a small but
significant reduction in $P$, $R$, and F1-score ($\sim 0.01$–0.03). Timed on a
mid-range laptop with a quad-core Intel i5-8350U CPU and 16 Gigabytes of RAM,
the pipeline used in fast mode takes at total of $\sim 2$ minutes to train its
classifier on a dataset with $\sim 120\,000$ examples and 70 features,
compared to $\sim 74$ minutes when using the default (5 base-learners)
pipeline configuration.

Our pipeline makes use of redshift information in one of several ways,
depending on the classification problem that is posed. When redshifts are
available, these can be included as an additional feature in the training and
test data. In addition, when the objective is to select quiescent galaxies in
a specific redshift interval, pre-binning can be used to discard objects that
lie outside the desired redshift interval.

In the absence of redshifts for the test sample, the pipeline first performs a
global selection of quiescent galaxies (without regard to redshift), then
trains a KNeighborsRegressor model to predict photometric redshifts for the
selected quiescent galaxies. The photometric redshift point estimates are
refined using our implementation of the semi-supervised ‘pseudo-labelling’
technique (Lee, 2013), which aims to use both labelled and unlabelled data to
learn the underlying structure of the data, thereby improving generalisation.
Finally, analogous to Fotopoulou & Paltani (2018); Singal et al. (2022), a
KNeighborsRegressor model is trained to predict whether a galaxy’s redshift
estimate is a catastrophic outlier, with a tunable probability threshold to
control the strength of the outlier removal.

Table 2: Impact on the F1-score from using one of several different imputation
strategies.

Learning algorithm | Imputed value: | $-99.9$ | mean |  $-99.9$ or mean | median | minimum
---|---|---|---|---|---|---
(1) |  | (2) | (3) | (4) | (5) | (6)
CatBoostClassifier |  | 0.633 | 0.632 | 0.633 | 0.644 | 0.621
LightGBMClassifier |  | 0.678 | 0.621 | 0.678 | 0.596 | 0.655
RandomForestClassifier |  | 0.607 | 0.561 | 0.607 | 0.576 | 0.607
MLPClassifier |  | 0.000 | 0.610 | 0.610 | 0.621 | 0.633
KNeighborsClassifier |  | 0.519 | 0.526 | 0.526 | 0.526 | 0.519
Meta-learner ensemble of the above |  | 0.667 | 0.600 | 0.656 | 0.623 | 0.610
XGBoostClassifier |  | 0.610 | 0.576 | 0.610 | 0.586 | 0.621

111111In this example, we have trained models to select quiescent galaxies
from the Int Wide mock catalogue in the redshift bin $2<z<2.5$, using $ugriz$,
Euclid, $W$1, $W$2, and 20 cm photometry and colours, and with pre-binning by
redshift, as described in Sect. 5.1.2. A single random seed is used for the
train/test split and base learners to allow a relatively controlled comparison
between methods. The columns are as follows: (1) The learning algorithm; also
shown are the final F1-scores after the models produced by the 5 default base-
learners have been ensembled using meta-learners; (2) F1-score when missing
values are imputed with the constant value $-99.9$; (3) F1-score when imputing
with the average value of a feature; (4) F1-score when missing values are
dynamically imputed with either the constant value $-99.9$ (tree-based
learners) or the mean of a feature (MLPClassifier and KNeighborsClassifier);
(5) F1-score when imputing with the median value of a feature; (6) F1-score
when imputing with the minimum value of a feature. Some F1-scores differ
significantly to those presented in Sect. 5, since here we use only a single
random seed instead of averaging results over multiple pipeline runs that use
different random seeds. Note that in this test, MLPClassifier was unable to
correctly identify any quiescent galaxies when missing values were imputed
with $-99.9$; nonetheless, this failure did not appear to be detrimental to
the final meta-learner ensemble.

Figure 6: Flow chart illustrating the overall learning algorithm used for the
ARIADNE classification pipeline.

Figure 7: Examples of feature importance derived from the RandomForest,
CatBoost, and LightGBM classifiers when selecting quiescent galaxies from
within the $0<z<3$ interval, without foreknowledge of galaxy redshifts. For
each learner, the feature importance values are normalised such that their sum
is 1.0. The $y$ axis labels correspond to feature names used by the pipeline
after the pre-processing steps outlined in Sect. 4.2 have been applied and
should be self-explanatory. For example, the feature named ‘u’ is derived from
the $u$-band magnitudes (i.e. after our pre-processing steps), the feature
‘VIS-20CM’ is derived from the $I_{\scriptscriptstyle{}\rm E}-20\,{\rm cm}$
colours, etc.

###  4.2 Feature engineering

####  4.2.1 Broadband colours

Before applying our algorithm to the data, we first performed a pre-processing
step known as ‘feature engineering’, whereby the data are enriched with
information to help the algorithm learn more efficiently. We start with the
broadband magnitudes and their $1\,\sigma$ errors as base features. Because
colours offer a potentially clearer description of the relative shape of the
broadband SED than magnitudes, we also calculated all possible (unique)
broadband colour permutations, and included them as features. In cases where
fluxes are given instead of magnitudes (e.g. VLA radio flux measurements), we
converted the values to magnitudes before deriving the related colours.

####  4.2.2 Missing data imputation strategy

One of the advantages of our machine-learning approach to galaxy
classification is the possibility to efficiently deal with missing data. A
subset of galaxies in the mock catalogues is undetected, or unobserved, in one
or more filters; we consider it highly desirable to include them in our
analysis, where possible, for several reasons: (i) such objects enlarge our
training dataset; (ii) non-detection in some bands is likely to carry useful
information for our learning algorithm (e.g. $u$-band drop-outs); and (iii)
the upcoming large surveys (e.g. Euclid, LSST, etc.) that motivate this work
will produce large datasets where many galaxies have missing data in one or
more bands, and such objects need to be utilised to make the most effective
use of the survey data.

Here, we impute values for missing data with a method that is independent of
the reason for it to be missing. When using tree-based learning algorithms
(CatBoostClassifier, LightGBMClassifier, RandomForestClassifier, or
XGBoostClassifier), we impute the missing values with the arbitrarily chosen
constant value $-99.9$. All broadband colours that would have used the missing
value are also set to $-99.9$. Because none of the measured magnitudes or
colours have this value (nor do they have similar values), information about
the presence of missing values is thus preserved such that the tree-based
learners can use non-detections to aid their classification of sources; this
information is essentially lost if the average, median, or minimum value would
be used instead for the imputation. Conversely, the MLPClassifier and
KNeighborsClassifier learning algorithms are generally more sensitive to the
normalisation of the input features, and imputing data with the value $-99.9$
is likely to create inappropriate and unhelpful artefacts in feature-space;
therefore, in the cases of MLPClassifier and KNeighborsClassifier we instead
impute missing values with the mean of the respective feature, computed across
the sample of galaxies using all the non-missing values.

We emphasise that the primary motivation behind our imputation strategy is to
flag non-detections such that the learning algorithms can deduce how to use
them most effectively; an added benefit of this strategy is that it allows the
use of objects with photometric SEDs that are missing one or more bands,
without necessarily having to discard them.

