# What can Gaussian Processes really tell us about supernova lightcurves?
Consequences for Type II(b) morphologies and genealogies

H. F. Stevance,1,2 A. Lee,1
1Department of Physics, The University of Auckland, Private Bag 92019,
Auckland, New Zealand
2Astrophysics Research Centre, School of Mathematics and Physics, Queen’s
University Belfast, N. Ireland, BT7 1NN, United Kingdom
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Machine learning has become widely used in astronomy. Gaussian Process (GP)
regression in particular has been employed a number of times to fit or re-
sample supernova (SN) light-curves, however by their nature typical GP models
are not suited to fit SN photometric data and they will be prone to over-
fitting. Recently GP re-sampling was used in the context of studying the
morphologies of type II and IIb SNe and they were found to be clearly distinct
with respect to four parameters: the rise time (trise), the magnitude
difference between 40 and 30 days post explosion ($\Delta m_{\rm 40-30}$), the
earliest maximum (post-peak) of the first derivative (dm1) and minimum of the
second derivative (dm2). Here we take a close look at GP regression and its
limitations in the context of SN light-curves in general, and we also discuss
the uncertainties on these specific parameters, finding that dm1 and dm2
cannot give reliable astrophysical information. We do reproduce the clustering
in trise–$\Delta m_{\rm 40-30}$ space although it is not as clear cut as
previously presented. The best strategy to accurately populate the trise–
$\Delta m_{\rm 40-30}$ space will be to use an expanded sample of high quality
light-curves (such as those in the ATLAS transient survey) and analytical
fitting methods. Finally, using the BPASS fiducial models, we predict that
future photometric studies will reveal clear clustering of the type IIb and II
light curve morphologies with a distinct continuum of transitional events.

###### keywords:

methods: statistical – supernovae: general – stars: evolution

††pubyear: 2015††pagerange: What can Gaussian Processes really tell us about
supernova lightcurves? Consequences for Type II(b) morphologies and
genealogies–A

##  1 Introduction

Massive stars, born with more than $\sim$8M⊙, will typically end their lives
as core-collapse supernovae (CCSNe). The chemical composition of the envelope
of the progenitor star will strongly influence the observed properties of the
explosion, and before the physical cause of these characteristics were known,
astronomers began using them to separate SNe into distinct categories. The
first divide split SNe with (type II) and without (type I) hydrogen signatures
in their spectra, and despite the “provisional" nature of the classification
at the time, it endured (Minkowski, 1941). Further subdivision in the
hydrogen-rich types was established on the basis of the shape of their light
curves: type IIP for those showing a plateau, type IIL for those showing a
linear decline. Sub-categories of hydrogen-poor SNe were also created
according to the presence (Ib) or absence (Ic) of helium in their spectra (for
a review of SN types see Filippenko 1997).

The divide between the classes however is not a sharp one: some type II
supernovae became type Ib later in life and were therefore called type IIb
(Filippenko, 1988), suggesting that their progenitors were stripped of most,
yet not all, of their hydrogen, retaining $\sim$0.5M⊙(Smith et al., 2011).
Additionally, in depth studies of the growing sample of type II SNe performed
in the last decade have revealed a correlation between the amount of hydrogen
in the ejecta of these SNe and the decay rate of their light curves (Anderson
et al., 2014; Sanders et al., 2015). These observations point to a SN spectrum
rather than a SN classification, Type IIP $\rightarrow$ Type IIL $\rightarrow$
Type IIb $\rightarrow$ Ib/c, where the deciding factor in the resulting
classification is the level of stripping of the hydrogen envelope of the
progenitor star.

Some studies however have highlighted sharper divides between type II and type
IIb SNe. In analysing type II SN light curves Arcavi et al. (2012) found that
type IIb consistently showed more rapid decline rates, however the sample used
at the time was rather small and there are now published examples of fast-
declining type II SNe (e.g. SN2013ai Davis et al. 2021). More recently Pessi
et al. (2019) – hereafter P19 – used innovative data analysis methods to
create parameter spaces where type II and IIb light curves could be compared.
They define the following features (for further detail see their section 4):

  * •

$t_{\rm rise}$: time between the explosion date and maximum light.

  * •

$\Delta m_{\rm 40-30}$: Magnitude difference between the phase 30 and phase 40
post explosion.

  * •

dm1: The earliest maximum (after maximum light) of the first derivative of the
light curve.

  * •

dm2: The earliest minimum (after maximum light) of the second derivative of
the light curve.

The resulting comparison yielded a very sharp divide between type II and IIb
SNe, which they interpreted as potentially indicating that the evolutionary
channels of type II and IIb SNe are separate.

One cause for concern however is the use of Gaussian processes (GP) to re-
sample their SN light curves before evaluating the features. GP are a
supervised machine method that can be used to perform regression on data sets
(Rasmussen & Williams, 2006) \- the trained algorithm can then be used to
interpolate between the data points. Although they have been used with some
success in fitting type Ia light-curves (Kim et al., 2013) and as a means to
augment photometric data to create photometric classifiers (Boone, 2019), GPs
are by their nature not well suited to fitting SN light-curves as they assume
unvarying timescale for the evolution of the function being fitted (see
Section 3.3 for an extended discussion). This is not consistent with the known
physics of SNe. Parametric methods such as Bazin fits (Bazin et al., 2009;
Villar et al., 2019) or the fits performed by Sanders et al. (2015) (see their
section 3) are physics-informed even if they are not physical simulations:
they are created to be able to reproduce known characteristics of SN II light
curves such as their plateaux and exponential decays. GP on the other hand may
have a tendency to over-fit the data.

Therefore in this paper we use a restricted sample of type II and IIb SNe (see
Section 2) to offer an extended discussion of what GPs are, what can be
achieved and where they fall short in the context of SN light-curve fitting
(Section 3). We then look at the parameter space defined by P19 and comment on
which parameters are reliable (Section 4), before discussing the implications
and expectations of future studies of type II(b) light curve morphologies in
the context of theoretical models of SN progenitors in the Binary Population
and Spectral Synthesis (BPASS) code (Eldridge et al. 2017; Stanway & Eldridge
2018 – Section 5). Finally we summarise and conclude in Section 6.

