rotate DTBLrotate

11institutetext:  Department of Physics and Astronomy, Aarhus University, Ny
Munkegade 120, DK-8000 Aarhus C, Denmark
(11email<EMAIL_ADDRESS>22institutetext: Observatories of the Carnegie
Institution for Science, 813 Santa Barbara St., Pasadena, CA 91101, USA
33institutetext: Carnegie Observatories, Las Campanas Observatory, Casilla
601, La Serena, Chile 44institutetext: Department of Physics, Virginia Tech,
Blacksburg, VA 24061, USA 55institutetext: Department of Physics, Florida
State University, 77 Chieftain Way, Tallahassee, FL, 32306, USA
66institutetext: Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer
de Can Magrans, s/n, E-08193 Barcelona, Spain 77institutetext: Institut
d’Estudis Espacials de Catalunya (IEEC), E-08034 Barcelona, Spain
88institutetext: Instituto de Astrofísica de La Plata (IALP), CONICET, Paseo
del Bosque S/N, 1900, Argentina 99institutetext: Facultad de Ciencias
Astronómicas y Geofísicas (FCAG), Universidad Nacional de La Plata (UNLP),
Paseo del Bosque S/N, 1900, Argentina 1010institutetext: Department of Physics
and Astronomy, University of Oklahoma, 440 W. Brooks, Rm 100, Norman, OK
73019, USA 1111institutetext: Finnish Centre for Astronomy with ESO (FINCA),
FI-20014 University of Turku, Finland 1212institutetext: Tuorla Observatory,
Department of Physics and Astronomy, FI-20014 University of Turku, Finland
1313institutetext: DTU Space, National Space Institute, Technical University
of Denmark, Elektrovej 327, DK-2800 Kgs. Lyngby, Denmark 1414institutetext:
George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and
Astronomy, Department of Physics and Astronomy, Texas A&M University, College
Station, TX 77843, USA

# The Carnegie Supernova Project I. Spectroscopic analysis of stripped-
envelope supernovae

S. Holmbo 11 M. D. Stritzinger 11 E. Karamehmetoglu 11 C. R. Burns 22 N.
Morrell 33 C. Ashall 44 E. Y. Hsiao 55 L. Galbany 6677 G. Folatelli 8899 M. M.
Phillips 33 E. Baron 1010 C. P. Gutiérrez 11111212 G. Leloudas 1313 T. E.
Müller-Bravo 66 P. Hoeflich 55 F. Taddia 11 N. B. Suntzeff 1414

(Received 31 October, 2022; Accepted March 13, 2023.)

An analysis leveraging 170 optical spectra of 35 stripped-envelope (SE) core-
collapse supernovae (SNe) observed by the Carnegie Supernova Project I and
published in a companion paper is presented. Mean template spectra were
constructed for the SNe IIb, Ib, and Ic subtypes, and parent ions associated
with designated spectral features are identified with the aid of the spectral
synthesis code SYNAPPS. Our modeled mean spectra suggest the $\sim 6150$ Å
feature in SNe IIb may have an underlying contribution due to silicon, while
the same feature in some SNe Ib may have an underlying contribution due to
hydrogen. Standard spectral line diagnostics consisting of pseudo-equivalent
widths (pEWs) and blue-shifted Doppler velocity were measured for each of the
spectral features. Correlation matrices and rolling mean values of both
spectral diagnostics were constructed. A principle component analysis (PCA)
was applied to various wavelength ranges of the entire dataset and suggests
clear separation among the different SE SN subtypes, which follows from trends
previously identified in the literature. In addition, our findings reveal the
presence of two SNe IIb subtypes, a select number of SNe Ib displaying
signatures of weak, high-velocity hydrogen, and a single SN Ic with evidence
of weak helium features. Our PCA results can be leveraged to obtain robust
subtyping of SE SNe based on a single spectrum taken during the so-called
photospheric phase, separating SNe IIb from SNe Ib with $\sim 80\%$
completion.

###### Key Words.:

supernovae: general, individual: SN 2004ew, SN 2004ex, SN 2004fe, SN 2004ff,
SN 2004gq, SN 2004gt, SN 2004gv, SN 2005Q, SN 2005aw, SN 2005bf, SN 2005bj, SN
2005em, SN 2006T, SN 2006ba, SN 2006bf, SN 2006ep, SN 2006fo, SN 2006ir, SN
2006lc, SN 2007C, SN 2007Y, SN 2007ag, SN 2007hn, SN 2007kj, SN 2007rz, SN
2008aq, SN 2008gc, SN 2008hh, SN 2009K, SN 2009Z, SN 2009bb, SN 2009ca, SN
2009dp, SN 2009dq, SN 2009dt – techniques: spectroscopic

##  1 Introduction

Between 2004 and 2009, the Carnegie Supernova Project I (hereafter CSP-I;
Hamuy et al. 2006) obtained optical and near-infrared light curves
(Stritzinger et al., 2018a, hereafter Paper 1) and visual-wavelength
spectroscopy (Stritzinger et al., submitted; hereafter Paper 4) of nearly
three dozen stripped envelope (SE) core-collapse supernovae (SNe). SE SNe are
associated with the deaths of massive stars that have lost the majority of
their hydrogen (and helium) envelopes prior to explosion. Within this context,
stars with increasing amounts of mass stripping lead to characteristic spectra
of different subtypes being either hydrogen (H) poor and helium (He) rich (SNe
IIb), hydrogen deficient with He features (SNe Ib), or objects that are
deficient of both H and He features (SNe Ic).

In a series of papers focusing on the CSP-I SE SN sample, we summarize key
facets of our contemporary knowledge of SE SNe. This includes the topics of
light curves and possible progenitor systems in Paper 1 (Stritzinger et al.,
2018a), which also presents the broadband photometry of three dozen SE SNe. In
Paper 2 (Stritzinger et al., 2018b), the photometry is used to devise improved
methods to estimate host-galaxy reddening, while Paper 3 (Taddia et al., 2018)
presents a detailed analysis of the light curves and inferred explosion
parameters. In Paper 4 a summary of the visual-wavelength spectroscopic
properties and spectral classification of SE SNe is presented, along with the
CSP-I SE SN spectroscopic sample consisting of 170 low-redshift (i.e.,
$z<0.1$) spectra of 35 SE SNe (see Stritzinger et al., 2023).

In a classic paper on the spectroscopic studies of SE SNe, Matheson et al.
(2001) studied 84 low-dispersion visual-wavelength spectra of 28 SE SNe
extending from early to late phases, and in doing so were the first to
characterize the heterogeneous nature of the different SE SN subtypes. Over
the years, numerous single object case studies have been published (for a
review see Branch & Wheeler, 2017, Chapters 15-17), while spectroscopic
samples and associated analysis papers have been published by the Center for
Astrophysics SN group (Modjaz et al., 2014; Liu et al., 2016; Williamson et
al., 2019), as well as the Palomar Transient Factory (PTF) and intermediate
PTF surveys (Fremling et al., 2018). The methods used in our analysis of the
CSP-I SE SN spectroscopy dataset were inspired by these previous sample
studies, as well as in part by a select number of papers analyzing the
spectroscopic datasets of thermonuclear SNe (e.g., Hsiao et al., 2007; Blondin
et al., 2012; Silverman et al., 2012; Folatelli et al., 2013).

The organization of this paper is as follows. First in Sect. 2, we focus on
spectral line identification, including the construction of mean template
spectra in Sect. 2.1, the calculation of synthetic spectra in Sect. 2.2, and
the association of spectral line features with parent ion(s) in Sect. 2.3. We
then turn to line diagnostics measurements and correlation matrices for the
line measurements of pseudo-equivalent widths (pEWs) in Sect. 3 and Doppler
velocity line measurements in Sect. 4. A principle component analysis (PCA) of
the dataset is presented in Sect. 5, which is then followed by the discussion
in Sect. 6 and a summary of our key findings in Sect. 7.

##  2 Spectral line identification

###  2.1 Construction of SE SN mean spectra

Close inspection of the CSP-I SE SN spectral sequences presented in Paper 4
reveals the presence of numerous spectral features superposed on a pseudo-
continuum with the shapes and strengths of the features being significantly
time dependent. Before we make direct measurements of the various spectral
features, we first set out to identify the key spectral features and determine
their commonality among the different subtypes. Once the locations of key
spectral features are identified we construct mean template spectra which are
compared with synthetic spectra computed using
SYNAPPS111https://c3.lbl.gov/es/ (Thomas et al., 2011), which enables us to
link the observed spectral features to parent ions (see Sect. 2.2). Then,
spectral line diagnostic measurements are made for the entire suite of
identified features.

Figure 1: Median spectrum for each SE SN subtype (black line) constructed from
all of the spectra plotted in gray and listed in Table 1. The vertical dashed
lines indicate 10 regions of interest. Regions 1–5, 9 and 10 are typically
present in each SE SN subtype, while regions 6–8 are typically found in SNe
IIb and SNe Ib and are usually most prevalent in the weeks after maximum
light. Green lines in the middle panel are spectra of the high-velocity Type
Ib SN 2004gq, and red lines plotted in the bottom panel correspond to the
spectra of the Type Ic-BL SN 2009bb (dotted) and the unusual Type Ic SN 2009ca
(dashed). These spectra were excluded in making the median spectra. Finally,
the number of spectra used to create each median spectrum plotted in each
panel is indicated within parenthesis located just below the given spectral
subtypes.

As a first step, a single median spectrum is constructed for each SE SN
subtype using all of the spectra listed in Table 1, except those of SNe
2004qv, 2009bb, and 2009ca. The resulting median spectrum of each SE SN
subtype are plotted in Fig. 1, along with the individual input spectra. In the
middle panel, the spectra of the Type Ib SN 2004gq that exhibit high-velocity
features are plotted in green, while in the bottom panel the spectra of the
broad-lined Type Ic SNe 2009bb and 2009ca are plotted in red. These spectra
are not included in construction of the mean spectrum plotted in the figure of
these two subtypes. A comparison between the three median spectra reveals a
number of common features as well as a handful of features that are typically
only present in SN IIb and/or SN Ib spectra. As indicated in Fig. 1, 10
different complexes of spectral features are assigned a specific number
(running from Feature 1 to Feature 10) and in some cases a feature may be
attributed to multiple ions. Key characteristics of these features are highly
dependent on the phase of the spectrum. An accurate study of the
spectroscopic. properties of our sample requires a larger grid of mean
template spectra. We therefore produce a coarse time-series of mean template
spectra for each subtype to be modeled with SYNAPPS.

Figure 2: Mean template spectra for the Type IIb SE SN subtype at epochs of
$-$7 d, 0 d, $+$7 d, $+$14 d, and $+$21 d. The number of spectra used to
construct each template is indicated within parenthesis below the
corresponding epoch. Each numerically indicated spectral feature is labeled
with its parent ion(s) as determined from SYNAPPS fits. Thin gray lines
indicate the 1-$\sigma$ error spectrum.

Figure 3: Mean template spectra for the Type Ib SE SN subtype at epochs of
$-$7 d, 0 d, $+$7 d, $+$14 d, and $+$21d. The number of spectra used to
construct each template is indicated within parenthesis below the
corresponding epoch. Each numerically indicated spectral feature is labeled
with its parent ion(s) as determined from SYNAPPS fits. Thin gray lines
indicate the 1-$\sigma$ error spectrum.

Figure 4: Mean template spectra for the Type Ic SE SN subtype at epochs of 0
d, $+$7 d, $+$14 d, and $+$21 d. The number of spectra used to construct each
template is indicated within parenthesis below the corresponding epoch. Each
numerically indicated spectral feature is labeled with its parent ion(s) as
determined from SYNAPPS fits. Thin gray lines indicate the 1-$\sigma$ error
spectrum. The vertical dashed line corresponds to the rest wavelength of C ii
$\lambda$6580.

Mean spectra were constructed for the SN IIb and SN Ib subtypes for the
epochs: $-7$ d, $+0$ d, $+$7 d, $+$14 d and $+$21 d. Due to a dearth of early
data we did not construct a $-$7 d mean spectrum for SNe Ic. The steps
required to construct a mean spectrum are several fold. First we identified an
input set of spectra that met the criteria of being obtained within $\pm$3.5 d
of the epoch under consideration. The spectra of each subsample were then
corrected for Milky Way and host reddening using values estimated by
Stritzinger et al. (2018b). Next the spectra were smoothed using a Fourier
Transform (FT) technique (see Marion et al., 2009), re-sampled, and then
combined yielding a mean spectrum. An associated error spectrum was also
computed using a semi-automated line fitting program written in PYTHON named
Measure Intricate Spectral Features In Transient Spectra (hereafter
misfits222https://github.com/sholmbo/misfits). misfits enables robust
measurements of standard line diagnostics, and in doing so estimates a
realistic error snake. This is done by computing 10,000 realizations of each
spectrum following a Monte Carlo approach. The resultant 1$\sigma$ standard
deviation of the Monte Carlo distribution for each spectrum then serves as the
error snake.

The resulting series of mean spectra and their 1-$\sigma$ error snakes of each
SE SN subtype are plotted in Fig. 2–4. Within Fig. 2 and Fig. 3 Features 1-10
are identified and labeled with their parent ion(s) as implied from SYNAPPS
modeling (see Sect. 2.2), while in Fig. 4 the features associated with He i
are excluded. Furthermore, a line is drawn connecting the red and blue edges
of each feature to define a pseudo-continuum. The area contained within each
feature is highlighted in color.

###  2.2 Computing synthetic spectra with SYNAPPS

Synthetic spectra were computed with SYNAPPS (Thomas et al., 2011) which is an
automated implementation of the highly parameterized spectral synthesis code
SYNOW (SYnthesis NOW; Fisher 2000). SYNOW is a synthetic spectrum code based
on a number of underlying assumptions, including spherical symmetry (see
Thomas et al., 2011). Despite these shortcomings, SYNOW and SYNAPPS have
proven to be effective tools to aid in spectroscopic studies of various
flavors of stripped core-collapse and thermonuclear supernovae (e.g., Deng et
al., 2000; Folatelli et al., 2006; Branch et al., 2007; Thomas et al., 2011;
Hsiao et al., 2015; Parrent et al., 2016). Holmbo (2018) discusses in detail
our SYNAPPS analysis, which is summarized in Appendix A for the interested
reader.

