Galaxy Properties from the Wisconsin Group

Principal Component Analysis (PCA) Method

Chen et al. (2012) model physical galaxy parameters based on a library of model spectra for which principal components (PCs) have been identified. The method is applied in the following steps.

Create library of model spectra

The library of model spectra is based on Bruzual & Charlot (2003) stellar population synthesis models. The model is parameterized by the following characteristics.

Star formation histories (SFHs)

Each SFH consists of three parts: an underlying continuous model + a series of super-imposed stochastic bursts + a random probability for star formation to stop exponentially (i.e. truncation). Figure 1 shows three examples of SFHs. The top panel is a continuous model, middle panel shows a continuous model with two random bursts, a truncation can be found in the bottom panel.

Figure 1: Three different examples of SFHs.

Metallicity

95% of the model galaxies in the library are distributed uniformly in metallicity from 0.2 - 2.5 Z_☉; 5% of the model galaxies are distributed uniformly between 0.02 and 0.2 Z_☉.

Dust extinction

Dust extinction is modeled using the two-component model described in Charlot & Fall (2000). The V-band optical depth has a Gaussian distribution over the range 0 < τ_V < 6. with a peak at 1.2 and 68% of the total probability distribution distributed over the range 0~2. This prior distribution of τ_V values is motivated by the observed distribution of Balmer decrements in SDSS spectra (Brinchmann et al. 2004). The fraction of the optical depth that affects stellar populations older than 0.01 Gyr is parameterized as μ, which is again modeled as a Gaussian with a peak at μ = 0.3, and a 68 percentile range of 0.1~1.

Velocity dispersion

Each of the model spectra is convolved to a velocity that is uniformly distributed over the range of values from 75 to 400 km/s.

Principal components (PCs) are identified from the model library

The regions around nebular emission lines are masked in the model spectra. We mask 500 km/s around the [OII]3726.03, [OII]3728.82, Hζ3889.05, [NeIII]3869.06, Hδ4101.73, Hγ4340.46, Hβ4861.33, [OIII]4959.91, and [OIII]5007.84 Å lines. Each spectrum in the masked library is normalized to its mean flux between 3700-5500 Å (this is the range we use for analysis). The mean spectrum of the masked library is calculated and subtracted from each of the model spectra.

Mean spectrum of model library and 7 eigenspectra

Figure 2: From top to bottom: the mean spectrum of the model library followed by the first to seventh eigenspectra.

PCA code is run on the "residual" spectra. Figure 2 presents the mean spectrum and the top seven PCs for the input model library.

Project BOSS data and models onto PCs

Figure 3 shows an example of projection. The black is the BOSS spectrum. The red is the PCA fit, it is a linear combination of the mean spectrum and the PCs, namely, fit = mean + C₁ × PC₁ + C₂ × PC₂ + … + C₇ × PC₇, where C_α (α = 1–7) are the coefficients of the projection.

Figure 3: An example of projection. The black is the BOSS spectrum. The red is the PCA fit.

Estimate galaxy physical parameters for BOSS

For an observed galaxy at redshift z, we select only models that have an age smaller than the age of the universe at that redshift. We step through the models one at a time, calculating the χ² as follows:

Figure 4: χ² formula.

where C_α (α = 1–7) represents the projection coefficients. The superscript "m" and "d" refer to model and data. i represents the i-th model. P_α,α′ is the inverse of the covariance matrix of C_α. The covariance matrix of C_α is calculated in the projection process.

A weight w_i = exp(-χ_i²/2) is defined to describe the similarity between the given galaxy and model i. A probability distribution function (PDF) is then built for each parameter P, by looping over all the model galaxies in the library and by summing the weights w_i at the value of P for each model. The final PDF is normalized, and the parameter values at the 2.5, 16, 50 (median), 84 and 97.5 percentiles of the cumulative PDF are calculated. The median is adopted as the nominal estimation of P and the 16 - 84 percentile range of the PDF as its ±1σ confidence interval.