Table 2 illustrates the impact on the F1-score from using one of several
different imputation strategies, using a fixed random seed for the train/test
split and base learners. Results for the following strategies are shown:
imputation with the mean, the median, the minimum, a constant value of
$-99.9$, or dynamically switching between $-99.9$ for tree-based learners and
the mean for MLPClassifier and KNeighborsClassifier. In each case, the
F1-scores are shown for the base learners and for the final stacked ensemble
classifier. While the results vary significantly depending on the choice of
random seed, the general outcome is that, as expected, the tree-based learning
algorithms (CatBoostClassifier, LightGBMClassifier, RandomForestClassifier, or
XGBoostClassifier) generally give higher F1-scores when using missing values
that are imputed with $-99.9$, whereas MLPClassifier and KNeighborsClassifier
generally produce higher F1-scores when using the mean, median, or minimum of
a feature for imputation.

To provide the learning algorithms with additional help to treat missing data,
we created a feature (n$\\_$missing) that counts the number of missing
magnitude values for each galaxy. Although not needed in the present study,
the ARIADNE pipeline has the capability to make use of categorical flags that
specify the reason for each missing data point in the input data (i.e. non-
detection vs. not observed or masked).

Standardisation was performed using the Standard Scaler from Scikit-Learn,
which removes the mean and scales to unit variance. Missing values flagged
with the value $-99.9$ are ignored during the standardisation procedure.

####  4.2.3 Target variable

We generate a target feature representing the binary classification of the
galaxies. This feature is dynamically filled, depending on the specific subset
of galaxies to be selected. At its simplest, the target variable is set to 0
for star-forming galaxies and 1 for quiescent galaxies. To select quiescent
galaxies in a specific redshift range, the target feature is set to 1 for all
quiescent galaxies in that redshift band, and 0 for all other galaxies.

It is important to note that in the case of the Int catalogues, the sSFR
label, and consequently the binary target variable, has an intrinsic
uncertainty. Depending on the nature of the uncertainties, it is entirely
possible that our classification methodology is outperforming the initial sSFR
evaluation. However, a detailed analysis of this potential effect is beyond
the scope of this paper.

####  4.2.4 Feature importance and selection

It is important to ensure that our models are trained using only features that
provide useful information for the prediction of the target variable. First,
we examined the feature importance information provided by three of the tree-
based learning algorithms that we use here (RandomForestClassifier,
CatBoostClassifier, and XGBoostClassifier). Feature importance provides a
general picture of the relative usefulness of each feature in the construction
of the resulting classification models. Each of the aforementioned learning
algorithms uses a slightly different method to calculate feature importance
values. RandomForestClassifier calculates the mean decrease in impurity when a
feature is used in a split. CatBoostClassifier provides several options, from
which we select PredictionValuesChange, which indicates the average change in
the predicted values that result from a change in the feature value. In the
case of XGBoostClassifier, we opt to use the gain, defined as the improvement
in accuracy resulting from the use of a feature in the branches it is on.

In Fig. 7 we show examples of the feature importances resulting from training
each of the three aforementioned tree-based learners to select quiescent
galaxies in the range $0<z<3$, without foreknowledge121212In other words, the
input features for the classification models do not contain redshift values,
nor are they binned or sorted by redshift. In cases where redshift information
is included among the input features, this will be stated. of galaxy
redshifts. Significant differences are evident among the results in terms of
the importance values themselves and feature importance rank, reflecting
differences among the learning algorithms, the fact that many of the features
are strongly correlated, and the somewhat different methods used to compute
feature importances.

In general, the broadband colours typically show some of the highest feature
importance values, indicating they are among the most informative, as one
would expect given the strong correlation between the shape of a galaxy’s SED
and its activity type. In addition, the broadband magnitudes are also clearly
useful to some degree. The feature that counts missing values (n_missing) also
appears to be useful, at least in some circumstances. However, the magnitude
errors (not shown) show very low importance values, implying they provide
little or no useful information. An important caveat is that feature
importance values do not necessarily give an accurate picture of how the
inclusion (or removal) of a particular feature affects the metric used to
quantify model performance. Moreover, our feature analysis applies only to the
tree-based learning algorithms we used, and there is no guarantee that it
applies to any of the other learning algorithms present in our pipeline.

Thus, to better understand the general usefulness of each of our features, we
tested how removing a feature affects the F1-score metric that we calculate
after execution of the complete pipeline. We conducted split-run tests in
which our algorithm is trained and cross-validated twice, once with a
particular feature removed, and once more with this feature reinstated. All
other parameters were kept constant between the two training runs, including
the training-test data split and all random seeds; this split-run test process
was repeated multiple times ($>10$) for each feature, each time using a new
random seed to ensure the split-run test results are not dependent on which
random seed is used. The resulting difference in the F1-scores between the two
split-run test runs then indicates whether the inclusion of a feature is
useful (higher F1-score) or not (lower or unchanged F1-score).

We find that removal of any of the broadband colours or magnitude values
results in a noticeable decrease in the F1-score, indicating they are all
useful for training our algorithm. While the broadband magnitudes technically
provide the learning algorithms with a full description of the broadband SED
enclosed by the respective wavelength range, it is clear that explicitly
providing spectral slopes in the form of broadband colours allows
significantly more accurate model training.

We also find that inclusion of n$\\_$missing results in a significant
improvement in the F1-score, although the size of the improvement appears to
depend somewhat on the classification objective, such as the redshift range of
galaxies under selection, and the base learner used. Conversely, removing any
(or all) of the magnitude error features results in a small but significant
improvement in the F1-score, indicating they are uninformative and merely add
noise to the training data. Therefore, our machine-learning pipeline trains on
the following features: (i) the magnitudes; (ii) the broadband colours; and
(iii) the n$\\_$missing feature.

Figure 8: Precision, recall, and F1-score for various methods of identifying
quiescent galaxies. We show results for the three cases discussed in Sect.
5.1: (i) Euclid Deep Survey photometry with supporting $ugriz$, $W1$, $W2$,
and 20 cm photometry; (ii) Euclid Wide Survey photometry with supporting
$ugriz$, $W1$, $W2$, and 20 cm photometry; and (iii) Euclid Wide Survey
photometry only. Two curves are shown for the Euclid-only case, corresponding
to results obtained either with or without foreknowledge of the galaxy
redshifts. All other results shown in this figure were obtained assuming
foreknowledge of redshifts. In addition, we show the result of applying the
$I_{\scriptscriptstyle{\rm E}}-Y_{\scriptscriptstyle{\rm E}}$,
$J_{\scriptscriptstyle{\rm E}}-H_{\scriptscriptstyle{\rm E}}$ colour-colour
selection method developed by B20, assuming foreknowledge of (photometric)
redshifts. In this and subsequent plots showing metrics vs. redshift, the $x$
axis represents the ‘ground truth’ photometric redshifts from COSMOS2015
(Laigle et al., 2016).