##  2 The sample

Because one of our main goals is to take a detailed look at the limitations of
GPs and what they can tell us about the morphology of type II SNe and their
progenitors, we create a restricted sample containing only high quality data.
There are two main criteria we select for: The data must be evenly sampled in
time (avoid large gaps in the light curve) and it must have a well studied
explosion dates. This will give our GP models the best chance to perform well,
although consequences of uneven sampling are already visible in some of the
SNe we selected (e.g. SN2003hn see Section 4) – we chose not to remove them
from the sample a posteriori. The smaller sample size also has the added
advantage that each object and its fits could be reviewed very thoroughly. We
make our full sample, as well as the codes used for the analysis available on
GitHub (see Data Availability)

We gather the light curves using the API of the Open Supernova Catalogue
(Guillochon et al., 2017)111Note that the front-end of the catalogue is no
longer available but the database remains available on GitHub. Unfortunately
it is no longer being updated as of April 8th, 2022. and most of the explosion
dates from Gutiérrez et al. (2017) (hereafter G17) and P19. The light curves
were visually inspected to select those which best fit our quality criteria.
Some of the late time data were cropped to avoid gaps in the light curve and
because the parameters defined by P19 do not need call for data beyond 80
days. We do none the less include data beyond 80 days as it is relevant to the
broader discussion of fitting type IIP SN light curves with GP.

In total we selected 8 type IIb (Figure 10) and 14 type II (Figure 11) SNe
which are listed in Table 1 including the time crop we applied and the
explosion dates we will be using in this study. For most SNe there are two
explosion date estimates: one from using the Supernova Identification software
(SNID; Blondin & Tonry 2007), which uses spectral template matching to derive
explosion dates based on a library of other SNe, and one from pre-explosion
non-detections, where the explosion date is taken as the mid-point between the
last non-detection and the first observation. The explosion data we used for
each target is summarised in Table 1.

In most cases both estimate were equivalent and we use the data derived from
observations (marked as [obs.]). When the two estimates differed slightly we
used the average between the two estimates ([avg.]) In a few cases, however,
the SNID estimate seemed more appropriate, such as SN 2013df (the
observationally derived explosion date would imply a shock-break-out and
cooling tail lasting about two weeks which is inconsistent with expectations
for type IIb SNe). In the case of SN 2006el, both dates seem to intersect with
the light curve, which is non-physical. In this case we made use of the 5-day
intrinsic error on the spectral fitting done by SNID quoted in P19 and took
the explosion date to be their SNID estimate – 5 days. A similar situation
arose with SN 2014cx, and the same solution is employed. Finally, in the case
of SN 2008aq the SNID estimate is much lower than that from observations, but
since the explosion epoch given by Stevance et al. (2016) is much more
consistent with the estimate from pre-explosion images, we do not take the
average with the SNID values as we are concerned this would lead to an
underestimate of the explosion date; we therefore use the literature value
from Stevance et al. (2016).

Table 1: Sample of SNe used in this study split by type and then in descending order of data quality (visual examination). The V-band light curve data were downloaded from the Open Source Supernova catalogue but not all downloaded data were included (e.g. spurious or noisy measurements were removed). In column 2 we show the references of for the data used in the final sample. Keys: T.1[obs.] – explosion date from table 1 ‘Observed’ column | T.1[SNID] – explosion date from table 1, ‘SNID’ column | T.1[avg.] – explosion date is the average of the ‘Observed’ and ‘SNID’ columns | T.A1 – explosion date from table 1 in appendix A P19 | T.A2[obs.] – explosion date from table 2 in appendix A of G17 column 7 | T.A2[SNID] – explosion date from table 2 in appendix A column 10 | T.A2[avg.] – explosion date from table 2 in appendix A of G17; average between column 7 and 10. For more details on the explosion dates of SN 2006el, SN 2008aq and SN 2014cx see main text.  Name |  Open Supernova Catalogue photometry reference(s) |  Dates included |  Exp. date used [MJD] |  Exp. date reference
---|---|---|---|---
Type IIb |  |  |  |
SN 2008ax |  Pastorello et al. (2008); Tsvetkov et al. (2009) |  < 54,700 [MJD] |  54528.3 |  T.1[obs.] P19
SN 2011dh |  Brown et al. (2014) |  all |  55712.4 |  T.1[obs.] P19
SN 1993J |  Okyudo et al. (1993); van Driel et al. (1993); Benson et al. (1994); Barbon et al. (1995); Richmond et al. (1996) |  < 49,175 [MJD] |  49073.1 |  T.1[avg.] P19
SN 2006T |  Bianco et al. (2014); Stritzinger et al. (2018) |  all |  53759.0 |  T.1[obs.] P19
SN 2004ex |  Stritzinger et al. (2018) |  all |  53287.9 |  T.1[avg.] P19
SN 2008aq |  Brown et al. (2014) |  all |  54513.0 |  Stevance et al. (2016)
SN 2013df |  Morales-Garoffolo et al. (2014); Brown et al. (2014) |  < 56,550 [MJD] |  56448.2 |  T.1[SNID] P19
SN 2006el |  Drout et al. (2011); Bianco et al. (2014) |  all |  53959 |  T.1[SNID - error] P19
– |  |  |  |
Type II |  |  |  |
SN2013ej |  Brown et al. (2014); Huang et al. (2015); Yuan et al. (2016); de Jaeger et al. (2019) |  < 56,700 [MJD] |  56497.5 |  T.A1 P19
SN2012aw |  Brown et al. (2014); Dall’Ora et al. (2014); de Jaeger et al. (2019) |  < 56,200 [MJD] |  56002.0 |  T.A2[SNID] G17
SN1999em |  Elmhamdi et al. (2003); Anderson et al. (2014); Faran et al. (2014); Galbany et al. (2016) |  < 51,700 [MJD] |  51476.5 |  T.A2[obs.] G17
SN2014cx |  Brown et al. (2014); Valenti et al. (2016) |  < 57,100 [MJD] |  56897.9 |  T.A1 P19 \- 5 days
SN2004er |  Anderson et al. (2014) |  all |  53271.8 |  T.A2[obs.] G17
SN2009ib |  Takáts et al. (2015) |  < 55,250 [MJD] |  55045.0 |  T.A1 P19
SN2013fs |  Valenti et al. (2016) |  < 56,650 [MJD] |  56570.8 |  T.A1 P19
SN2009N |  Takáts et al. (2014); Brown et al. (2014); Anderson et al. (2014); Hicken et al. (2017); de Jaeger et al. (2019) |  all |  54845.4 |  T.A2[col.10] G17
SN2014G |  de Jaeger et al. (2019) |  < 56,750 [MJD] |  56669.6 |  T.A1 P19
SN2003hn |  Galbany et al. (2016) |  all |  52866.5 |  T.A2[obs.] G17
SN2008aw |  Anderson et al. (2014) |  < 54,600 [MJD] |  54520.3 |  T.A2[obs.] G17
SN1999gi |  Leonard et al. (2002) |  all |  51518.2 |  T.A1 P19
SN1992ba |  Anderson et al. (2014); Galbany et al. (2016) |  all |  48884.9 |  T.A2[SNID] G17
SN2013ai |  Valenti et al. (2016) |  all |  56339.7 |  T.A1 P19

