# Corpora Compared: The Case of the Swedish Gigaword & Wikipedia Corpora

Tosin P. Adewumi

Foteini Liwicki

Machine Learning group, EISLAB
Luleå University of Technology, Sweden.
<EMAIL_ADDRESS>
Marcus Liwicki

Corresponding author — Presented at the Eighth Swedish Language Technology
Conference (SLTC)

###### Abstract

In this work, we show that the difference in performance of embeddings from
differently sourced data for a given language can be due to other factors
besides data size. Natural language processing (NLP) tasks usually perform
better with embeddings from bigger corpora. However, broadness of covered
domain and noise can play important roles. We evaluate embeddings based on two
Swedish corpora: The Gigaword and Wikipedia, in analogy (intrinsic) tests and
discover that the embeddings from the Wikipedia corpus generally outperform
those from the Gigaword corpus, which is a bigger corpus. Downstream tests
will be required to have a definite evaluation.

##  1 Introduction

It is generally observed that more data bring about better performance in
Machine Learning (ML) tasks Adewumi et al. (2019); Stevens et al. (2020). What
may not be very clear is the behaviour of variance of homogeneity in datasets.
It is always better to have a balanced or broad-based dataset or avoid an
overly-represented topic within a dataset Stevens et al. (2020). Furthermore,
noise (or contamination) in data can reduce performance Hagan et al. (1997).
However, not all noise is bad. Indeed, noise may be helpful Stevens et al.
(2020).

In this work, we compare embeddings (in analogy test) from two Swedish
corpora: The Gigaword and Wikipedia. The Gigaword corpus by Rødven Eide et al.
(2016) contains data from different genre, covering about 7 decades since the
1950s. Meanwhile the Wikipedia is a collection of articles on many, various
subjects Wikipedia (2019).

Word similarity or analogy tests, despite their weaknesses, have been shown to
reveal somewhat meaningful relationships among words in embeddings, given the
relationship among words in context Mikolov et al. (2013); Pennington et al.
(2014). It is misleading to assume such intrinsic tests are sufficient in
themselves, just as it is misleading to assume one particular extrinsic
(downstream) test is sufficient to generalise the performance of embeddings on
all NLP tasks Gatt and Krahmer (2018); Faruqui et al. (2016); Adewumi et al.
(2020b).

The research question being addressed in this work is: does bigger corpus size
automatically mean better performance for differently-sourced Swedish corpora?
The contribution this work brings is the insight into the differences in the
performance of the Swedish embeddings of the Gigaword and Wikipedia corpora,
despite the over 40% additional size of the Gigaword corpus. Furthermore, this
work will, possibly, enable researchers seek out ways to improve the Gigaword
corpus, and indeed similar corpora, if NLP downstream tasks confirm the
relative better performance of embeddings from the Wikipedia corpus. The
following sections include related work, methodology, results & discussion and
conclusion.

##  2 Related Work

Rødven Eide et al. (2016) created the Swedish corpus with at least one billion
words. It covers fiction, government, news, science and social media from the
1950s. The sentences of the first six lines of the content of this Gigaword
corpus are:

> 1 knippa dill
> patrik andersson
> TV : Danska Sidse Babett Knudsen har prisats på tv-festivalen i Monte Carlo
> för rollen
> i dramaserien Borgen .
> Hon sköts med ett skott i huvudet , men tog sig fram till porten och ringde
> på .
> I början av juni tog hon examen från den tvååriga YH-utbildning , som hon
> flyttade upp till huvudstaden för att gå .
> Det blev kaos , folk sprang fram för att hjälpa , någon började filma …

The content of the Wikipedia corpus is a community effort, which began some
years ago, and is edited continually. It covers far-reaching topics, including
those of the Swedish Gigaword corpus, and in addition, entertainment, art,
politics and more. The sentences of the first seven lines of the content of
the pre-processed version of the Wikipedia corpus are given below. It would be
observed that it contains a bit of English words and the pre-processing script
affected non-ascii characters. However, these issues were not serious enough
to adversely affect the models generated, in this case, as the embedding
system seems fairly robust to handle such noise.

> amager r en dansk i resund ns norra och v stra delar tillh r k penhamn medan
> vriga delar upptas av t rnby kommun och drag rs kommun amager har en yta p
> nine six two nine km och befolkningen uppg r till one nine six zero four
> seven personer one one two zero one eight en stor del av bebyggelsen har f
> rortspr gel men ven tskilliga innerstadskvarter finns i k penhamn samt i
> drag r p den stra delen av n finns kastrups flygplats amager r delvis en
> konstgjord delvis en naturlig s dan n r mycket l g och vissa delar ligger
> under havsytan framf r allt det genom f rd mning.

Adewumi et al. (2020a) created the Swedish analogy test set, which is similar
to the Google analogy test set by Mikolov et al. (2013). This was because
there was no existing analogy test set to evaluate Swedish embeddings Fallgren
et al. (2016); Précenth (2019). The analogy set has two main sections and
their corresponding subsections: the semantic & syntactic sections. Two native
speakers proof-read the analogy set for any possible issues (with percentage
agreement of 98.93% between them), after valuable comments from the reviewers
of this paper. It is noteworthy that some words can have two or more possible
related words. For example, based on the dictionary, the Swedish word man can
be related to kvinna and dam in very similar ways. Four examples from the
gram2-opposite sub-section of the syntactic section are:

> medveten omedveten lycklig olycklig
> medveten omedveten artig oartig
> medveten omedveten härlig ohärlig
> medveten omedveten bekväm obekväm

Faruqui et al. (2016) correctly suggest there are problems with word
similarity tasks for intrinsic evaluation of embeddings. One of the problems
is overfitting, which large datasets (like the analogy set in this work) tend
to alleviate Stevens et al. (2020). In order to have a definite evaluation of
embeddings, it’s important to conduct experiments on relevant downstream tasks
Faruqui et al. (2016); Faruqui and Dyer (2014); Lu et al. (2015); Gatt and
Krahmer (2018).

##  3 Methodology

Table 1 gives the meta-data of the two corpora used. The Gigaword corpus was
generated as described by Rødven Eide et al. (2016) while the Wikipedia corpus
was pre-processed using the recommended script by Grave et al. (2018). This
script returned all text as lowercase and does not always retain non-asci
characters. This created noise in the corpus, which may not necessarily be
harmful, as it has been shown in a recent work that diacritics can adversely
affect performance of embeddings unlike their normalized versions Adewumi et
al. (in press). A portion of the pre-processed text (given in the previous
section) was also tested for coherence on Google Translate and the English
translation returned was meaningful, despite the noise. Hence, the noise issue
was not serious enough to adversely affect the models generated in this case,
as the embedding system seems fairly robust to handle such noise.

Meta-data | Gigaword | Wikipedia
---|---|---
Size | 5.9G | 4.2G
Tokens | 1.08B | 767M
Vocabulary | 1.91M | 1.21M
Year | 2016 | 2019
Table 1: Meta-data for both Swedish Corpora

The authors made use of the fastText C++ library (with default hyper-
parameters, except where mentioned) by Grave et al. (2018) to generate 8
word2vec models and 8 subword models from each corpus, based on the optimal
hyper-parameter combinations demonstrated by Adewumi et al. (2020b). Each
model was intrinsically evaluated using the new Swedish analogy test set by
Adewumi et al. (2020a) in a Python-gensim program Řehůřek and Sojka (2010).
The hyper-parameters tuned are window size (4 & 8), neural network
architecture (skipgram & continuous bag of words(CBoW)) and loss (heirarchical
softmax and negative sampling). The subword models used lower & upper
character n-gram values of 3 & 6, respectively.

Although each model in the first set of experiments, with default (starting)
learning rate (LR) of 0.05, was run twice and average analogy score
calculated, it would have been more adequate to calculate averages over more
runs per model and conduct statistical significance tests. Nonetheless, the
statistical significance tests can be conducted for the downstream tasks,
which usually are the key tests for the performance of these embeddings. It
should also be noted that deviation from the mean of each model performance
for their corresponding two runs is minimal. Due to the observation of one
model (of Gigaword-CBoW-hierarchical softmax) failing (with Encountered NaN
error) when using the default LR of 0.05, another set of experiments with the
LR of 0.01 was conducted but with single run per model, due to time
constraint.

##  4 Results & Discussion

Table 2 gives mean analogy scores for LR 0.05 of embeddings for the two
corpora and table 3 for LR of 0.01. It will be observed that the skipgram-
negative sampling combination for both corpora for word2vec and subword models
performed best in both tables, except one in table 3, confirming what is known
from previous research Mikolov et al. (2013); Adewumi et al. (2020b, a). From
table 2, the highest score is 60.38%, belonging to the word2vec embedding of
the Wikipedia corpus. The lowest score is 2.59%, belonging to the CBoW-
hierarchical softmax, subword embedding of the Gigaword corpus. The highest
score in table 3 also belongs to the Wikipedia word2vec model. Among the 8
embeddings in the word2vec category in table 2, there are 6 Wikipedia
embeddings with greater scores than the Gigaword while among the subword,
there are 5 Wikipedia embeddings with greater scores. Nearest neighbour
qualitative evaluation of the embeddings for a randomly selected word is given
in table 4.

| Skipgram (s1) | CBoW (s0)
---|---|---
| H. S. (h1) | N. S. (h0) | H. S. (h1) | N. S. (h0)
window (w) | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8
Word2Vec % |
Wikipedia | 47.02 | 44.09 | 60.38 | 60.38 | 29.09 | 30.09 | 54.39 | 56.81
Gigaword | 40.26 | 44.23 | 55.79 | 55.21 | 26.23 | 27.82 | 55.2 | 55.81
Subword % |
Wikipedia | 46.65 | 45.8 | 56.51 | 56.36 | 28.07 | 24.95 | 38.26 | 35.92
Gigaword | 41.37 | 44.7 | 58.31 | 56.28 | 2.59 | - | 46.81 | 46.39

Table 2: Mean Analogy Scores for Swedish Gigaword & Wikipedia Corpora with
LR=0.05

| Skipgram (s1) | CBoW (s0)
---|---|---
| H. S. (h1) | N. S. (h0) | H. S. (h1) | N. S. (h0)
window (w) | 4 | 8 | 4 | 8 | 4 | 8 | 4 | 8
Word2Vec % |
Wikipedia | 48.92 | 49.01 | 51.71 | 53.48 | 32.36 | 33.92 | 47.05 | 49.76
Gigaword | 39.12 | 43.06 | 48.32 | 49.96 | 28.89 | 31.19 | 44.91 | 48.02
Subword % |
Wikipedia | 45.16 | 46.82 | 35.91 | 43.26 | 22.36 | 21.1 | 14.31 | 14.45
Gigaword | 39.13 | 43.65 | 45.51 | 49.1 | 31.67 | 35.07 | 28.34 | 28.38

Table 3: Analogy Scores for Swedish Gigaword & Wikipedia Corpora with LR=0.01

Figure 1: Word2Vec Mean Scores, LR:0.05

Figure 2: Subword Mean Scores, LR:0.05

Nearest Neighbor | Result
---|---
Wiki: syster | systerdotter (0.8521), systern (0.8359), ..
Gigaword: syster | systerdotter (0.8321), systerdottern (0.8021), ..

