STANDARDIZING TYPE Ia SUPERNOVA ABSOLUTE MAGNITUDES USING GAUSSIAN PROCESS DATA REGRESSION

We begin by training a parameterized model for the multi-band light curves that make up our data set. For a single blueshifted filter set, the multi-band light curves underlying supernovae are modeled as arising from a Gaussian process,

$$\begin{equation} m_{(t,\lambda)} \sim {\rm GP}(\bar{m}(t,\lambda ;\mathbf {\bar{m}}_0),k_m(t,\lambda,t^{\prime },\lambda ^{\prime };l_{k_m}, \boldsymbol {\sigma }_{k_m})), \end{equation} \tag{ 1 }$$

where m_{(t, λ)} is the photometric magnitude at epoch t relative to the B-band peak in the filter indexed by λ. (The definition of a Gaussian process and associated equations used in the analysis of this article are given in the Appendix.) Subscripted (t, λ) represent points whereas those in unsubscripted parentheses are function variables. This notation means that for a set of coordinates {(t, λ)}, the magnitude values are drawn from a normal distribution

$$\begin{equation} m_{\lbrace \left(t,\lambda \right)\rbrace } \sim \mathcal {N}\left(\lbrace \bar{m}(t,\lambda ;\mathbf {\bar{m}}_0)\rbrace,K \right), \end{equation} \tag{ 2 }$$

where the elements of the model covariance matrix are

$$\begin{equation} K_{\left(t,\lambda \right),\left(t^{\prime },\lambda ^{\prime }\right)} = k_m(t,\lambda,t^{\prime },\lambda ^{\prime };l_{k_m}, \boldsymbol {\sigma }_{k_m}). \end{equation} \tag{ 3 }$$

Measurements of a realized supernova are then also described by a Gaussian process,

$$\begin{eqnarray} m^{\star }_{\left(t,\lambda \right)} &\sim& {\rm GP}(\bar{m}(t,\lambda ;\mathbf {\bar{m}}_0),k_m(t,\lambda,t^{\prime },\lambda ^{\prime };l_{k_m}, \boldsymbol {\sigma }_{k_m})\nonumber\\ &&+ n_m(t,\lambda,t^{\prime },\lambda ^{\prime };\boldsymbol {\sigma }_{n_m})). \end{eqnarray} \tag{ 4 }$$

Given a set of measurements $m^{\star }_{\left(t,\lambda \right)}$, the likelihood that the data arise from a Gaussian process can be expressed analytically, as well as the mean and covariance of the probability distribution function of a finite set of underlying points m_{(t, λ)} (see the Appendix).

For our model, we choose as mean function $\bar{m}(t,\lambda ;\mathbf {\bar{m}}_0)$ the synthetic photometry of band λ from the updated templates of Hsiao et al. (2007) offset by zero points $\mathbf {\bar{m}}_0$ for each band. The mean function could include a parameter for the date of peak B magnitude but the spectral time series we work with has been preprocessed and is already expressed in terms of days relative to peak determined by SALT2. A zero mean function would lead to expectations of zero flux in temporal regions lacking data; the Hsiao mean is used to avoid abrupt deviations toward zero where there are temporal gaps in the light curves, whose locations differ from supernova to supernova.

The kernel represents the covariance of the underlying flux: we use the square exponential kernel

$$\begin{equation} k_m(t,\lambda,t^{\prime },\lambda ^{\prime };l_{k_m}, \boldsymbol {\sigma }_{k_m}) = \sigma _{k_m}^2(\lambda)\delta _{\lambda \lambda ^{\prime }} \exp { \left[ -\left(\frac{t-t^{\prime }}{l_{k_m}}\right)^2 \right] }, \end{equation} \tag{ 5 }$$

where each band has its own normalization factor $\sigma _{k_m}(\lambda)$ but shares a common timescale $l_{k_m}$. The $l_{k_m}$ does not represent the timescale of the light curves themselves, but rather those of deviations from the mean model. The kernel does not correlate the light curves in different bands. Although supernova flux across bands is correlated, for this stage of the analysis we preferred imposing no prior assumptions on color. We opt for the square exponential kernel as it is frequently used in the Gaussian process literature; model testing comparing the likelihoods of other kernels is deferred to future study.