Figure 9: Precision, recall, and F1-score for quiescent galaxy identification
methods when applied to the subset of galaxies detected in $u$,
$I_{\scriptscriptstyle{\rm E}}$, and $J_{\scriptscriptstyle{\rm E}}$, using
features derived from the $ugriz$, Euclid, $W$1, $W$2, and (for the Int
catalogues) the 20 cm bands. For comparison, we show the F1-scores when only
features derived from the $u$ and Euclid bands are used. We also show curves
representing the $u-I_{\scriptscriptstyle{\rm E}}$, $I_{\scriptscriptstyle{\rm
E}}-J_{\scriptscriptstyle{\rm E}}$ colour-colour selection method proposed by
B20. Table 3: Results from global selection of quiescent galaxies at $0\leq
z\leq 3$, and photometric redshift estimation.

|  | $\overbrace{\rule{100.00015pt}{0.0pt}}^{\text{\small classification statistics}}$ | $\overbrace{\rule{170.00026pt}{0.0pt}}^{\text{\small photometric redshift statistics}}$
---|---|---|---
Catalogue | Detections Required | $P$ | $R$ | F1-score | NMAD | $f_{\rm out}$ | Bias | $f_{\rm rej}$
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
Int Wide |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | 0.85 | 0.75 | 0.80 | 0.027 | 0.022 | 0.0031 | 0.14
SED Wide |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | 0.88 | 0.76 | 0.82 | 0.033 | 0.017 | 0.0000 | 0.22
Wide (averaged) |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | 0.87 | 0.76 | 0.81 | 0.030 | 0.019 | 0.0016 | 0.18
Int Deep |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | 0.77 | 0.60 | 0.67 | 0.030 | 0.022 | 0.0002 | 0.44
SED Deep |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | 0.88 | 0.79 | 0.83 | 0.022 | 0.015 | $-0.0008$ | 0.29
Deep (averaged) |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | 0.83 | 0.70 | 0.75 | 0.026 | 0.018 | $-0.0003$ | 0.37
Int Wide |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | 0.88 | 0.85 | 0.86 | 0.023 | 0.020 | 0.0024 | 0.20
SED Wide |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | 0.96 | 0.91 | 0.94 | 0.024 | 0.009 | 0.0008 | 0.01
Wide (averaged) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | 0.92 | 0.88 | 0.90 | 0.024 | 0.014 | 0.0016 | 0.10
Int Deep |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | 0.80 | 0.63 | 0.70 | 0.030 | 0.020 | 0.0013 | 0.39
SED Deep |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | 0.96 | 0.92 | 0.94 | 0.020 | 0.010 | 0.0000 | 0.01
Deep (averaged) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | 0.88 | 0.78 | 0.82 | 0.025 | 0.015 | 0.0007 | 0.20

131313The columns are as follows: (1) Mock catalogue; (2) bands in which
galaxies are required to be detected; (3) precision $P$ for quiescent galaxy
selection; (4) recall $R$ for quiescent galaxy selection; (5) the F1-score for
quiescent galaxy selection; (6) normalised median absolute deviation (NMAD)
for the photometric redshifts; (7) catastrophic outlier fraction ($f_{\rm
out}$) for the photometric redshifts; (8) bias of the photometric redshifts;
(9) fraction of quiescent galaxy redshifts rejected as potential catastrophic
outliers ($f_{\rm rej}$). The typical uncertainties on the $P$, $R$, and
F1-score values herein are $\leq 0.01$.

##  5 Results

###  5.1 Selection in redshift bins

We now apply our classification pipeline to the problem of selecting quiescent
galaxies from the mock Euclid photometry catalogues. The objective is to
examine the suitability of our method for the separation of quiescent and
star-forming galaxies, to facilitate expected Euclid legacy science related to
quiescent galaxies. We assume that prior to application of our pipeline, the
following steps have been performed: (i) correction for Galactic extinction;
(ii) pre-classification into star, active galaxy, and non-active galaxy
classes; and (iii) photometric or spectroscopic redshifts have been
determined, where applicable.

To allow a direct comparison with the colour-colour methods of B20, we select
quiescent galaxies in redshift bins delimited by the values $z=0$, 0.25, 0.5,
0.75, 1.0, 1.25, 1.5, 2.0, 2.5. We include an additional bin covering the
range $2.5\leq z<3$. The binning is performed using the 30-band photometric
redshifts from Laigle et al. (2016), and assumes that high quality photometric
(or spectroscopic) redshifts will be available for all the galaxies.
Hereinafter, all redshift values correspond to the 30-band photometric
redshifts from Laigle et al. (2016), with the obvious exception of those
derived using our pipeline in Sect. 5.2.

A significant fraction of the Euclid survey area is expected to have deep,
overlapping ground-based imaging observations from optical surveys (e.g.
LSST), but in some areas these observations may be sparse or non-existent.
Therefore, we test the performance of our pipeline under the three main
expected cases in terms of photometric depth and coverage: (i) Euclid Deep
Survey photometry with supporting $ugriz$, $W1$, $W2$, and 20 cm photometry;
(ii) Euclid Wide Survey photometry with supporting $ugriz$, $W1$, $W2$, and 20
cm photometry; and (iii) Euclid Wide Survey photometry only. Results for
these, and additional cases, are shown in Fig. 8 and 9.

####  5.1.1 Deep survey: Euclid, $ugriz$, $W1$, $W2$, 20 cm

When selecting from the Int Deep catalogue using features derived from the
Euclid, $ugriz$, WISE, and VLA photometry (blue points, second row of Fig. 8),
the F1-score shows a general rise from values of $\sim 0.7$ at low-$z$, before
peaking at a value of 0.86 in the $1.0<z<1.25$ bin and declining towards
higher redshifts, reaching a value of 0.48 in the $2.5<z<3.0$ bin.

Selecting from the SED Deep catalogue results in a broadly similar F1-curve
(blue points, fourth row of Fig. 8), albeit with a broad plateau over the
range $0.75\la z\la 2.0$, with maximum and minimum values of 0.97 and 0.75,
respectively. The F1-scores are systematically higher by $\sim 0.1$–0.3
compared to values obtained in the same redshift bins using the Int Deep
catalogue.

Interestingly, at $z\ga 1$ there is only a marginal reduction in F1-score when
the selection is performed without the $ugriz$, WISE, and 20 cm photometry. In
other words, provided the redshifts are known beforehand, these bands are
largely superfluous for the selection of quiescent galaxies, presumably due to
the fortuitous positioning of the 4000-Å break within the Euclid broadband
SED.

####  5.1.2 Wide survey: Euclid, $ugriz$, $W1$, $W2$, 20 cm

The situation is broadly similar when selecting from the Wide catalogue using
features derived from the Euclid, $ugriz$, WISE and 20 cm photometry (blue
points in the first and third rows of Fig. 8). The F1-curve shows a gradual
increase from $z=0$ to a broad peak or plateau at $0.75\la z\la 2.0$, after
which there is a gradual decline towards higher redshifts. In the case of the
Int Wide selection, the maximum and minimum F1-scores are 0.87 and 0.42,
respectively. For the SED Wide selection, these values are 0.89 and 0.69.