###  2.3 Line identification

####  2.3.1 SNe IIb

  * •

Feature 1 is characterized by a W-shape line profile (see Fig. 1, panel a).
SYNAPPS fits suggest this feature is produced from a blend of H$\gamma$
(forming the bluer of the two dips in the W), a forest of Fe lines including
that of Fe II $\lambda$4550, a contribution from Ti II $\lambda\lambda$4395,
4444, 4469, and also He i $\lambda$4471 which mostly produces the red dip of
the W-shape profile.

  * •

Feature 2 is produced by a blend of lines. At early phases the feature is
primarily formed by H$\beta$ with a contribution from Fe ii $\lambda\lambda
4924,5018$, while at later phases a contribution due to He i $\lambda$4922
emerges (see the $+$21 d mean SN IIb spectrum in Fig. 2).

  * •

Feature 3 is largely produced by the third member of the Fe II multiplet 42,
$\lambda$5169\. This feature is well defined in each of the SE SN subtypes,
and is used as a proxy for the bulk velocity of the SN ejecta.

  * •

Feature 4 is produced by Na i $\lambda\lambda$5890, 5896 and He i $\lambda
5876$.

  * •

Feature 5 is ubiquitous to SNe IIb and is formed by H$\alpha$. The absorption
profile of the feature in the mean spectra show an extended blue wing, which
as indicated by the SYNAPPS fits shown in Fig. LABEL:fig:synapps(a), could be
due to a contribution from the Si ii $\lambda$6355 doublet. Indeed, the
velocity would match that of Fe ii, as we show for SNe Ib and Ic below. If
Feature 5 was purely hydrogen, the pEW of this line would be expected to
decrease with time. However, as can be seen in Fig. 5, the pEW is constant
with time. This is likely due to both the contribution from the blue feature
(which is most likely Si ii).

  * •

Features 6, 7, 8 are attributed to He i $\lambda$6678, He i $\lambda$7065 and
He i $\lambda$7281, respectively. In general, the strength of these features
grows over time becoming most prominent several weeks past maximum.

  * •

Feature 9 is attributed to O I $\lambda 7774$. This line is often contaminated
by the telluric O2 A-band absorption feature. As demonstrated in Paper 4, we
were able to successfully correct for telluric absorption for many spectra in
the sample, particularly those obtained with the du Pont telescope equipped
with the Wide Field re-imaging CCD (WFCCD) camera.

  * •

Feature 10 is attributed to the Ca II NIR triplet $\lambda\lambda
8498,8542,8662$. In some objects, for example, the SNe Ib/c (flat-velocity SNe
IIb) 2005bf and SN 2007Y, their early spectra exhibit a distinct high-velocity
component that promptly vanishes over several days of evolution, while a
photospheric component emerges and grows in strength (Folatelli et al., 2006;
Stritzinger et al., 2009).

####  2.3.2 SNe Ib and SNe Ic

  * •

Feature 1 is produced (similar to SNe IIb) by a forest of Fe II and Ti II
lines, with an additional contribution from He i $\lambda$4471 in SNe Ib,
which turns the characteristic SNe IIb W-shape profile to a Y-shape profile.
As SNe Ic lack both H$\gamma$ and He i the feature takes on a U-shape profile
(see Fig. 1).

  * •

Feature 2 is formed by Fe ii $\lambda\lambda 4924,5018$, with SNe Ib having an
additional contribution from He i $\lambda$4921.

  * •

Feature 3 is largely produced by Fe II $\lambda$5169\. In the case of SNe Ic
this feature may also contain a contribution from Co II $\lambda 5526$, which
increases in strength during the post-maximum evolution.

  * •

Feature 4 is attributed to the Na i $\lambda\lambda$5890,5896 doublet in SNe
Ic, while in in SNe Ib a significant contribution comes from He i $\lambda
5876$.

  * •

Feature 5, unlike in Type IIb SNe, is not hydrogen by the traditional spectral
classification system. Although in the past it has been linked to the Si ii
$\lambda$6355 doublet (e.g., Harkness et al., 1987; Branch et al., 2002),
various different ions have been proposed for this feature. Other than Si ii,
these include high-velocity H$\alpha$ (Liu et al., 2016; Parrent et al.,
2016), as well as Fe ii, Co ii, C ii, and Ne i (see Gal-Yam, 2017, and
references therein).

Close inspection of the template spectra reveals evidence of a blend of at
least two features contained within Feature 5, which we refer to as the red
and blue portions. The red portion is stronger than the blue portion,
especially up to +14 days in both SN subtypes. It is unlikely that the red
portion, which is stronger, is produced by Si II, as its position of maximum
absorption would imply a red-shifted Doppler velocity. However, assuming the
blue portion is due to Si II brings the velocities in line with other ions,
which is demonstrated in Fig. 6.

With the blue portion attributed to Si II, the synthetic spectra suggest that
the red portion of the feature could be formed by a residual amount of H
detached from the photosphere. In fact, it has been identified as H in so-
called transitional SNe Ib/c (e.g., SN 1999ex, Hamuy et al. 2002), which
Folatelli et al. (2014) refer to as flat-velocity SNe IIb. However,
identifying Feature 5 as H possibly contradicts theoretical and observational
evidence showing SNe Ic to be deficient in H and He (Taddia et al., 2018;
Hachinger et al., 2012).

As previously mentioned, the red portion could instead be one of the other
proposed lines (Fe ii, Co ii, C ii, and Ne i), or a blend. Shahbandeh et al.
(2022) found evidence of strong C i features in NIR spectra of SNe Ic,
indicating the optical feature could at least partially be due to C ii. In our
analysis and figures, we assume the blue portion to be Si II for SNe Ib and Ic
when it can be reliably measured, while the red portion is not used.

  * •

Features 6, 7, 8 are, as in the case of SNe IIb, attributed to He i
$\lambda$6678, He i $\lambda 7065$, and He i $\lambda$7281, respectively. By
definition SNe Ic contain no He features, but see Sect. 5.3.2.

  * •

Feature 9 is attributed to O I $\lambda 7774$.

  * •

Feature 10 is attributed to the Ca II NIR triplet.

##  3 Analysis of pseudo-equivalent width measurements

In this section pEW measurements for Features 1–10 for the entire sample of
spectra are computed and used to construct correlation matrices for various
pairs of Features 1-10. The strength and evolution of pEWs (and Doppler
velocities, see below) provides a wealth of information related to the
progenitor stars (Branch & Wheeler, 2017). For example, the spectral features
themselves provide a window to the ionizaton state and chemical content of the
ejecta above the photosphere. As the SN ejecta expand and cool, the
photosphere recedes into the inner ejecta enabling a direct view to the
otherwise opaque inner regions of the progenitor stars. In addition, the
presence (or lack thereof), strength and time-evolution of features produced
from certain ions (e.g., H and/or He) provides information on the spectral
type of the progenitors, their mass-loss history and even their explosion
physics, while the expansion velocities provides a measure of the explosion
energy (see Paper 3, and references therein).

###  3.1 Measuring pseudo-equivalent widths

Armed with the spectral line identifications of Features 1–10, we conduct a
quantitative analysis of the line strength and evolution of the various
features via pEW measurements. This is a common line diagnostic having been
utilized to study large spectroscopic samples of thermonuclear supernovae
(e.g., Branch et al., 2006; Garavini et al., 2007; Blondin et al., 2012;
Silverman et al., 2012; Folatelli et al., 2013). Liu et al. (2016) and
Fremling et al. (2018) have also followed suit, using both pEW and Doppler
velocity measurements in their analysis of the CfA and PTF SE SN spectroscopic
datasets, respectively. The use of pEW measurements serves as a flexible and
accurate line diagnostic, particularly compared to fitting a Gaussian
function, which is often not appropriate given the asymmetric and time-
dependent spectral features inherent to SE SNe.

The term pseudo in pEW highlights the difficulties faced when attempting to
separate continuum flux from absorption and emission flux of time-dependent
spectral features that also suffer significant Doppler broadening. Following
Folatelli (2004), we define the pseudo-continuum to be a straight line
connecting two maxima defining a spectral absorption feature:

$pEW=\sum_{i=0}^{N-1}\left(1-\frac{f(\lambda_{i})}{f_{c}(\lambda_{i})}\right)\Delta\lambda_{i}.$
(1)

Here $\lambda_{i}$ corresponds to the wavelength of each pixel contained
within the spectral range located between the blue and red edges of a specific
feature. Furthermore, $N$ is the number of pixels contained between the red
and blue edges, the parameter $\Delta\lambda_{i}$ represents the width of
pixel $i$, $f(\lambda_{i})$ corresponds to the observed flux at $\lambda_{i}$,
$f_{c}(\lambda_{i})$ is the pseudo-continuum at $\lambda_{i}$, and
$\Delta\lambda_{i}$ is the sum over the defined wavelength interval.

Given the number of spectra in the CSP-I sample and having up to 10 features,
it quickly becomes cumbersome to measure the pEWs by hand. We therefore
developed the misfits spectral analysis package, and in doing so, created a
resource that minimizes user bias while enabling efficient measurements in a
standard manner. To make pEW measurements of Features 1–10 for a given
spectrum we adopted the following steps. First misfits smooths an observed
spectrum following the use of a Fourier Transform (FT) smoothing technique
(Marion et al., 2009) and then identifies the highest peak located between the
defined boundaries at the blue and red end of Features 1–10. The wavelength
ranges for each blue/red boundary of each of the Features measured are listed
in Table 2. With boundary end points identified, a pseudo-continuum is defined
by connecting the boundary points with a straight line. This is demonstrated
in Fig. 2 where the spectral regions of each feature are identified. These
regions are used to infer pEW values using Eq. (1).

Each pEW measurement also has an associated 1-$\sigma$ uncertainty estimated
via the following recipe. The error spectrum is first multiplied by a random
number taken from a normal distribution and the resulting product is added to
the FT smoothed observed spectrum. pEW measurements are computed from the
altered spectrum, and following a Monte Carlo approach, this is done for
10,000 realizations. The 1-$\sigma$ value of the resulting pEW distribution
then serves as the pEW 1-$\sigma$ measurement uncertainty. This method
accounts for the uncertainty associated with the ability of the algorithms to
accurately determine the heights of the blue/red boundaries of the spectral
line features.

Figure 5: Pseudo-equivalent width values for Features 1–10 plotted vs days
relative to the epoch of $T(B)_{max}$. The measurement symbols are encoded by
SN subtype with SNe IIb appearing as blue triangles, SNe Ib as yellow squares,
and SNe Ic as magenta squares. Filled shaded regions correspond to the mean
$\pm$1-$\sigma$ standard deviation of a rolling window characterized with a
size of ten days. The rolling windows are evaluated on epochs containing at
least three distinct pEW measurements and two pEW measurements in the
preceding and subsequent epochs.

###  3.2 pEW measurement results for Features 1-10

The definitive pEW measurements of Features 1–10 computed for all of the
photospheric phase spectra are plotted in Fig. 5 versus phase relative to the
epoch of $B$-band maximum. Over-plotted in each of the panels are filled
shaded regions color-coded by spectral subtype representing the mean
1-$\sigma$ standard deviation computed using a rolling window with a size of
ten days. The rolling windows were evaluated only on epochs containing a
minimum of three measurements and have at least two preceding and subsequent
measurements.

Inspection of Fig. 5 reveals that the pEWs for most of the features generally
increase in strength over the week prior to maximum. Depending on the
particular feature, this behavior continues, or they at least remain constant
over the duration of $\approx 3-6$ weeks past maximum. The pEW values are
found to range from as little as $\sim$ 10 Å (e.g., ${He\textsc{i}}$ features
at early times) all the way up to $\gtrsim$600 Å in the case of the
${Ca\textsc{II}}$ NIR triplet. We now briefly summarize the nature of the pEW
measurement of Features 1–10 and in passing compare and contrast the results
amongst the different subtypes.

  * •

Feature 1 (Fe II, Ti II in each subtype with additional contributions from
H$\gamma$ and He i $\lambda 4471$ in SNe IIb and He i in SNe Ib). The pEW
values typically increase within a few weeks from maximum and then remain
relatively constant over the duration of our observations. pEW values range
from $\sim 100$ Å up to $\sim 250$ Å. At early times SNe Ic tend to exhibit
smaller values as compared to SNe IIb and SNe Ib.

  * •

Feature 2 (Fe II in each subtype with an additional contribution from H$\beta$
and He i in SNe IIb and only He i in SNe Ib) The pEW values generally increase
over time for the SNe IIb while the increase in SNe Ib exhibits a shallower
slope. pEW values typically reach peak values in excess of 100 Å within 2 to 3
weeks past maximum and then remain constant and/or slowly decrease. The
feature is less prevalent in SNe Ic with pEW values reaching around
$50\leavevmode\nobreak\ \AA$ at all epochs.

  * •

Feature 3 (Fe ii and Co II in SNe Ic) consists of a forest of Fe ii lines,
often displaying a prominent P-Cygni Fe ii emission profile, making it a
somewhat troublesome feature for misfits to accurately identify the red
boundary point automatically (see Fig. 2). This can lead misfits to compute
negative values at early phases for some of the objects. Nevertheless within a
week past maximum the majority of objects exhibit pEW values between $\sim$ 20
Å to 70 Å, and these continue to increase reaching $\sim$ 50 to 150 Å by a
month and half past maximum. In SNe Ic there may be at later times a
contribution from Co ii, however, the mean pEW values are consistent with
those inferred from the SNe IIb and SNe Ib, indicating this feature is largely
dominated by Fe ii.

  * •

Feature 4 (Na I and/or He i) in the majority of objects exhibits pEW values
that increase with time. In the week leading to maximum the SNe IIb and SNe Ib
exhibit very different pEW values with the former below 50 Å, while the later
are in excess of 75 Å (indicated by a dashed line) with the rolling mean
values around $\sim$100 Å. In the SNe Ib at early times, the feature appears
to slightly decrease and at around $+10$ d begins to increase and thereafter
follow the same trend as the SNe IIb. Over time the pEW values for both SNe
IIb and SNe Ib continue to increase reaching similar values out to $+$50 d. On
the other hand, the SNe Ic exhibit an average pEW value of $\sim 40\pm 30$ Å
at maximum and generally exhibit lower pEW values as time goes on relative to
both SNe IIb and SNe Ib. Given the significant difference between the
premaximum pEW measurements of this feature in SNe Ib relative to the both SNe
IIb, the pEW value of this feature could serve as a useful aid in spectral
classification (see Sect. 6.2.1).