Measurements of m include a nugget term that contains all other sources of variance in the photometry. Our nugget model is

$$\begin{equation} n_m(t,\lambda,t^{\prime },\lambda ^{\prime };\boldsymbol {\sigma }_{n_m})=\left(\sigma _{n_m}^2(\lambda) + \mathrm{var}(m_{(t,\lambda)}) \right)\delta _{t,t^{\prime }}\delta _{\lambda,\lambda ^{\prime }}, \end{equation} \tag{ 6 }$$

where within the scope of our analysis there are no two measurements with the same (t, λ) so the nugget applies per measurement. Each band has its own intrinsic dispersion and the SNfactory photon noise has variance var(m_{(t, λ)}). The measurement covariance is very small given the small overlap of the griz bands, and so is ignored.

The kernel includes an additional term that accounts for radial peculiar velocities (Bonvin et al. 2006):

$$\begin{equation} \sigma _{\alpha \beta }^2=\left[\sigma _{z}^{{\rm pec}}\frac{5}{\ln {(10)}}\frac{(1+z_\alpha)}{\chi H}\right]^2\delta _{\alpha \beta }, \end{equation} \tag{ 7 }$$

where the peculiar velocity dispersion, $\sigma _{z}^{{\rm pec}}$, can be either a fixed or fit parameter, and z_α is the actual redshift of the supernova; in this paper, we fix $\sigma _z^{{\rm pec}}=300$ km s⁻¹. Here α and β are the supernova indices, so the δ_αβ reflects the common realization of peculiar velocity for a single supernova. At low redshifts, the conformal distance can be conveniently approximated as χ ≈ z − 0.75Ω_Mz² for a flat ΛCDM cosmology.

For each of four training/validation runs, 75% of the SNfactory supernovae are designated as the training set. Photometry with S/N < 50 are culled as often being associated with poor extractions and a supernova must have at least 16 light-curve points total over all bands to be included. Cuts are applied after the division into training and validation sets, so each of the analyses do not have exactly the same sample sizes: after the S/N and number-of-point cuts there remains 80–90 training supernovae per run.

All training-set supernovae are used to fit the free parameters (and hyperparameters) that give the maximum likelihood of the Gaussian process model. The hyperparameters $l_{k_m}$, $\boldsymbol {\sigma }_{k_m}$, and $\boldsymbol {\sigma }_{n_m}$ are common to all SNe Ia and are fit simultaneously to the full ensemble of data, whereas a distinct $\mathbf {\bar{m}}_0$ is fit for each supernova. The best-fit hyperparameters for $l_{k_m}$, $\boldsymbol {\sigma }_{k_m}$, and $\boldsymbol {\sigma }_{n_m}$ for one of the four-fold training runs are shown in Table 1; these values do not differ significantly between the different realizations. The strength of the model covariance between points in a single light-curve range from 0.08 to 0.14 mag with correlation-length scales of around 6 days. The intrinsic dispersion is 0.05–0.07 mag except in the rest-frame UV where the dispersion is 0.14 mag.

Table 1. Best-fit Hyperparameters of the Gaussian Process that Models the Supernova Multi-band Light Curves, Given in Equation (1)

System z	$l_{k_m}$	$\sigma _{k_m}(g)$	$\sigma _{k_m}(r)$	$\sigma _{k_m}(i)$	$\sigma _{k_m}(z)$	$\sigma _{n_m}(g)$	$\sigma _{n_m}(r)$	$\sigma _{n_m}(i)$	$\sigma _{n_m}(z)$
0.00	5.63	0.09	0.09	0.12	...	0.05	0.06	0.07	...
0.25	5.87	0.13	0.08	0.09	0.13	0.14	0.05	0.06	0.06
0.50	6.18	...	0.11	0.08	0.10	...	0.06	0.05	0.05
0.75	6.56	...	0.10	0.11	0.11	...	0.14	0.06	0.05

Notes. Given are the results of one of the four training sets for each of the blueshifted DES photometric systems. All values have magnitude units except $l_{k_m}$, given in supernova-frame days.

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The regressed g-band date of maximum has a distribution with mean and standard deviation of 0.058 ± 0.76 days. The temporal alignment of our model is consistent with that of SALT2.

To summarize this first step in the analysis, supernova light curves are described by a parameterized model for the covariance in their residuals about a parameterized mean function. Light-curve photometry of a supernova training set is used to optimize the parameters that best describe their scatter.

We now turn to the supernova parameters used to describe the light curve of a single event, in analogy to the light-curve shape and color parameters used in classical light-curve fitters. The parameters adopted in this article are the (linearly transformed) values of interpolated light curves.