Table 4: Example qualitative assessment of Swedish subword w4s1h0 models

We hypothesize that the general performance difference observed between the
embeddings of the two corpora may be due to a) the advantage of wider domain
coverage (or corpus balance in topics) of the Wikipedia corpus - which is the
most plausible reason, b) the small noise in the Wikipedia corpus or c) the
combination of both earlier reasons.

Since it’s preferable to have more than one criterion for the difference
between the two corpora, future work will focus, particularly, downstream
tasks to confirm this Faruqui et al. (2016); Gatt and Krahmer (2018).
Implementation without using the pre-processing script by (Grave et al., 2018)
on the original Wikipedia corpus will also be attempted.

##  5 Conclusion

This work has shown that better performance results from differently sourced
corpora of the same language can be based on reasons besides larger data size.
Simply relying on larger corpus size for performance may be disappointing. The
Wikipedia corpus showed better performance in analogy tests compared to the
Gigaword corpus. Broad coverage of topics in a corpus seems important for
better embeddings and noise, though generally harmful, may be helpful in
certain instances. Future work will include other tests and downstream tasks
for confirmation.

##  6 Acknowledgement

The authors wish to thank the anonymous reviewers for their valuable
contributions and the very useful inputs from Carl Borngrund and Karl Ekström,
who proof-read the analogy set. The work on this project is partially funded
by Vinnova under the project number 2019-02996 ”Sprkmodeller fr svenska
myndigheter”.

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The relation between globular cluster systems and supermassive black holes in spiral galaxies III. The link to the M_∙ - M_∗ correlation.

Rosa A. González-Lópezlira

0000-0003-1557-4931]Rosa A. González-Lópezlira
Instituto de Radioastronomia y Astrofisica, UNAM, Campus Morelia,
Michoacan, Mexico, C.P. 58089

0000-0003-2127-2841]Luis Lomelí-Núñez
Instituto Nacional de Astrofísica, Óptica y Electrónica, Luis Enrique Erro 1, Tonantzintla, Puebla, C.P. 72840, Mexico
Instituto de Radioastronomia y Astrofisica, UNAM, Campus Morelia,
Michoacan, Mexico, C.P. 58089

0000-0001-7966-7606]Yasna Órdenes-Briceño
Instituto de Astrofísica, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, 7820436 Macul, Santiago, Chile

0000-0002-5635-3345]Laurent Loinard
Instituto de Radioastronomia y Astrofisica, UNAM, Campus Morelia,
Michoacan, Mexico, C.P. 58089

0000-0001-8221-8406]Stephen Gwyn
Herzberg Institute of Astrophysics, National Research Council of Canada, Victoria, BC V9E 2E7, Canada

0000-0002-5897-7813]Karla Álamo-Martínez
Departamento de Astronomía, Universidad de Guanajuato, Apdo. Postal 144, Guanajuato 36000, Mexico

0000-0002-6971-5755]Gustavo Bruzual
Instituto de Radioastronomia y Astrofisica, UNAM, Campus Morelia,
Michoacan, Mexico, C.P. 58089

0000-0002-7214-8296]Ariane Lançon
Observatoire Astronomique de Strasbourg, Université de Strasbourg, CNRS, UMR 7550, 11 rue de l'Université,  67000 Strasbourg, France

0000-0003-0350-7061]Thomas H. Puzia
Instituto de Astrofísica, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, 7820436 Macul, Santiago, Chile

We continue to explore the relationship between globular cluster total number,
$N_{\rm GC}$, and central black hole mass, $M_\bullet$, in spiral galaxies. We
present here results for the Sab galaxies NGC 3368, NGC 4736 (M 94) and
NGC 4826 (M 64), and the Sm galaxy NGC 4395.  The globular cluster (GC)
candidate selection is based on the ( - )  versus ( - )
color-color diagram, and -band shape parameters.  We determine the
$M_\bullet$ versus $N_{\rm GC}$ correlation for these spirals, plus NGC 4258,
NGC 253, M 104, M 81, M 31, and the Milky Way. We also redetermine the
correlation for the elliptical sample in Harris et al., 2014, with updated galaxy
types from Sahu et al., 2019.  Additionally, we derive total stellar galaxy mass,
$M_\ast$, from its two-slope correlation with $N_{\rm GC}$ [Hudson et al., 2014], and
fit $M_\bullet$ versus $M_\ast$ for both spirals and ellipticals. We obtain log
$M_\bullet \propto$ (1.01 $\pm$ 0.13) log $N_{\rm GC}$ for ellipticals, and log
$M_\bullet \propto$ (1.64 $\pm$ 0.24) log $N_{\rm GC}$ for late type galaxies
(LTG).  The linear $M_\bullet$ versus $N_{\rm GC}$ correlation in ellipticals
could be due to statistical convergence through mergers, but not the much
steeper correlation for LTG.  However, in the $M_\bullet$ versus total stellar
mass ($M_\ast$) parameter space, with $M_\ast$ derived from its
correlation with $N_{\rm GC}$, $M_\bullet \propto$ (1.48 $\pm$ 0.18) log
$M_\ast$ for ellipticals, and $M_\bullet \propto$ (1.21 $\pm$ 0.16) log
$M_\ast$ for LTG.  The observed agreement between ellipticals and LTG in this
parameter space may imply that black holes and galaxies co-evolve through
“calm” accretion, AGN feedback, and other secular processes.

§ INTRODUCTION

It is widely accepted that all massive galaxies contain a supermassive black hole (SMBH). In
spheroidal systems, the masses of the SMBH, $M_{\bullet}$, correlate with other
properties of their host galaxies: the bulge luminosity <cit.>, the bulge mass <cit.>, the bulge
stellar velocity dispersion <cit.>, the dark matter halo mass [Spitler & Forbes, 2009].  Today, the
$M_\bullet$–$\sigma_\ast$ relation is known to have a $\approx$ 0.6 dex
scatter and depend on the galaxy's merger history [Bogdán et al., 2018, Sahu et al., 2019], and
the $M_\bullet$–$M_{\rm bulge}$ relation depends on the morphology of the
galaxy [Savorgnan et al., 2016, van den Bosch, 2016, Sahu et al., 2019].  Graham, 2012, Graham & Scott, 2013, Scott et al., 2013, Davis et al., 2019
find that there are correlations between $M_\bullet$, and the luminosity and
mass of the bulge, but that the slope is steeper for spiral bulges with a
Sersic profile than for early type galaxies (ETG), irrespective of whether the
latter are better fitted by a Sersic or a core-Sersic profile.

Exciting as these correlations are because of the clues that they may provide
regarding galaxy formation and assembly, and about their possible co-evolution with
the central black hole, they are not without controversy.  While
Ferrarese, 2002, Baes et al., 2003, Davis et al., 2019 state that $M_\bullet$ correlates with the disk
maximum rotation velocity and hence with the dark matter halo mass,
Kormendy & Bender, 2011 retort that black holes do not correlate directly with dark
matter haloes, and that a correlation appears only if the galaxy has a bulge.
Likewise, Kormendy et al., 2011 contend that black holes do not correlate with either
disks or pseudobulges. Davis et al., 2018 argue that $M_\bullet$ actually
correlates with disk mass, albeit with a low correlation coefficient, $r = $

Investigating the limits and deviations of these scaling relations, i.e., at
high and low masses, may be particularly useful to uncover the
processes that are involved in the formation and evolution of galaxies and their black holes.
For example, the log $M_\bullet$ versus log $M_\ast$ correlation, with $M_\ast$
the total galaxy stellar mass, has been explored in both ETG and late type
galaxies (LTG).  van den Bosch, 2016, Davis et al., 2019, Sahu et al., 2019 find that
the slope for LTG is between 70% and 100% steeper than the nearly linear one
followed by ETG. Davis et al., 2019 rules out that such a steep correlation in LTG could
be due to statistical convergence through hierarchical galaxy formation
[Peng, 2007, Jahnke & Macciò, 2011]; in addition, since the LTG in their sample sit on the same
correlation, regardless of whether they contain or not an active galactic
nucleus (AGN), they call into question the relevance of AGN feedback to the
relation. Reines & Volonteri, 2015 analyze two samples, one of ETG and classical bulges,
mostly without active AGN, and another of LTG with active AGN. They find
similar slopes (respectively, 1.4 and 1.0, i.e., less steep for LTGs), but with
an offset of more than one order of magnitude at log $M_\ast/M_\odot \sim$ 10.
In the same parameter space, Simmons et al., 2017 determine that the slope for a
sample of AGN in disk-dominated, merger-free, galaxies is 1.2, roughly equal
to that of the bulge sample of Häring & Rix, 2004, and with a small offset of
$\sim$ 0.2 dex at log $M_\ast/M_\odot =$ 10; they hence conclude that mergers
are inconsequential for the co-evolution of black holes and galaxies, and that
secular processes, like “calm" accretion and AGN feedback, are more
fundamental for galaxy assembly and black hole growth.

An intriguing correlation, given the extremely disparate scales, is the one
between central black hole mass and the total number of globular clusters
($N_{\rm GC}$) [Burkert & Tremaine, 2010, Harris & Harris, 2011, Harris et al., 2014]. The correlation can be expressed
as $N_{\rm GC} \propto M_\bullet^{1.02\pm0.10}$, and spans over 3 orders of
magnitude.  Possible causal links have been proposed, e.g., feedback by the
jets of AGN <cit.> and
cannibalization of GCs by black holes
<cit.>; since it is roughly linear for
ellipticals, it has been argued that it could be due to statistical convergence through
merging.  On the other hand, GC systems are known to be very good tracers of
galaxy virial mass <cit.>. This
correlation is good over 7 orders of magnitude in halo mass, from dwarf
galaxies to galaxy clusters, with a strongly declining dispersion with
increasing mass, $\delta N_{\rm GC}$/$N_{\rm GC} \propto$ $M_{\rm vir}^{1/2}$
[Forbes et al., 2018, Burkert & Forbes, 2020].