As before, the F1-scores are usually higher when selecting from the SED Wide
catalogue compared to the Int Wide catalogue, with values that are up to $\sim
0.3$ higher. Again, at $z\ga 1$ there is only a marginal reduction in F1-score
when the selection is performed without the $ugriz$, WISE, and VLA photometry.

As discussed in Sect. 4.1, when selecting from the $2.5<z<3.0$ bin we used a
single base-learner, XGBoost, together with the generalised stacking algorithm
(see also Fig. 14, right panel). When selecting quiescent galaxies from the
Wide catalogues in this redshift bin, this pipeline setup provides
significantly higher F1-scores compared to the default pipeline configuration
where five base-learners are employed. Although it is not immediately clear
why this is the case, we suggest that its ability to understand which values
are missing, and its subsequent use of missing values when performing splits,
allows the XGBoost algorithm to build stronger classifiers than other learners
when there is a high fraction of (informative) missing values in the dataset,
as is the case here (see Fig. 4).

It is also interesting to note that the Int Wide catalogue contains only 16
quiescent galaxies in the range $2.5\leq z\leq 3$ with detections in all of
the Euclid bands. As such, the training set contains on average 10.7 quiescent
galaxies, and the test set 5.3, making this a ‘few-shot learning’ problem.
Remarkably, despite the small number of examples in this redshift band, our
pipeline is able to obtain $P$, $R$, and, F1-score of $\sim 0.43$.

####  5.1.3 Wide survey: Euclid only

We also examine the performance of our pipeline when only Euclid observations
are available and for which a reliable redshift is not available (black points
in Fig. 8). These conditions are likely to pertain to a small, but potentially
significant number of galaxies in the Wide survey. In this case, our
classification pipeline must learn to place galaxies simultaneously into the
correct activity class and into the correct redshift bin.

The quality of the classification varies substantially across the redshift
range, with a peak F1-score of 0.71 in the $z=0.75$–1 bin, and with minima of
$\sim 0.2$–0.3 at either end of the range. The results are largely independent
of whether the Int Wide or SED Wide catalogue is used.

Without $ugriz$ photometry or redshifts, the classification problem becomes
much more challenging, and unsurprisingly the resulting selection is of
reduced quality compared to the cases discussed in Sects. 5.1.1 and 5.1.2. In
particular, the F1-scores are consistently lower than those obtained when also
using $ugriz$ photometry and redshifts, with differences of a factor of 2 or
more occurring near endpoints of the considered redshift range. Thus, a key
result is that redshift information allows for a significantly more accurate
selection of quiescent galaxies.

###  5.2 Global selection and redshift estimation

An alternative to selection in bins (Sect. 5.1) is first to perform a global
selection of quiescent galaxies, ignoring redshift information, and
subsequently derive photometric redshifts for the selected galaxies. In this
approach, we set the Target variable to 1 for all quiescent galaxies in the
range $0\leq z\leq 3$. The Target is set to 0 for star-forming galaxies at
$z\leq 3$, and also for all galaxies at $z>3$, regardless of their sSFR. This
analysis is performed on two different subsets of the mock data. Casting a
relatively wide net, we use all galaxies from the subset of each mock
catalogue for which there is a detection in all of the Euclid bands. In
addition, the analysis is performed for the subset of galaxies for which there
is a detection in each of the $ugriz$ and Euclid bands.

The results are shown in Table 3. The F1-score, $P$, and $R$ metrics can vary
significantly depending on which mock catalogue is used and whether galaxies
with a non-detection in an optical band are included or rejected. When
detections are required in Euclid $I_{\scriptscriptstyle{\rm E}}$,
$Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, and
$H_{\scriptscriptstyle{\rm E}}$ bands only, we obtain $P=0.85$, $R=0.75$, and
an F1-score of 0.80 for the Int Wide catalogue, or $P=0.88$, $R=0.76$, and an
F1-score of 0.82 for the SED Wide catalogue. Using the Int Deep catalogue, we
obtain the metric values $P=0.77$, $R=0.60$, and an F1-score of 0.67, which
are significantly lower than those obtained with Int Wide. On the other hand,
when using the SED Deep catalogue, the metrics are $P=0.88$, $R=0.79$, and an
F1-score of 0.83.

When detection in Euclid $I_{\scriptscriptstyle{\rm E}}$,
$Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$,
$H_{\scriptscriptstyle{\rm E}}$, and all of $ugriz$ is required, the metrics
are substantially improved, with $P=0.88$, $R=0.85$, and an F1-score of 0.86
for the Int Wide catalogue, or $P=0.96$, $R=0.91$, and an F1-score of 0.94 for
the SED Wide catalogue. These improvements are predominantly due to the fact
that requiring a detection in each of the optical and NIR bands reduces the
input data to a substantially smaller subset with relatively well constrained
broadband SEDs (see Fig. 5 and Table 7). Under these conditions, for the Int
Deep catalogue, we obtain $P=0.80$, $R=0.63$, and an F1-score of 0.70; for SED
Deep, $P=0.96$, $R=0.92$, and an F1-score of 0.94 are obtained.

To evaluate the quality of the photometric redshift estimates, the 30-band
photometric redshifts derived by Laigle et al. (2016) were used as the ground
truth. In Table. 3, we also give the values of NMAD, bias, and the
catastrophic outlier fraction ($f_{\rm out}$) for the quiescent subset of
galaxies whose photometric redshifts were not flagged as outliers by our
pipeline.

Removal of likely catastrophic outliers was performed as described in Sect.
4.1, using a relatively stringent probability threshold of 0.15. In other
words, we rejected photometric redshift estimates that were assigned a
probability of $\geq 0.15$ of being a catastrophic outlier. As with most
outlier removal methods, the removal of genuine catastrophic outliers usually
comes at the cost of also removing cases that are not catastrophic outliers.
The fraction of quiescent galaxies whose photometric redshifts were rejected
as catastrophic outliers ($f_{\rm rej}$), and thus were not used to calculate
NMAD, bias, or $f_{\rm out}$, is also shown.

Like the classification metrics, the metrics of photometric redshift quality
show variation depending on which mock catalogue (or subset thereof) is used.
We obtain values for NMAD between 0.020 and 0.033, catastrophic outlier
fractions between 0.009 and 0.022, and values of bias in the range $-0.0008$
to $0.0031$. While these values appear to improve on the results of the recent
Euclid Photometric Redshift Challenge (Euclid Collaboration: Desprez et al.,
2020), it is important to recognise that the mock Euclid photometry catalogue
used therein has a significantly different construction to those we have used
herein, making the inter-comparison of photometric redshift metrics
potentially unreliable.

Figure 10: Comparison between our machine-learning method, LePhare SED
fitting, and $UVJ$ colour-colour method, for the selection of quiescent
galaxies from the SED Wide mock Euclid catalogue. The following three
configurations of input data were used: (i) Euclid photometry only, with no
redshift information; (ii) Euclid photometry only, this time with photometric
redshifts from Laigle et al. (2016); and (iii) Euclid, $ugriz$, $W1$ and $W2$
photometry, again using photometric redshifts from Laigle et al. (2016).