  * •

Feature 5 (H$\alpha$ and/or Si II) pEW values show diversity dependent on
spectral subtype and the phase of the spectrum. The majority of SNe IIb
exhibit pEW values in excess to those characterizing SNe Ib and SNe Ic and
this trend appears to increase over time. As indicated by the dashed line in
the Feature 5 panel of Fig. 5 there is a clear separation at pEW of $\approx
140$ Å between the pEW values measured in SNe IIb compared to the values
measured in SNe Ib and SNe Ic. Quantitatively, the pEW values of SNe IIb take
on high values (pEW $\sim 140-200$ Å) from early phases and onward. On the
other hand, the SNe Ib and SNe Ic exhibit smaller pEW values ranging between
$\approx 60-100$ Å. In the earliest observations of SNe Ib, Feature 5 shows
higher values and this is likely due to absorption associated with a small
shell of hydrogen material retained by their progenitor stars. As the
photosphere recedes the pEW value in these cases drops to values similar to
the SNe Ic, indicating that the feature becomes dominated by Si ii and has
little to no contribution from H$\alpha$. In other words, the fact that SNe Ib
and SNe Ic exhibit similar pEW values around maximum light and beyond,
provides an argument that at these phases the red component of Feature 5 in
SNe Ib is largely produced by Si ii. We quantify this separation in a PCA
analysis presented below. The finding that SNe IIb exhibit larger pEW values
of Feature 5 at early times as compared to SNe Ib is consistent with the idea
that their progenitor stars retain higher amounts of hydrogen relative to the
progenitors of SNe Ib.

  * •

Features 6-8 (He I) increase in strength from early phases out to weeks past
maximum. As expected by the atomic data of He i, Feature 7 is significantly
more prominent than Features 6 and 8 at all epochs. Quantitatively, Feature 7
reaches pEW values in excess of 100 Å while Features 6 and 8 extend between
10-80 Å.

  * •

Feature 9 (O I) in the SNe IIb and SNe Ib exhibits pEW values that generally
increase from $\leq 50$ Å around maximum to in excess of 100 Å a month past
maximum. SNe Ic show more premaximum diversity with pEW values ranging from
$\sim$ 25 Å up to $\sim$100 Å. The rolling means indicate that beyond +30 d
the pEW values level off or even decrease to values $<100$ Å. During the same
phases, SNe IIb and SNe Ib exhibit mean pEW values $\gtrsim$100 Å.

  * •

Feature 10 (Ca II) overall exhibits an increase in the pEW value for each SE
SN subtype. Prior to maximum the SNe IIb and SNe Ib exhibit pEW values of
around $160\pm 50$ Å that evolve up to $\sim$400 Å by $+$40 d. Around maximum,
SNe Ic tend to reach somewhat larger pEW values compared to SNe IIb and SNe
Ib, while by a month past maximum the three SE SN subtypes show similar
values.

Figure 6: Time evolution of the Doppler velocity at maximum absorption
($-v_{abs}$) for Features 1–10 and color-coded by SE SN subtype. Shaded
regions correspond to the standard deviation of the rolling mean computed
using a window size of ten days and color-coded by spectroscopic subtype as
indicated. The rolling windows are evaluated on epochs having a minimum of
three measurements and at least two measurements before and after. We note
that the peculiar SNe Ic 2009bb and 2009ca are excluded.

##  4 Analysis of Doppler velocity measurements

The wavelength of maximum absorption for a given spectral feature provides an
estimate on the bulk velocity of the line-forming material. Although prominent
features may be produced from the blending of numerous lines, in general, the
Doppler velocity at maximum absorption ($-v_{abs}$) provides a useful measure
of the kinematics of the line-forming region, and depending on the spectral
features used, a constraint on the explosion energy of the supernova (see,
e.g., Branch et al., 2002; Fremling et al., 2018; Taddia et al., 2018).

###  4.1 Measuring Doppler velocity

Determining a value of $-v_{abs}$ based on a spectral feature in a 1D
supernova spectrum is done using the relativistic Doppler approximation (see
Blondin et al., 2006, their Eq. 6). Measuring the observed wavelength
($\lambda_{obs}$) of a feature from a 1D spectrum is relatively
straightforward, however, the use of an automatic detection of the exact
position of a feature’s minimum becomes increasingly difficult in low signal-
to-noise spectra due to an accompanying increase in the number of local
minima. To overcome this problem an algorithm was developed to detect
groupings of minima within a certain threshold and treat each of the groupings
as a single minimum.

###  4.2 Doppler velocities and evolution

Plotted in Fig. 6 are the $-v_{abs}$ values measured for Features 1–10 using
all of our spectra obtained prior to $+$100 days relative to the epoch of
$B$-band maximum. We note that in SNe IIb Feature 1 and Feature 2 have an
additional absorption component attributed to H$\gamma$ and H$\beta$,
respectively. We therefore separate each feature into two features denoted as
Feature (a) and Feature (b). We now summarize the overall trends for Features
1–10, and then examine the correlation matrices computed for the various pairs
of the $-v_{abs}$.

  * •

Feature 1a (mostly H$\gamma$ in SNe IIb) can be problematic to measure given
the spectral range of this feature is often noisy. Feature 1a plotted in the
top left panel of Fig. 6 exhibits, as do the majority of other features,
$-v_{abs}$ values that decrease over the first two weeks of evolution. The
feature exhibits $-v_{abs}\sim 10,000-14,000$ km s-1 in the week prior to
maximum, and by $+$21 d decreases to $\sim 8000-10,000$ km s-1.

  * •

Feature 1b (Fe ii, Ti ii & He i) is similar in SNe IIb and SNe Ib with
$-v_{abs}$ values between 3,000-6,000 km s-1 from $-10$ d to $+10$ d. Upon
reaching a minima $-v_{abs}$ at around $+10$ d, it experiences an upturn,
reaching within three weeks, similar (in SNe IIb) or even higher values (SNe
Ib) compared to those inferred from spectra in the week leading up to maximum.
By $+$40 d the velocity evolution decreases in SNe Ib leveling off to mean
$-v_{abs}$ values of $\sim 6,000$ km s-1, which is similar to the SNe IIb mean
value of $\sim 4,000$ km s-1. Coverage of the SNe Ic begins a week after the
SNe IIb/Ib with a mean $-v_{abs}$ value between $+0$ d to $+$10 d of $\sim
7,000$ km s-1. Between $+$10 d to $+$25 d the mean $-v_{abs}$ value drops to
$\sim 2,500$ km s-1, similar to the mean values exhibited by the SNe IIb and
SNe Ib when they reached their initial minima occurring 10-15 days earlier.
The (delayed) upturn in the SNe Ic extends through $+$30 d, and then again,
turns over declining to values around a factor of 2 or more less than inferred
from the SNe IIb and SNe Ib at similar epochs (i.e., around $+$45 d).

  * •

Feature 2a (H$\beta$) rolling mean $-v_{abs}$ values extend between $\sim
11,000\pm 2,000$ km s-1 down to $\sim 9,000\pm 1,000$ km s-1 from $-10$ d to
$+50$ d. The velocities and evolution of this feature are quite consistent
with that of Feature 1a.

  * •

Feature 2b (Fe II $\lambda$5018 & He i) in SNe Ib exhibits rolling mean
$-v_{abs}$ values that are $\sim 2,000$ km s-1 higher than the SNe IIb. At the
earliest phases covered, SNe Ib reach rolling mean $-v_{abs}$ values of $\sim
13,000$ km s-1, while SNe IIb never exceed $12,000$ km s-1. SNe Ic mean values
lie between the SNe IIb/Ib until $+20$ d, and then subsequently drop to $\sim
6,000$ km s-1, slightly below the SNe Ib mean values. Close inspection of
panel F2b also reveals an SN Ib at early times exhibiting a $-v_{abs}$ value
$\sim 17,000$ kms-1. This object is SN 2004gv and also exhibits high
$-v_{abs}$ values in its other Features, particularly those attributed to He i
(see F3, F4, F6, F7). SN 2004gv is not included in the computation of the
various rolling mean $-v_{abs}$ values plotted in the figure.

  * •

Feature 3 (Fe II & possibly Co ii in SNe Ic) exhibits at most epochs similar
rolling mean $-v_{abs}$ values in both SNe Ib and SNe IIb. At the earliest
phases of coverage SNe Ib show $-v_{abs}$ values extending $\gtrsim 4,000$ km
s-1 higher than that of the SNe Ib rolling mean 1-$\sigma$ dispersion.
Quantitatively, in the week leading to maximum, SNe Ib exhibit mean
$-v_{abs}\sim 13,000\pm 2,000$ km s-1 while SNe IIb exhibit mean $-v_{abs}\sim
10,000\pm 2,000$ km s-1. Between $\sim+0$ d to $+10$ d the SNe Ic exhibit mean
$-v_{abs}$ values fully consistent with the SNe IIb, while beyond $+10$ d
their values are typically a few 1,000 km s-1 higher than the mean values of
both SNe IIb and SNe Ib.

  * •

Feature 4 (Na i & He i in SNe IIb and SN Ib) exhibits at all phases similar
mean $-v_{abs}$ values and 1-$\sigma$ error snakes. At early phases a large
range of velocities are measured extending from $\sim 6,000$ km s-1 up to and
in excess of $\sim 14,000$ km s-1, with SNe Ib exhibiting somewhat more
dispersion than SNe IIb. Turning to SNe Ic, the rolling mean $-v_{abs}$ values
are consistent with the SNe IIb/Ib though their error snake is characterized
by larger dispersion.

  * •

Feature 5a (Si ii in SNe Ib and SNe Ic) $-v_{abs}$ values were measured for
the objects in the sample having a clear feature likely produced by Si ii.
Between maximum and $+$14 d the rolling mean $-v_{abs}$ values of the SNe Ib
decrease from $\sim 13,000\pm 1,000$ km s-1 to $\sim 9,000$ km s-1 , whereupon
they slow to just above 8,000 km s-1 out to $+45$ d. Our temporal coverage of
this feature is limited in the SNe Ic sample, with mean $-v_{abs}$ values
ranging between 13,000 km s-1 to 9,000 km s-1 between $+$0 d to $+$21 d.

  * •

Feature 5b (H$\alpha$) exhibits a smoothly evolving rolling mean $-v_{abs}$
values in the SNe IIb sample ranging from $\sim 16,000\pm 2,000$ km s-1 down
to a value of $\sim 10,000\pm 1,000$ km s-1 at $+30$ d. Subsequently, the
feature remains constant for weeks, consistent with the idea H gas is not
mixed with the heavier, lower velocity elements. The purported H$\alpha$
feature measured in our sample of SNe Ib has rolling mean values and evolution
similar to that of SNe IIb, though its 1-$\sigma$ error snake is larger and
our sample does not extend in phase as far. The feature vanishes from the
majority of the SNe Ib by $+$21 d (see Fig. 3). This accounts for the rolling
mean of the SNe Ib ending around $+$30 d, while that of the SNe IIb extends to
later phases, and is consistent with the idea that the former retain smaller
amounts of hydrogen compared with the latter.

  * •

Features 6, 7 and 8 (He I) slowly evolve over time. The $-v_{abs}$ values and
evolution of Features 6 and 7 are completely consistent with one another. In
the case of SNe Ib both features exhibit rolling mean $-v_{abs}$ values that
extend from $\sim 8,000\pm 3,000$ km s-1 a week before maximum to $\sim
6,000\pm 1,000$ km s-1 by $+40$ d. We note that the rolling mean $-v_{abs}$
values for the SNe IIb sample begin several days post maximum. Feature 8
consistently shows lower $-v_{abs}$ values compared to Features 6 and 7 with
both subtypes exhibiting rolling mean $-v_{abs}$ values of $\sim 6,000\pm
2,000$ km s-1 between $+0$ d to $+$40 d.

  * •

Feature 9 (O I) in general emerges first in SNe Ic followed soon after by SNe
Ib, while for at least our small sample, this feature typically emerges in SNe
IIb more than a week past maximum. SNe Ic exhibit high rolling mean $-v_{abs}$
values with significant dispersion at all epochs. Around maximum the SNe Ic
mean values are around $\sim 9,500\pm 2000$ km s-1 and then subsequent slowly
evolve over a period of a month. Rolling mean $-v_{abs}$ values of SNe IIb are
consistently $\sim 2,000$ km s-1 less than the SNe Ic mean values, and
typically $\sim 2,000$ km s-1 higher compared with the SNe IIb.

  * •

Feature 10 (Ca II) shows an exponentially declining rolling mean $-v_{abs}$
evolution for each SE SN subtype. Already by $-7$ d the SNe IIb and SNe Ib
exhibit a high degree of similarity with rolling mean values of $\sim
14,000\pm 1,000$ km s-1. SNe Ic exhibit higher rolling mean $-v_{abs}$ values
compared to the He-rich subtypes at all phases though the associated mean
error snake is large at early phases mostly due to SN 2009dp. This is a
noteworthy object as it is as bright as SN 2009bb and shows high $-v_{abs}$
values yet no broad-line features.

###  4.3 Doppler velocity correlation coefficients

Spearman’s rank correlation coefficients were computed for different pairs of
$-v_{abs}$ values for Features 1-10, and examined and shown in the Appendix B.
SNe Ic and SNe Ib are found to only have a handful of pairs that are
correlated, and those that are correlated typically show a low-to-moderate
degree of correlation. In the case of the SNe Ib, the He i features only show
correlations of low statistical significance as their $-v_{abs}$ values evolve
very little, and do not monotonically change as required to produce a
statistically significant correlation. On the other hand, SNe IIb show a
larger number of moderately to highly correlated pairs, particularly for the
photospheric phase subset of the data. This is due to the spectral features
appearing more prominent at earlier phases compared to in the SNe Ib and SNe
Ic, and as a result of the rapid early evolution, a number of moderately to
highly correlated pairs were computed.

##  5 Principal component analysis

Principal Component Analysis (PCA; Pearson 1901) provides a means to
reconstruct the multidimensional information contained within the CSP-I SE SN
spectral library using just a few variables. This can be achieved through the
use of the Singular Value Decomposition (SVD) of a data matrix (see Hsiao et
al. 2007; Cormier & Davis 2011 and Holmbo 2018; Williamson et al. 2019;
Shahbandeh et al. 2022 for the applications of PCA to SNe Ia and SE SN
spectral datasets, respectively). PCA is applied to summarize the large amount
of information contained within an extended dataset by reducing its
dimensionality while using only the most informative explanatory variables
that can be derived from the dataset.