The study of GC systems in spiral galaxies has been hampered by the sparsity of
their globular clusters, compared to ellipticals, and has been mostly limited
to edge-on objects, given the difficulty of detecting individual GCs projected
on a spiral disk.  Until relatively recently, there were only 5 spiral galaxies
with precise measurements of both  $N_{\rm GC}$ and $M_\bullet$: the Milky Way
(MW, Sbc), M 104 (Sa), M 81 (Sab), M 31 (Sb) and NGC 253 (Sc).
Burkert & Tremaine, 2010 only included the first four, and readily noticed that, while
M 31, M 81, and M 104 fall right on the $N_{\rm GC} - M_\bullet$ correlation
for ellipticals, our Galaxy has a black hole that is about one order of
magnitude lighter than expected from its $N_{\rm GC}$. They also observed that,
compared to M 31, M 81 and M 104, the MW has both a later Hubble type and
possibly a pseudobulge, and hypothesized these factors could be related to its
non-compliance with the correlation.

We are in the process of studying the $N_{\rm GC}$ versus $M_\bullet$
correlation in all nine spirals with precise black hole measurements within 16
Mpc: NGC 1068, NGC 3368, NGC 4051, NGC 4151, NGC 4258, NGC 4395,
NGC 4736, NGC 4826, and NGC 5055, although so far we have full data sets
only for NGC 3368, NGC 4258, NGC 4395, NGC 4736, and NGC 4826.  Bulge type
was not considered when selecting the sample, but with one exception all of
them seem to have a pseudobulge [Kormendy et al., 2010, Kormendy & Ho, 2013, Kormendy, 2013], while
NGC 4258, NGC 4826, and NGC 5055 may also have a classical bulge. The single
exception, NGC 4395, is a pure disk, bulgeless, spiral [Filippenko & Ho, 2003, Kormendy & Ho, 2013].
For completeness, we should mention that M 31 and M 81 have classical bulges,
NGC 253 and the  MW have pseudobulges, and M 104 has both a bulge and a
pseudobulge [Kormendy & Ho, 2013].  Pseudobulges are widely considered the product of
secular evolution with few major mergers <cit.>,
although according to, for example, Sauvaget et al., 2018, they can also be formed by
major mergers of gas-rich progenitors in the distant universe.

In this paper, we present new measurements and analyses of the globular cluster
systems of the Sab galaxies NGC 3368 (M 96), NGC 4736 (M 94) and NGC 4826
(M 64), and the Sm galaxy NGC 4395.  Their surface brightness fluctuations
(SBF) distances are 10.4$^{+1.1}_{-1.0}$ Mpc for NGC 3368, 5.0$\pm$ 0.4 Mpc
for NGC 4736, and 7.3$\pm$ 0.7 Mpc for NGC 4826 [Tonry et al., 2001, Blakeslee et al., 2010].
NGC 4395 is completely devoid of a bulge, and hence lacks an SBF measurement;
we adopt a Cepheid distance of 4.3$\pm$ 0.4 Mpc [Thim et al., 2004].  The masses of
their central supermassive black holes are (3.6 $\pm$ 1.1)$\times 10^5\
M_\odot$ for NGC 4395 [Peterson et al., 2005]; (7.5 $\pm$ 1.5)$\times 10^6\ M_\odot$
for NGC 3368; 6.8$^{+1.5}_{-1.6} \times 10^6\ M_\odot$ for NGC 4736, and
(1.6$\pm$0.4)$\times 10^6\ M_\odot$ for NGC 4826 [Kormendy & Ho, 2013]. Their
inclinations to the line of sight are, respectively, 46$\fdg$2 (NGC 3368),
33$\fdg$7 (NGC 4395), 35$\fdg$6 (NGC 4736) and  57$\fdg$5 (NGC 4826)
[de Vaucouleurs et al., 1991]. We take the following position angles (PA; North through East):
5$\degr$ (NGC 3368), 147$\degr$ (NGC 4395), 115$\degr$ (NGC 4736 and
NGC 4826).

In order to identify globular cluster candidates (GCCs) in these galaxies, we
apply the ($u^*-i'$) vs. ($i'-K_s$) diagram technique ( hereafter). It
is extremely powerful to isolate, in different regions and according to their
stellar populations and star formation histories, foreground Galactic stars,
background galaxies, young clusters in the disk, and GCCs. In addition, objects
in such separate areas of the plot also differ in their light concentration
parameters, at least up to a distance of $\sim$ 16 Mpc [Muñoz et al., 2014]. The
efficacy of the procedure has been corroborated by Powalka et al., 2016 in M87, and
by González-Lópezlira et al., 2019, with OSIRIS observations of the GC candidates in the
megamaser prototype NGC 4258. The method allows for a very low contamination
of the samples from foreground stars, background galaxies, and young stellar
clusters, even in disk galaxies that are not completely edge-on, and in the
absence of radial velocity measurements.

§ DATA

All the data for this work were obtained with the
Canada-France-Hawaii-Telescope (CFHT), the optical images with MegaCam
[Boulade et al., 2003], and the -band ones with the Wide-field InfraRed Camera
<cit.>. MegaCam has a field-of-view (FOV) of 0$\fdg96 \times
0\fdg$94, with a plate scale of 0$\farcs$186 pixel$^{-1}$, while the FOV of the
WIRCam is $\sim 21^\prime \times 21^\prime$, with a plate scale of 0$\farcs$307
pixel$^{-1}$.  Brief descriptions of the detectors' layouts, cameras'
performances, and data reduction procedures are given in González-Lópezlira et al., 2017.
Table <ref> gives a summary of the observations.

Except for NGC 4395, the , , and  data are archival, and were
originally secured through programs 11AC08 (P.I. G. Harris), and 13AS03
(P.I. Zhao-Yu Li). The  and -band images, as well as all the NGC 4395
data, are ours, and were acquired through programs 17AF12, 18AF99 (P.I. K.Álamo-Martínez), and 19AF03 (P.I. Y. Órdenes-Briceño).

To avoid saturation, individual  exposures of the galaxies were only 10 s
long. For NGC 3368, one image was taken in each of the four quadrants of the
WIRCam, before nodding the telescope to obtain a sky exposure 2$\fdg$5 apart.
The other three galaxies were observed in an approximately circular pattern
with a radius $\sim$ 1$\farcm$6. Given their large diameters compared to the
WIRCam FOV <cit.>, separate (also
10  s long) sky frames were taken, respectively, 2$\fdg$5 (NGC 4395),
1$\fdg$5 (NGC 4736) and 3$\fdg$5 (NGC 4826) away, with the same circular
pattern, using the target(T)-sky(S) sequence STTSSTTSS...TTS.

Before combining them into mosaics, the individual optical images were
recalibrated with the MegaPipe pipeline [Gwyn, 2008], in order to achieve
astrometric internal and external accuracies of 0$\farcs$04 and 0$\farcs$15,
respectively, and a photometric accuracy of 0.03 mag.  The -band mosaics
were produced with the WIRwolf pipeline [Gwyn, 2014], which sky-subtracts
individual images and achieves an internal astrometric accuracy of 0$\farcs$1.
Both MegaPipe and WIRwolf render images with photometry in the AB system
[Oke, 1974], with zero points ($zp$) set to 30 mag.
Figure <ref> shows the images of the four galaxies in the

CFHT -band images, asinh scale. Top left: NGC 3368; top
right: NGC 4395; bottom left: NGC 4736; bottom right: NGC 4826.

§ DETECTION AND PHOTOMETRY

Source detection and photometric measurements in all the stacked images were
carried out with SExtractor [Bertin & Arnouts, 1996] and PSFEx [Bertin, 2011]. The
detection was performed on images from which their median-filtered version was
subtracted; faint sources are detected more easily in a median-subtracted
image. All median images were produced with a 31 pix$\times$31 pix median
filter. Conversely, the photometry measurements were made on the original
stacked images.  PSFEx employs point sources detected in a first pass of
SExtractor to build a point-spread function (PSF) model that SExtractor can
then apply in a second pass to obtain PSF magnitudes of sources.  We chose
adequate PSF stars manually, based on their brightness versus compactness, as
measured by SExtractor parameters MAG_AUTO <cit.> and FLUX_RADIUS (similar to the effective
radius); their values for each filter and galaxy are given in
Table <ref>. A detailed description of this procedure is presented
in González-Lópezlira et al., 2017. The spatial variations of the PSF were modeled with
polynomials of degree 3.  To create the PSF, the flux of each star was measured
in an 18 pixel aperture in all bands (equivalent to $3\farcs3$ in the optical
and $5\farcs5$ at $K_s$); such aperture, determined through the growth-curve
method for each passband, is large enough to measure the total flux of the
stars, but small enough to reduce the likelihood of contamination by extraneous

PSF stars parameters

2cFLUX_RADIUS    2cMAG_AUTO

Galaxy    Filter    Min     Max    Min    Max    $N$ PSF stars

2cpixel     2cmag

NGC 3368     u^*             2.10      2.60       16.0        23.0         1998

g^'      1.60      2.00       17.5        25.5         3876

i^'      1.30      1.60       17.0        23.0         5306

K_s              1.10      1.60       15.5        20.0         1162

NGC 4395     u^*             1.80      2.40       17.0        23.0         2379

g^'      1.70      2.20       17.0        23.0         3937

i^'      1.60      2.10       15.0        21.5         3741

K_s              1.10      1.60       16.0        20.0          449

NGC 4736     u^*             2.00      2.70       15.5        22.0         1296

g^'      3.10      3.70       16.0        22.9         3112

r^'      3.20      3.70       15.5        22.0         4302

i^'      2.50      2.80       16.0        22.0         5047

K_s              1.30      1.70       15.0        19.5          320

NGC 4826      u^*             2.50      3.10       15.0        23.0         2300

g^'      2.00      2.40       16.0        23.0         2379

r^'      3.00      3.60       16.0        22.0         5115

i^'      1.80      2.20       16.0        22.0         4805

K_s              1.10      1.40       15.5        20.5          519

§ COMPLETENESS

Completeness tests were only performed on the -band mosaics; these are the
smallest and shallowest images, and in fact set our object detection limits. In
order to determine the GC detection completeness as a function of magnitude,
65,000 artificial point sources were generated for each galaxy, based on their
respective  PSF model, in the interval  19 mag $< m_{Ks} <$ 24 mag, and
with a uniform, or box-shaped, magnitude distribution.  Due to the WIRCam pixel
size,[0$\farcs$307 or $\sim$ 6 pc, $\gtrsim$ 7 pc, $\sim$ 11 pc, and
$\sim$ 14.5 pc  at the distances of, respectively, NGC 4395, NGC 4736,
NGC 4826, and NGC 3368.] GCCs are mostly unresolved in the -band images.
The artificial objects were added only 500 at a time, to avoid creating
artificial crowding; that is, a total of 130 images with simulated sources were
produced for each galaxy.  The positions of the added sources were random, and
assigned  without regard to the positions of other artificial objects or of
real sources.