##  6 Comparison with other methods

We test the performance of our quiescent galaxy selection pipeline against
four different colour-colour selection methods. We make like-for-like
comparisons, such that our pipeline and the colour-colour method under
consideration are applied to photometrically identical subsets of the mock
catalogues. In summary, our machine-learning method outperforms each of the
colour-colour methods we tested; full details are given in the following
subsections.

###  6.1 $I_{\scriptscriptstyle{\rm E}}-Y_{\scriptscriptstyle{\rm E}}$,
$J_{\scriptscriptstyle{\rm E}}-H_{\scriptscriptstyle{\rm E}}$ selection

To perform a like-for-like comparison with the $I_{\scriptscriptstyle{\rm
E}}-Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm
E}}-H_{\scriptscriptstyle{\rm E}}$ selection method, we select from the mock
catalogues those galaxies that have a detection in all of the Euclid bands
(i.e. $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$,
$J_{\scriptscriptstyle{\rm E}}$, and $H_{\scriptscriptstyle{\rm E}}$). We then
bin the galaxies by redshift as described in Sect. 5.1, and for all bins
(except $z=2.5$–3.0 where there are too few galaxies; see B20), we select
quiescent galaxies using the $I_{\scriptscriptstyle{\rm
E}}-Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm
E}}-H_{\scriptscriptstyle{\rm E}}$ criteria given by B20. The results are
shown by the green points in Fig. 8.

Our selection pipeline outperforms the $I_{\scriptscriptstyle{\rm
E}}-Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm
E}}-H_{\scriptscriptstyle{\rm E}}$ for almost every combination of redshift
and mock catalogue, with the greatest improvements in the F1-score occurring
near each endpoint of the considered redshift range. When our pipeline uses
all the available photometry (blue line in Fig. 8), the improvement in the
F1-score ranges from negligibly small at $z\sim 1.5$, to a factor of $\sim 2$
in the $z=0$–0.25 and $z=2$–2.5 bins.

While the inclusion of additional photometry bands is clearly part of the
reason for the improvements we have obtained, it is not the whole story. We
find that, even when our pipeline has access to exactly the same photometry
bands as the $I_{\scriptscriptstyle{\rm E}}-Y_{\scriptscriptstyle{\rm E}}$,
$J_{\scriptscriptstyle{\rm E}}-H_{\scriptscriptstyle{\rm E}}$ method (i.e. all
four Euclid bands; orange points in Fig. 8), it still significantly
outperforms the colour-colour method. This is because the other Euclid colours
($I_{\scriptscriptstyle{\rm E}}-J_{\scriptscriptstyle{\rm E}}$,
$I_{\scriptscriptstyle{\rm E}}-H_{\scriptscriptstyle{\rm E}}$,
$Y_{\scriptscriptstyle{\rm E}}-J_{\scriptscriptstyle{\rm E}}$,
$Y_{\scriptscriptstyle{\rm E}}-H_{\scriptscriptstyle{\rm E}}$) carry
information regarding the sSFR that is not present in
$I_{\scriptscriptstyle{\rm E}}-Y_{\scriptscriptstyle{\rm E}}$ or
$J_{\scriptscriptstyle{\rm E}}-H_{\scriptscriptstyle{\rm E}}$.

When our pipeline is configured to select quiescent galaxies in the same set
of redshift bins, using Euclid photometry only, and in the absence of redshift
information (black points in Fig. 8, we find that, in roughly half of the
redshift bins, the resulting F1-scores are similar to those obtained using the
$I_{\scriptscriptstyle{\rm E}}-Y_{\scriptscriptstyle{\rm E}}$,
$J_{\scriptscriptstyle{\rm E}}-H_{\scriptscriptstyle{\rm E}}$ method (which
has the benefit of foreknowledge of galaxy redshifts). This is generally the
case for $z\la 1$, while in the range $1\la z\la 1.75$ our pipeline yields
significantly lower F1-scores in this configuration. Interestingly, the
F1-scores obtained using our pipeline to select quiescent galaxies in the
$z=2$–2.5 bin are practically identical to, or slightly above, those obtained
using the $I_{\scriptscriptstyle{\rm E}}-Y_{\scriptscriptstyle{\rm E}}$,
$J_{\scriptscriptstyle{\rm E}}-H_{\scriptscriptstyle{\rm E}}$ method.

###  6.2 $u-I_{\scriptscriptstyle{\rm E}}$, $I_{\scriptscriptstyle{\rm
E}}-J_{\scriptscriptstyle{\rm E}}$ selection

For our comparison with the $u-I_{\scriptscriptstyle{\rm E}}$,
$I_{\scriptscriptstyle{\rm E}}-J_{\scriptscriptstyle{\rm E}}$ method, we use
only galaxies that are detected in each of the $u$, $I_{\scriptscriptstyle{\rm
E}}$, and $J_{\scriptscriptstyle{\rm E}}$ bands. Due to the increasing
sparsity of the $u$-band data at high redshifts, B20 were only able to derive
$u-I_{\scriptscriptstyle{\rm E}}$, $I_{\scriptscriptstyle{\rm
E}}-J_{\scriptscriptstyle{\rm E}}$ selection criteria in redshift bins
delimited by the values $z=0$, 0.25, 0.5, 0.75, 1.0 in the case of the Wide
survey mock catalogues, or $z=0$, 0.25, 0.5, 0.75, 1.0, 1.25, 1.5 for the Deep
catalogues. To these bins we applied our quiescent galaxy selection pipeline.
The results are shown in Fig. 9, together with the result of applying the
$u-I_{\scriptscriptstyle{\rm E}}$, $I_{\scriptscriptstyle{\rm
E}}-J_{\scriptscriptstyle{\rm E}}$ method of B20.

Once again, our pipeline (blue or orange points in Fig. 9) outperforms the
colour-colour selection method (green points). The improvement is minimal when
using the Wide survey mock catalogues, but is more substantial in the case of
the Deep catalogues (up to a factor of $\sim 1.4$), mainly due to improved
recall. The largest improvements occur at the upper endpoint of the considered
redshift ranges. There is little to no reduction in the performance of our
pipeline when only the $u$, $I_{\scriptscriptstyle{\rm E}}$,
$Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, and
$H_{\scriptscriptstyle{\rm E}}$ bands are used (orange points in Fig. 9),
compared to when the full suite of photometry is used (blue points in Fig. 9).
Once again, the improvement obtained from using our machine-learning pipeline
is largely due to its ability to make use of a larger colour- and magnitude-
space.

###  6.3 $BzK$ selection

In addition, we compare the quality of our quiescent galaxy selection method
against the $BzK$ selection method, which is designed to select quiescent
galaxies at $1.4<z<2.5$ using the $z-K$ and $B-z$ broadband colours (Daddi et
al., 2004). For this comparison, we used the observed $B$, $z$ and $K$
magnitudes from the COSMOS2015 catalogue, for galaxies in the Int Wide
catalogue, adopting the criteria of Daddi et al. (2004) of $(z-K)-(B-z)<-0.2$
and $z-K>2.5$ to select quiescent galaxies. To evaluate the success of the
$BzK$ method, we use the sSFR values from the Int Wide catalogue, adopting the
same ${\rm sSFR}<10^{-10.5}{\rm yr}^{-1}$ threshold to define quiescence.