##  3 Gaussian Process regression

###  3.1 A Brief introduction to Gaussian Processes

GP are at the heart of this study, so we provide a brief introduction to the
method that should be accessible to the broader SN community (also see
appendix A of Boone 2019). For an extended introduction aimed at astronomers
we suggest consulting Chapter 3.4 of McAllister (2017) and for an in depth
statistical view we refer the reader to the textbook by Rasmussen & Williams
(2006)222Official PDF of the textbook is freely available at
http://gaussianprocess.org/gpml/chapters/RW.pdf.

GP are a powerful machine learning method which can approximate a function of
unknown form with a multivariate Gaussian distribution. In the context of
fitting SN light-curves, the function underlying our data has one independent
variable (time), and one dependent variable (magnitude). Then the
approximation of the observed magnitudes (m) given our observation dates
(t333These variables are vectors each containing all the values for time and
magnitude in the data) made by the GP model can be written with the
probability distribution function:

$P(\textbf{m}|\textbf{t})=\mathcal{N}(\mu(\textbf{t}),\textbf{K})$ (1)

where $\mu(\textbf{t})$ is the mean function, which describes how the expected
value of m varies as a function of t, and K is the covariance matrix.

In essence, each data point is its own Gaussian, and training the GP model is
understanding how they all relate to each other by optimizing
$\mu(\textbf{t})$ and K to best fit the data. Subsequently the optimised model
of Eq. 1 can be used to infer new magnitude values for our choice of dates.
This makes GP a powerful fitting and re-sampling method on two accounts:
firstly we do not need a physical or analytical model; secondly it naturally
provides uncertainties on the inferred data points. A number of
implementations exist which make the use of GP very accessible and easy to
use. Two of the most popular are the GPy (GP specific) and the the sklearn
(general machine learning) python packages (The GPyOpt Authors, 2016;
Pedregosa et al., 2011).

When training the GP algorithm, the most important consideration of the user
is the choice of the covariance kernel. There are a number of commonly used
kernels with a variety of properties (for an in depth introduction see chapter
2 of Duvenaud (2014)444For an informal summary of this chapter consults
https://www.cs.toronto.edu/~duvenaud/cookbook/ ); the essential take away here
is that the kernel is a function chosen to reflect characteristics expected to
be seen in the functions we are approximating: e.g. an eclipsing binary light
curves would be fit with a periodic kernel.

When fitting SN light curves the preferred kernels in the literature have been
the squared-exponential kernel (Kim et al., 2013; Fakhouri et al., 2015; Pessi
et al., 2019) and the Matèrn–3/2 kernel (Boone, 2019). The former is the
prototypical kernel used in many non-periodic applications of GP modelling;
the latter results in much sharper variations in the model, which could be
more suitable for type II plateau SNe light curves where their sharp drop off
the plateau could be smeared by the squared-exponential kernel. Boone (2019)
mention that they find little difference in fits performed with the Matèrn–3/2
and the squared-exponential kernel, but we do not find that to be the
consistently the case (see Section 3.2.2).

In a first instance, we will use the squared-exponential kernel to perform our
fits. It is characterised by two hyper-parameters: the length scale and the
variance. In theory, and crudely speaking, a smaller length scale results in a
mean function $\mu(\textbf{t})$ with more frequent "wiggles" and a larger
variance allows the data to deviate more from its mean (potentially increasing
the amplitude of said "wiggles"). Depending on the GP implementation used and
the data, the starting values chosen by the user may have a dramatic impact on
the final fit: in practice we found that the optimizers in GPy converged on
the same solution when given a large range of input length scale and variance,
whereas with sklearn fits required more careful consideration of the boundary
values for the length scale (see Section 3.4).

Another type of kernels used by P19 were those meant not to reproduce the
physical patterns in the data but the stochastic ones: White noise or bias
kernels can be added to other kernels when building the GP model. A white
noise kernel is one which adds variance to model data without changing the
mean function, whereas the bias kernel can also affect the meant function
$\mu(\textbf{t})$. As we will see, whether a noise kernel is included and how
it is implemented can have an influence on the resulting model and whether it
fits (or over-fits) the data and there are other avenues to taking into
account noise that may not be taken into account in the error bars (see below
the case of SN 2004ex).

###  3.2 Kernel choice and kernel ‘engineering’

Boone (2019) in their section 5.5 mention that further exploration of which
kernels are most suitable to SN light curve fitting should be conducted. The
choice of kernel is indeed the most important consideration in the
construction of a GP model and it is not uncommon for different kernels to be
combined in order to better reflect the behaviour of a data set – this is
sometimes called kernel ‘engineering’. It is beyond the scope of this paper to
systematically test how specific kernel combinations behave on a large sample
of SN data, however we found that even within our constrained set of light-
curves distinct kernel choices, and at times some basic kernel engineering,
allowed for better fits.