PCA is essentially a linear decomposition of a collection of data by a change
of basis defined by the principal components (PCs; also known as eigenvectors)
of the covariance matrix and the amplitudes (also known as projections)
defined by the inner product between the data and the new basis. PCs are
sorted/ranked by the degree to which they contribute to the variance within
the data. This effectively means that the first basis vector, that is PC1,
accounts for the largest variation within the dataset, PC2 the second largest,
and so on and so forth as the dimensionality increases. Since PCA requires
little human intervention and is algorithmically performed by a computer in a
matter of seconds, it provides a means to explore data in a much less labor
intensive manner as compared to the line diagnostics visited in the previous
sections.

In the following we examine the PCs contributing to the largest variations
within various segments of our spectral data library. Our analysis makes use
of the scikit-learn PCA decomposition toolbox (Pedregosa et al., 2011) and
adheres to standard procedures as described in detail by Holmbo (2018, 2020).
First, each observed spectrum is normalized to a common scale such that its
mean flux is equal to zero, and a common spectral range is used. A mean
spectrum is then determined for these input spectra, from which PCs and the
amount of variance they account for are calculated using SVD. In practice this
means that, for each input spectrumj used to estimate the mean spectrum, a set
of PCi are obtained, along with their corresponding Amplitudesji that reflect
the degree to which PCi contributes to the variations of the data used to
determine the mean spectrum. This is represented by the formalism:

${\rm Spectrum}_{j}={\rm Mean\leavevmode\nobreak\ Spectrum}+\sum_{i=1}^{N}{\rm
Amplitude}_{j,i}\cdot{\rm PC}_{i}.$ (2)

We now examine our PCA results obtained using a large subset of the CSP-I SE
SN spectral library before turning our focus toward more nuanced aspects of
the data by inspecting sets of PCs determined from particular phase and
spectral wavelengths ranges that effectively trace the strength and temporal
evolution of the spectral features associated with H i and He i.

Figure 7: Results from PCA. Top: Solid black line is the mean spectrum
computed using all observed (i.e., 111) spectra obtained out to $+40$ d and
over the spectral range of 3950–8700 Å. The shaded regions correspond to Mean
spectrum + Amplitudei $\cdot$ PCi, yielding the range of amplitudes shown in
the panel below. Here the blue shaded regions correspond to negative values,
and the red regions correspond to positive values. Bottom: Amplitude1 versus
Amplitude2, color-coded by spectral subtype and with the intensity of colors
corresponding to the phase of the observed spectra following the multicolumn
colorbar located to the right.

###  5.1 PCs associated with color and spectral line strength

Plotted in the top of Fig. 7 is the mean spectrum (solid black line) computed
using all CSP-I SE SNe spectra, which was used for this portion of our PCA.
This includes 111 spectra covering the temporal phase out to $+40$ d and over
the spectral wavelength range 3950–8700 Å.333The phase range was limited to
$+40$ d since there are only a few SNe Ic spectra afterwards. Over-plotted on
the mean spectrum with shaded red and blue coloring is the full range of
Amplitudei $\cdot$ PCi covered by the input data, with PC1 shown in the upper
half-panel and PC2 in the lower half-panel. Here the blue shading indicates a
negative while the red indicates a positive contribution. PC1 accounts for 61%
of the total spectral variations of the data, PC2 accounts for 11%, and PC3
(not shown) accounts for 7%. The significant amount of variation associated
with PC1, particularly at the blue end of the spectral wavelength range, is
consistent with the observed broadband photometric colors of SE SNe (see Paper
2), while the positions and amplitudes of PC2 suggests a connection to the
depth/height of the spectral features. As shown in Paper 4, the spectral
colors are directly correlated to the photometric colors, with a typical rms
uncertainty of $\lesssim$ 0.1 mag.

Turning to the bottom of Fig. 7, we plot the values of Amplitude1 and
Amplitude2 as determined from our application of PCA. Each point in the figure
is computed from a single spectrum with the color-coding of the points
differentiating the subtype, and with the intensity of the coloring providing
an indication of its temporal phase relative to the epoch of $B$-band maximum.
Here, negative Amplitude1 values correspond to a subtraction from the mean
spectrum and are coded blue. On the other hand, points with positive
Amplitude1 values correspond to adding a larger contribution of PC1, and are
coded red. This explains why the cluster of SNe IIb and SNe Ib points located
within the left-half of the plot are associated with early phase spectra when
the spectral energy distributions of their associated SNe are hot and bluer.
Subsequently, as the SN ejecta expands and cools, their broadband colors
evolve to longer (red) wavelengths. This therefore explains why the right-half
side of the plot is populated predominately with points associated with post-
maximum phase spectra. Unfortunately, there is a dearth of SNe Ic spectra in
the days leading up to maximum light, preventing a more rigorous comparison
between their early phase spectra with those of the He-rich SE SNe.

Figure 8: Results from PCA. Top: Solid black line is the mean PCA spectrum
computing using all observed spectra obtained out to $+$70 d and covering the
wavelength region 3950–7350 Å. Blue and red shaded regions correspond to the
full range of Amplitudei with blue (red) corresponding to negative (positive)
values. Vertical gray regions correspond to He i (PC2 panel) and H i (PC3
panel) related features (same as in Fig. 2), indicating that variation in PC2
corresponds strongly to Helium and in PC3 to Hydrogen. Bottom: Amplitude2
versus Amplitude3, color-coded by spectral subtype with the intensity of
colors corresponding to the phase of the observed spectra following the
multicolumn colorbar located to the right.

Further inspection of the distribution of points indicates significant
diversity among the Amplitude2 values inferred from the post-maximum phase
spectra of the different SE SN subtypes. Interestingly, the vast majority of
post-maximum phase SNe IIb are found to preferentially cluster within the
upper-right quadrant of the figure contained within the parameter space of
Amplitude${}_{1}>0$ and Amplitude${}_{2}\gtrsim 1$. This is fully consistent
with the temporal evolution of the pEWs of SNe IIb previously documented in
Sect. 3.2, which indicated a strengthening of most spectral features (e.g.,
those associated with He i) over time. On the other hand, the SNe Ib (and to a
lesser extent the SNe Ic) exhibit a range of Amplitude2 values that produces
the triangular distribution of points within the bottom of Fig. 7. For
example, some maximum and post-maximum spectra of both subtypes exhibit little
change in their Amplitude2 relative to premaximum epochs. However, other
points associated with SN Ib exhibit Amplitude2 values ranging between $-$6
all the way up to 9, while in the case of SNe Ic the Amplitude2 parameter
space ranges between $\sim-9$ up to $2$. In short, the wide diversity among
the Amplitude2 parameter space is largely inherent to the difference in the
suite of lines present in the various SE SN subtypes and the phase of the
spectra. In order to assess the level of variations within the PCs that are
largely associated with the various Balmer and He i features, we now turn to
PCA results obtained by examining more limited spectral wavelength regions.

###  5.2 PCs associated with H i and He i

####  5.2.1 PC2 versus PC3

Figure 8 presents the PCA results obtained using all (128) of the spectra
contained within the CSP-I SE SN library extending up to $+$70 d and over the
spectral wavelength range of 3950-7350 Å. The temporal range ensures that the
line strength evolution of the features are well sampled, while the spectral
wavelength range excludes features associated with O i $\lambda 7773$, Ca ii
H&K and the Ca ii NIR triplet.444While $+$40 d was used in Sect. 5.1, we now
extend it to cover line evolution, since the PCA analysis worked well and the
mean epoch of the template is slightly after peak. However, the trade-off is
that the number of SNe Ic spectra from late epochs included in the template is
minimal. By using a more limited wavelength range that excludes these common
SE SN lines, the top PCs will contain more of the variance in H and He
features, thereby avoiding diluting our results. As in Fig. 7, the black solid
line in the top of Fig. 8 corresponds to the mean spectrum and the over-
plotted blue/red (negative/positive) shaded regions display the full range of
Amplitude${}_{i}\cdot$ PCi covered by the data, where $i=2$ is shown in the
upper panel and $i=3$ in the lower panel. PC1 (not plotted) accounts for 65%
of the spectral variations and is linked to color, as previously discussed.
Meanwhile, PC2 and PC3 account for 9% and 6% of the overall spectral
variations, respectively. The majority of the blue/red shaded regions of PC2
are located bluewards of the He i rest-wavelengths (i.e., F1b, F2b, F4, F6,
F7, F8). Similarly, PC3 exhibits clear variations from the mean spectra at
locations just bluewards of the rest wavelengths of Balmer features (i.e.,
F1a, F2a, F5b). However, both PCs also likely have contamination from other
(non-He or non-H) features (interested readers are referred to Sect. 2.3.2).

We first consider the comparison between Amplitude2 versus Amplitude3, which
is presented in the bottom of Fig. 8. As before, the points are color-coded
based on SE SN subtype with the intensity of the color corresponding to the
temporal phase. Interestingly, the comparison of PC2 and PC3 exhibits
groupings separating SNe IIb from SNe Ib and SNe Ic, and also SNe Ib from SNe
Ic, although the latter is less well differentiated. We now examine some
specifics of the groupings.

Relative to the coordinate origin (0,0) located at the center of the plot, the
SNe IIb points overwhelmingly populate the Amplitude${}_{3}\gtrsim 0$
parameter space, that is, they essentially cover the entire top-half region of
the figure. On the other hand, the points associated with SNe Ib and SNe Ic
tend to populate the entire bottom-half region (i.e., Amplitude${}_{3}\lesssim
0$) with SNe Ib mostly located in the bottom-right quadrant (i.e.,
Amplitude${}_{2}\gtrsim 0$) and the SNe Ic in the bottom-left quadrant (i.e.,
Amplitude${}_{2}\lesssim 0$). These groupings are essentially dictated by the
presence (or lack thereof), strength and time-dependence of the Balmer and He
i features, as traced by $\text{PC}_{2}$ and $\text{PC}_{3}$. The temporal
dependence of the Balmer features explains why the SNe IIb points associated
with premaximum and around maximum spectra generally exhibit high Amplitudes3
values as this is when the Balmer features are most prevalent. Similarly, the
points associated with the post-maximum SNe IIb points (when Balmer lines
typically weaken as the photosphere recedes into deeper layers of the ejecta
devoid of H) are found to group among the SNe Ib with much lower Amplitudes3
values and higher Amplitude2 values.

Turning to SNe Ic, its main grouping is generally located within the bottom-
left quadrant of the Amplitude2 versus Amplitude3 diagram, reflecting the
traditional SE SN spectroscopic taxonomy where SNe Ic lack both H i and He i
features. However, both the SN Ic and SN Ib subtypes do have several members
that are mixed together along the Amplitude2 (primarily He i) axis. In
addition, to the mixture between the SN Ib and SN Ic subtypes in Amplitude2,
there are two early points located in the top-left quadrant among the
peripheral grouping of young SNe IIb spectra with positive Amplitude3 values
(i.e, $\sim 2$) that are associated with SN 2009ca. SN 2009ca is a
superluminous SN Ic that is more than 2 magnitudes brighter than the rest of
the sample (Taddia et al., 2018). Therefore, due to the expected spectral
differences between SLSN-Ic and normal SNe Ic, its position separated from the
rest of the SNe Ic is not surprising.

Figure 8 suggests that the SN IIb, SN Ib and SN Ic subtypes can be
differentiated based solely on PC2, PC3, and phase. The strongest dichotomy
appears to exist among SNe IIb and the SNe Ibc, which is investigated more
deeply in the next section. The difference in SNe IIb is primarily from
Amplitude3 with a smaller contribution from Amplitude2 and the phase, although
at late times they do evolve toward lower Amplitude3 and more closely resemble
SNe Ib. As discussed below, these findings are consistent with pEW
measurements of H$\alpha$ post maximum (see, e.g., Liu et al. 2016, and our F5
panel in Fig. 5).

Figure 9: Results from PCA. Top: Full range of the linear combination of
Amplitude2 and Amplitude3 determined after applying a $\theta=30^{\circ}$
rotation following Eq. (3). Over-plotted the mean spectrum (black line)
determined from all spectra extending to $+45$ d and covering the wavelength
range of 3950–7000 Å. Blue and red shaded regions correspond to the full range
of the first rotated PC
($0.87\cdot\text{Amplitude}_{3}-0.50\cdot\text{Amplitude}_{2}$) with the blue
corresponding negative values and red to positive. Balmer series features are
indicated with gray regions as in Fig. 2. Bottom: The corresponding linear
combination of Amplitude2 and Amplitude3, plotted versus phase. This
particular linear combination separates particularly well between $+$20 d to
$+$40 d the SNe IIb from the SNe Ib and SNe Ic.

####  5.2.2 Linear combinations of PC2 and PC3

In order to explore whether PCA can be used to differentiate SNe Ibc from SNe
IIb, we use the fact that PCs are orthogonal, linear combinations of the input
variables. A linear combination of a linear combination is another linear
combination. Therefore, we search for a rotation of PC2 and PC3 (a rotation is
just a linear combination) that creates a large separation between SNe IIb and
the SNe Ibc. As demonstrated below, by introducing linear combinations as a
change of basis we achieve greater separation between the subtypes. This aids
both in the visualizations and the interpretability of the PCA results. This
is performed by applying the 2D rotation matrix to the basis defined by PC2
and PC3 following:

$\displaystyle\text{Rotated}\leavevmode\nobreak\ \text{PC}_{ij}(\theta)$
$\displaystyle=\begin{bmatrix}cos(\theta)&-sin(\theta)\\\
sin(\theta)&cos(\theta)\end{bmatrix}\begin{bmatrix}\text{PC}_{i}\\\
\text{PC}_{j}\end{bmatrix}$ (3)
$\displaystyle=\begin{bmatrix}cos(\theta)\text{Amplitude}_{i}-sin(\theta)\text{Amplitude}_{j}\\\
sin(\theta)\text{Amplitude}_{i}+cos(\theta)\text{Amplitude}_{j}\end{bmatrix}.$

We reiterate that the resulting Rotated PCs are orthogonal, linear
combinations of the original PCs. Instead of giving them new names, for
clarity, we explicitly refer to them by their linear combination expressions.