First, we take the subset of the Int Wide catalogue that lies in the redshift
range $1.4<z<2.5$, and apply the $BzK$ selection method. We obtain $P$, $R$,
and F1-score values of 0.51, 0.50 and 0.50, respectively. Applying the ARIADNE
pipeline to the same subset of galaxies, using features derived from the
Euclid, $ugriz$, $W1$, $W2$, and 20 cm bands, we obtain $P$, $R$, and F1-score
values of 0.79, 0.80, and 0.79, respectively. Repeating the same tests over
the full redshift range of the Int Wide catalogue, the $BzK$ method results in
$P$, $R$, and F1-score of 0.44, 0.07, and 0.13, compared to 0.85, 0.75, 0.80
from ARIADNE. In summary, our machine-learning method strongly outperforms the
$BzK$ selection method.

###  6.4 SED fitting with LePhare and $UVJ$ selection

We also tested, though not exhaustively, our selection pipeline against an SED
fitting method wherein the galaxies are separated into quiescent and star-
forming on the basis of the sSFR estimated from the LePhare best fit to the
mock photometry. The fitting procedure is identical to that described in Sect.
2, except that the fitting is performed on SED Wide mock photometry, instead
of on observational data. Three slightly different approaches are taken, as
follows: (i) Only Euclid photometry is used, with redshift being a free
parameter; (ii) only Euclid photometry is used, but redshift is fixed to the
value given in Laigle et al. (2016); and (iii) Euclid, $ugriz$, $W1$, and $W2$
photometry is used, including upper limits, with redshift fixed. We derive
rest-frame $UVJ$ colours from the best-fitting spectral templates, and use the
$UVJ$ selection criteria of Whitaker et al. (2011). The approach used here may
overestimate how well LePhare and the $UVJ$ method can recover the galaxy
class, since the creation of the SED Wide mock catalogue and our subsequent
fitting were in both cases performed using LePhare, with an identical set of
base templates.

The F1-scores as a function of redshift are shown in Fig. 10 for the LePhare
fitting and $UVJ$ methods. Also shown is the F1-score for our machine-learning
method when applied to the same subset of data, with identical redshift
information.

Our machine-learning method significantly outperforms both approaches, most
noticeably at $z<1$ and $z\ga 2.5$. While the LePhare SED fitting method
sometimes comes close to reaching the F1-scores obtained by our machine-
learning method within the $z\sim 1$–2.5 range, in most redshift bins its
F1-scores are dramatically lower; in the case of the $UVJ$ method, the
F1-scores are always dramatically lower, typically by $\sim 0.2$, than those
obtained with our machine-learning method. The superior performance of our
method is at least partly due its ability to learn how to optimally weight the
different bands and colours in different regions of feature-space, unlike the
LePhare fitting and $UVJ$ methods.

##  7 Further analysis and tests

Here we summarise some additional analysis and tests we have conducted. Full
details are provided in Appendix B.

###  7.1 Stacking versus individual learners

Our implementation of the generalised stacking method demonstrably improved
classification performance. With few exceptions, the stacking method
consistently outperforms each individual base-learner, as well as
outperforming model averaging and hard-voting (Fig. 14). The method is robust
against pollution by multiple low-quality classifier models, and can be used
as a form of model selection. Finally, when applied to a single classifier
model, the meta-learner often makes a substantial improvement over the
original model.

###  7.2 The nature of the false positives

We also examined the distribution of false positives within sSFR space, with
the following three main conclusions. First, as expected, the incorrectly
classified objects cluster around the class threshold value ($10^{-10.5}{\rm
yr}^{-1}$), with a density that is highest in the bins immediately adjacent to
the class boundary (Figs. 15 and 16). Second, the precise distribution of the
incorrect classifications differs between the different mock catalogues.
Finally, our pipeline offers a significant improvement in $\bar{I}_{\rm FP}$
over the $I_{\scriptscriptstyle{\rm E}}-Y_{\scriptscriptstyle{\rm E}}$,
$J_{\scriptscriptstyle{\rm E}}-H_{\scriptscriptstyle{\rm E}}$ and
$u-I_{\scriptscriptstyle{\rm E}}$, $I_{\scriptscriptstyle{\rm
E}}-J_{\scriptscriptstyle{\rm E}}$ colour-colour methods, reducing the
degeneracy between quiescent galaxies and dusty, star-forming galaxies as
shown in Fig. 17 (in addition to improving on the P, R, and the F1-score
metrics as described above).

Table 4: Global selection of quiescent galaxies at $0\leq z\leq 3$ for
different pipeline and data configurations.

Catalogue | Bands Used | Detections Required | Redshifts | $P$ | $R$ | F1-score
---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7)
Int Wide |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$, 20 cm |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | none | 0.85 | 0.75 | 0.80
Int Wide (fast mode) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$, 20 cm |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | none | 0.84 | 0.74 | 0.79
SED Wide |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$ |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | none | 0.88 | 0.76 | 0.82
SED Wide (fast mode) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$ |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | none | 0.88 | 0.75 | 0.81
Wide (averaged) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$, 20 cm |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | none | 0.92 | 0.88 | 0.90
Wide (averaged) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$, 20 cm |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | $100\%$ | 0.92 | 0.91 | 0.92
Wide (averaged) |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | none | 0.81 | 0.68 | 0.74
Wide (averaged) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$, 20 cm |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | none | 0.87 | 0.76 | 0.81
Wide (averaged) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$, 20 cm |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | $50\%$ | 0.86 | 0.79 | 0.82
Wide (averaged) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$, 20 cm |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | $100\%$ | 0.86 | 0.83 | 0.85
Wide (averaged, $\sigma_{z}=0.025$) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$, 20 cm |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | $100\%$ | 0.86 | 0.83 | 0.84
Wide (averaged, $\sigma_{z}=0.05$) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$, 20 cm |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | $100\%$ | 0.86 | 0.82 | 0.84
Wide (averaged, $\sigma_{z}=0.075$) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$, 20 cm |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | $100\%$ | 0.86 | 0.81 | 0.83
Int Deep |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$, 20 cm |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | none | 0.77 | 0.60 | 0.67
Int Deep (fast mode) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$, 20 cm |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | none | 0.77 | 0.59 | 0.67
SED Deep |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$ |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | none | 0.88 | 0.79 | 0.83
SED Deep (fast mode) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$ |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | none | 0.88 | 0.78 | 0.83
Deep (averaged) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$, 20 cm |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | none | 0.88 | 0.78 | 0.82
Deep (averaged) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$, 20 cm |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | $100\%$ | 0.89 | 0.85 | 0.87
Deep (averaged) |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | none | 0.73 | 0.54 | 0.62
Deep (averaged) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$, 20 cm |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | none | 0.83 | 0.70 | 0.75
Deep (averaged) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$, 20 cm |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | $50\%$ | 0.83 | 0.74 | 0.78
Deep (averaged) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$, 20 cm |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | $100\%$ | 0.85 | 0.81 | 0.83
SED Wide (train), Int Wide (test) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$ |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | $100\%$ | 0.76 | 0.86 | 0.81
Int Wide (train), SED Wide (test) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$ |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | $100\%$ | 0.89 | 0.76 | 0.82
Int Deep (train), Int Wide (test) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$, 20 cm |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | none | 0.76 | 0.72 | 0.74
SED Deep (train), SED Wide (test) |  $ugriz$, $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$, $W1$, $W2$ |  $I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$, $J_{\scriptscriptstyle{\rm E}}$, $H_{\scriptscriptstyle{\rm E}}$ | none | 0.71 | 0.83 | 0.76