Since we are ultimately interested in determining the magnitude at peak brightness, both training and validation sets are pared down to include only those supernovae having a first measurement at least two days before B peak to avoid extrapolating the value of $M_{(\lambda _0, t_0)}$. This leaves 49–57 training supernovae.

Observed supernovae have disparate temporal sampling; those data must be transformed into a set of parameters that can be directly compared between supernovae. The Gaussian process model of the light curve is used to predict the fluxes on a common set of epochs, daily from −10 to 35 rest-frame days relative to B peak, in the blueshifted DES bands that are spanned by SNfactory data. (The choice of daily sampling of the light curves was arbitrary and probably finer than necessary.) The predictions (Equation (A5)) have covariance (Equation (A6)) so 50 light-curve grids are realized per supernova to include uncertainties in the training and propagate uncertainties in the inferences. Predictions and realizations of regressed light curves from sample supernova photometric measurements are shown in Figures 1–3.

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We want to calibrate the supernova absolute magnitude in some band λ₀ at the phase of peak B brightness t₀: this is referred to as the "true" absolute magnitude and is estimated by the $M_{(\lambda _0, t_0)}$ predicted by our model. This information is removed from the predicted light curves by subtracting $M_{(\lambda _0, t_0)}$ from all other magnitudes. What remains is a light-curve shape in the selected band λ₀ and colors for the other bands, giving 137 (46 × 3 − 1 for the case of three bands) or 183 (46 × 4 − 1 for four bands) potential parameters to describe a supernova realization.

For practical reasons, we re-express the light-curve shapes and colors in terms of their principal components ordered by the dimensions that carry the leading population variance by applying a principal component analysis (PCA) on all 50 realizations of all training supernovae. The coordinates of the leading eigenvectors that account for 95% of the variance are used as the supernova parameters, and are denoted by x. This reduces the dimensionality of the problem down to ∼15 in anticipation that the truncated components carry little real signal that correlates with absolute magnitude. As a result of this transformation each parameter no longer represents a specific color at a specific phase, but rather the contribution to the temporal evolution of all colors from a single PCA eigenvector. Note that due to the correlated noise in the regressed light curves, PCA is not encapsulating pure supernova diversity efficiently in a minimal number of components.

As a representative example of one of the training sets, we show results from one of the runs calibrating the i band of the z = 0.25 filter set. The first several PCA eigenvectors describing light-curve shapes and colors (after subtraction of the mean) are shown in Figure 4. These eigenvectors capture 54%, 15%, 9%, and 5% of the population variance.

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The distribution of PCA coefficient values for the training set is shown in Figure 5. For each component i, the coefficients have standard deviation $\sigma _{x_i}$. In solid (red) are the coefficients for the mean light curves and colors for each training-set supernova. Surrounding each of those points is a cloud of gray points of the coefficients from the 50 realizations; the size of the cloud is reflective of the dispersion in the Gaussian process model as well as the data quality. The ∼50 training-set supernovae do not fill a continuous distribution in this space; there are clear outliers due to realizations of a single supernova. The distinct blobs containing multiple supernovae seen in the x(0)–x(2) plot are not significant as they are not as prominent in the three other training runs.

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Mean and perturbed light curves from 1$\sigma _{x_j}$ and 2$\sigma _{x_j}$ changes in the x(j) parameter are shown for j = 0, 1, 2 in Figures 6–9. The j = 0 eigenvector (Figure 6) behaves as if it were due to dust: the correction is (mostly) phase independent and monotonically decreases from blue to redder bands. The j = 1 eigenvector (Figure 7) generates light curves that vary in width, with supernovae with relatively skinny blueshifted-g (observer-frame central wavelength of 380 nm) light curves having relatively broader light curves in the redder bands. The g − X colors vary but the other colors do not vary. The j = 2 eigenvector (Figure 8) has a particularly strong influence in the shape and colors in the near-UV band. It has the property that the curves intersect at later phases; the effect is pronounced in the blueshifted i band where the phase and height of the secondary maximum vary. Light-curve shapes and all color combinations also are influenced by this component. The j = 3 eigenvector (Figure 9) is similar to j = 2 and has a particularly exaggerated influence on the shape of the near-UV band.

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The PCA coefficients are correlated with the SALT2 color and x₁ parameters. Figure 10 contains scatter plots of the first four principal components with the SALT2 color and x₁ parameters for one of the runs for the z = 0 filter set: the first PCA component is strongly correlated with color and the second and fourth are correlated with x₁.