####  3.2.1 RBF and RBF+Bias

Here we present how results can be effected by including a noise kernels
alongside a squared-exponential kernel, also known as the Radial Basis
Function (RBF) kernel – this configuration is what was used by P19. We focus
on SN 2011dh as an example (see Figure 1) but note that other SNe show similar
behaviour (e.g. SN 2008ax). Following P19 the prior length scale and variance
used for the RBF kernel are the average time interval between observed epochs
and the standard deviation of that difference; the noise kernels were set up
using the dot product of the errors on the magnitude as their variance. Note
that the results presented in this section were obtained with the GPy
implementation and as mentioned above, these starting values have actually
little bearing on the final model when the data is high quality and the kernel
appropriate: in the case of SN 2011dh the models converge to a length scale of
$\sim$35.5 when giving a starting length scales of 1000 or 0.01. This is not
true of fits produced with sklearn see Section 3.4.

Figure 1: GPy fits of SN 2011dh using three different kernels. Note that on
the first and second panels the variances on the fits are not shown to make
the variations in the means easier to see.

We can see in Figure 1 that, as expected, the RBF solutions and the RBF+white
noise solutions for $\mu(\textbf{t})$ are identical, but including the bias
kernel can have both positive and negative effects on the fit. The late time
light curve is over fit by the RBF+bias kernel (it is also slightly over fit
by the RBF kernel but not to the same degree). At the same time, the
asymmetric peak of SN 2011dh is better reproduced by the RBF+bias kernel
whilst the RBF kernel alone smooths it out, resulting in a slightly
underestimated time at maximum (still within a couple of days).

This leads to an interesting question: how are we to judge goodness of fit?
Should one use a solution which can better reproduce the patterns of the early
light curve at the risk of over-fitting the late times, or should one prefer
smoother fit that risks under estimating the maximum light by a few days? The
answer obviously depends on the application of the fit – if the purpose is to
create model SN light curves over the full light-curve of the object over-
fitting would be highly problematic whilst a systematic error on the maximum
light could be relatively easy to account for. In this specific case we opted
for the model with less over-fitting. Generally these considerations should be
made transparent as the choices made by one team may not be applicable to
another.

Additionally we note that the goodness of fit was judged through visual
examination. Tools such as least-squares or $\chi_{2}$ statistics are not
useful here as a model that over-fits will perform best. Generally speaking
methods of regularisation do exist to allow best-fit statistics to account for
model complexity and discriminate against over-fitting, but the issue is that
these methods are pure mathematical applications that do not take into account
the known physics or behaviour of these time series, and what “over-fit” or
“under-fit” is (not) acceptable. This is one of those few occasions where
human expertise is hard to replace and for the visual examination we use two
key criteria: Can the model fit the peak? Does it over-fit the linear
decline/plateau?

We also use the Figure 1 to re-emphasise a typical word of caution when using
machine learning methods: it is not appropriate to use a trained model outside
the range of data it was trained on. When using the RBF kernel in GP, the
length scale is typically the limit of reasonable extrapolation, however in
the case of our SN light-curves it is clear that well within one length scale
of the data the model already deviates form physical expectations.
Consequently GP can be powerful interpolation tools, but similarly to splines
they cannot reliably be used for extrapolation in the same way a physically
motivated analytical model would.

Figure 2: Examples of light curves where the Matèrn kernel performs better
than the RBF (squared exponential) kernel.

####  3.2.2 Matèrn 3/2

In Figure 2 we show 6 light curves which were not well fitted by a classic RBF
kernel and for which the Matèrn 3/2 kernel was more appropriate. Although in
the case of the Boone (2019) study they found that the RBF kernel performed
practically as well as the Matèrn kernel, we find that for quite a few
objects, particularly type IIP SNe, the latter is better suited. This is not
unexpected, as type IIP are characterised by very slow evolving light curves
with a significant drop in magnitude as the plateau phase ends. We can see in
cases such as SN 1999gi how the RBF kernel is too shallow at the drop of the
plateau, an in all cases how much the mean function fluctuates in magnitude
when it is physically (and observationaly) expected to be flat. The Matèrn
kernel still displays some unphysical wobble but to a much smaller extent.

Once again, not all cases are clear cut and the goodness-of-fit question we
posed with SN 2011dh is relevant with SN 2004er: the Matèrn 3/2 kernel is
better at fitting the early rise and peak of the light curve but subsequently
over fits the late time light curve. The RBF kernel is less prone to over
fitting since it does not allow for sharp enough variations in the mean
function, but subsequently it cannot fit the rapid rise to peak and in the
case of this study where trise is a parameter of interest, it would be
profoundly unsuitable.

####  3.2.3 RBF+RBF

Another case where the typical RBF kernel failed was SN 2004ex as the GPy
converged to a short length scale of 2.9 days leading to large oscillations in
the late time light curve and over fitting at early days. We suggest that the
sparser sampling is to blame in this case, which gives us a window into the
difficulties that can be encountered on larger samples of SN light curves with
varying quality.

We find that the Matèrn 3/2 kernel provided a much more suitable solution, and
a similar fit was obtained when combining two RBF kernels that were optimised
by the solver to length scales of 27 days and 0.9 days respectively. This
solution has an interesting physical interpretation: the 27 days length scale
kernel relates to the evolution of the light curve from the SN whereas the
short length scale kernel accounts for noisy deviations in the individual
observations (which are separated by a day or more in these data).

This is a very interesting prospect for fitting noisier and more sporadic
light curves - especially if observing conditions vary from night to night,
the level of noise in each data point could vary. An additional RBF kernel
would be much better suited to capture this point-by-point variability than a
white noise kernel.

###  3.3 Full sample fits and length scale considerations

We now look at the results for our full sample; we will discuss the parameters
of interest ($t_{\rm rise}$, $\Delta m_{\rm 40-30}$, dm1, dm2), their
uncertainties and astrophysical implications in the following sections. In
Table 2 we record which kernels were used for each SN and in Figure 3 we show
the fits and resulting first and second derivatives. Note that for the type
IIbs SN 2013df and SN 1993J the adiabatic cooling part of the light curve is
fit well by our models but not showed in full here as it is the radioactive
decay peak that is used to calculate trise.