141414No pre-binning of the data by redshift was performed. The columns are as
follows: (1) Mock catalogue; where the metrics from using the SED and Int
catalogues have been averaged, this is indicated in parentheses; where the
pipeline has been used in its fast mode, this is also indicated; in the case
of transfer-learning, we specify separately the catalogues from which the
training and test data are drawn; (2) bands used for galaxy selection, which
includes missing values; (3) bands in which galaxies are required to be
detected; (4) redshift information included in the input data; (none, or a
percentage of redshifts included in a feature); (5) precision $P$ for
quiescent galaxy selection; (6) recall $R$ for quiescent galaxy selection; (7)
the F1-score for quiescent galaxy selection. The typical uncertainties on the
$P$, $R$, and F1-score values herein are $\leq 0.01$.

###  7.3 Reconciling the Int and SED results

Generally speaking, our classification results differ depending on whether the
mock catalogue was constructed using the Int or SED method. Firstly, we argue
that results obtained using the Int catalogues are likely to be pessimistic
with regard to the performance of our pipeline when applied to real Euclid
data; this is because the Euclid photometry is expected to have significantly
higher signal-to-noise ratios, and because the method of interpolating
photometry to simulate the Euclid bands is also likely to introduce errors.
Conversely, we expect that results obtained with the SED catalogues are likely
to be somewhat optimistic, because the construction of the SED catalogues
involves forcing the photometry to conform to one of a limited range of galaxy
templates.

Thus, we argue that results obtained with the Int and SED catalogues will
bracket the real-Universe performance of our pipeline. As a result, we show
performance metrics averaged over the Int catalogue and the corresponding SED
catalogue (i.e. Int Wide and SED Wide; Int Deep and SED Deep) in Tables 3 and
4, where appropriate.

###  7.4 Tuning the probability threshold

We investigated the impact on the precision and recall of tuning the value of
the class probability threshold, instead of adopting the default threshold
value of 0.5 (see Fig. 18). Our two main findings are as follows. First, there
exists a trade-off between precision and recall such that one may be
increased, but at the cost of reducing the other; tuning the probability
threshold allows a balance to be struck between $P$ and $R$ that is suitable
for different scientific needs.

Second, using the case shown in Fig. 18 (bottom panel) as an example, adopting
a probability threshold of 0.85 yields a very pure sample of quiescent
galaxies (P = 0.98) but with moderate incompleteness (R = 0.56); conversely, a
probability threshold of 0.05 results in high completeness (R = 0.97) but
moderate purity (P = 0.61).

###  7.5 Impact of including redshift as a feature

We examined the usefulness of including the Laigle et al. (2016) COSMOS2015
photometric redshifts as a feature in the data used for model training (e.g.
Simet et al., 2021). Selected results are included in Table 4, and are shown
in Fig. 19. Our two main findings are as follows. First, the inclusion of
redshifts as a feature in the data significantly improves the classification
metrics by reducing the degeneracy between redshift and sSFR, particularly for
galaxies at $z\leq 0.5$ or $z\geq 2.5$.

Second, even when only a subset of the redshifts are included, the
classification metrics are still improved compared to the case where no
redshift are included. In other words, it is beneficial to include any
available redshift information, even if it is somewhat sparse, when training
classification models.

###  7.6 The impact of noise

The impact of different types of noise has been explored. The main results are
summarised as follows. As expected, adding extra noise to the photometric data
results in a reduction in the F1-score when selecting quiescent galaxies.
However, even when the data becomes extremely noisy (${\rm S/N}\sim 3$), our
classification pipeline remains nominally functional with an F1-score of $\sim
0.67$ (see Fig. 20).

Our classification pipeline is robustly resistant to labelling errors when
these are random. However, systematic labelling errors tend to propagate into
the final classification results.

Finally, when photometric redshifts are included as a feature, adding Gaussian
noise to their values generally results in little or no reduction in the
classification metrics; we find that including noisy redshift values still
gives generally better classification results compared to the case where no
redshifts are included.

###  7.7 Transfer learning experiments

####  7.7.1 Training on templates and predicting on real SEDs

We also experimented with using classification models trained on spectral
templates to select quiescent galaxies from catalogues of observed photometry
(see Fig. 21). We find that classifiers trained on the SED Wide catalogue are
indeed able to select quiescent galaxies from the Int Wide catalogue, albeit
with marginally lower F1-scores compared to models trained on the Int Wide
catalogue itself. Thus, machine-learning models trained on synthetic galaxy
SEDs are a potential alternative to traditional methods used selecting
quiescent galaxies at redshifts where there are few (or no) known examples
(e.g. Girelli, Bolzonella, & Cimatti, 2019; Cecchi et al., 2019).

###  7.8 Which observables are useful to select quiescent galaxies?

We investigated the impact of adding one of $u$,$g$,$r$,$i$,$z$, $W$1, $W2$,
and 20 cm to the Euclid bands, when selecting quiescent galaxies. The main
results are summarised as follows.

Generally speaking, this results in a significant improvement in the F1-score
when selecting quiescent galaxies. The improvement varies depending on which
band is added, and the redshift interval in which the selection is performed
(Figs. 22, 23, 24, and 25).

As expected, the addition of a longer-wavelength optical band is typically
more useful for selecting quiescent galaxies at higher redshift. Conversely,
shorter wavelength optical bands are more useful for low-redshift selection.

Interestingly, the 20 cm radio band provides a significant improvement in
F1-score at $1.75\la z\la 2.5$, despite these data being very sparse.

##  8 Summary and concluding remarks

We have introduced the ARIADNE machine-learning pipeline for the
classification of galaxies. It uses a novel architecture with meta-learning to
combine the strengths of tree, nearest-neighbours, and deep-learning methods,
resulting in significantly higher classification accuracy compared to the
individual learning algorithms. The most relevant technical conclusions from
this study are as follows:

  * •

We have applied the tree-ensemble methods CatBoostClassifier and LightGBM to
the selection of quiescent galaxies, and find that both offer significant
performance improvements over the commonly used Scikit-Learn
RandomForestClassifier method, in terms of model quality and training
efficiency.

  * •

Providing our pipeline with sparsity awareness, by quantifying the sparsity of
the photometry for each galaxy, improves the classification performance. In
addition, the sparsity-aware method XGBoostClassifier was found to be well
suited for selecting quiescent galaxies at high redshifts ($z>2.5$), which
tend to have many missing photometry values.

  * •

We have shown that our implementation of the ‘generalised stacking’ method can
be used to perform error correction on individual machine-learning-based
galaxy classification models, sometimes turning a mediocre model into a
significantly better one.