Figure 9 presents the PCA results obtained for the first Rotated PC,
($0.87\cdot\text{Amplitude}_{3}-0.50\cdot\text{Amplitude}_{2}$), corresponding
to a Rotation of $\theta=30^{\circ}$. As SNe IIb and SNe Ib generally evolve
toward each other at late times (see Fig. 8), here PCA was restricted to data
obtained to $\sim+40$ d and the spectral range was limited to 3950–7000 Å.
Plotted in the top panel of Fig. 9 is the mean SE SN spectrum with the full
range of the first rotated PC
($0.87\cdot\text{Amplitude}_{3}-0.50\cdot\text{Amplitude}_{2}$) over-plotted
and indicated by shaded regions, while the bottom panel contains the linear
combination of Amplitude2 and Amplitude3 versus phase. Inspection of the
bottom panel reveals that with a Rotation of $\theta=30^{\circ}$, we obtain
good separation between the SNe IIb and SNe Ibc. This is attributed to the
larger contribution of Amplitude3, particularly during the post-maximum
phases. This is also fully consistent with the results of Fig. 8 where PC3 was
identified as primarily (but not entirely) associated with H Balmer features.
Meanwhile Amplitude2, which is primarily made up of He features, contributes
less. Clearly, a linear combination of PC2 and PC3 can reliably identify SNe
IIb and differentiate them from SNe Ib between $\sim+20$ to $+40$ d. There is
only a single interloping point that breaks this dichotomy, and we
subsuquently return to the question of classifying and differentiating SNe IIb
from SNe Ib (see Sect. 6.2). Such interloping points also suggest that the
differences among groupings are not just due to uncertainties (see Sect.
5.3.2).

Figure 10: Results from PCA. Top: Linear combinations of the first two PCs
determined from PCA applied to the spectra extending to $+$70 d and between
6000–7000 Å. Bottom: A linear combination of Amplitude1 and Amplitude2 plotted
versus its orthogonal (30∘) rotation. In addition to the CSP-I SE SN sample,
PCA resutls are shown for the normal Type Ia SN 2013gy (Holmbo et al., 2019)
and unpublished CSP-II spectra of the Type Ic superluminous SN 2015bn
(Phillips et al., 2019). Solid curved lines highlight subtypes of He i-rich SE
SNe (see Sect. 5.3). GMM clusters are indicated with one and two sigma
contours for each cluster using dashed and dot-dashed lines (see Sect. 5.3.1
and Table 3), respectively. To guide the eye, curved arrows are over-plotted
highlighting the temporal evolution of SNe IIb and SNe Ib.

###  5.3 PCA between 6000–7000 Å

As the rotated PCs in Sect. 5.2.2 are able to separate SNe IIb from the SNe Ib
and SNe Ic, we now search for a manner to better separate the groupings of all
three subtypes using a similar methodology. Previously, the biggest
uncertainty when attempting to do this with PC2 and PC3 was contamination from
spurious spectral features associated with ions other than H and He, which
tautologically should be the defining distinction between the SE SN subtypes.
In order to ascertain whether the mixture among groupings of SNe IIb, SNe Ib
and SNe Ic as seen in Fig. 8 are an effect of the spectral wavelength range
adopted in the PCA, we apply PCA to the even more restricted spectral range of
6000–7000 Å. This spectral wavelength range should primarily contain
H$\alpha$, and if present, features associated with
${He\textsc{i}}\leavevmode\nobreak\ \lambda 6678$ and possibly
${He\textsc{i}}\leavevmode\nobreak\ \lambda 7065$. In addition, the narrow
wavelength range ensures that the amplitude of any PCs primarily corresponding
to the shape of the SED (like PC1 encountered in Sect. 5.1) are limited.

Following Sect. 5.2, we identified the linear combinations of PCs (i.e., PC1
and PC2) that represent a rotation creating separation among the different SE
SN subtypes. Following Eq. (3), we once again identified a rotation angle of
$\theta=30^{\circ}$ to be ideal. The results of this portion of our analysis
are presented in Fig. 10. The mean spectrum is plotted in the two upper panels
as a black line, while the left panel shows the first rotated PC
$0.87\cdot\text{Amplitude}_{1}-0.50\cdot\text{Amplitude}_{2}$ and the right
panel displays the (orthogonal) second rotated PC
$0.50\cdot\text{Amplitude}_{1}+0.87\cdot\text{Amplitude}_{2}$. The two rotated
PCs are compared in the bottom panel of Fig. 10, with each point color-coded
by spectral subtype and with the intensity of the shading indicating the
temporal phase.

Figure 10 reveals a separation between the groupings (similar to Fig. 8). SNe
IIb clearly occupy the right half side of the figure with rotated PC
$0.87\cdot\text{Amplitude}_{1}-0.50\cdot\text{Amplitude}_{2}$ $\gtrsim$ 1,
while the SNe Ib and Ic occupy the left-half of the panel, with the SNe Ib in
the upper-quadrant and the SNe Ic in the lower-quadrant. Between the right and
left sides a clear separation appears within the plotted parameter space
devoid of objects and which is highlighted in gray. This region suggests that,
at least for this sample, there is not a clear continuum over the parameter
space covered by these PCs.

Considering the evolution of SE SN subtypes with phase, we first notice in
Fig. 10 that the young SNe IIb located toward the lower/middle right of the
figure evolve in time toward the right side of the quadrant reflected by
increasing values of the second rotated PC
$0.50\cdot\text{Amplitude}_{1}+0.87\cdot\text{Amplitude}_{2}$. Beginning
around maximum and extending over a month, the SNe IIb then follow one of two
distinct tracks, which are highlighted in Fig. 10 with blue-curved arrows. One
track is populated with objects exhibiting PC
$0.87\cdot\text{Amplitude}_{1}-0.50\cdot\text{Amplitude}_{2}$ $\gtrsim$
2.1–3.0 and the other track exhibits values $\lesssim 2.1$. However, after a
month past maximum the two tracks begin to overlap, and as the SNe IIb
continue to evolve with decreasing values of the rotated PCs they are found to
populate the same region of parameter space as old SNe Ib. On the other hand,
the SNe Ib located on the left quadrant of the figure (except for a few
interlopers) first appear with
$0.50\cdot\text{Amplitude}_{1}+0.87\cdot\text{Amplitude}_{2}$ $\lesssim 0.5$.
As time evolves through maximum and beyond, these objects migrate upward with
ever increasing $0.50\cdot\text{Amplitude}_{1}+0.87\cdot\text{Amplitude}_{2}$
of $\gtrsim 1.6$, and then later exhibit lower PCs values. Turning to the SNe
Ic, they occupy the lower left quadrant of Fig. 10, and evolve further left
with phase (although there are a handful of SNe Ib mixed in at early and late
times).

Based on the groupings reflected in Fig. 10, in the case of the SNe IIb and
SNe Ib, the first rotated PC
$0.87\cdot\text{Amplitude}_{1}-0.50\cdot\text{Amplitude}_{2}$ mainly traces H,
while the second rotated PC
$0.50\cdot\text{Amplitude}_{1}+0.87\cdot\text{Amplitude}_{2}$ mainly traces
He. As SNe Ic by definition lack both H and He features, they occupy a
relatively narrow region of the parameter space, especially among the second
rotated PC (He). The evolution along the first rotated PC, tracing mainly H,
can be explained by contamination from an additional feature. The origin of
this feature in SNe Ic spectra remains under debate (see discussion in Sect.
2.3.2).

####  5.3.1 Clustering analysis

In order to explore the inherent clustering in our PCA data and to
statistically test our qualitative identification of the three main groupings,
we used unsupervised learning making use of the scikit-learn package
(Pedregosa et al., 2011) to perform clustering analysis using K-means and
Gaussian Mixture Modeling (GMM). In other words, rather than adopting a by-eye
approach, we wish to quantitatively assess how well PCA can distinguish
between the SE SN subtypes using the original (single template) PCs.

Using three component models and as described with some detail at the end of
Appendix C, both K-means and GMM readily identified three clusters centered at
the qualitatively identified location of each grouping from the previous
section. We tabulate the completeness of each algorithmically derived cluster
compared to the known labels (i.e., IIb, Ib, Ic) in Table 3, and which are IIb
$\sim 80\%$, Ib $\sim 50\%$, and Ic $\sim 95\%$, respectively. Results from
this analysis consisting of the one and two sigma contours for each Gaussian
component of the GMM fit are included in Fig. 10.

Figure 11: Time-evolution of the Doppler velocity measured from the minimum of
the He i $\lambda 5876$ feature. Each object is plotted with a unique
combination of symbol and color where the color indicate subtype, with SN Ib
plotted in blue and SN Ib in green. The three SN Ib plotted with filled
symbols are SNe 2005bf, 2007Y, and 2007kj, and are identified as ‘flat-
velocity’ SNe IIb by Folatelli et al. (2014), while others have referred to
such objects as transitional SNe Ib/c (e.g., Hamuy et al., 2002; Stritzinger
et al., 2009). Using classification criteria based on the pEW of Features 4
and 5 as discussed in this paper these objects are consistent with a Type Ib
classification (see also Liu et al. 2016 and Prentice & Mazzali 2017 for
additional discussion).

####  5.3.2 Interlopers among SNe Ib, IIb, and Ic groupings

Upon examination of interlopers appearing among the groupings revealed in the
bottom panel of Fig. 10, we notice some interesting patterns. Most noticeably,
a handful of SNe Ib at early times intrude within the SNe IIb grouping and
have rotated PC $0.87\cdot\text{Amplitude}_{1}-0.50\cdot\text{Amplitude}_{2}$
$\gtrsim$ 1\. These are SN 2004ew (single spectrum at $+26$ d) and SN 2007kj
(spectra at $+9$ d and +14 d). Two more SNe Ib including SN 2005bf and SN
2007Y also appear close, but do not violate our arbitrary limit. With the
exception of SN 2004ew, the rest of these objects are part of a sub-group of
SNe Ib that exhibit a flat-velocity evolution of the He i $\lambda 6678$ line,
as illustrated in Fig 11. This sub-group appears to show evidence of a weak
feature at around 6250 Å, which could be high-velocity H$\alpha$ (see, e.g.,
Folatelli et al., 2014). Although SN 2004ew does not belong to this flat-
velocity SN Ib sub-group, it also initially shows the same absorption feature
at $\sim 6250$ Å. By the second spectrum at $+40$ d, this feature disappears
and SN 2004ew returns to being firmly among the other SNe Ib in Fig. 10. The
absorption related to this feature seems to drive the positive evolution along
the x-axis (first rotated PC) in Fig. 10. As can be seen in the top left panel
of the Fig. 10, there is a deep red trough at around this position, which is
positively correlated with the first rotated principal component
$0.87\cdot\text{Amplitude}_{1}-0.50\cdot\text{Amplitude}_{2}$ and explains the
interlopers. It is important to point out that, where we do have spectral
coverage, they are found to evolve back over time toward the SNe Ib grouping,
and do not remain among the SNe IIb.

Figure 12: Spectroscopic time-series from $+$7 d to $+$23.2 d of the Type Ic
SN 2005aw. The spectra are plotted in velocity space with the segments of data
containing the He I $\lambda\lambda$5876, 6678, and 7065 features plotted on
top of each other. As revealed from the PCA results and consistent the data,
SN 2005aw exhibits discernible features likely formed by He i with
$-v_{abs}\sim 8,500$ km s-1.

Looking at the interlopers belonging to the SNe Ic subtype, the most
noticeable are SN 2005aw and SN 2009ca having, respectively, the highest and
most negative values of the second rotated PC, corresponding to He i. As
discussed previously, the origin of the features in this region is under
debate, but an association with H and He is not expected, especially a strong
one. Nevertheless, as can be seen in Fig. 12, SN 2005aw shows evidence of
early He features. Therefore, it is not surprising to find it as an interloper
among the SNe Ib and having a high value of the second rotated PC. In the case
of the other interloper, SN 2009ca, it is known to be an unusually bright SN
Ic with peculiar features (see Taddia et al., 2018, Paper 3). Hence, its
separation from the rest of the grouping is not surprising.

Looking at the interlopers belonging to SNe IIb, we primarily see a few early
phase points near the early SNe Ib and SNe Ic groupings, which all belong to
the type IIb SN 2009K. The earliest spectrum of SN 2009K taken on $-18.4$ d
actually lacks Balmer features, however they do emerge by $-13.5$ d, which
explains the early divergence.

Switching gears, we note that the two very late-phase SNe IIb points appearing
in the same region as the SNe Ib are in fact expected as the former evolve to
resemble the latter during the post-maximum epochs as the H features dissipate
in SNe IIb. A similar convergence between SNe IIb and SNe Ib was seen in the
late-time pEW evolution of the Feature 5. For this reason, spectral epochs
later than $+$40 d were not included in Fig. 9. On the whole, SNe IIb seem to
be the most well separated, which we subsuquently return to in the discussion.

The fact that distinguishing spectral features can explain the interlopers
suggests that the differences are real, and not merely driven by uncertainty
of the spectra or PCA analysis. Therefore, we have not considered the
uncertainties of the PC amplitudes in the discussion, which is a nontrivial
undertaking.

##  6 Discussion

###  6.1 Comparison with literature findings

Here we examine our line diagnostic results presented in Sect. 3 and Sect. 4
to average behavior of key features reported in the literature by the Berkeley
SN group (Matheson et al., 2001; Shivvers et al., 2019), the CfA SN group
(Modjaz et al., 2014; Liu & Modjaz, 2014), the (i)PTF survey (Fremling et al.,
2018), and also Prentice & Mazzali (2017) who compiled a (mostly) literature-
based sample. Considering features studied in common by the literature, we
find broadly consistent results which we now summarize.

####  6.1.1 ${Fe\textsc{ii}}$ & ${Co\textsc{ii}}$: Feature 3

Mean $-v_{abs}$ velocities of Feature 3 reported by Liu et al. (2016) are
systematically higher in SNe Ib than in SNe IIb. We find for the CSP-I sample
a similar trend up to maximum (see Fig. 6), however during post maximum
phases, the sample shows no bifurcation between the two subtypes. Turning to
SNe Ic, we find that around maximum they exhibit similar mean $-v_{abs}$
values as the SNe IIb, while SNe Ib mean values are $\gtrsim 2,000$ km s-1
higher. Due to the lack of early SN Ic spectra in the CSP-I sample, we are
unable to comment on premaximum phases. However beginning around a week past
maximum, the average values in our SNe Ic tend to be larger than in SNe Ib. We
remind the reader that Feature 3 in SNe Ic may have a nonnegligible
contribution from Co ii, and therefore this comparison should be approached
with caution. Finally, to our knowledge, no measurements on the pEW of Feature
3 are available in the literature for us to compare.