Artificial sources were then recovered by running SExtractor on each one of the
simulated images, with the same parameters used for the original  galaxy
images, and cross-matching the positions of all detections with the known input
coordinates of the added objects.  Sources with SExtractor output parameter
FLAGS $\neq$ 0 were eliminated; this operation excludes artificial objects
falling on top of other artificial or real sources, and hence discards
preferentially added objects in crowded regions. The restriction FLAGS=0 was
also applied in the selection of true sources (see Section <ref>).

As done previously with NGC 4258 by González-Lópezlira et al., 2017, for all the galaxies we
estimated completeness in four different regions within 1.7 $R_{25}$: an
ellipse with semimajor axis equal to 0.5 $R_{25}$, and three elliptical annuli,
respectively, from 0.5 to 1.0 $R_{25}$, from 1.0 to 1.4 $R_{25}$, and between
1.4 and 1.7 $R_{25}$. With the exception of the innermost ellipse of NGC 3368,
all regions have the axis ratios and PA assumed for the observed disks of the
galaxies (respectively, 0.69 and 5$\degr$ for NGC 3368, 0.82 and 147$\degr$
for NGC 4395, 0.81  and 115$\degr$ for NGC 4736, and 0.54 and 115$\degr$ for
NGC 4826); the PA of the central ellipse of NGC 3368 follows the alignment of
the surface brightness there, and hence differs from the orientation of the
outer isophotes.  With the goal of estimating completeness with statistically
equivalent samples, for each region we considered a number of added sources
inversely proportional to its area; in other words, for the inner ellipses the
artificial sources from all 130 images were taken into account, whereas only
43, 34, and 39 were included for the first, second, and third annuli,
respectively.   On average, for each region we consider $\sim$ 2000 artificial
sources in the cases of NGC 4395 and NGC 4736, and $\sim$ 1000 for NGC 3368
and NGC 4826.  Figure <ref> shows all recovered sources in
black, and those from the artificial images actually considered in colors; in
the innermost ellipse (within 0.5 $R_{25}$), all recovered sources were used,
hence all dots appear colored in both black and red.

Completeness tests, recovered artificial point sources. Top
left: NGC 3368; top right: NGC 4395; bottom left: NGC 4736;
bottom right: NGC 4826.  Colors indicate points from artificial images
that were actually used in the fits, to ensure the samples of all regions were
statistically equivalent.

In the case of a box-shaped magnitude distribution, the fraction of recovered
artificial sources as a function of magnitude is often well described by the
Pritchet function <cit.>:

\begin{equation}
f^P(m) = \frac{1}{2}  \left[ 1\ -\ \frac{\alpha_{\rm cutoff} (m - m_{\rm lim})}{\sqrt{1 + \alpha_{\rm cutoff}^2(m - m_{\rm lim})^2}}  \right];
\label{eq:pritchet}
\end{equation}

$m_{\rm lim}$ is the magnitude at which completeness is 50%, and $\alpha_{\rm
cutoff}$ determines the steepness of the cutoff.  However, in the case of
crowded regions, where even at bright magnitudes the recovered fraction is flat
but $< 1$, better fits are obtained with a modified Pritchet function:

\begin{equation}
f^{mP}(m) = \frac{1}{2}  \left[ \gamma - \delta\ \frac{\alpha_{\rm cutoff} (m - m_{\rm lim})}{\sqrt{1 + \alpha_{\rm cutoff}^2(m - m_{\rm lim})^2}}  \right];
\label{eq:pritchmod}
\end{equation}

here, $\gamma$ is the asymptotic value of the fraction at bright
magnitudes, and $\delta$ is the fraction at the cutoff.

For each one of the galaxies, the quotient of recovered to added sources as a
function of output PSF magnitude was fit with equation <ref>,
separately for each region. The recovered fractions (dots) and their fits
(solid lines) versus  mag are displayed in Figure <ref>. It is
quite obvious that the regions within 0.5 $R_{25}$ are extremely affected by
crowding and confusion due to partially resolved sources.  This is true also
for NGC 4395, in spite of the absence of a conspicuous lack of recovered
sources in the central region.  The values of the fit parameters $m_{\rm lim}$,
$\alpha_{\rm cutoff}$, $\gamma$, and $\delta$  are shown in
Table <ref>.

Completeness tests. Top left: NGC 3368; top right:
NGC 4395; bottom left: NGC 4736; bottom right: NGC 4826.  Fits
(solid lines) to recovered fractions (dots) versus $K_s$ mag with
eq. <ref>. Colors (red, blue, green, gold) refer to
centermost ellipse, and inner, middle, and external annuli, respectively. Blue dotted lines indicate the 90% and 50% completeness values.

Completeness Fit Parameters

Galaxy   Region     $m_{\rm lim}$     $\alpha_{\rm cutoff}$    $\gamma$   $\delta$

$R_{25}$     $K_s$ AB mag

NGC 3368       0.0$-$0.5       22.89$\pm$0.09      2.22$\pm$0.71          0.69$\pm$0.05      0.64$\pm$0.06

0.5$-$1.0       22.98$\pm$0.03      2.24$\pm$0.20          0.94$\pm$0.20      0.96$\pm$0.20

1.0$-$1.4       22.69$\pm$0.05      2.72$\pm$0.83          0.99$\pm$0.83      1.03$\pm$0.83

1.4$-$1.7       22.65$\pm$0.02      3.68$\pm$0.66          0.97$\pm$0.66      1.03$\pm$0.66

NGC 4395      0.0$-$0.5       22.14$\pm$0.05      3.09$\pm$1.15          0.46$\pm$0.02      0.46$\pm$0.03

0.5$-$1.0       22.10$\pm$0.01      4.77$\pm$0.26          0.97$\pm$0.26      0.98$\pm$0.26

1.0$-$1.4       22.23$\pm$0.01      5.08$\pm$0.23          0.98$\pm$0.23      1.00$\pm$0.23

1.4$-$1.7       22.16$\pm$0.01      4.26$\pm$0.36          0.97$\pm$0.36      1.01$\pm$0.36

NGC 4736       0.0$-$0.5       22.52$\pm$0.23      0.90$\pm$0.45          0.50$\pm$0.08      0.62$\pm$0.14

0.5$-$1.0       22.63$\pm$0.03      2.72$\pm$0.32          0.95$\pm$0.32      0.92$\pm$0.32

1.0$-$1.4       22.73$\pm$0.05      2.36$\pm$0.43          0.98$\pm$0.43      0.98$\pm$0.43

1.4$-$1.7       22.73$\pm$0.01      5.01$\pm$0.41          0.99$\pm$0.41      0.98$\pm$0.41

NGC 4826       0.0$-$0.5       21.87$\pm$0.11      1.86$\pm$0.65          0.24$\pm$0.01      0.21$\pm$0.02

0.5$-$1.0       22.48$\pm$0.05      2.33$\pm$0.19          0.83$\pm$0.19      0.74$\pm$0.19

1.0$-$1.4       22.57$\pm$0.04      3.64$\pm$0.98          1.01$\pm$0.98      0.98$\pm$0.98

1.4$-$1.7       22.67$\pm$0.02      3.82$\pm$0.68          0.97$\pm$0.68      1.01$\pm$0.68

§ SELECTION OF GLOBULAR CLUSTER CANDIDATES

The main tool used for the selection of GCCs is the ($u^*\ -\ i^\prime$) versus
($i^\prime\ -\ K_s$) color-color diagram [Muñoz et al., 2014].  The combination of
optical and near infrared (NIR) data in this parameter space constitutes the
most powerful photometric-only method for the selection of a clean sample of GC
candidates.  It probes simultaneously the main-sequence turnoff, and the red
giant and horizontal branches of stellar populations.  In this plane, GCs
occupy a region that is distinct from the areas inhabited by background
galaxies, on the one hand, and foreground Galactic stars, on the other. Objects
in these various regions not only have different stellar populations, but light
concentration parameters that roughly distinguish GCs, which appear marginally
resolved in the optical, from both point and extended sources.

Figure <ref> shows the  diagrams for all the sources
detected in the images of NGC 3368 (top row), NGC 4395 (second row),
NGC 4736 (third row) and NGC 4826 (bottom row).  All the data have been
corrected for Galactic extinction, using the values from Schlafly & Finkbeiner, 2011 given
for the Sloan Digital Sky Survey (SDSS) filters by the NASA Extragalactic
Database,[The NASA/IPA Extragalactic Database (NED) is operated by the
Jet Propulsion Laboratory, California Institute of Technology, under contract
with the National Aeronautics and Space Administration.]respectively,
for NGC 3368, $A_u = 0.107$, $A_g = 0.083$, $A_i = 0.043$, $A_{Ks} = 0.005$;
for NGC 4395, $A_u = 0.073$, $A_g = 0.057$, $A_i = 0.029$, $A_{Ks} = 0.008$;
for NGC 4736, $A_u = 0.075$, $A_g = 0.059$, $A_r = 0.041$, $A_i = 0.030$, $A_{Ks} = 0.005$;
for NGC 4826, $A_u = 0.175$, $A_g = 0.137$, $A_r = 0.095$, $A_i = 0.070$, $A_{Ks} = 0.012$.

The sources are color-coded by their light concentration in the -band, as
gauged by the FWHM (left column), and 100$\times$ the dimensionless SExtractor
parameter SPREAD_MODEL (right column).[For each object, the
SPREAD_MODEL value results from the comparison between its best fitting PSF,
and the convolution of such PSF with an exponential disk whose scale length is
equal to the FWHM of said PSF [Desai et al., 2012].] In these panels, reddish-brown
sources are extended, blue sources are pointlike, and cyan sources are
marginally resolved. The color limits have been chosen to highlight these
differences. Objects in the reddish-brown cloud are mostly galaxies, while
those in the blue band are main-sequence stars in our Galaxy.