Table 2: Kernels used to obtain best fit to each SNe. $\lambda$ is the length scale of convergence for the squared-exponential (RBF) kernels – see Section 3.3 for a discussion of the length scales. Name |  GPy kernels |  sklearn kernels
---|---|---
Type IIb |  |
SN 2008ax |  RBF ($\lambda=32.4$) |  RQ+White
SN 2011dh |  RBF ($\lambda=35.5$) |
SN 1993J |  RBF ($\lambda=11.3$) |  RQ+White
SN 2006T |  RBF ($\lambda=34.2$) |  RQ+RQ
SN 2004ex |  RBF ($\lambda=27.9$) + RBF($\lambda=0.87$) |  RQ+RQ
SN 2008aq |  RBF ($\lambda=32.6$) |  RBF
SN 2013df |  RBF ($\lambda=17.7$) |  RBF
SN 2006el |  RBF ($\lambda=43.0$) |  –
– |  |
Type II |  |
SN 2013ej |  RBF ($\lambda=13.4$) |  RBF
SN 2012aw |  RBF ($\lambda=37.2$) |  RBF
SN 1999em |  Matèrn–3/2 |  RBF
SN 2014cx |  RBF ($\lambda=21.6$) |  RQ + White
SN 2004er |  RBF ($\lambda=16.2$) + Bias |  RBF
SN 2009ib |  RBF ($\lambda=25.1$) |  RBF
SN 2013fs |  RBF ($\lambda=23.1$) |  RBF
SN 2009N |  Matèrn–3/2 |  RQ+ White
SN 2014G |  RBF ($\lambda=32.7$) |  RBF
SN 2003hn |  Matèrn–3/2 |  RQ + White
SN 2008aw |  RBF ($\lambda=56.7$) |  RBF
SN 1999gi |  Matèrn–3/2 |  RBF
SN 1992ba |  Matèrn–3/2 |  RBF
SN 2013ai |  RBF ($\lambda=42.9$) |  RBF
Figure 3: Best GPy regressions of our SN sample – the kernels used are
summaries in Table 2. The data is shown in magenta (with error bars which are
sometimes smaller than the data points), the mean function of the fit
$\mu(\textbf{t})$ is show in black and 100 samples of P(m|t) (Eq. 1) are shown
in grey. We also show the first (red) and second (blue) derivatives of
$\mu(\textbf{t})$ and of the 100 samples of P(m|t).

When looking at Figure 3 it is clear that GP models have a tendency to over-
fit the SN light curves. In the case of type IIb SNe that is mostly the case
at later times on the radioactive decay tail; in type II SNe it occurs both on
the plateau and the tail. This is particularly true for some of the more
difficult to fit data sets such as SN 2014cx, but even in the case of SN
2008ax which is generally very successful before 150 days, a slight "wobble"
can be seen in the data. This type of oscillation can be seen in figure 1 of
P19 as well as in their figure 2 (although the over plotting makes is harder
to distinguish), and it is not user error.

This is a direct and inevitable consequence of the commonly used kernels in GP
regression: they are stationary kernels and as a result intrinsically fit for
a function with unchanging length scale over the course of the event. We know
that not to be the case with SNe. In light curves with a clear radioactive
peak and decay tail, the evolution of the peak powered by Nickel-56
($\tau_{1/2}=6.1$ days) will be faster than that of the Cobalt-56
($\tau_{1/2}=77$ days) powered tail. In type IIP SNe the plateau phase can
last for a few months, but the hydrogen recombination and fall off the plateau
lasts a few weeks at most. By adding kernels together we can account for the
behaviour of complex functions that evolve on several timescales
simultaneously (e.g. the mixture of two RBF kernels for SN 2004ex) but our GP
models cannot appropriately reproduce a function that has timescales which
change over time.

In Table 2 we report the convergence length scales from the GPy fits. We can
see that they are typically between 15 and 40 days, which is the time window
in the early light curve or in the drop off the plateau where the most change
will occur. These phases show the most amplitude variation in the light curve
and are the most well sampled (at least in this data set) – consequently
during the optimization phase and the maximisation of the log likelihood, the
model will be penalised the most if it does not reproduce these portions of
the light curve well. That is why they will be the dominant driver of the
length scale of convergence.

Now in general our GPs and the basic kernels perform sufficiently well when
the data is well sampled (which is the case for most of our SNe by design) as
it anchors the fits: a good example of this is SN 2013ej, which has a
convergence length-scale of 13.4 days which is much shorter than that time
scales on which most of its light curve varies in smoothness, but the dense
sampling anchors the models and the small length scale only results in minor
oscillations that are withing uncertainties. But cases such as SN 2003hn and
SN 1999gi show what the impact of too small a length scale can have on re-
sampling too sparse a data-set: a ‘bloated’ variance between the data points,
which does not reflect the likely variations in the light curve at these
epochs.

In P19 the average time between observations is used as the prior length
scale: as can be seen from Table 2, this does not reflect the typical length
scales of our SN light curves. The issue with using the average time between
observations as the length scale is not simply that it is not related to the
evolution of our SNe, it is also because in practice light curve sampling is
most dense at early days. Therefore the average time between samples will
typically be lower than the time separation of the data in the tail (which is
already prone to over-fitting even with dense sampling), and if the length
scale is smaller than the separation between the data points very large
oscillations can occur in the GP solution as we saw above in the cases of SN
2003hn and SN 1999gi. Therefore this length scale prescription has the
potential to worsen the over-fitting of the late time light curve depending on
the implementation of GP used (see Section 3.4). We surmise that P19
encountered no major issues because they used GPy which converges to
reasonable values most of the time even with unphysical initial guesses, as we
tested above.

Strictly speaking, the fitting of SN light curves with GP should require the
use a non-stationary kernels which are capable of fitting functions with
variable smoothness (e.g. see Figure 1 of Paciorek & Schervish 2003). Such and
implementation is beyond the needs or scope of this paper, but we emphasise
that this detail in the kernel choice could be highly relevant for teams
looking to fit large samples of SN light curves of varied types and they are
set on using GP regression.