####  6.1.2 He i: Features 4, 6, 7

He i lines corresponding to Features 4, 6, and 7 are the most commonly studied
SE SN spectral features, with broad consensus in the literature on their
characterization. In general, and as demonstrated in Fig. 5, the He i features
emerge earlier in SNe Ib and consistently exhibit higher pEW mean values. As
highlighted in the Feature 4 panel of the figure with a horizontal dashed
line, the differences between the pEW values of the two subtypes prior to
maximum is significant, while mean pEW values for Features 6, 7 (and 8) reveal
somewhat less significant differences. In the weeks following maximum
brightness and as the photospheres of the SNe IIb recede into deeper He-rich
layers of ejecta, prominent He i features emerge with mean pEW values fully
consistent with those of the SNe Ib. Again, this is in agreement with findings
in the literature (see Liu et al., 2016; Prentice & Mazzali, 2017; Fremling et
al., 2018; Shivvers et al., 2019).

Turning to Doppler velocity, work by Liu et al. (2016), Prentice & Mazzali
(2017) and Fremling et al. (2018) indicated SNe Ib tend to exhibit higher
values prior to maximum than SNe IIb, while during post-maximum phases their
mean values are similar. Similarly, and as demonstrated in Fig. 6, we also
find agreement between their mean values beginning from maximum and beyond.
Unfortunately, do to a dearth of data, we are unable to comment on premaximum
phases.

####  6.1.3 Feature 5: Si ii and/or H$\alpha$

Liu et al. (2016) and Prentice & Mazzali (2017) report distinct pEW values
between the SNe IIb and SNe Ib at all phases. As shown in Fig. 5 and
highlighted by a horizontal dashed line, the SNe IIb in the CSP-I sample do
exhibit consistently higher pEW values than the SNe Ib, however, given the
large dispersion of values measured for the SNe Ib there is some overlap
between $-$5 d to $+$15 d. In the case of SNe Ic, we computed mean pEW values
significantly less than those of the SNe IIb at all phases, while between $+$0
d to $+$10 d they appear somewhat less than SNe Ib, and then between $+$10 d
to $+$30 d they are found to overlap.

Liu et al. (2016) and Prentice & Mazzali (2017) report higher $-v_{abs}$
values in their SNe Ib samples relative to SNe IIb. This is contrary to the
results of Feature 5 presented in Fig. 6, which indicates quite consistent
rolling mean values between the two subtypes. In fact, at the earliest days of
our coverage, the SNe IIb exhibit somewhat higher mean values, though we note
this is based on smaller sample size than that considered by Liu et al. and
Prentice & Mazzali. Beyond maximum, the SNe IIb and SNe Ib exhibit similar
rolling mean $-v_{abs}$ values for around a month.

####  6.1.4 Feature 9: O i

Between maximum and ${\sim}$two weeks post maximum, Fig. 5 reveals that the
mean pEW values of Feature 9 are $\gtrsim 50\leavevmode\nobreak\ \AA$ higher
in the SNe Ic, followed by SNe Ib and then SNe IIb. Similarly, Fig. 6
indicates Feature 9 systematically exhibits higher mean $-v_{abs}$ values in
SNe Ic by a few thousand km s-1, followed by SNe Ib and then SNe IIb. These
results are in line with findings presented in the literature (cf. Matheson et
al., 2001; Liu et al., 2016; Fremling et al., 2018; Shivvers et al., 2019).

####  6.1.5 Feature 10: Ca ii

Previous sample studies of SE SN spectra do not consider the Ca ii NIR
triplet. Only Matheson et al. (2001), who previously reported (based on a
handful of spectra taken at different, mostly post-maximum phases) a mean
velocity in SNe Ic of $10,800\pm 800$ km s-1, which is significantly less than
what we find for CSP-I SNe Ic. Even upon removal of a single object in our
sample with very high velocities, we find mean $-v_{abs}$ around maximum light
on the order of $\gtrsim 14,000$ km s-1.

###  6.2 Classification

####  6.2.1 Line diagnostics

Here we consider a few rules of thumb to follow that others may find useful in
their quest to subtype SE SN based on either premaximum and/or post-maximum
phase spectra based on a few of the key findings presented in Sect. 3.2.
Considering the pEW trends of the CSP-I SE SN shown in Fig. 5, we suggest that
the pEW values of Features 4 and 5 in premaximum spectra can reliably
differentiate between SNe IIb and SNe Ib. As indicated by the dashed line in
the panel of Feature 4, SNe Ib consistently exhibit pEW values in excess of
$\gtrsim 75$ Å, while the other subtypes typically exhibit pEW values below
this cutoff. The panel of Feature 5 shows a clear bifurcation at all phases
between SNe IIb and SNe Ic with the latter consistently exhibiting pEW values
below 100 Å, while the former exhibit values in excess of 120 Å. We suggest
Feature 5 can serve as a reliable indicator to differentiate SNe IIb from SNe
Ib, beginning from $+$20 d onward. This is contrary to the advice of Liu et
al. (2016) who advocated that Feature 5 provides a clear indication between
SNe IIb and SNe Ib at all phases, and this is due to those SNe Ib that do
exhibit (modest) H$\alpha$ at early times. These objects are responsible for
the significant variance of the premaximum rolling mean pEW values. Feature 5
being a robust discriminator during post-maximum phases is in agreement with
our PCA analysis (see below). Finally, we note that Feature 9 (O i) also
provides some indication of the SE SN subtype as discussed in Sect. 6.1.4. In
particular, the velocity of Feature 9 in SNe Ic is consistently higher than
exhibited in the SNe Ib and IIb, and at early times, SNe Ic show larger pEW
values as compared with the other subtypes.

####  6.2.2 PCA

The PCA analysis presented in Sect. 5.2 and the results obtained in Fig. 9
demonstrate we are able to reliably classify SNe IIb distinctly from SNe Ib
using a combination of phase, PC2, and PC3, with results being strongest at
phases between $+$20 d to $+$40 d. This result mirrors previous findings of
Liu et al. (2016) and Prentice & Mazzali (2017) whom pointed out that the pEW
value of H$\alpha$ is a suitable diagnostic to distinguish between SNe IIb and
Ib. Appendix C contains a practical guide on how to leverage PCA for those
wishing to distinguish between SNe IIb and Ib using one or more spectra taken
at any phase. Whether SNe Ib discovered during these post-maximum epoch are
SNe Ib, or are simply SNe IIb that evolved to become SNe Ib has been
questioned previously (e.g., Milisavljevic et al., 2013). We note that all but
one SN Ib in our sample (SN 2004ew) have early spectra indicating they are not
traditional SNe IIb with strong H features. This suggests there is a clear
separation of these observables between the two subtypes. Indeed, as estimated
in Sect. 5.3.1, interlopers from the SNe IIb into SNe Ib subtype can be found
with $\sim 80\%$ completeness.

Contrary to Modjaz et al. (2014) and Liu et al. (2016) who report a similar
separation, we did not re-classify any of our SNe, nor did we exclude
transitional or peculiar SNe from our PCA as was done by Williamson et al.
(2019). Nevertheless, we obtain a similar result and find a clear separation
between SNe IIb and SNe Ib, indicating that the re-classification in these
papers can not account for the observed differences.

As discussed in Sect. 5.3.2 and depicted in Fig. 10, we naturally find more
scatter and some outliers within group properties such as PCs, pEWs, and
$-v_{abs}$ values, even though they are accountable by peculiar outlier SNe.
Specifically, for separating between SNe IIb and SNe Ib, the main exceptions
to the PCA based classification, which were transitory outliers are: 1) SNe Ib
with early and weak high-velocity H features all but one of which exhibit He i
lines that slowly evolve over time, and 2) SNe IIb with H features that are
stronger than SNe Ib, but relatively weak compared to the rest of the SNe IIb.
Aside from these two interloping _sub_ -subtypes, our results indicate that
SNe Ib, IIb, and Ic are distinct. It can be argued that these interlopers
represent a continuum from SN IIb with weak H to SN Ib with weak H, or they
can be interpreted as distinct subtypes themselves. Future PCA results from
larger datasets and detailed study of these interlopers are required to
disentangle the two possibilities.

To serve as a comparison, rotated PCs of the normal Type Ia SN 2013gy relative
to the SE SN mean spectrum are also plotted in Fig. 10 (gray stars). The PCs
of this normal SN Ia are clearly separated from the SE SNe, and therefore,
such objects are not expected to serve as a source of confusion. However, PCs
of superluminous SNe Ic (gold stars), as exemplified by SN 2015bn, do appear
within the same region as less luminous SNe Ic.

Shahbandeh et al. (2022) showed that PCA of NIR SE SN spectra can
differentiate SNe Ib from Ic. While a good degree of differentiation was also
seen in our un-tuned optical results (Sect. 5.3), the difference between SNe
Ib and Ic was not as robust as in the NIR, nor as the difference between SNe
IIb and Ibc (although we specifically delved deeper on the latter). One reason
for this could be that in the optical, variance among SNe Ib and Ic is not as
high as variance among SNe IIb and Ibc. Since we did not consider higher order
PCs, we may have missed PCs which better differentiate SNe Ib from SNe Ic. As
Shahbandeh et al. (2022) notes, NIR PCA is very good at this task due to
features at 1 and 2 micron produced by He and/or C, while the optical is
better for H. Similar to our investigation of H and He when aiming to classify
SNe IIb, our results could be extended to focusing on optical regions
containing He and C, and extending to higher order PCs. This may tease out to
search for an optical counterpart to the NIR results of Shahbandeh et al.
(2022).

The significance of our PCA based result is that it is not reliant upon the
particular methodology used to construct pEWs measurements or infer
velocities. As noted by Fremling et al. (2018), the choice of how to make
these measurements can be the most significant uncertainty in the analysis.
Our PCA methodology is fully reproducible and agnostically applicable to any
past or future SE SN samples. Unlike spectral template matching methods widely
used in the literature, the groupings can be entirely re-constructed from a
given sample of SE SNe without relying on external data. Thus, it can be used
as an independent verification for classification of SE SNe, particularly for
SNe IIb and Ib, and especially if the initial classification is done via
another method such as spectral template matching.

Future efforts related to PCA and SE SNe spectroscopic samples could focus on
applying a similar analysis using an expanded sample. A guide to using PCA for
this purpose, including a discussion of the practical considerations, is
provided in Appendix C.

##  7 Summary

We presented a detailed analysis of the CSP-I SE SN optical spectroscopic
sample. Key completed tasks and highlights of the analysis include:

  * •

The construction of mean spectra for each SE SN subtype at distinct phases.
Prevalent spectral line features were then identified in the mean spectra of
each SE SN subtype. Spectral synthesis modeling using SYNAPPS enabled the
identification of the parent ions associated with the designated features.
This includes the potential inclusion of: Si II $\lambda$6355 in some SNe IIb,
H$\alpha$ in some SNe Ib, and a contribution to the $\sim 6150$ Å feature in
SNe Ic by an unknown species.

  * •

Pseudo-equivalent width (pEW) width and Doppler absorption velocity
($-v_{abs}$) measurements were measured for the spectral features in all
photospheric phase spectra. With these measurements rolling mean values for
both spectral indicators were determined and Spearman’s rank correlation
coefficient matrices constructed.

  * •

Adopting a PCA formalism, we devise a method to reliably classify SE SNe using
a single spectrum taken during the photospheric phase. Using linear
combinations of key principle components, we identify distinct groupings
between the different SE SN subtypes. Moreover, based on a single post maximum
spectrum we demonstrate the ability of PCA to provide a robust means to
disentangle SNe IIb and Ib. This finding reflects results already in the
literature suggesting the pEW of H$\alpha$ can be used as a proxy to
distinguish between SNe IIb and Ib (see Liu et al., 2016).

  * •

Outliers among the distinct SE SN groupings are identified and considered in
detail. In summary we find: (i) two SNe IIb subtypes, (ii) two SNe Ib subtypes
and (iii) a single SN Ic with He i features all exhibiting similar $-v_{abs}$
values. The difference between the two SNe IIb subtypes is the strength of the
H$\alpha$ feature. In the case of the SNe Ib, along with those objects that
fall within the traditional classification, our PCA analysis indicates the
existence of a second type of SNe Ib exhibiting weak, high-velocity H$\alpha$
along with He i features characterized by a flat-velocity evolution. However,
as indicated by our PCA results those SNe Ib that exhibit H$\alpha$ eventually
evolve to appear as a normal SNe Ib weeks after maximum.

In this paper, we demonstrated PCA provides a avenue to gain deeper insights
into the different SE SN subtypes and a means to classify SE SNe free of human
bias. Further efforts should aim to study the full public SE SN sample and
include NIR spectroscopy.

###### Acknowledgements.

The Aarhus supernova group is funded by the Independent Research Fund Denmark
(IRFD, grant numbers 8021-00170B and 10.46540/2032-00022B), and by the VILLUM
FONDEN (grant number 28021). L.G. acknowledges financial support from the
Spanish Ministerio de Ciencia e Innovación (MCIN), the Agencia Estatal de
Investigación (AEI) 10.13039/501100011033, and the European Social Fund (ESF)
”Investing in your future” under the 2019 Ramón y Cajal program
RYC2019-027683-I and the PID2020-115253GA-I00 HOSTFLOWS project, from Centro
Superior de Investigaciones Científicas (CSIC) under the project PIE
20215AT016 and LINKA20409, and the program Unidad de Excelencia María de
Maeztu CEX2020-001058-M. The CSP has received support from the National
Science Foundation (USA) under grants AST–0306969, AST–0607438, AST–1008343,
AST–1613426, AST–1613455, and AST-1613472. This research used resources from
the National Energy Research Scientific Computing Center (NERSC), which is
supported by the Office of Science of the U.S. Department of Energy under
Contract No. DE-AC02-05CH11231. This research has made use of the NASA/IPAC
Extragalactic Database (NED), which is operated by the Jet Propulsion
Laboratory, California Institute of Technology, under contract with the
National Aeronautics and Space Administration.
Lead authors contribution statement: Maximilian Stritzinger (MS) wrote the
abstract, the majority of Sections 1 to 4, significant portions of Sect. 5,
all of Sect. 6.1, portions of 6.2, and all of Sect. 7. Emir Karamehmetoglu
(EK) performed the clustering analysis, co-wrote Sect. 5 (PCA analysis) and
the discussion in Sect. 6.2, and the entirety of Appendix C. The rest of the
paper, including data preparation, data analysis, SYNAPPS calculations and the
initial figures, were created by Simon Holmbo (SH). SH had input on the text
and wrote several paragraphs. SH led all aspects of the paper during his PhD
thesis, while iteratively working through data analysis, figures, and
discussion in each section with MS and EK. EK and MS made changes to and re-
plotted Figs. 3, 5, 6, 7, 8, and 9, as well as revising the draft following
co-author comments.