Morphology variations across the  diagram. Top: NGC 3368;
second row: NGC 4395; third row: NGC 4736; bottom:
NGC 4826. Indicators of shape in the -band are coded as shown by the color
bars: FWHM (left) and 100$\times$SPREAD_MODEL (right).

In between the galaxy cloud and the stellar band, the selection region for GCCs
is indicated by the gray contour.  The selection region was determined using
data of the giant elliptical M 87 [Muñoz et al., 2014, Powalka et al., 2016]; it has the shape of a
drop or pennon, and contains more than 2000 spectroscopically confirmed GCs in
that galaxy. The technique has already been applied to the spiral megamaser
prototype NGC 4258 [González-Lópezlira et al., 2017], and its effectiveness in spirals assessed
by González-Lópezlira et al., 2019.  One difference with the observations of M 87 and our
previous work on NGC 4258 is that a new  filter was used for the
observations presented in this paper.  As a consequence, the  luminosity
function turn-over (LFTO) is slightly fainter, and the GC selection region
shifted 0.24 mag to the red in ( - ).

The spectroscopic observations of the NGC 4258 GCCs reported by González-Lópezlira et al., 2019
found no false negatives, and a sample contamination of, conservatively, 7/23
or 30%, fully consistent with the 50–100 contaminants found by Powalka et al., 2016
for the GCCs of M 87, in a FOV 9 times larger. Like M 87 and NGC 4258, the
fields of NGC 3368, NGC 4395, NGC 4736 and NGC 4826 lie at high Galactic
latitudes, respectively, $b=57\fdg0$, $b=81\fdg5$, $b=76\fdg0$, and $b

González-Lópezlira et al., 2019 also determined that the contaminants could be reduced by
setting the light concentration parameters SPREAD_MODEL $\leqslant$ 0.015 and
FWHM $\leqslant$ 29 pc ($0\farcs58$ for NGC 3368, $1\farcs40$ for NGC 4395,
$1\farcs20$ for NGC 4736 and $0\farcs82$ for NGC 4826).  Moreover, we only
kept objects within $\pm\ 3\sigma$ of the expected GC LFTO magnitude in every
filter, except at $K_s$, since González-Lópezlira et al., 2019 realized that objects more than
1.7$\times \sigma$ brighter than the turnoff at that wavelength were potential
contaminants. Other restrictions were an axis ratio $\geqslant$ 0.7 and an
error MAGERR_PSF $< 0.2$ mag, both in the -band, as well as FLAGS = 0 in
all bands.  The standard deviation of the GCLF depends on galaxy luminosity
[Jordán et al., 2007]; based on their eq. 18, we have assumed 0.9 for NGC 4395, 1.1
for NGC 4826, and 1.2 for NGC 3368 and NGC 4736.[The MW GCLF has
$\sigma_V =$ 1.15 $\pm$ 0.1 [Jordán et al., 2007], also consistent with their eq.18.] Values of the LFTO in the optical were derived by combining the absolute
LFTO magnitude in the $g$ band, $M^0_g = -7.2$ mag [Jordán et al., 2007], with the
(AB) colors given in the MegaCam filter system by <cit.> models for an SSP with $Z = 0.0004$, an age of 12 Gyr, and a
Chabrier initial mass function <cit.>. The LFTO in the $V$-band
would be $M^0_V = -7.4$ mag.  The LFTO magnitude in the -band was taken from
Wang et al., 2014.  TO absolute magnitudes, observed LFTO magnitudes, and
magnitude ranges of the GCLF are shown in Table <ref>.

Turnover magnitudes and GCLF ranges
2cNGC 3368    2cNGC 4395    2cNGC 4736    2cNGC 4826

Filter     $M^0$     $m^0_{\rm TO}$     $m^0_{\rm TO} \pm\ 3\sigma$     $m^0_{\rm TO}$     $m^0_{\rm TO} \pm\ 3\sigma$    $m^0_{\rm TO}$     $m^0_{\rm TO} \pm\ 3\sigma$     $m^0_{\rm TO}$     $m^0_{\rm TO} \pm\ 3\sigma$

mag     mag     mag     mag    mag      mag     mag     mag    mag

-5.96      24.12     $20.52  < m^0_{u^*}      <27.72$     22.21       $19.51 < m^0_{u^*}      < 24.91$   22.53      $18.93 < m^0_{u^*}      < 26.13$     23.36      $20.06 < m^0_{u^*} < 26.66$

-7.20       22.88     $19.28  < m^0_{g^\prime} <26.48$     20.97       $18.27 < m^0_{g^\prime} < 23.67$   21.29      $17.69 < m^0_{g^\prime} < 24.89$    22.12      $18.82 < m^0_{g^\prime} < 25.42$

-7.69       22.40     $18.80  < m^0_{r^\prime} <26.00$     20.48       $17.78 < m^0_{r^\prime} < 23.18$   20.80      $17.20 < m^0_{r^\prime} < 24.40$    21.63     $18.33 < m^0_{r^\prime} < 24.93$

-7.92       22.16     $18.56  < m^0_{i^\prime} <25.76$     20.25       $17.55 < m^0_{i^\prime} < 22.95$   20.57      $16.97 < m^0_{i^\prime} < 24.17$    21.40     $18.10 < m^0_{i^\prime} < 24.70$

-8.1        22.0      $19.9   < m^0_{Ks}       <25.6$     20.1         $18.5  < m^0_{Ks}       < 22.8$   20.4        $18.4   < m^0_{Ks}       < 24.0$    21.2     $19.3 < m^0_{Ks} < 24.5$

Figures <ref>, <ref>, <ref>,
and <ref>, left column, show the  diagram, FWHM (top
rows), and 100$\times$ structural parameter SPREAD_MODEL (bottom rows) of
sources in and closely around the GCC selection region (gray contour) for
NGC 3368, NGC 4395, NGC 4736 and NGC 4826, respectively. The GCC final
samples for the galaxies are displayed in the right column.

( - ) versus ( - ) colors and structural parameters of
sources in and closely around selection region of NGC 3368.  Top: FWHM;
bottom: 100 $\times$ SPREAD_MODEL.  Left column: all sources; right column: final sample.  The gray contour outlines the GCC selection
region.  GCCs appear with shades of blue and cyan in both compactness

( - ) versus ( - ) colors and structural parameters of
sources in and closely around selection region of NGC 4395.  Top: FWHM;
bottom: 100 $\times$ SPREAD_MODEL.  Left column: all sources; right column: final sample.  Symbols as in Fig. <ref>.

( - ) versus ( - ) colors and structural parameters of
sources in and closely around selection region of NGC 4736.  Top: FWHM;
bottom: 100 $\times$ SPREAD_MODEL.  Left column: all sources; right column: final sample.  Symbols as in Fig. <ref>.

( - ) versus ( - ) colors and structural parameters of
sources in and closely around selection region of NGC 4826.  Top: FWHM;
bottom: 100 $\times$ SPREAD_MODEL.  Left column: all sources; right column: final sample.  Symbols as in Fig. <ref>.

The spatial distributions of the GCC samples are shown in
Figure <ref>.  The members of the samples are displayed as green
circles on the -band images of NGC 3368 (top left), NGC 4395 (top right),
NGC 4736 (bottom left), and NGC 4826 (bottom right). The dark blue ellipses
outline the regions where source confusion is highest, inferred from the
completeness simulations and the dearth of GCC detections. This boundary is
located at 0.375 $R_{25}$ for NGC 3368, 0.25 $R_{25}$ for NGC 4395, and
0.5$R_{25}$ for both NGC 4736 and NGC 4826.  The cyan circles indicate

Spatial distributions of the GCC samples. Top left: NGC 3368;
top right: NGC 4395; bottom left: NGC 4736; bottom right:
NGC 4826. The GCCs are shown as green filled circles. Dark blue ellipse:
boundary of areas where source confusion is highest (see text); solid cyan
line: $R_{25}$.

Given the distances to the galaxies, as well as the resolution of the MegaCam
data (1 MegaCam pixel = 0$\farcs$186, i.e., $\simeq$ 3.9 pc, 4.5 pc, 6.6 pc,
and 9.4 pc, respectively, for NGC 4395, NGC 4736, NGC 4826, and NGC 3368),
we have also been able to calculate the half-light radii, $r_e$, in the
-band of all the candidates.  To this end, we used the program ishape
[Larsen, 1999] in the BAOLAB[<http://baolab.astroduo.org>.]
software package.  ishape measures the size of a compact source by
comparing its observed light profile with models generated by convolving
different analytical profiles with the image PSF.  For data with S/N $\ga$ 30,
ishape can measure $r_{\rm eff}$ reliably down to $\sim$ 0.1 the PSF
FWHM, or $\sim$ 1.7 pc for NGC 4736, 2.3 pc for NGC 4826, 2.4 pc for
NGC 3368, and 2.5 pc for NGC 4395; objects smaller than this are effectively
unresolved [Larsen, 1999, Harris et al., 2009].  We fitted all the sources with King
[King, 1962, King, 1966] and Sérsic [Sérsic, 1963] profiles.  For KING profiles,
we tried fixed concentration indices ($C \equiv r_{\rm tidal}/r_{\rm core}$) of
30 and 100; in the case of Sersic, we chose a fixed exponent of
2.[According to Turner et al., 2012, based on the measurements of
Graham & Spitler, 2009 and Seth et al., 2010, a Sersic parameter of $\sim$2 represents
Galactic GCs and nuclear clusters well.] For each object, the model providing
the fit to the data with the smallest $\chi^2$ residuals was then used to
derive the effective radius, by applying the conversion factors between $r_{\rm
eff}$ and FWHM given in the ishape manual.  For each galaxy, Tables
<ref>, <ref>, <ref>,
and <ref> list shape parameters in the -band: FWHM,
100$\times$SPREAD_MODEL,  $r_e$, the type and index of the best-fitting
profile, the SNR of the data, and the $\chi^2$ of the fit.