###  3.4 Different implementations, different kernels

Figure 4: Red curve: sklearn GP regression with RBF kernel with length scale
bounds (mean time interval between epochs; 100 days). Black curve: sklearn GP
regression with RBF kernel with length scale bounds (30, 100).

A final point of discussion related to Gaussian processes that we want to
address is that different implementations can yield different results. To test
this we fitted our whole sample with sklearn and recorded which kernels
performed best in Table 2 \- we do not show the final mean fits themselves as
for the most part they are similar (although not the same) as those presented
in Figure 3. However as an example we show GP regression performed on the
light curve of SN 2008aq in two different configurations (see Figure 4). The
first (red) fit was performed using a length scale bounds starting at the mean
time interval between epochs (a few days) as we did throughout our GPy fits,
whilst the latter had a lower bound of 30 days, which is more typical as seen
in Table 2. In the GPy implementation we have seen that the start length scale
had little influence on the final value of convergence, but here the lower
length scale bound resulted in a model that severely over-fit our data – this
behaviour was noted across our sample.

Generally we find that the RBF kernels in GPy and sklearn behave slightly
differently and in the latter fits are sometimes more successful when
performed with a Rational Quadratic (RQ) kernel (essentially a sum of RBF
kernels). Additionally the Matèrn–3/2 kernel in sklearn never performed better
than the RBF or the RQ kernels. It is unclear why that is the case and in
terms of analysis and results it is not particularly important: so long as the
fits are close enough to the data (within uncertainties) the re-sampling will
be successful. This consideration does have an impact on how we share our
findings and what ‘reproducibility’ means for this type of work when using so-
called black boxes: it is important to be specific about the packages employed
and, if possible, to release publicly the codes and scripts used for analysis.

##  4 Ligth curve Parameters

We now turn our attention to the light curve morphology parameters $t_{\rm
rise}$, $\Delta m_{\rm 40-30}$, dm1 and dm2. In this section we review the
uncertainties associated with these parameters in the context of GP fits, and
discuss which ones can reliably be used for astrophysical interpretation .

###  4.1 dm1, dm2 and uncertainties

The issues of over fitting and the presence of oscillations in the model that
are not present in the light curve which we discussed above are causes for
concern for the values and uncertainties on the first and second derivatives.
In Figure 3 we show the first and second derivatives of 100 re-samplings of
the models for each SN over 400 epochs - we also show the position of the
maximima of the first and second derivatives of the mean function
$\mu(\textbf{t})$. It is this mean function that P19 use for their calculation
and that they plot in their figure 3, but it is important to re-sample the
models using the posterior co-variances calculated by the optimizer. As we can
see the level of noise is such that the position of the first maxima in the
first and second derivatives of the mean functions are not at all obvious.

Figure 5: Firs (red) and second (blue) derivatives of the mean functions
$\mu(\textbf{t})$ of the GP regressions performed with two sets of kernels
(see legends) for SN 2009, SN 2004er and SN 2004ex. The parameters dm1 and dm2
are shown as vertical lines of the style corresponding to the kernels used for
the interpolation, and the thin grey lines show the time at maximum light.

Additionally we want to highlight how the derivatives can vary with kernel
choice and how the impact can be unpredictible even when only considering mean
values of the parameters. In Figure 2 we can see the fits of SN 2004er, SN
2004ex and SN2009N resulting form different kernels, in particular a
Matèrn–3/2 kernel and an RBF(+bias) kernel resulting in fits that are very
similar. But in Figure 5 we show the first and second derivatives of the mean
functions for these different kernels and they are strikingly different. These
differences can have no impact on the parameters as in SN 2004ex where the
amplitude of the derivatives changes but their maxima and minima are very
similar particularly at early days when the parameters are calculated.
However, this is not always the case. For SN 2004er, the first derivatives are
very similar and their resulting dm1 parameters indistinguishable, but the dm2
parameter varies drastically. Finally in the case of SN 2009N, both the dm1
and dm2 parameter are affected by the choice of kernel, despite both fits
lying on top of each other until 80 days post-explosion. Overall we conclude
that using the derivatives of SN light curve fits done with GP is not a
reliable way to look for morphological differences related to their
progenitors as the results are very unstable to the choice of kernel.

###  4.2 $t_{\rm rise}$, $\Delta m_{\rm 40-30}$ and uncertainties

Although dm1 and dm2 suffer too much intrinsic uncertainty, the rise time
$t_{\rm rise}$ and magnitude difference between phase 40 and 30 $\Delta m_{\rm
40-30}$ are well determined given sufficient sampling.

Figure 6: $t_{\rm rise}-\Delta m_{\rm 40-30}$ parameter space showing the
values of the parameters for 100 re-sampling of each SN model. The type II SN
are show in red and the type IIb in blue. The darker points show the values of
$t_{\rm rise}$ and $\Delta m_{\rm 40-30}$ for the mean functions of each SN
model. Figure 7: Distribution of the value of trise in days for 100 samples of
the model of SN1999em. The mean is shown as the dashed line.

In Figure 6 we show the $t_{\rm rise}-\Delta m_{\rm 40-30}$ parameter space.
We do not use error-bars to represent our uncertainties as they are for the
most part non-Gaussian. Instead we re-sample the GP models 100 times per SN
and record the two parameters. We also plot the values of $t_{\rm rise}$ and
$\Delta m_{\rm 40-30}$ calculated for the mean functions of each SN. It is
worth noting that the mean will not necessarily be found were the density of
points is highest in many cases – an example is SN 1999em where the mean is
very offset from the majority of the data due to the shape of its distribution
(see Figure 7). It is because of this behaviour that we chose re-sampling over
the plotting of the mean and standard deviation.