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Table 1: Spectroscopic sample of CSP-I SE SNe.

SN | Redshifta | Spectral | No. of | Phases
---|---|---|---|---
| $z$ | Type | Spectra |
2004ew | 0.0218 | Ib | 4 |  $+$26.1, $+$58.3, $+$74.9, $+$80.7
2004ex | 0.0176 | IIb | 5 |  $-$3.9, $+$19.7, $+$28.4, $+$36.3, $+$41.2
2004fe | 0.0179 | Ic | 4 |  $+$8.6, $+$9.7, $+$18.5, $+$35.1
2004ff | 0.0227 | IIb | 3 |  $+$13.9, $+$22.6, $+$30.4
2004gq | 0.0065 | Ib | 3 |  $-$5.1, $+$0.8, $+$23.6
2004gt | 0.0055 | Ic | 4 |  $-$1.6, $+$21.3, $+$45.2, $+$49.2
2004gv | 0.0199 | Ib | 1 |  $-$6.6
2005Q | 0.0224 | IIb | 2 |  $-$0.5, $+$7.3
2005aw | 0.0095 | Ic | 5 |  $+$7.3, $+$11.3, $+$16.3, $+$19.2, $+$23.2
2005bf | 0.0189 | Ib | 9 |  $-$7.1, $-$2.2, $+$0.8, $+$4.6, $+$22.3, $+$27.2, $+$46.0, $+$47.8, $+$51.8
2005bj | 0.0222 | IIb | 2 |  $+$0.9, $+$3.9
2005em | 0.0260 | Ic | 1 |  $-$0.1
2006T | 0.0081 | IIb | 11 |  $+$0.2, $+$14.1, $+$20.0, $+$22.9, $+$28.9, $+$36.8, $+$38.7, $+$47.6, $+$61.6, $+$68.4, $+$70.3
2006ba | 0.0191 | IIb | 3 |  $+$3.7, $+$20.3, $+$28.1
2006bf | 0.0239 | IIb | 5 |  $+$7.2, $+$10.1, $+$29.6, $+$30.5, $+$32.6
2006ep | 0.0151 | Ib | 3 |  $+$19.0, $+$32.8, $+$33.8
2006fo | 0.0207 | Ib | 3 |  $+$1.6, $+$3.5, $+$18.3
2006ir | 0.0200 | Ic | 2 |  $+$19.1, $+$42.7
2006lc | 0.0162 | Ib | 1 |  $+$1.2
2007C | 0.0056 | Ib | 9 |  $-$1.4, $+$3.5, $+$16.5, $+$28.5, $+$35.4, $+$41.4, $+$48.3, $+$63.1, $+$92.0
2007Y | 0.0046 | Ib | 12 |  $-$12.4, $-$6.4, $+$0.6, $+$7.5, $+$10.5, $+$15.4, $+$22.4, $+$40.3, $+$230.8, $+$232.7, $+$257.6, $+$271.5
2007ag | 0.0207 | Ib | 2 |  $+$7.23, $+$10.1
2007hn | 0.0273 | Ic | 4 |  $+$9.8, $+$25.2, $+$26.2, $+$56.3
2007kj | 0.0179 | Ib | 5 |  $-$4.2, $-$2.2, $+$8.6, $+$13.5, $+$40.0
2007rz | 0.0130 | Ic | 2 |  $+$7.4, $+$31.9
2008aq | 0.0080 | IIb | 12 |  $-$5.3, $+$8.6, $+$13.6, $+$14.6, $+$22.5, $+$25.4, $+$37.3, $+$51.2, $+$54.1, $+$66.0, $+$76.9, $+$308.2
2008gc | 0.0492 | Ib | 6 |  $+$9.1, $+$9.9, $+$14.7, $+$22.4, $+$28.9, $+$117.3
2008hh | 0.0194 | Ic | 3 |  $+$2.1, $+$16.8, $+$31.4
2009K | 0.0117 | IIb | 8 |  $-$18.4, $-$13.5, $+$3.4, $+$6.3, $+$20.1, $+$38.8, $+$50.7, $+$282.3
2009Z | 0.0248 | IIb | 8 |  $-$6.0, $-$3.0, $+$0.9, $+$9.7, $+$10.6, $+$11.7, $+$28.2, $+$40.9
2009bb | 0.0099 | Ic-BL | 18 |  $-$1.4, $-$0.5, $+$4.6, $+$8.4, $+$16.6, $+$18.4, $+$19.3, $+$23.3, $+$24.3, $+$27.5, $+$31.3, $+$32.2, $+$45.0, $+$53.9, $+$61.9, $+$282.9, $+$283.9, $+$307.5
2009ca | 0.0957 | Ic-BL | 3 |  $+$2.5, $+$11.6, $+$16.2
2009dp | 0.0232 | Ic | 3 |  $+$2.0, $+$3.0, $+$34.2
2009dq | 0.0047 | IIb | 2 |  $-$8.9, $+$14.0
2009dt | 0.0104 | Ic | 2 |  $-$6.4, $+$16.4

aafootnotetext: Redshifts were retrieved from NED, or as determined from host-
galaxy emission lines in the optical spectra of SNe 2007hn, 2008gc, 2009ca,
and 2009dq.

Table 2: Refined spectral feature IDs and line diagnostic fitting parameters. Feature ID | Ion(s) | Average $\lambda_{rest}$ | Detection boundaries
---|---|---|---
|  | [Å] | intervals [Å]
SNe IIb
F1a | Ti II, H$\gamma$, Fe II | 4540 | 3850-4100, 4400-4700
F1b | Ti II, Fe II, He I | 4472 | 3850-4100, 4400-4700
F2a | H$\beta$ | 4861 | 4400-4700, 4800-5100
F2b | He I, Fe II | 5018 | 4400-4700, 4800-5100
F3 | Fe II | 5169 | 4800-5100, 5100-5700
F4 | He I, Na I | 5887 | 5000-5700, 5650-6050
F5a | Si II(?) | 6355 | 5650-6050, 6050-6750
F5b | H$\alpha$ | 6563 | 5650-6050, 6050-6750
F6 | He I | 6678 | 6250-6520, 6520-6750
F7 | He I | 7065 | 6350-6750, 6850-7100
F8 | He I | 7281 | 6850-7100, 7100-7450
F9 | O I | 7774 | 7100-7450, 7500-7900
F10 | Ca II | 8567 | 7500-7900, 8450-8800
SNe Ib
F1b | Ti II, Fe II, He i | 4472 | 3850-4100, 4400-4700
F2b | He i, Fe II | 5018 | 4400-4700, 4800-5100
F3 | Fe II | 5169 | 4800-5100, 5100-5700
F4 | He I, Na I | 5887 | 5250-5610, 5700-6050
F5a | Si II | 6355 | 5650-6050, 6050-6750
F5b | H$\alpha$(?) | 6563 | 5650-6050, 6050-6750
F6 | He I | 6678 | 6250-6520, 6520-6750
F7 | He I | 7065 | 6350-6750, 6850-7100
F8 | He I | 7281 | 6850-7100, 7100-7450
F9 | O I | 7774 | 7100-7450, 7500-7900
F10 | Ca II | 8567 | 7500-7900, 8450-8800
SNe Ic
F1b | Ti II, Fe II | 4472 | 3850-4100, 4400-4700
F2b | Fe II | 5018 | 4400-4700, 4800-5100
F3 | Fe II, Co ii | 5169 | 4800-5100, 5100-5700
F4 | Na I | 5887 | 5250-5610, 5700-6050
F5a | Si II | 6355 | 5650-6050, 6050-6750
F9 | O I | 7774 | 7100-7450, 7500-7900
F10 | Ca II | 8567 | 7500-7900, 8450-8800
Table 3: Results of K-means and GMM clustering analysis. Subtype | Total | K-means | GMM | K-means | GMM
---|---|---|---|---|---
| counts | matches | matches | completeness | completeness
IIb | 49 | 37 | 40 | 0.76 | 0.82
Ib | 46 | 21 | 20 | 0.46 | 0.43
Ic | 43 | 41 | 40 | 0.95 | 0.93

##  Appendix A Spectral line identification with SYNAPPS

Here we summarize the results presented by Holmbo (2018), who computed SYNAPPS
synthetic spectra to match the mean spectra discussed in Sect. 2.1, in order
to obtain plausible line identifications. Following the standard protocols
(see Thomas et al. 2011), SYNAPPS requires a number of basic input parameters
including: a velocity range encompassing the outer and lower bounds of the
photosphere ($v_{ph}$), a black-body temperature ($T_{BB}$), and an input list
of ions each with their own set of input parameters. The input list of ions
include: H i, He i, C ii, O i, Na i, Mg ii, Si ii, S ii, Ca ii, Ti ii, Fe ii,
Co ii and Ni ii. All but S ii and C ii are found to most likely contribute to
the formation of at least one or more of Features 1–10 (see Fig. 1) for at
least one of the SE SN subtypes (see below). We note that there is also
evidence for C ii appearing as a notch in the mean SN Ic spectra. This is
highlighted with a vertical dashed line in Fig. 4, though the line
identification in our opinion is not significant enough to warrant a Feature
designation in the present analysis.

As for an initial $v_{ph}$ value we adopted the Doppler velocity at maximum
absorption ($-v_{abs}$) measured from the Fe ii $\lambda 5169$ (i.e., Feature
3) in the maximum light mean spectra, while the lower bound was taken to be
5,000 km s-1 and the upper bound was set to 30,000 km s-1. Fe ii $\lambda
5169$ provides a reasonable proxy for the velocity of the bulk of the SN
ejecta (see Paper 3, and references therein). SYNAPPS tunes the $v_{ph}$ value
in the fitting process and overall found values between 7,000–10,000 km s-1
for all of the mean spectra, except the $+$14 d SN Ic mean spectrum, which has
a best-fit spectrum characterized by $v_{ph}\approx 12,600$ km s-1. $T_{BB}$
was initially set to 7,000 K and then it was fine tuned by SYNAPPS to values
between 6,000 K to 8,000 K for all of the best-fit mean spectra, except the
$+$21 d SN Ic spectrum, which best-fit model has $T_{BB}$ $\approx$ 5,000 K.

Each input ion also has an accompanying set of input parameters. These include
the line opacity ($\tau$) at a specified reference velocity ($v_{ref}$), upper
and lower velocity limits ($v_{max}$ and $v_{min}$), a value for the
parameterization of the opacity profile (here with an exponential e-folding
length $v_{e}$), and an excitation temperature ($T_{exc}$). Values of these
parameters can vary significantly between the various ions within a single
synthetic spectrum, as well as significant variation of any given ion over the
range of the spectral evolution covered by our set of mean spectra. SYNOW was
used to obtain initial values for the input parameter set used in our SYNAPPS
calculations.

We found the most efficient manner to perform the calculations was to
initially begin with the $+$21 d mean spectrum of each subtype and then work
backwards to the earlier phases, where for each successive spectrum the
results from the previous spectrum were used to guide the range of the various
input parameters. Once the SNe IIb spectra were modeled we continued with the
SNe Ib mean spectra, which were modeled including all of the same ions as in
the case of the SNe IIb. This was followed by modeling of the mean SNe Ic
spectra omitting both H i and He i.

The SYNAPPS spectra for each of the SE SN subtypes are plotted in Figs. 13–15.
This includes the SYNAPPS fit to each of the template spectra, as well as the
model spectrum of each of the individual ions. Comparison of the templates to
the synthetic spectra reveals reasonable matches for the majority of Features
1–10, though in some cases there is some ambiguity. The ions identified
contributing in whole or partially to Features 1–10 are listed in Table 4.

Table 4: Spectral features 1 through 10. Feature ID | Ion |  $\lambda_{rest}$ [Å]
---|---|---
SNe IIb
F1 | H$\gamma$ |  $\lambda$4340
| Fe II | forest of lines
| Ti II | forest of lines
| He I |  $\lambda$4471
F2 | H$\beta$ |  $\lambda$4861
| Fe II |  $\lambda$4924, $\lambda$5018
| He I |  $\lambda$4922
F3 | Fe II |  $\lambda$5169
F4 | He I |  $\lambda$5876
| Na I |  $\lambda\lambda$5890, 5896
F5 | Si II(?) |  $\lambda$6355
| H$\alpha$ |  $\lambda$6563
F6 | He I |  $\lambda$6678
F7 | He I |  $\lambda$7065
F8 | He I |  $\lambda$7281
F9 | O I |  $\lambda$7774
F10 | Ca II |  $\lambda\lambda$8498, 8542, 8662
SNe Ib
F1 | Fe II | forest of lines
| Ti II | forest of lines
| He I |  $\lambda$4471
F2 | Fe II |  $\lambda$4924, $\lambda$5018
| He I |  $\lambda$4922
F3 | Fe II |  $\lambda$5169
F4 | He I |  $\lambda$5876
| Na I |  $\lambda\lambda$5890, 5896
F5 | Si II |  $\lambda$6355
| H$\alpha$(?) |  $\lambda$6563
F6 | He I |  $\lambda$6678
F7 | He I |  $\lambda$7065
F8 | He I |  $\lambda$7281
F9 | O I |  $\lambda$7774
F10 | Ca II |  $\lambda\lambda$8498, 8542, 8662
SNe Ic
F1 | Fe II | forest of lines
| Ti II | forest of lines
F2 | Fe II |  $\lambda$4924, $\lambda$5018
F3 | Fe II, Co ii |  $\lambda$5169, $\lambda$5526
F4 | Na I |  $\lambda\lambda$5890, 5896
F5 | Si II |  $\lambda$6355
F9 | O I |  $\lambda$7774
F10 | Ca II |  $\lambda\lambda$8498, 8542, 8662
Figure 13: SYNAPPS fits (red lines) computed for mean template spectra (black
lines) representing epochs of $-7$ d, $+0$ d, $+7$ d, $+14$ d and $+21$d. Each
mean template spectrum was computed using data obtained within $\pm 3.5$ days
relative to its specific epoch. Spectral features attributed to each ion are
also plotted in black. Figure 14: SYNAPPS fits (red lines) computed for SE SN
template spectra (black lines) representing epochs of $-7$ d, $+0$ d, $+7$ d,
$+14$ d and $+21$d. Each mean template spectrum was computed using data
obtained within $\pm 3.5$ days relative to its specific epoch. Spectral
features attributed to each ion are also plotted in black. Figure 15: SYNAPPS
fits (red lines) computed for SE SN template spectra (black lines)
representing epochs of $+0$ d, $+7$d, $+14$ d and $+21$ d. Each mean template
spectrum was computed using data obtained within $\pm 3.5$ days relative to
its specific epoch. Spectral features attributed to each ion are also plotted
in black.