We present in Figure  <ref> a few exposures of  GCCs in
NGC 3368 and NGC 4736 that we were able to find in the Hubble Space Telescope
(HST) Legacy Archive; their visual appearance is consistent with their
classification.  J104635+114912 was identified in the Advance Camera for
Surveys (ACS) frame hst_10433_02_acs_wfc_f555w.fits (filter F555W,
proposal GO 10433, P.I. B. Madore); J104651+114819, in the Wide Field Camera
3 (WFC3) image hst_13364_07_wfc3_uvis_f555w.fits (F555W, proposal GO
13364, P.I. D. Calzetti).  All the NGC 4736 images were acquired with the
ACS.  J125039+410913 and J125045+410950 were identified in the frame
hst_10402_09_acs_wfc_f555w_drz.fits (proposal GO 10402, P.I. R.Chandar); J125131+410849, in hst_9480_u8_acs_wfc_f775w_drz.fits (filter
F775W, proposal GO/PAR 9480, P.I. J. Rhodes).

Grayscales of GC candidates, from HST images. Top, left to
right: J104635+114912 and J104651+114819, in NGC 3368.  Bottom:
J125039+410913, J125045+410950, and J125131+410849, in NGC 4736.

§ PROPERTIES OF THE GC SYSTEMS

§.§ Color distribution

The distributions of our final samples of GCCs in the colors ( - ),
( - ), ( - ), (- ), (- ), and (- ), fit
with two heteroscedastic Gaussians, are shown in
Figure <ref>. The means and dispersions of the Gaussian
fits are presented in Table <ref>. The colors of the individual
candidates, shown in
tables <ref>, <ref>, <ref>
and <ref>, have been corrected for foreground extinction in
the Milky Way (MW), as already mentioned.

Color distributions of GC systems. Top: NGC 3368 GCCs; second row:
NGC 4395; third row: NGC 4736; bottom: NGC 4826.  From left to
right: ($u^\ast\ - g^\prime$), ($u^\ast\ - i^\prime$), ($g^\prime - r^\prime$),
($g^\prime - i^\prime$), ($g^\prime - K_s$), ($r^\prime - i^\prime$).

González-Lópezlira et al., 2017 found that the colors of GCs and GCCs in the Milky Way, M31, and
NGC 4258 were remarkably consistent, once internal extinction was taken into
account. The same is true for NGC 3368, NGC 4395, NGC 4736 and NGC 4826.
Their GCC colors have not been corrected for internal extinction, and the bluer
Gaussians fit to their color distributions are bracketed by the values of
single Gaussian fits to the GC system of M 31, with and without such
correction. Both these and the colors of the MW GC system are shown in
Table <ref>.

Single Gaussian Fits to the Colors of the GC Systems of the MW and M 31

System     2c$(u^\ast - g')$     2c$(u^\ast - g')_0$     2c$(u^\ast - i')$     2c$(u^\ast - i')_0$
2c$(g' - r')$     2c$(g' - r')_0$     2c$(g' - i')$     2c$(g' - i')_0$
2c$(g' - K_s)$     2c$(g' - K_s)_0$     2c$(r' - i')$     2c$(r' - i')_0$

$\mu$     $\sigma$     $\mu$     $\sigma$     $\mu$     $\sigma$     $\mu$     $\sigma$
$\mu$     $\sigma$     $\mu$     $\sigma$     $\mu$     $\sigma$     $\mu$     $\sigma$
$\mu$     $\sigma$     $\mu$     $\sigma$     $\mu$     $\sigma$     $\mu$     $\sigma$

M 31                       1.30     0.26     1.05              2.29     0.53     1.72              0.64     0.17     0.45              1.00     0.28     0.68              1.24     0.61     0.56                0.36     0.12     0.23

MW                                            1.01     0.15                       1.67     0.26                       0.45     0.09                       0.66     0.12                                                             0.21     0.03

We remind the reader that colors using the  new  filter are $\sim$ 0.2 mag
redder. The colors of the M 31 GCs were transformed by González-Lópezlira et al., 2017 to the
previous CFHT system.

§.§ Luminosity functions

The -band luminosity functions of the final samples of GCCs of the four
galaxies are shown in Figure <ref>.  Given the exclusion of the
regions close to the centers of the galaxies, i.e., the areas most affected by
crowding and confusion, the corrections for incompleteness would be barely
significant.  The solid red lines are Gaussians with the expected
characteristics of the luminosity functions: mean = 22.0 mag, $\sigma = 1.2$
mag for NGC 3368; mean = 20.1 mag, $\sigma = 0.9$ mag for NGC 4395; mean =
20.4 mag, $\sigma = 1.2$ mag for NGC 4736; mean = 21.2 mag, $\sigma = 1.1$ mag
for NGC 4826.  The curves have been scaled such that their integrals are equal
to the expected GCCs for each galaxy, respectively, 35, 15, 46, and 69, for
NGC 3368, NGC 4395, NGC 4736, and NGC 4826 (see Section <ref>
and Table <ref>.)

We also take the opportunity here to stress that we do not use the
extrapolation over the luminosity function to derive the total number of
globular clusters in the system. We perform the completeness and GCLF
corrections implicitly, as we explain in the next section.

-band luminosity functions of the GCCs. Top left: NGC 3368;
top right: NGC 4395; bottom left: NGC 4736; bottom right:
NGC 4826.  Histograms: numbers of detected objects; solid red
lines: Expected Gaussian GCLFs (see Table <ref> and
Section <ref>.

MW GC system, at the distance and orientation of studied galaxies, projected
coordinates $+X_{\rm proj},+Y_{\rm proj}$. Top: NGC 3368; bottom:
NGC 4395. Left: 3-D rotation about Galactic $X^\prime$ axis before
projection on the plane of the sky.  Right: 3-D rotation about Galactic
$Y$ axis. Solid black line: mask; red crosses: sources visible in
the WIRCam FOV; yellow crosses: recovered in the simulation, but outside
of the WIRCam FOV; black crosses: masked-out GCs.

MW GC system, at the distance and orientation of studied galaxies, projected
coordinates $+X_{\rm proj},+Y_{\rm proj}$. Top: NGC 4736; bottom:
NGC 4826.  Symbols as in Figure <ref>.

§ TOTAL NUMBER OF GLOBULAR CLUSTERS

§.§ Correction for incomplete spatial coverage

Given the small number of clusters detected, and the lack of constraints in the
centermost regions of the galaxies due to confusion, here we derive the total
number of GCs for each galaxy through a comparison with the GC system of the
Milky Way. The procedure was introduced by Kissler-Patig et al., 1999, who applied it to
edge-on spirals; its results are consistent with those from a fit to the radial
number density, when both methods can be applied to the same object
[Kissler-Patig et al., 1999, Goudfrooij et al., 2003].  González-Lópezlira et al., 2017 adapted it for use on any galaxy,
regardless of orientation. With this method, the correction for incomplete
spatial coverage implicitly takes care of the completeness correction and the
extrapolation over the LF.

In brief, the method consists in estimating the number of GCs that one would
detect if the MW were at the distance and orientation of the galaxy of
interest, and were observed with the same instrument and to the same depth. By
comparison, the total number of GCs would be:

\begin{equation}
N_{\rm GC} = N_{\rm GC} {\rm (MW)} \frac{N_{\rm obs}}{N_{\rm FOV}}.
\label{eq:kissler}
\end{equation}

$N_{\rm GC}$(MW) is the total number of clusters in the MW, including objects
invisible to us behind the bulge; $N_{\rm obs}$ is the number of clusters
observed in the target galaxy, and $N_{\rm FOV}$ is the number of objects
recovered in the artificial observation of the MW.  We take $N_{\rm GC}$ (Milky
Way) = 160$\pm$10 [Harris et al., 2014].

In order to estimate $N_{\rm FOV}$ we use the catalog by Harris 1996 , 2010, which
provides  $X,Y,Z$ coordinates for 144 MW GCs. The origin of this coordinate
system is at the position of the Sun; the $X$ axis points towards the Galactic
center, the $Y$ axis in the direction of Galactic rotation, and $Z$ is
perpendicular to the Galactic plane. We define a new coordinate system,
$X^{\prime}YZ$, with origin in the Galactic center, assuming a Galactocentric
distance for the Sun R${_0}$ = 8.34 kpc [Reid et al., 2014].  For galaxies that are
not edge-on, we first rotate the MW GC system in 3-D by the inclination angle
$i$ of the target galaxy (i.e., 46$\fdg$2 for NGC 3368, 33$\fdg$7 for
NGC 4395, 35$\fdg$6 for NGC 4736 and 57$\fdg$5 for NGC 4826) with respect to
either the $X^\prime$ or $Y$ axis, then project it on the plane of the sky; the
rotated pairs ($X^\prime_{{\rm rot},X^\prime}, Y_{{\rm rot},X^\prime}$),
($X^\prime_{{\rm rot},Y}, Y_{{\rm rot},Y}$), where the subscript indicates the
Galactic axis of 3-D rotation, are directly the projected coordinates. The
easiest way to place the projected system on the WIRCam FOV is to then apply a
rotation around the line of sight equal to the PA of the target galaxy
(5$\degr$ for NGC 3368, 147$\degr$ for NGC 4395, and 115$\degr$ for both
NGC 4736 and NGC 4826, always North through East).

To ensure a fair comparison with the MW, we then mask out elliptical regions in
the centers of the galaxies.  For NGC 4395, NGC 4736, and NGC 4826, they
have the same axis ratios [i.e., cos($i$)] and PA as the galaxies. Their
semimajor axes are equal to, respectively, 98$\farcs9$, 168$\farcs3$, and
150$\arcsec$, or 0.25 $R_{25}$ for NGC 4395, and 0.5 $R_{25}$ for both
NGC 4736 and NGC 4826. For NGC 3368, we exclude an ellipse with axis ratio
0.69 and PA=155$\degr$, like the region in the galaxy with the highest surface
brightness, and semimajor axis equal to 85$\farcs6$, or 0.375 $R_{25}$
[de Vaucouleurs et al., 1991].  These are the areas most affected by confusion (see
Section <ref> and Figures <ref>
and <ref>).  We also need to set detection limits for these
simulated observations. To this end, we combine the real limiting magnitude of
the $K_s$ images, with the difference between the $K_s$ and $V$ bands TO
magnitudes; the results are $V$ = 22.5, 21.6, 23.4, and 23.1 mag, respectively,
for NGC 3368, NGC 4395, NGC 4736, and NGC 4826.  Finally, the $V$ mag of
each MW cluster is corrected for the Galactic extinction given in the
Harris 1996 , 2010 catalog, and subsequently dimmed by the foreground extinction in
the direction of each galaxy, i.e., 0.069 mag (NGC 3368), 0.047 mag
(NGC 4395), 0.049 mag (NGC 4736), and 0.113 mag (NGC 4826), respectively.

