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Welcome to the SESNspectraPCA wiki! This page will be dedicated to presenting finished versions of plots with the goal of being a first draft of a paper.
We use SNID-ified spectra that have been cleaned for atmospheric effects and continuum normalized. Since we are only interested in large features in the spectra, we reduce the resolution by combining wavelength bins. This reduces the dimension of each spectra. Spectra with large gaps are thrown out.
Figure 1 shows the SNID Spectra used at phase = 15 +/- 5 Days after V max. The spectra are filtered to the wavelength range 4000 to 7000, and spectra with gaps have been thrown away.
Here is a link to the Plotly version of Figure 1 for interacting: Plotly Figure 1
We use PCA to reduce the dimension of the spectra phase space. PCA finds the directions in phase space that contain the most information on the spectra variation. Below we show the first few eigenspectra of the phase = 15 +/- 5 set of spectra.
Figure 2 shows the first 8 principal components. These are the most important components for explaining spectra variation. The label on each component shows the cumulative percent of variation that is explained by the components up to the one in question. As we can see, using the first 8 components is enough to preserve 80% of the information in our spectra sample.
Using the eigenspectra, we can reconstruct the original spectra. Using more eigenspectra will result in better reconstructions of the original spectra.
Figure 3 shows an example spectrum that has been reconstructed using varying numbers eigenspectra.
We now are interested in analyzing the spectra in PCA space. For the time being, we consider the first 5 principal components, which contain about 75% of the spectra variance. One goal is to find clustering that could be used for classification, or understanding how different the current SNe categories really are.
Figure 4 shows the 2D marginalizations of the PCA components for the phase = 15 +/- 5 spectra sample. The translucent larger circles show the centroids of the different SNe types. There are four important things to note:
- Component 0 separates Ic broad from the other types.
- Component 3 separates IIb from the other types.
- Component 2 separates a subsample of Ib from the rest of the Ib.
- There are some rogue SNe that are found near the centroids of other types, far from their own centroid.
We will investigate each of the above points.
First consider bulletin 3. This separation of a subsample of type Ib SNe is a little unexpected. We are immediately interested in which SNe make up this subsample
We present Figure 4 in Plotly so that it is interactive. Figure 4 Plotly
Figure 5 shows PCA component 2 vs PCA component 4, and zooms in on the Ib subtype.
For the following, we ignore the type IIb SN2011ei and focus on the other five SNe that are classified as type Ib. In order to understand how these SNe spectra differ from the other Ib spectra, we average the main Ib spectra and compare the average Ib spectra to the subgroup.
Figure 6 shows the average type Ib spectrum compared to each of the subgroup Ib spectra. We include a scaled version of PCA Component 2 in each plot to see which feature this component is singling out. It appears as if the Ib subgroup spectra are systematically shifted to the left.
We now consider the rogue SNe mentioned in bulletin 4. Specifically, we will consider the plot of PCA component 1 vs PCA component 3. There are some rogue SNe near the Ib locus, and also some rogue SNe near the Ic locus.
Figure 7 identifies SN2004gt as a type Ic which is far from the Ic locus and very close to the Ib locus.
Figure 8 identifies SN2009jf and SN1990I as type Ib SNe near the Ic locus, and SN1997ef and SN2003jd as type Ic broad near the Ic locus.
We present a series of plots comparing the first 8 PCA eigenspectra to the Liu, Modjaz et al templates.
Figure 9 shows the average spectra that make the templates.
Figure 10 shows the Ib template with the first 8 PCA eigenspectra overplotted.
Figure 11 shows the Ic template with the first 8 PCA eigenspectra overplotted.
Figure 12 shows the IIb template with the first 8 PCA eigenspectra overplotted.
We are interested to understand how effective PCA is as a function of time after V max. Certain differences, like the He absorption lines that distinguish Ib/Ic SNe from Liu, Modjaz et al, are only apparent about 10 days after Vmax. Here we present the 2D marginalizations of the PCA components for phase = 0 +/- 5 days and compare them to the above Figure 4.
Figure 13 shows the phase = 0 +/- 5 PCA 2D marginalizations. Notice that component 1 still does a reasonable job of separating the type IIb SNe. PCA component 0 separates Ic broad very well. Interestingly, the subgroup of type Ib SNe that was identified at phase = 15 +/- 5 in Figure 5 is no longer distinguishable.
Figure 14 presents the eigenspectra corresponding to the phase = 0 +/- 5 sample.
Here we present plots showing how many PCA components are necessary to reconstruct each SNe type of spectrum to a given accuracy. We also show plots for a given number of PCA components, what accuracy is achieved in the reconstruction.
Figure 15 shows how many PCA components are needed to achieve 50% accuracy in reconstructing the spectra.
Figure 16 shows how many PCA components are needed to achieve 80% accuracy in reconstructing the spectra.
Figure 17 shows the percent accuracy of the reconstruction using 5 components.
Figure 18 shows the percent accuracy of the reconstruction using 10 components.
Figure 19 shows the percent accuracy of the reconstruction using 20 components.
Here we show the results of using ICA on spectra that have been reduced in dimension using PCA. The following plot uses 8 PCA components. Every component in ICA is weighted equally, so it is not helpful to only look at a subset of the ICA components.
Figure 20 shows ICA 2D marginalizations using 8 components. The original spectra are reduced using the first 8 PCA components.
Here we repeat our analysis using smoothed spectra for the IcBL supernovae. Figure 21 shows the result of smoothing the IcBL spectra.
Figure 22 shows the eigenspectra using the smoothed IcBL spectra.
Figure 23 shows one of the 2D PCA marginalization plots using the smoothed IcBL spectra.
Figure 24 shows the distribution of HEI 5876 line velocities of the SNe from Liu et al 2016 Table 3, for the SNe with phases 15 +/- 5 days. The vertical lines mark the line velocities for the unusual cluster of type Ib SNe found in Figure 5.
This work has primarily focused on analyzing spectra with phase in the range [10,15]. The eigenbasis that PCA returns is a complete basis for the phase space of spectra. Therefore, it is reasonable to decompose a spectrum with a phase outside of the range [10,15] in this basis. It is impossible to make any inferences about a single spectrum with phase outside the relevant range decomposed in the basis calculated by PCA using only spectra with phase in [10,15]. However, there may be a useful application. In the following figures, I use many phases for each type Ib and Ic spectra and plot the trajectory of each SNe in the PCA space as a function of phase. This could be informative for example, if every type Ib spectrum happened to orbit the Ib centroid, except for one that has a dramatically different trajectory.
Figure 25 shows an animation of the trajectories of all of the type Ic SNe in PCA space. Most of the SNe begin far away from the Ic centroid, then get closer as the phase approaches 15 days. Many of the SNe end up approaching the Ib centroid, but only at very large phase, ie near 100 days. However sn2007gr makes a loop in PCA space that brings it close to the Ib centroid much faster than the other SNe. Also, we see that it is very unlikely that the outlier sn2004gt started near the Ic centroid.
Figure 26 shows an animation of the trajectories of all of the type Ib SNe in PCA space.