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To summarize this second step of the analysis, the mean multi-band flux and covariance of the underlying light curve of an individual supernova are calculated for a set of epochs and filters using the sparsely measured photometry combined with the trained light-curve Gaussian process model. Each realization of the light curves is decomposed into one "true" absolute magnitude and light-curve shapes and color curves. Anticipating that absolute-magnitude variations are encoded in SN Ia heterogeneity, for the light-curve shapes and colors we transform to a new coordinate system that is defined with a PCA on the training set.

The light-curve and shape coordinates in the PCA basis are correlated with true absolute magnitude (modulo a constant offset for the arbitrary distance at which the SNe are placed; see Section 2.2), as seen in Figure 11.

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In this section, we relate light-curve and color parameters to absolute magnitudes with a linear and a Gaussian process model. The linear model makes an explicit choice for the relationship between light-curve parameters and absolute magnitude that can be well constrained by our small data set. The Gaussian process methodology lets the data drive the model. While having the advantage of not requiring an explicit model, results can be sensitive to statistical fluctuations in the training set or large interpolations in sparsely populated regions. An analogous contrast can be made between SALT (Guy et al. 2005) which used parameterized functions, and MLCS (Riess et al. 1996) which used data averages to construct light-curve models.

Linear Model. The absolute magnitude at phase t₀ and band λ₀ is a linear function of the light-curve parameters $M_{\left(t_0, \lambda _0\right)} (\mathbf {x};p_{\bar{M}}) = \bar{M}_0+\sum _{j=0}^{n_{{\rm max}}-1} p_j x(j)$, where $\bar{M}_0$ and p_j are fit parameters. Most of the over 100 supernova parameters capture little of the population variance: as is justified in Section 2.4, n_max = 4 is adopted in this article.

Figure 12 shows the linear-model solution for the z = 0.00 filter set calibrating the g band, which closely corresponds to the bands for which SALT2 is directly trained. As in SALT2, there are clear correlations between the first two parameters, which in turn are correlated with color and x₁, respectively (as shown in Figure 10). There is no correlation for the third parameter, but a correlation with the fourth parameter is apparent.

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Gaussian Process Model. While previous analyses have modeled linear and quadratic correlations between light-curve parameters and absolute magnitude, it is well known that such simple relationships are not representative of the full range of light-curve shapes (Hamuy et al. 1996). Here we infer the absolute magnitude through regression in the high-dimensional PCA parameter space.

The fiducial magnitude at phase t₀ and band λ₀, $M_{\left(t_0, \lambda _0\right)}$, is modeled as a Gaussian process dependent on light-curve shapes and colors expressed by the parameters x,

$$\begin{eqnarray} &&M_{(t_0, \lambda _0)} \sim {\rm GP} (\bar{M}(\mathbf {x};\bar{M}_0),k_M(\mathbf {x},\mathbf {x^{\prime }};\mathbf {l}_{k_M},\sigma _{k_M})+n_M(\mathbf {x},\mathbf {x^{\prime }};\sigma _{n_M})). \nonumber\\ \end{eqnarray} \tag{ 8 }$$

The mean function $\bar{M}(\mathbf {x};p_{\bar{M}}) = \bar{M}_0+\sum _{j=0}^{n_{{\rm max}}-1}p_j x(j)$ is linear and has free parameters $\bar{M}_0$ and p_j. The kernel function is again a square exponential

$$\begin{eqnarray} &&k_M(\mathbf {x},\mathbf {x^{\prime }};\mathbf {l}_{k_M},\sigma _{k_M}) = \sigma _{k_M}^2 \exp { \left[ -\sum _{j=1}^n \left(\frac{x(j)-x^{\prime }(j)}{l_{k_M}(j)}\right)^2 \right] }, \nonumber\\ \end{eqnarray} \tag{ 9 }$$

where n is the number of elements in x, i.e., the number of eigenvectors capturing 95% of the population variance. There is an independent per-point nugget, $n_M(\mathbf {x},\mathbf {x^{\prime }};\sigma _{n_M})= \sigma _{n_M}^2 \delta _{\mathbf {x},\mathbf {x^{\prime }}}$.