For every galaxy, there are four possible orientations of the FOV and mask that
preserve its alignment; they can be seen as mirror reflections of the projected
coordinates ($+X_{\rm proj},+Y_{\rm proj}$;$+X_{\rm proj},-Y_{\rm proj}$;
$-X_{\rm proj},+Y_{\rm proj}$; $-X_{\rm proj},-Y_{\rm proj}$, with two sets of
four pairs, i.e., one for each of the two possible rotation axes.)

We show in Figures <ref> and <ref> the results of the
artificial observations of the MW GC system, for the rotation around the
$X^\prime$ (left) and $Y$ (right) axes, respectively, and $+X_{\rm
proj},+Y_{\rm proj}$. Results for NGC 3368 and NGC 4395  are displayed,
respectively, in the top and bottom panels of Figure <ref>, whereas
outcomes for NGC 4736 and NGC 4826 are presented in the top and bottom panels
of Figure <ref>.  Angular distances in arcsec are measured relative
to the center of the galaxies, with the horizontal axis increasing in the
direction of decreasing RA.  The ellipses delineated with the black solid line
mark the outer edges of the masks; the red crosses are clusters visible in the
WIRCam FOV, whilst sources that have been masked out are represented by black

In the case of NGC 3368, for the 4 realizations rotating around the $X^\prime$
axis we detect, respectively, 50 ($+X_{\rm proj},+Y_{\rm proj}$), 50 ($+X_{\rm
proj},-Y_{\rm proj}$), 50 ($-X_{\rm proj},+Y_{\rm proj}$), and 50 ($-X_{\rm
proj},-Y_{\rm proj}$) sources; for rotating around the $Y$ axis, the numbers
are 46, 46, 46, 46.  With these numbers, there would be, on average, 48$\pm$2
simulated clusters visible in NGC 3368, and eq. <ref> gives a total
$N_{\rm GC} = 50\pm13$. This error is statistical only; it includes errors in
the assumed number of total GCs in the MW, Poisson errors in the observed
number of GCCs in NGC 3368, and Poisson errors in the number of simulated
clusters in the WIRCam FOV. To this error, we add potential systematics. Errors
in the distance to NGC 3368 result in uncertainties in the detection limiting
magnitude, and in the effective areas of the galaxy covered by the FOV and
excluded by the mask. For the given uncertainty in the distance to NGC 3368
($^{+1.1}_{-1.0}$ Mpc), $N_{\rm GC}$ would vary by +21/-14 if the random and
systematic errors are added in quadrature. Another potential concern is that
the MW and NGC 3368 likely have a different number of obscured clusters. To
account for this possibility, we include an additional uncertainty of 25%
($\pm$ 12). We end up with $N_{\rm GC} = 50^{+24}_{-19}$, or
50$\pm13^{+20}_{-13}$, with the first error statistical and the second
systematic (from distance and differential obscuration).

For NGC 4395, rotations around $X^\prime$ yield 80, 80, 80, 79; for the $Y$
axis, the results are: 73, 71, 75, 71. Therefore, the average simulated
clusters visible would be 76$\pm$4, and a total $N_{\rm GC} = 21\pm5$ from
eq. <ref>.  The distance uncertainty of $\pm$ 0.4 Mpc results in an
additional error of $\pm$1 in  $N_{\rm GC}$. Including a 25% ($\pm$6)
uncertainty for differential obscuration, the census for NGC 4395 is $N_{\rm
GC} = 21\pm8$, or 21$\pm5\pm6$.

For NGC 4736, we find, respectively, 62, 62, 62, and 61 for rotations around
$X^\prime$; for the $Y$ axis, the results are: 60, 60, 61, and 60. Hence,  the
average simulated clusters visible would be 61$\pm$1, and a total $N_{\rm GC} =
66\pm14$ from eq. <ref>.  The distance uncertainty of $\pm$ 0.4 Mpc
results in an additional error of +6/-3 in  $N_{\rm GC}$. Including a 25%
($\pm$16) uncertainty for differential obscuration, the tally for NGC 4736 is
$N_{\rm GC} = 66^{+22}_{-21}$, or 66$\pm 14^{+17}_{-16}$.

Finally, for NGC 4826, rotations around $X^\prime$ give 49, 49, 50, 49
detections; for the $Y$ axis, the results are: 48, 47, 47, 48. Accordingly, the
average simulated clusters visible would be 48$\pm$1, and a total $N_{\rm GC} =
99\pm19$ from eq. <ref>.  The distance uncertainty of $\pm$ 0.7 Mpc
results in an additional error of +20/-10 in  $N_{\rm GC}$. Including a 25%
($\pm$25) uncertainty for differential obscuration, the count for NGC 4826 is
$N_{\rm GC} = 99^{+37}_{33}$, or 99$\pm 19^{+32}_{21}$.

We make a final, downward, correction of 30% to these numbers, based on the
percentage contamination found for NGC 4258 by González-Lópezlira et al., 2019, from
spectroscopic observations of its GC candidates. Hence, we get $N_{\rm GC} =
35^{+20}_{-16}$ for NGC 3368, $N_{\rm GC} = 15\pm7$ for NGC 4395, $N_{\rm GC}
= 46\pm18$ for NGC 4736, and $N_{\rm GC} = 69^{+31}_{-28}$ for NGC 4826.

The correlation of $N_{\rm GC}$ with virial mass provides an additional sanity
check, at least for NGC 4736 and NGC 4826. From their rotation curves
(respectively, Jalocha et al. 2008 and Saburova et al. 2009), both galaxies
are quite light, with total halo masses $\sim$ 4$\times 10^{10} M_\odot$; their
log $N_{\rm GC} \sim $ 1.7 is fully consistent with the larger dispersion of
the correlation at low masses (see Burkert & Forbes 2020, their Figure
1).[We note here that, from their analysis of the NGC 4736 rotation
curve, Jałocha et al., 2008 conclude that it violates the sphericity condition at
large radii (i.e., mass ceases to increase with distance). Hence, the galaxy
would not have a massive spherical halo and its mass distribution would be best
fit with a disk model.]

§ SPECIFIC FREQUENCY, AND THE $N_{\RM GC}$ VS. $M_\BULLET$ RELATION

From the distance to NGC 3368 and the values of $B_{\rm T,0} = 9.80\pm 15$ mag
and ($B\ - V$)$_{\rm T,0} = 0.79 \pm 0.01$  given in de Vaucouleurs et al., 1991, we find $M_V
= -21.08 \pm 0.36$ mag, and a specific frequency $S_N = N_{\rm GC} \times
10^{0.4 \times [M_V + 15]} = 0.13 \pm 0.06$, if we consider random errors only,
and $S_N = 0.13\pm 0.07$, if we include the uncertainty in the number of
obscured clusters.[The systematic error in the distance basically
cancels out, because the absolute magnitude of the galaxy also changes.] For
NGC 4395, de Vaucouleurs et al., 1991 report $B_{\rm T,0} = 10.57 \pm 0.54$ mag and ($B\ -
V$)$_{\rm T,0} = 0.46 \pm 0.08$ mag; this results in $M_V = -18.06 \pm 0.59$
mag and $S_N = 0.90\pm 0.55$, with statistical errors only, and $S_N = 0.90 \pm
0.65$ if the uncertainty in the number of obscured clusters is included.  For
NGC 4736, the values are $B_{\rm T,0} = 8.75 \pm 0.13$ mag and ($B\ -
V$)$_{\rm T,0} = 0.72\pm0.01$ mag, from which $M_V = -20.46\pm 0.21$ mag, and
$S_N = 0.30 \pm 0.10$, with random errors only, and $S_N = 0.30 \pm 0.13$, if
we include systematics.  For NGC 4826, de Vaucouleurs et al., 1991 give $B_{\rm T,0} = 8.82
\pm 0.24$ mag and ($B\ - V$)$_{\rm T,0} = 0.71\pm0.01$ mag; $M_V = -21.21 \pm
0.32$ mag, and $S_N = 0.23\pm0.09$ or $S_N = 0.23 \pm 0.11$, respectively,
without and with the uncertainty in the number of obscured clusters. For
comparison, the Milky Way has $S_N = 0.5\pm0.1$ [Ashman & Zepf, 1998].

Before placing our four galaxies on the $N_{\rm GC}$ – $M_\bullet$
correlation, we redetermine it in the Appendix for the sample in
Harris et al., 2014, plus NGC 4258 and the spirals in this paper, albeit with
updated galaxy classifications from Sahu et al., 2019.

Figures <ref> and <ref>, and Table <ref>
display the Bivariate Correlated Errors and Intrinsic Scatter
<cit.> fits for, respectively, ellipticals and spirals. The
black solid lines represents the best fit, and the translucent gray lines
illustrate the range of possible solutions. Each gray line corresponds to a
realization of the fit, obtained by adding a random variable to each the slope
$\beta$ and the intercept $\alpha$ of the best fit, drawn from the normal
distributions of errors,  $\sigma_\beta$ and $\sigma_\alpha$. Since 1,000
realizations were obtained for each fit, roughly 50 gray lines ($5\%$ of the
total 1,000) belong to fits separated by $2\sigma$ or more from the best
values.  In the figures, ellipticals appear as solid red circles and
lenticulars as solid dark green ones; the spirals M 31, M 81, and M 104 are
the blue crosses. The locations of the MW, NGC 253, NGC 3378, NGC 4258,
NGC 4395, NGC 4736, and NGC 4826 are shown as solid stars, respectively,
blue, light olive, violet, purple, light brown, dark brown, and light blue.  We
correct the position of NGC 253, whose number of GC had been read erroneously
by Harris et al., 2014. Indeed, Olsen et al., 2004 report $N_{\rm GC} = 21$, but
with a completeness of 0.21, i.e., they really implied $N_{\rm GC} \sim 100$.
They also give a specific frequency in the $K$-band $S_{N,K} = N_{\rm GC}
\times 10^{0.4 \times [M_K + 18]} = 0.30 \pm 0.07$, or $N_{\rm GC} = 92\pm21$.
Previously, Blecha, 1986 had found $N_{\rm GC} = 80 \pm 40$ for NGC 253.