##  Appendix B Spearman’s rank pEW and Doppler velocity correlation
coefficients

With our pEW measurements in hand, we examined the extent of the correlations
between different pairs of Features 1–10. To visualize the large amount of
information encoded within the pEW measurements and to obtain quantitative
measurements of the strength of correlation among various pairs of pEW
parameters, Spearman’s rank correlation coefficients ($\rho$) were computed.
The results of this analysis are summarized in Fig. 16. In this case, each SE
SN subtype has its own panel. Within the off-diagonal triangle containing the
$\rho$ values determined from spectra obtained up to the first three weeks
relative to the epoch of $B$-band maximum. Color-coding provides an indication
of the degree of correlation with lighter colors indicating higher degrees of
correlation or anticorrelation. Quantitatively, pairs with $\rho$ values
greater than $0.4$ are considered to be moderately to highly correlated, while
those with $\rho$ values less than $-0.4$ are considered to be moderately to
highly anticorrelated. Furthermore, pairs with $\rho$ values between $-0.4$ to
$0.4$ are of low correlation, while pairs with $\rho$ values characterized by
p-values below 0.1 are considered to be of low statistical significance and
are shown in Fig. 16 by gray.

Examination of the three panels indicates that a handful of the features are
correlated, however determining whether or not any correlation is due to a
particular physical relationship between a given pair of features is
difficult. We note the following findings:

  * •

In SNe Ic the pair of features that are most correlated are Features 1 (Fe ii,
Ti II) and Feature 2 (Fe ii), in SNe Ib they are Feature 7 (He i) and Feature
8 (He i), and in SNe IIb, Feature 1 (Fe ii, H$\gamma$, Ti ii, He i) and
Feature 3 (Fe ii) display the highest correlation.

  * •

The diagonal of the SNe IIb correlation coefficients panel indicates that the
Feature 6 (He i) presents a high degree of correlation with Features 7 and 8.
A similar trend is also seen in the SNe Ib, but to a lesser degree, while as
mentioned previously, Features 7 and 8 are highly correlated.

  * •

Feature 4 (He i $\lambda$5876 and Na i $\lambda\lambda$5890, 5896) and Feature
7 (He i $\lambda$7065) are highly correlated with high statistical
significance in SNe IIb at all phases, while SNe Ib exhibit moderate to low
correlation. The high degree of correlation between these two features
therefore suggests He i is a significant contributor to Feature 4 rather than
Na i in SNe IIb, since Na i does not, at all, contribute to Feature 7.

  * •

The pEW measurements of Feature 4 versus those of Feature 5 (H$\alpha$, Si ii)
at early times show a positive (though low) correlation for the SNe Ic and no
correlation among the SNe IIb and SNe Ib. This could indicate that the
correlation in the SNe Ic is driven by Na i and Si ii, where both features
become somewhat more prevalent over time (see Fig. 5). The lack of correlation
in the SNe IIb and SNe Ib may be due to He i and H$\alpha$ being susceptible
to nonthermal affects, and Na i contributing less to the formation of Feature
4.

  * •

SNe Ic show only a handful of correlations with statistical significance. Most
notable are Feature 1 and Feature 2 which are moderately correlated at early
phases in all 3 subtypes.

  * •

There is little evidence of anticorrelations among the various pEW pairs.
However, the diagonal in the SNe IIb panel of Fig. 16 does reveal some
anticorrelation between Feature 5 and Feature 9 in SNe IIb. The physical
causes for this anticorrelation could be related to the fact that SNe IIb are
less stripped than the other SE SN subtypes, which may then result in low pEW
values inferred for Feature 9 from spectra obtained around maximum light.

In the spirit of completeness, the Spearman’s rank correlation coefficients
($\rho$) between pairs of velocity ($-v_{abs}$) measurements for the different
SE SN subtypes are given in Fig. 17.

Figure 16: Spearman’s rank correlation coefficients ($\rho$) between pairs of
pEW measurements. The colors of the boxes is associated with the correlation
coefficient with numeric values indicated by the colorbar at the top of the
figure. The lighter the color, the stronger the correlation. Contained within
the triangle of each panel are the correlation coefficients between pairs of
pEW measurements computed using spectra obtained up to three weeks post the
epoch of $B$-band maximum. The relative significance of $\rho$ for each pair
is indicated through the use of white lines. Here white-dotted, white-dashed
and white-solid lines correspond to pEW pairs having low ($0.4<$ $|\rho|$
$\leq 0.6$), moderate ($0.6<$ $|\rho|$ $\leq 0.8$), or high ($0.8<$ $|\rho|$)
correlation coefficients, respectively. Colored boxes with no white lines
exhibit minimal correlation. Gray squares have low statistical significance
and are characterized by probability (p) values $p>0.1$.

Figure 17: Spearman’s rank correlation coefficients ($\rho$) between pairs of
velocity ($-v_{abs}$) measurements. The color indicates the correlation
coefficients with numeric values connected to colors as indicated by the
mapping of the colorbar at the top of the figure. The lighter the color, the
stronger the correlation. Contained within the triangle of each panel are the
correlation coefficients between $-v_{abs}$ pairs computed from spectra
obtained within the first three weeks relative to the time of $B$-band
maximum. To highlight the relative significance of the Spearman’s rank
correlation coefficient each of the squares are marked with a different white
line where the white dotted, white dashed and white solid lines indicate
$-v_{abs}$ pairs of low ($0.4<$ $|\rho|$ $\leq 0.6$), moderate ($0.6<$
$|\rho|$ $\leq 0.8$), and high ($0.8<$ $|\rho|$) correlation coefficients,
respectively. Gray squares are correlations of low statistical significance
characterized by probability values ($p$-values) $>0.1$.

##  Appendix C Classification and distinguishing SNe IIb and Ib several weeks
past maximum

As seen in Fig. 10 and discussed in Sect. 6.2, our results indicate that it is
possible to distinguish between SNe IIb and SNe Ib, even if the spectra are
obtained a few weeks after maximum light. Previously, in the literature and in
practice, there has been an appreciation that SNe Ib classified with spectra
taken after maximum could be SNe IIb in disguise. Essentially, if the
prevalent H features have disappeared, it is difficult to tell whether the
objects in question are SNe IIb or SNe Ib. This is an important consideration
for population level studies of SE SNe, or their progenitors, since many SNe
Ib may either be excluded or contribute significantly to the final error
budget.

Our PCA results provide a blueprint for overcoming this practical difficulty.
Therefore, we outline an algorithm for applying our PCA results to past and
future SE SN samples. Fundamentally, there are two different approaches. The
first is seemingly harder one of creating and using a new template spectrum
from a new or expanded sample, and determining appropriate PCs to use in the
PCA. Second, using an established template from a literature sample (such as
the template in this work), adding the new SNe only during the PCA phase (not
into the template), and using the previously determined PCs of importance.
Although the latter seems simpler, care must be taken to verify and keep the
same PCs by not “overloading” the template sample, but also not washing out
the new spectra. The simplest way to achieve this is by calculating the PCs
for each new spectrum or SN, or small sets of spectra/SNe (a good rule of
thumb is $\lesssim 10\%$ of the template sample), and by verifying that the
PCs can appropriately recreate the new input spectrum when added onto the
template. Since both approaches have parts of the same basic steps, we now
provide practical guidance on each step.

Create a template (mean) spectrum. Using spectra of similar wavelength range
and not too dissimilar resolving power (i.e., not mixing high-, medium-, and
low-resolution spectra), a common template for all SE SNe should be created.
SNe with uncertain classification should be excluded if contamination by SNe
Ia or SLSNe is possible, or if they make balancing the classes difficult. The
template is a mean spectrum, defined by time and wavelength bins. We show that
the wavelength range between 6000 to 7000 Å is an appropriate choice, and the
limited range has the benefit of significantly aiding in including the maximum
number of SNe for consideration, while excluding PCs borne of the SED shape
instead of spectral features. Under ideal conditions, the time bins would be
small, but our results show that even binning the entire sample together, from
$-20$ to $+40$ days, is adequate.

Most importantly, care should be taken to not over-represent individual SNe
with dense spectral coverage by ensuring that one or two SNe do not dominate
the template at any time or wavelength bin. Similarly, bin sizes should be
chosen to provide a relatively even mix of SN subtypes and a relatively
uniform distribution in time, if a large time bin is chosen. The most
appropriate choice of bin size is an investigation we leave to future work,
but our results can be replicated using the previously mentioned 6000 to 7000
Å bin in the optical with a time bin covering the typical bell shaped light
curves of SE SN up to 5-6 weeks past maximum. All spectra must be normalized
to a common scale, where we have chosen this to be such that the mean flux is
zero, which is a mathematically advantageous choice. We note that well
sampled, diverse, high N literature template(s) could be constructed and used
in the future.

Perform PCA on all of the spectra. The spectra to perform PCA on can be
included in the template sample, or an appropriately robust literature
template could be used. Our analysis in Sect. 5 already contains a detailed
practical guide and software recommendation. The main point is that for each
spectrum, a set of PCs is obtained, where the top few PCs correspond to the
majority of the observed variation in the input spectrum compared to the
template. A good check of having used enough PCs, and the appropriateness of
PCA, is if the input spectrum can be approximately recreated by summing the
PCs and adding them to the template spectrum. This verification gains
increased importance if the input spectrum is not a part of the sample used to
make the template. As discussed, it can be desirable to leave out the new
input spectrum from the template sample to avoid the next step of selecting
appropriate PCs.

It is important to highlight that, even when using a previously calculated
template, _all_ spectra, including the ones that went into the template,
should be included in the PCA! Otherwise, the outcome will live in an entirely
different PC space each time. However, this is not a requirement if the new
spectra are numerous enough, and contain many secure classifications.
Moreover, the most appropriate PCs will have to be re-derived, and systematic
differences which make this more difficult are possible.

Select the strongest PCs corresponding to spectral features. Because the
variability among SE SN subtypes is not random, the top PCs likely represent
independent orthogonal processes. In our work, we found that in the limited
range between 6000 Å to 7000 Å, the top PCs roughly corresponded to H and He
features, while in the wider wavelength range the top PC represented SED
color. However, the correlated nature of SN spectra might mean that the PCs
constructed from a different template or sample can correspond to another
feature, a mix of characteristics, or something even more difficult to
interpret physically. To replicate our results and differentiate SNe IIb from
SNe Ib, care should be taken to exclude the PC, if any, that corresponds to
the overall SED shape/color, and focus on PCs which seem to correspond to
spectral line features, even if the precise line cannot be identified. A plot
like the top panels of our PCA Figs. 9 and 10 can be helpful, showing the
contribution of each PC to the observed variability about the template.

In theory, the number of PCs can be anywhere from two to N. However, we show
that two are enough for this task and allow a visually verifiable PC space to
investigate. Multidimensional clustering and classification is difficult to
visually interpret, but if employed, should be able to boost these results.

Optionally, using the strongest PCs corresponding to spectral features, the PC
space which creates the greatest variation among known (template) SNe IIb, Ib,
and Ic groupings should be created. One way of doing this is using rotated PCs
as we have shown in Fig. 10. However, simple relations between the strongest
PCs, or between PCs and epoch as in Fig. 9 can also work, as long as the
separation between subtypes in observed.

Classification with clustering. Using the newly created or existing PC space,
SNe Ib with spectra only at late time should be compared to the SN IIb and SN
Ib groupings. In our work presented in Sect. 5.3.1, interlopers from the SNe
IIb into SNe Ib class can be found with more than 80% completeness. The
essential task is to create a decision boundary for classification in the PC
space. This can be performed either by eye, or by using clustering analysis.
Ultimately, this step will improve with increasing data size, allowing for
more robust decision boundaries which take into account not just clustering,
but the paths that various subtypes and sub-groups among the SN subtypes take
in the PC space as they evolve. If using our template, a reasonable decision
boundary is $>1$ in the first rotated PC space represented by the x-axis of
Fig. 9.

Technical details related to the clustering analysis. The K-means fitting was
done robustly using 50 random restarts, and identifies the three most tightly
bound clusters in the dataset without assuming any distribution. Following
standard procedure by using the K-means results to initialize our GMM, we used
the Expectation Maximization algorithm to find the best fit maximum log-
likelihood for our GMM model. Unlike K-means which forces a single label on
every item, GMM is a soft clustering algorithm which assigns a probability for
each item being inside a cluster or not based on well known properties of
Gaussian distributions. Therefore, the results in Table 3 assumes the label
with the highest probability to be the correct one for calculating
completeness of our GMM. While the means (centers) of our clusters are well
separated, there is significant overlap at the two-sigma level for every
cluster. This can be due to noise from the relatively low number of points or
it could represent the fact that there is a continuum of classification
between SE SN subtypes. However, the center of each cluster is well separated
from the other. Only a minority of objects overlap between groupings or are
interlopers. We note that our findings should be regarded with caution as (i)
the sample is limited in size and wavelength range ($\sim 1000$ Å), and (ii)
assumes no time-dependency among the sample, which is incorrect given the
time-dependent nature of the spectral energy distributions of SE SNe.