  * •

We have used the pseudo-labelling technique (Lee, 2013) to improve the quality
of our photometric redshift estimates, with improvements in NMAD and the
catastrophic outlier fraction of $\sim 1$–3 per cent. When applied to the
high-volume of data that will come from future very large surveys, such as
Euclid or LSST, we expect pseudo-labelling to have a much greater effect.
Further exploration of the application of pseudo-labelling to the estimation
of galaxy physical properties is presented in Humphrey et al. (2022).

We have applied our pipeline to the selection of quiescent galaxies from mock
Euclid photometric catalogues, using simulated Euclid
$I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$,
$J_{\scriptscriptstyle{\rm E}}$, and $H_{\scriptscriptstyle{\rm E}}$
photometry, optionally using ancillary optical, infrared, or radio
measurements. The main results are as follows:

  * •

We have shown that our classification pipeline is able to efficiently select
quiescent galaxies from within the redshift range $0<z<3$, using mock Euclid
$I_{\scriptscriptstyle{\rm E}}$, $Y_{\scriptscriptstyle{\rm E}}$,
$J_{\scriptscriptstyle{\rm E}}$, and $H_{\scriptscriptstyle{\rm E}}$
photometry and somewhat sparse supporting data at other wavelengths. The
precision (purity), recall (completeness), and F1-scores vary substantially
with redshift, and between the various mock catalogues and subsets thereof.

  * •

We find that including ancillary $ugriz$, mid-infrared (WISE), and radio (20
cm) photometry yields substantial improvement in the selection of quiescent
galaxies at $z\la 1$. Smaller, but nonetheless significant, improvements were
found at $z\ga 1$.

  * •

In like-for-like comparisons, our machine-learning pipeline strongly
outperforms the $UVJ$ method (Whitaker et al., 2011) when derived from Euclid
(and ancillary) survey mock photometry, and usually outperforms the Euclid-
specific $I_{\scriptscriptstyle{\rm E}}-Y_{\scriptscriptstyle{\rm E}}$,
$J_{\scriptscriptstyle{\rm E}}-H_{\scriptscriptstyle{\rm E}}$, and
$u-I_{\scriptscriptstyle{\rm E}}$, $I_{\scriptscriptstyle{\rm
E}}-J_{\scriptscriptstyle{\rm E}}$ colour-colour selection methods (B20). The
improvement we obtain over the colour-colour methods can exceed a factor of 2
in terms of completeness and F1-score, with the greatest improvements
occurring at $z\la 1$ and $z\ga 2$.

  * •

In addition to being fewer in number, the false positives resulting from our
classification pipeline are less extreme than those resulting from the
$I_{\scriptscriptstyle{\rm E}}-Y_{\scriptscriptstyle{\rm E}}$,
$J_{\scriptscriptstyle{\rm E}}-H_{\scriptscriptstyle{\rm E}}$, and
$u-I_{\scriptscriptstyle{\rm E}}$, $I_{\scriptscriptstyle{\rm
E}}-J_{\scriptscriptstyle{\rm E}}$ methods, in that their actual sSFR values
are typically closer to the boundary between quiescent and star-forming
galaxies.

  * •

The significantly improved classification compared to the colour-colour,
$UVJ$, and template-fitting methods is likely the result of the more efficient
use of the available data by our machine-learning methodology. Compared to the
colour-colour methods, the ability to perform the classification in a higher-
dimensional colour-magnitude space clearly helps. More generally, machine-
learning methodologies have the ability to automatically weight the different
colours and filters according to their usefulness for the classification task
at hand, whereas traditional methods often take a somewhat ‘blind’ approach to
data weighting.

  * •

Our pipeline is able to derive photometric redshifts for galaxies selected as
quiescent, aided by the pseudo-labelling semi-supervised method, also using an
outlier detection algorithm to identify and reject likely catastrophic
outliers. Our pipeline achieves a normalised mean absolute deviation of $\la
0.03$ and a fraction of catastrophic outliers of $\la 0.02$ when measured
against the COSMOS2015 photometric redshifts of Laigle et al. (2016). These
values appear to improve on the results of the Euclid Photometric Redshift
Challenge (Euclid Collaboration: Desprez et al., 2020), but we emphasise that
the mock photometry catalogue used therein is of a significantly different
construction to those we have used herein, making any inter-comparison of
photometric redshift metrics potentially unreliable.

  * •

The inclusion of galaxy redshifts among the train and test datasets offers
significant improvement in the quality of our classification models, even when
the redshifts are relatively noisy or incomplete.

  * •

We have investigated the potential impact of various systematics. Most
notably, we find that our pipeline results are robust against the presence of
random errors in the class labels of the training data, for label error rates
of up to $\sim 33$ per cent.

This work has added to the growing body of evidence supporting the importance
of machine-learning techniques (or artificial intelligence) in astronomy and
astrophysics. In particular, we have demonstrated that machine learning
usually outperforms colour-colour methods for the selection of quiescent
galaxies; while part of this improvement is due to the ability to make use of
a larger number of bands and colours, we have also shown that machine-learning
methods still perform a superior selection even when the training data
contains only the bands used by the respective colour-colour method. In future
publications the methods presented herein will be further developed and
applied to other related problems in extragalactic astrophysics.

## Acknowledgments

This work was supported by Fundação para a Ciência e a 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.
LB acknowledges financial support by the Agenzia Spaziale Italiana (ASI) under
the research contract 2018-31-HH.0. KIC acknowledges funding from the European
Research Council through the award of the Consolidator Grant ID
681627-BUILDUP. AH acknowledges support from the NVIDIA Academic Hardware
Grant Program. AH also thanks colleagues Jean Gomes, João Pedroso, Catarina
Lobo, Tom Scott, Ana Afonso, Patricio Lagos, Israel Matute, Stergios
Amarantidis, Jose Afonso, Rodrigo Carvajal, and Ciro Pappalardo for useful
discussions or comments. The Euclid Consortium acknowledges the European Space
Agency and a number of agencies and institutes that have supported the
development of Euclid, in particular the Academy of Finland, the Agenzia
Spaziale Italiana, the Belgian Science Policy, the Canadian Euclid Consortium,
the Centre National d’Etudes Spatiales, the Deutsches Zentrum für Luft- und
Raumfahrt, the Danish Space Research Institute, the Fundação para a Ciência e
a Tecnologia, the Ministerio de Economia y Competitividad, the National
Aeronautics and Space Administration, the National Astronomical Observatory of
Japan, the Netherlandse Onderzoekschool Voor Astronomie, the Norwegian Space
Agency, the Romanian Space Agency, the State Secretariat for Education,
Research and Innovation (SERI) at the Swiss Space Office (SSO), and the United
Kingdom Space Agency. A complete and detailed list is available on the Euclid
web site (http://www.euclid-ec.org). In the development of our pipeline, we
have made use of the Pandas (McKinney, 2010), Numpy (Harris et al., 2020),
Scipy (Virtanen et al., 2020) and Dask (Rocklin, 2015) packages for Python.

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