Training-set supernovae are used to find the best-fit values for $\bar{M}_0$, $\mathbf {l}_{k_M}$, p_j, $\sigma _{k_M}$, and $\sigma _{n_M}$. The values of $\sigma _{k_M}$, the strength of magnitude correlations, are found to be around 0.18 mag. The residual absolute magnitude dispersion after training is given by the $\sigma _{n_M}$s, which range from 0.04 to 0.07 mag except for the z = 0.75 bands that have a range from 0.07 to 0.11 mag.

The hyperparameters for the length scale of the correlation $\mathbf {l}_{k_M}$ alone do not provide much physical insight. A more interesting statistic is the ratio of the correlation scale and a typical value of the parameter in the training set; the metric represents a typical scale for each parameter in the exponent in Equation (9), which in turn gives the relative contribution to the correlation strength. Parameters with low values in Figure 13 contribute more to the correlation. Figure 13 shows the best-fit $l_{k_M}(j)/\sigma _{x_j}$, where $\sigma _{x_j}$ is the standard deviation of the jth coefficient of all realizations in the training set of a representative run.

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The relationship between each parameter and absolute magnitude is illustrated for a slice in parameter space for the same run shown for the linear model; Figure 12 shows the most-likely absolute magnitude as a function of parameter value while holding the coordinates for all other dimensions at zero. In this example, the coordinate range spanned by the supernova set decreases with the dimension order: −6 < x(0) < 4, −2 < x(1) < 2, −1.8 < x(2) < 1.8, and −1 < x(3) < 1. The correlation between parameters and absolute magnitude is nonlinear.

The Gaussian process and linear solutions for the first two parameters are in general agreement, consistent with the SALT2 model. X(2) does not have a monotonic trend in the region and extends a small range of influence on luminosity. The fourth parameter has agreement in X(3) within the range where there are training-set data.

To summarize this step of the analysis, supernova absolute magnitudes are taken to be a function of the light-curve shapes and colors. In one analysis a linear fit is performed. In the other the functional form is not explicit; instead its stochastic behavior is modeled. Training-set supernovae provide several measurements of this function and we use these to train a Gaussian process description of their correlation. The trained Gaussian process provides a prescription for interpolating an "inferred" absolute magnitude from the light-curve shape and color. It should be emphasized that the training procedure is not optimized to minimize the dispersion in the residuals of absolute magnitudes but rather to maximize the likelihood of the data for the Gaussian process model.

The SNfactory data do not have perfect temporal sampling or S/N. As a selection criterion, we study those validation supernovae with synthetic photometry in at least eight distinct phases in each of the bands. This leaves 13–23 validation supernovae per run except in the z = 0.75 filter set, which has 7–11 SNe per run. Measurement uncertainties contribute to uncertainties in the interpolated absolute magnitude and light curves. To illustrate, the distribution of light curves realized from the Gaussian-process-predicted mean and covariance shown in Figures 1–3 indicates the size of the uncertainties. Figure 15 contains a histogram of the standard deviations of the measured true absolute magnitude at B peak for the i band of the z = 0.25 filter set; here the absolute-magnitude uncertainties per supernova range from 0.02 to 0.06 mag.

As a representative example of the light-curve regression, we present results for one random supernova from one of the runs with the z = 0.25 filter set. Figure 1 shows the synthetic photometry, the predicted mean of the underlying light curves in solid, and 10 random realizations in dashed lines.

We identify two odd supernovae, SN 13 and SN 19, that are culled from the sample. SN 19 data and predicted light curves for the z = 0 filter set are shown in Figure 2. This supernova does not have the wavelength coverage to make the z = 0.25 sample and its PCA coefficients fail the distance cut for z = 0.75. It is an extremely slow decliner with a g band Δm₁₅ = 0.7 derived from the Gaussian process prediction. The PCA coefficients from the z = 0 case are shown among those of the training set in Figure 14; it is clear that SN 19 does not have a corresponding analog in the training set. With z = 0.0055, this supernova has the lowest cosmic microwave background-frame redshift of the full sample of which >96% has z > 0.015; a peculiar velocity of 300 km s⁻¹ would contribute a 0.40 mag residual. SN 13 data and predicted light curves for the z = 0.25 filter set are shown in Figure 3. The discrepancy is accounted for by one spectrum near the peak that is fainter than expected; indeed this discrepancy disappears in subsequent versions of the spectral reduction.