Overall the $\Delta m_{\rm 40-30}$ distributions for type II and IIb SNe are
distinct but adjacent. The small sample of SNe here, and the use of only one
light curve re-sampling method does not allow us to draw definitve conclusions
on the shape of these distributions, but it suggests the divide may not be as
marked as presented in P19. Calculating these parameters for a much more
exhaustive sample will be necessary and we suggest that data from the ATLAS
transient survey (Smith et al., 2020) would be ideal for this purpose: the <1
day cadence in the light curves would provide dense enough sampling that
complex fitting and re-sampling would not be required to obtain values of
$t_{\rm rise}$ and $\Delta m_{\rm 40-30}$ with small uncertainties. The
consistent cadence would make these data easy to fit with GP models as done
here, although alternatives, such as Bazin analytical models (Bazin et al.,
2009; Villar et al., 2019), would likely be preferable due to the known over-
fitting issue discussed in earlier sections.

Finally we note that Figure 6 does not take into account uncertainties on the
explosion date, which would affect $t_{\rm rise}$. This was deliberate to
focus on the natural spread that arises from GP modeling, and given the
present results our conclusions would be unchanged. Indeed, the explosion date
uncertainty is at most a few days (see Figures 10 and 11) and the peaks of the
$t_{\rm rise}$ distributions for the type II and IIb supernovae are $\sim$10
days apart, so they would remain clear; additionally those distributions
already overlap, and the explosion date uncertainty would only strength this
overlap. We do caution that taking into account the explosion date uncertainty
in this type of analysis would be non-trivial: the explosion date would need
to be sampled, but the appropriate distribution is not clear as the errors on
the explosion dates are systematic rather than stochastic. An extended
discussion of this challenge is beyond the scope of this publication.

##  5 Discussion

The parameters defined by P19 trise, $\Delta m_{\rm 40-30}$, dm1 and dm2 each
are intended to capture a characteristic of light curve morphology which can
lead to physical interpretation. dm1 and dm2, the first maxima post peak of
the first and second derivatives of the light curve, should indicate the
maximum rate at which the SN fades and how much the curvature of the light
curve changes where it changes the most. As we have seen in the previous
section, the over-fitting from GP and the uncertainty in the re-sampled
gradients make these parameters impractical for astrophysical interpretation
in this particular context, but they may be more successful when calculated
with an alternative fitting method.

trise and $\Delta m_{\rm 40-30}$, the time from explosion to maximum light and
the magnitude difference between phase 40 and 30 (post-explosion)
respectively, have much more stable values and we do see in our results a
clear clustering of the type II and IIb. This separation between type II and
IIb SNe in parameter space was more severe in P19 and interpreted as
potentially caused by separate progenitor channels as the light curve decline
in particular is correlated with how much hydrogen remains in the envelope at
the time of explosion: a lesser degree of envelope stripping leaves behind
more hydrogen which can contribute more light on recombination providing an
excess compared to a pure radioactive decay tail.

A distinction between the two classes had been suggested by Arcavi et al.
(2012) who reported that rapidly declining light curves in their sample solely
belonged to type IIb SNe. However the sample was small (20 SNe) and
transitional events showing characteristics of both type II and IIb SNe have
now been reported (SN 2013ai; Davis et al. 2021). Additionally, Anderson
(2019) performed a meta-analysis of CCSNe, their Nickel-56 yields, and
implications for their progenitors; they reviewed existing evidence from the
literature which point to an interesting dichotomy: Based on direct imaging of
the progenitors, type II and stripped envelope SNe do not seem particularly
difference, whereas from environment studies seem there may be a decreasing
age sequence from type II to more stripped sub-types.

In this context, should we expect further studies of trise and $\Delta m_{\rm
40-30}$ to show type II and IIb as clear spectrum or should we expect two
distinct clusters? Here we flip the problem on its head and use results from
theoretical models of binary stellar evolution to qualitatively assess what
morphologies we expect to see. To do so we use the BPASS fiducial models which
have been extensively tested against observations and are capable of
reproducing a large array of observable phenomena particularly in massive
stars (e.g. Eldridge et al. 2017; Stanway & Eldridge 2018; Rosdahl et al.
2018; Eldridge & Xiao 2019; Stanway et al. 2020b; Stanway et al. 2020a; Massey
et al. 2021; Ghodla et al. 2022; Briel et al. 2022; Stevance et al. 2022). We
use hoki (Stevance et al., 2020) to search for type II and IIb progenitors
using similar criteria for the hydrogen-rich/hydrogen-poor boundary as in
Stevance & Eldridge (2021): H-rich progenitors are taken as containing more
than 0.001 M⊙of hydrogen at time of death with a hydrogen mass fraction
greater than 10 percent (Dessart et al., 2012). Also we only consider models
which will explode as SNe by searching for models with a CO core mass at death
be greater than 1.38 M⊙and that the remnant be less than 3M⊙555baryonic mass.
Although the type IIb classification itself is unlikely to have a sharp
threshold in envelope hydrogen mass, we use the commonly used 0.5M⊙values
(Smith et al., 2011) as a boundary between type II and type IIb.

Figure 8: Hydrogen mass remaining in the envelope at the time of explosion for
the type IIB and II progenitors in BPASS at Z=0.020.

In Figure 8 we show the number of type II and IIb SN in a 106M⊙stellar
population born with Z=0.020 (solar metallicity) as function of remaining
hydrogen mass. We see that their is no gap between the type II and IIb SNe,
but there is a large number of systems with very low hydrogen envelope masses
($<$0.05M⊙) whereas the majority of type II explosions have $>3.5$M⊙of
hydrogen left in their envelope as they explode. This would explain why with
low number of observations the two classes could look quite distinct and with
growing sample intermediary events are beginning to emerge.

Figure 9: Zero Age Main Sequence mass of the type II and IIb SN progenitors in
BPASS at Z=0.020

Although it is not the object of this paper to draw any firm conclusions on
progenitor differences we can begin to explore our model population by looking
at the distribution of ZAMS masses (see Figure 9). We can see that type II
ZAMS masses steadily decline as we move to high masses as we expect from the
point of view of the initial mass function,the type IIb on the other hand show
a density peak around 12 solar masses. This behaviour could explain the
dichotomy in the observations: host environments could tend to indicate that
type IIb SNe come from more younger stars because their ZAMS mass distribution
is skewed to higher values (and therefore lower ages), but since the type II
ZAMS mass distribution spans that of the type IIb direct detection of
progenitors would be virtually identical because of low-number statistics.