The Figures show $M_\bullet$ versus $N_{\rm GC}$ in the top row; from left to
right, fits without extra “cosmic" scatter, fits with additional $\epsilon_y$,
and fits with additional intrinsic scatter in both $x$ and $y$ ( see Appendix).
The bottom row illustrates $N_{\rm GC}$ versus $M_\bullet$, only for fits with
additional $\epsilon_y$, which are not symmetrical.  We find that the slope of
the correlation is roughly 60% steeper for spirals than for ellipticals (see
Table <ref>).

§ THE $M_\BULLET$ VERSUS $M_\AST$ CORRELATION FROM $N_{\RM GC}$

In order to investigate whether the discrepant slopes of the $M_\bullet$ versus $N_{\rm
GC}$ correlations for elliptical and spiral galaxies reflect the slopes between
$M_\bullet$ and $M_\ast$, we derive $M_\ast$ from its correlation with $N_{\rm GC}$, for the ellipticals
and LTGs in our sample. Hudson et al., 2014 have obtained log $M_{\rm GC}$ versus
log $M_\ast$ for the 422 galaxies (248 ellipticals, 93 lenticulars, and 81
spirals and irregulars) in the Harris et al., 2013 sample, where $M_{\rm GC}$ is
the total mass of the GC system of a galaxy. In actuality, $N_{\rm GC}$ is the
observable, while $M_{\rm GC}$ has the advantage of a smaller sensitivity to
undetected faint GCs, since 99% of the total GC system mass is roughly
contained in the clusters brighter than 1 mag below the GCLF turnover
[González-Lópezlira et al., 2017].[The trends of $N_{\rm GC}$ with total $K$-band
luminosity, a good tracer of mass, are remarkably similar for all galaxy
types.] Hudson et al., 2014 assume a relation between total number and total mass of
GC log $N_{\rm GC} = 0.90 \times M_{\rm GC}$ - 4.65.  The relations between log
$M_{\rm GC}$ or $N_{\rm GC}$ and $M_\ast$ are broken power laws, with log
$M_{\rm GC} = 0.53\ \times$ log $M_\ast$ + 2.00, log $M_\ast = 2.11\ \times$
log $N_{\rm GC}$ + 5.99, for lower galaxy masses, and $M_{\rm GC} = 1.60
\times$ log $M_\ast$ - 9.22, log $M_\ast =  0.69\ \times$ log $N_{\rm GC}$ +
8.96, for higher galaxy masses, with the inflection point at  log
$M_\ast/M_\odot \sim$ 10.4, log $N_{\rm GC} \sim$  2.1.

Figure <ref> and Table <ref> show the
BCES fits to $M_\bullet$ versus $M_\ast$ for, respectively, ellipticals (red-dashed and pink translucent lines) and spirals (black solid
and gray translucent lines). Their slopes are, roughly, 1.47$\pm$0.16 and
1.22$\pm$15, respectively, i.e., the same within the errors, with the “best"
fits as close or closer than those found by Reines & Volonteri, 2015, also less steep for
spirals than for ellipticals, and with an offset of $\sim$ 0.2-0.3 dex at log
$M_\ast/M_\odot$ = 10. A fit to both ellipticals and spirals together
(Figure <ref>, Table <ref>) yields a slope
$\sim 1.5\pm 0.15$, and a cosmic scatter $\epsilon \sim$ 0.4, of the order of
half that found by van den Bosch, 2016.

§ DISCUSSION AND CONCLUSIONS

We have now successfully applied the  diagram GC selection technique
[Muñoz et al., 2014] to five spiral galaxies: NGC 4258 [González-Lópezlira et al., 2017], NGC 3368,
NGC 4395, NGC 4736, and NGC 4826 (this work). We complement the  plot
with GCC structural parameters; this combination of techniques constitutes a
very efficient tool to study GC systems of nearby galaxies. The sample of
spirals with measurements of both the GC system and the SMBH mass has now been
increased to 10, or 100% more than before we embarked on this project. Our
full spiral working sample comprises all galaxies observable from the northern
hemisphere with a direct measurement of their central SMBH mass, and within 16
Mpc, regardless of orientation in the sky, i.e., a total of 9; so far, we have
analyzed the 5 for which we have complete , , and  data. We have
also taken the opportunity to correct the position of NGC 253 in the
$M_\bullet$ versus $N_{\rm GC}$ diagram.

The spiral galaxies for which we have determined $N_{\rm GC}$, as well as
NGC 253, seem to have undergone few major mergers.  Like the MW, they all
belong to small groups, i.e., Leo I, Canes I, Canes II, and Sculptor. They also
have pseudobulges, except for NGC 4395, which is a bulgeless, purely disk
galaxy. Additionally, the GC systems of both NGC 4258 and NGC 253 seem to lie
in rotating disks [González-Lópezlira et al., 2019, Olsen et al., 2004].  Together with simulations
<cit.> and observations of clumpy star-forming
disks at $z \sim 2$ <cit.>, the discovery of GC disks in
the local universe supports the idea that GCs probably formed in disks, and
came to populate halos through multiple mergers

We have investigated the correlation between $M_\bullet$ and $N_{\rm GC}$, for
both ellipticals and LTG.  We find log $M_\bullet \propto$ (1.01
$\pm$ 0.13) log $N_{\rm GC}$ for ellipticals, and log $M_\bullet \propto$ (1.64 $\pm$
0.24) log $N_{\rm GC}$ for LTG, in both cases including cosmic scatter in both variables.
Hence, they are not consistent with each other,
and the slope of the correlation for LTG, in contrast with that for ellipticals, is not
linear. This, together with the likely scarcity of major interactions for LTG,
may argue in favor of statistical convergence through mergers as the cause of
the correlation in elliptical galaxies.

On the other hand, we have derived $M_\ast$, for the ellipticals and LTG in our sample,
from its relation with $N_{\rm GC}$, and measured the correlations with $M_\bullet$. Far from
mirroring the discrepant slopes of the $M_\bullet$ versus $N_{\rm GC}$ fits, they are
quite similar, i.e., $M_\bullet \propto$ (1.48 $\pm$
0.18) log $M_\ast$ for ellipticals, and $M_\bullet \propto$ (1.21 $\pm$ 0.16) log
$M_\ast$ for LTG, with a small offset of $\sim$ 0.2-0.3 dex at log
$M_\ast/M_\odot = 10$, in both cases including cosmic scatter in both variables.  These
results are in line with those obtained by Reines & Volonteri, 2015
and Simmons et al., 2017, and argue for a mostly non-merger driven $M_\bullet$
growth, and galaxy-black hole co-evolution through secular processes, like calm
accretion and feedback.  [Simmons et al., 2017, Martin et al., 2018, Smethurst et al., 2019].

Researching GC systems and scaling relations in spirals is very important.
While these relations in ellipticals seem to be showing the end result of star
formation and galaxy assembly, in lower mass galaxies we may be witnessing the
fingerprints of the individual steps of the process.

R.A.G.L. acknowledges the financial support of DGAPA, UNAM, project IN108518.
R.A.G.L. and L.L.N. acknowledge the support of CONACYT, Mexico, project
A1-S-8263.  L.L. acknowledges the support of DGAPA PAPIIT grants IN112417 and
IN112820, CONACYT-AEM grant 275201, and CONACYT-CF grant 263356.  Y.O.-B.
acknowledges support from FONDECYT Postdoctoral Fellowship project No.3210442.  We would like to thank the anonymous referee, for their pertinent
criticisms and suggestions, and Daniel Díaz-González, for his help with
the routine we used to calculate two-Gaussian fits.

This research has made use of the VizieR catalogue access tool, CDS,
Strasbourg, France (DOI: 10.26093/cds/vizier). The original description of the
VizieR service was published in A&AS 143, 23.

daophot [Stetson, 1987, Stetson, 2011],
galsim [Rowe et al., 2015],
iraf [Tody, 1986, Tody, 1993],
ishape [Larsen, 1999],
matplotlib [Hunter, 2007],
numpy [Harris et al., 2020],
PSFEx [Bertin, 2011],
saoimage [Joye & Mandel, 2003],
scipy [Virtanen et al., 2020],
SExtractor [Bertin & Arnouts, 1996],
supermongo [Lupton & Monger, 1991]

We determine the $M_\bullet$ versus $N_{\rm GC}$ correlation for the sample in
Table <ref>. We use the galaxy types in [Sahu et al., 2019] (Col. 3),
which are based on the classification grid of [Graham, 2019].  We obtain
separate fits for ellipticals (E), lenticulars (L) and spirals (S), with the
FITEXY regression routines [Press et al., 1992], as modified by Tremaine et al., 2002 and
Novak et al., 2006 to allow for the addition of intrinsic (or “cosmic") scatter in
the parameters.  Summarizing, if data pairs ($x_i,y_i$), with individual
measurement uncertainties ($\sigma_{x,i},\sigma_{y,i}$) are related by the
linear equation $y = \alpha + \beta x$, the expression to minimize is

\begin{equation}
\chi^2 = \Sigma \frac{y_i - \alpha - \beta (x_i - \langle x \rangle)^2}{(\sigma^2_{y,i} + \epsilon^2_y) + \beta^2(\sigma^2_{x,i} + \epsilon^2_x)},
\label{eq:chisq}
\end{equation}

where $\langle x \rangle \equiv N^{-1} \Sigma^N_{i=1} x_i$ is the sample mean,
and $\epsilon_x$ and $\epsilon_y$ represent the additional intrinsic scatter
over the formal measurement errors.  If additional “cosmic" scatter indeed
exists, as implied by a reduced $\chi^2_\nu > 1$, the epsilons can be increased
until $\chi^2_\nu = 1$.  We searched for solutions both assuming $\epsilon_x =
0$, with all additional scattering in $y$, and distributing the potential
supplemental scattering between $\epsilon_x$ and $\epsilon_y$. In the absence
of other constraints, when epsilons are added simultaneously to the $M_\bullet$
and $N_{\rm GC}$ values, their squares are kept in the same ratio observed when
only $\epsilon_y$ is appended to either variable.

Correlation solutions with the FITEXY routines are shown in
Table <ref>.  We also carry out fits with the Bivariate Correlated
Errors and Intrinsic Scatter (BCES) method by Akritas & Bershady, 1996, with results for
the bisector line of the forward ($Y|X$) and inverse ($X|Y$) regressions
presented in Table <ref>.  The solutions have the form $Y = \alpha +
\beta (X - \langle x \rangle)$.  To note: lenticular galaxies do show a

We perform analogous fits to $M_\bullet$ versus $M_\ast$, for E, S, and E+S
galaxies. The solutions are listed in Tables <ref>
and <ref>.

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