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Validation supernovae may not have an analog in the training set. We approximate the PCA coefficient distribution as Gaussian, described by the standard deviations $\sigma _{x_i}$ of each dimension. A validation supernova is included in the analysis if it lies within the core of the distribution

$$\begin{equation} \sum _j{\left(\frac{x_j}{\sigma _{x_j}}\right)}^2 < \chi ^2(p,n_{{\rm PCA}}), \end{equation} \tag{ 10 }$$

where χ²(p, n) is the value where the cumulative χ² distribution has p-value 0.05 and n_PCA is the number of principal components that capture 95% of the variance. This cut criterion was made after the distributions were constructed and is therefore not blind. However, 0.05 is selected as a nice round number that should capture 95% of the population and ensuing results are insensitive to departures from this value. This cut typically removes 0–2 objects from the sample.

The covariance in the regression is accounted for in our analysis by using 50 light-curve realizations of each supernovae. Each of the 50 multi-band light-curve realizations of each supernova is temporally sampled from −10 to 35 days using the best-fit Gaussian process model of the training set. The light curves are normalized by the true $M_{\left(t_0,\lambda _0\right)}$ and expressed in the coordinate system defined by the PCA eigenvectors. The PCA coefficients are used to infer $M_{\left(t_0,\lambda _0\right)}$. The standard deviation of its 50 realizations provides the measurement uncertainty of the $M_{\left(t_0,\lambda _0\right)}$ of a supernova; this value depends on the quality of the measured light curve and ranges. The means and standard deviations of the 50 realizations of true and inferred absolute magnitudes from the validation procedure (modulo a constant offset) are given in Tables 2–5 for the Gaussian process model. (In the tables, omitted supernovae were used for light-curve training but did not pass the selection criteria for the peak-magnitude analysis. In italics are supernovae that have data before two-day pre-maximum with light curves that have at least eight points. In bold are supernovae used for validation that have data before the two-day pre-maximum and at least eight points in all light curves.) Figure 15 shows the histogram of the standard deviations of the inferred absolute magnitude at the B peak for the i band of the z = 0.25 filter set: the inferred absolute magnitude measurement uncertainties per supernova mostly fall within the range 0.04–0.10 mag.

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The accuracy to which true absolute magnitudes are predicted by absolute magnitudes inferred from SNfactory synthetic light curves is the root mean square of the difference between the true and inferred $M_{\left(t_0,\lambda _0\right)}$ over all realizations of all supernovae. The wrms of the differences of each validation set, and then the mean and standard deviation of those wrms' over the four training/validations ($\overline{\rm wrms}$ and σ(wrms), respectively) are calculated. The wrms weights the squared residual by the inverse measurement variance visualized in Figure 15 (the expression for the wrms is given in Blondin et al. 2011). The mean shows how well the algorithm standardizes the distance indicator over all supernovae. The standard deviation of the wrms of the four runs indicates the sensitivity to the sample selection. The wrms' have contributions from measurement errors, peculiar velocities, and the intrinsic ability that the methodology has to calibrate supernovae, σ_int. Assuming a peculiar velocity dispersion of 300 km s⁻¹, for each run we report the mean and standard deviations of the fit value the intrinsic dispersion σ_int.

The wrms calculated above represents an estimate of the absolute-magnitude uncertainty when using 75% of the data training, such that none of the validation dispersions are calculated with supernovae used in the training. Our analysis lends to easy calculation of the wrms from K-fold cross analysis (as done by Blondin et al. 2011) for K = 4, where the dispersion is calculated for all supernovae together despite the fact that they are based on different training sets; these results are given in Tables 6 and 7. We provide two numbers for the K-fold analysis; the first using the subset after applying the sample cut of Equation (10) and the second using the full sample of objects. The former statistic is calculated anticipating that the supernova parameter space is not densely sampled and considering that our absolute-magnitude model may not extrapolate well to objects outside the training regime.