On the whole the answer to the question "are type II and IIb progenitors
different?" is likely a mixture of ‘yes’ and ‘no’, if only because the
classification is observational and unlikely to represent a systematic divide
in the stellar evolution. Nevertheless we can look to our fiducial BPASS
stellar population to find differences of note in the genealogies of the type
IIb and II SNe in our simulations. We find here that 100 percent of the type
IIb were at one point in a binary system, whereas the type IIs come from
single stars in $\sim$30 percent of cases. Interestingly we also find that
nearly 50 percent of the type IIb progenitors are merger products, whereas
this value is only $\sim$16 percent for type IIs. The importance of binary
interactions in order to strip the progenitors of type IIbs has long been
established (e.g. Podsiadlowski et al. 1992, 1993; Nomoto et al. 1993; Young
et al. 2006; Stancliffe & Eldridge 2009; Maund & Smartt 2009; Claeys et al.
2011; Yoon et al. 2017), but further analysis will be required to understand
how these differences in the proto-typical evolutionary channels relate to the
observable morphologies of type II(b) light curves, and it may also be the
case that the genealogies of the most heavily stripped type IIbs differ from
that of events which are closer to the type II boundary: we may see a spectrum
of differences in the stellar evolution as well as in the morphologies.

##  6 Conclusions

In this paper we used 22 of the best sampled type II(b) SNe to carefully
investigate the suitability of Gaussian Process regression in the context of
SN light-curves generally and in re-sampling the data to determine the
morphological parameters trise, $\Delta{\rm mag}_{\rm 40-30}$ defined by Pessi
et al. (2019).

We find that due to the the way GPs are trained the light curve peak or the
drop of the plateau will be dominant factors in the length scale optimisation.
Since these parts of the light curve evolve on shorter times scales than
others (e.g. late tail, plateau), the slower phases will systematically be
over fit. On the whole GPs with traditional kernels (squared exponential Kim
et al. 2013; Pessi et al. 2019; Matèrn–3/2 Boone 2019) are technically
speaking not suitable to reproduce SN light curves due to the varying
timescale of the phenomenon. An avenue would be to implement GP regression
with non-stationary kernels (Paciorek & Schervish, 2003) that can reproduce
data with varying smoothness.

In general if the main goal of GP regression is to fit solely the early time
light curve rather than the time series in full, the issue of varying time
scales remains minor. Additionally if implementing non-stationary kernels is
not feasible we would none the less recommend testing for the best kernel
combination, which will be dependent on the sampling and shape of the light
curves in a given data set. For example we found the Matèrn–3/2 kernel to
perform better than the typical squared-exponential (RBF) kernel on several
occasions. Quite notably we discovered that the combination of two RBF kernels
could be powerful in capturing the SN light curve behaviour as well as point-
to-point deviations (see Section 3.2.3) – since the noise in the data is
likely to be different from white noise in most cases this kernel combination
could be useful. Finally it is worth pointing out that the best set-up for GP
regression is to a large degree dependent on the implementation (see Section
3.4).

On the whole, good fits can be found with GP regression and re-sampling given
that the data are well sampled to anchor the fit. In the context of this study
this means that for parameters like trise and $\Delta{\rm mag}_{\rm 40-30}$
the slight over fitting does not increase the uncertainties in most cases (in
the case of type IIPs whose early light curve would be best fit by a straight
line however the trise values from GPs are mostly determined by the over-
fitting). The main issue is in the calculation of the gradients from the
models. We find that the large uncertainties and the tendency for the
gradients to be sensitive to the kernel even when the mean fits are
indistinguishable makes these parameters unreliable for astrophysical
interpretation in the present case. They might however be worth considering if
analytical fits not prone to "wobbliness" were performed instead.

We do find clear clustering of the type II and IIb SNe in trise – $\Delta
m_{\rm 40-30}$ space, although the division between the two classes is not as
clear as in P19, possibly because of the use of re-sampling rather than error
bars in our visualisation. Further study of trise and $\Delta m_{\rm 40-30}$
using a larger sample of events will be required to draw firm conclusions on
this parameter space. We propose that the ATLAS transient survey (Smith et
al., 2011) will provide the ideal data set for this purpose given the dense
sampling (<1–2 days between observations over the whole light curve). We will
also test the scalability of fitting with analytical models (e.g. Villar et
al. 2019) to derive the light curve morphologies.

Given the type II and IIb progenitors predicted by the BPASS fiducial models
we expect the trise – $\Delta m_{\rm 40-30}$ parameter space to show a bimodal
behaviour with a number of transitional objects bridging the gap between the
clear type IIb SNe with hydrogen envelope masses $<$0.1M⊙and clear type II(P)
progenitors with hydrogen envelope masses $>$3.5M⊙.

## Acknowledgements

The authors thank the referee for their careful review of our manuscript. HFS
is very thankful to P.J. Pessi for forwarding data and code which allowed a
smoother comparison to their original study. HFS acknowledges the Marsden Fund
Council managed through the Royal Society of New Zealand Te Apārangi as well
as the University of Auckland Summer Scholarship No SCI008 which funded the
work of AL. HFS would like to thank S. Littlefair for very helpful and
animated discussions about Gaussian Processes and J.J.Eldridge for her support
and mentoring.

## Data Availability

The supernovae light curves are presently available at the Open Supernova
Catalogue. The data and code used to perform the analysis presented in this
publication can be found on GitHub in the following repository:
https://github.com/HeloiseS/CODE_AND_DATA_MN_22_2485_MJ

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##  Appendix A Sample Light curves

Figure 10: Light curves in our sample of type IIb SNe. Note that we show both
the observationally constrained (grey) and the SNID (blue) explosion dates
presented in P19 in all light curves to illustrate the similarities (or
differences) – the explosion date we use in practice is shown in Table 1.
Figure 11: Light curves in our sample of type II SNe. Note that when both the
observationally constrained (Grey) and the SNID (Red) explosion dates are
reported in the literature they are shown here to illustrate the similarities
(or differences) – the explosion date we use in practice is shown in Table 1.
The explosion dates used by P19 are shown with a solid black line.