Table 6. Results from the Linear Model

Filter System z	Band	σ0	$\overline{\rm wrms}$	σ(wrms)	$\overline{\sigma _{{\rm int}}}$	σ(σint)	Four-fold Cross-validation	NSN
0.00	g	0.322	0.146	0.026	0.123	0.049	0.154 (0.158)	50
0.00	r	0.229	0.118	0.014	0.049	0.036	0.120 (0.122)	49
0.00	i	0.265	0.137	0.018	0.099	0.021	0.136 (0.140)	50
0.25	g	0.437	0.141	0.039	0.119	0.052	0.155 (0.169)	44
0.25	r	0.301	0.119	0.039	0.068	0.052	0.122 (0.147)	43
0.25	i	0.236	0.091	0.019	0.025	0.030	0.090 (0.102)	43
0.25	z	0.250	0.136	0.032	0.111	0.032	0.130 (0.132)	44
0.50	r	0.391	0.153	0.033	0.126	0.042	0.156 (0.157)	50
0.50	i	0.301	0.149	0.023	0.123	0.021	0.148 (0.149)	48
0.50	z	0.248	0.135	0.025	0.093	0.027	0.132 (0.131)	49
0.75	r	0.320	0.140	0.036	0.093	0.062	0.148 (0.184)	35
0.75	i	0.237	0.129	0.042	0.088	0.059	0.133 (0.184)	35
0.75	z	0.210	0.131	0.035	0.080	0.058	0.133 (0.160)	35

Notes. For each filter system and calibrated band: σ₀, the raw standard deviation of $M_{(t_0,\lambda _0)}$ for the input sample; $\overline{\rm wrms}$ and σ(wrms), the mean and standard deviation over the four validation samples of the wrms difference between true and inferred magnitude; the mean intrinsic dispersion and its standard deviation over the four validation samples, the four-fold cross-validation error estimate with (and without) rejection of objects with extreme parameter values; N_SN is the number of validated supernovae.

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Table 7. As in Table 6, but for the Gaussian Process Model

Filter System z	Band	σ0	$\overline{\rm wrms}$	σ(wrms)	$\overline{\sigma _{{\rm int}}}$	σ(σint)	Four-fold Cross-validation	NSN
0.00	g	0.322	0.129	0.040	0.099	0.044	0.134 (0.145)	50
0.00	r	0.229	0.120	0.041	0.087	0.039	0.123 (0.130)	49
0.00	i	0.265	0.131	0.040	0.100	0.045	0.136 (0.146)	50
0.25	g	0.437	0.101	0.017	0.065	0.018	0.102 (0.141)	44
0.25	r	0.301	0.100	0.021	0.068	0.026	0.102 (0.139)	43
0.25	i	0.236	0.093	0.013	0.064	0.017	0.094 (0.128)	43
0.25	z	0.250	0.113	0.025	0.076	0.024	0.113 (0.158)	44
0.50	r	0.391	0.122	0.019	0.085	0.033	0.121 (0.118)	50
0.50	i	0.301	0.118	0.013	0.077	0.021	0.116 (0.121)	48
0.50	z	0.248	0.114	0.005	0.073	0.006	0.113 (0.112)	49
0.75	r	0.320	0.129	0.046	0.061	0.072	0.136 (0.189)	35
0.75	i	0.237	0.131	0.054	0.068	0.050	0.147 (0.212)	35
0.75	z	0.210	0.120	0.032	0.047	0.039	0.123 (0.163)	35

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For the linear model, the number of light-curve parameters related to absolute magnitude is set as n_max = 4. This choice is made as follows: we calculate $\overline{\rm wrms}$ for the linear model, varying n_max from 1 to 8 for the i-band calibration of the z = 0.25 filter set, obtaining 0.097, 0.099, 0.104, 0.091, 0.092, 0.094, 0.093, and 0.095 mag. There is a step in the $\overline{\rm wrms}$ when four parameters are used. In the Gaussian process model, n_max is set to the number of principal components that capture 95% of the variance.

Calculated dispersions from the validation are shown in Table 6 for the linear fit, and in Table 7 for the Gaussian process model. The weighted mean of each validation set is ≲ 0.01 mag.

The statistics in Efron (1983) are calculated for the case of the i band from the z = 0.25 filter set: the apparent error rate is determined from the residuals of the training-set supernovae run through the same processing as the validation supernovae. The linear model gives 0.089 mag. The resulting 0.025 mag from the Gaussian process model is an over-optimistic error estimate: the best-fit $\sigma _{n_M}=0.060$ mag nugget parameter (introduced after Equation (9)) gives an estimate of the absolute magnitude dispersion of the training supernovae. The bootstrap estimate is calculated from 50 realizations of training sets generated using random draws with replacement from the full set and the ∼37% unselected supernovae forming the validation set; this error estimate is 0.107 (0.117) mag for the linear (Gaussian process) model. The .632 estimator is not rigorously motivated but is empirically exhibiting small biases in the estimator of the true prediction error in certain cases; it is a function of the apparent and bootstrap estimators for which we get 0.100 (0.083) mag for the linear (Gaussian process) model.