Galaxy Evolution fromMachine Learning
Tsutomu T. TAKEUCHIDivision of Particle and Astrophysical Science,
Nagoya University, Japan
6th Galaxy Evolution Workshop, Kashiwa, 5-7 June, 2019

1. Introduction: What is the Galaxy Evolution?
In those days
Galaxies evolve.

Beatrice Tinsley1. Introduction: What is the Galaxy Evolution?
In those days
Galaxies evolve.

In galactic astronomy of 70-80’s, themeaning of galaxy evolution seemed tobe unambiguous.
Beatrice Tinsley
Galaxy evolution= evolution of stellar population and chemical composition

TodayGalaxies evolve in various aspects:

TodayGalaxies evolve in various aspects:x = x(T|T > t)

TodayGalaxies evolve in various aspects:

This is the formal and ultimate goal of the studies ongalaxy evolution, but clearly it is a substantiallycomplicated problem. It is time to define the evolution ofgalaxies with more objective point of view.

2. Galaxy Evolution from Luminosities
2.1 Galaxy evolution in multiband luminosity space

Flux (normalized)

Star formation history (SFH)
is one of the key factors ofgalaxy evolution.
SFH is directly reflected to thespectral luminosity of galaxies.

Wavelength [Å]Bruzual & Charlot (1993)

2. Galaxy Evolution from Luminosities
2.1 Galaxy evolution in multiband luminosity space

Flux (normalized)

Star formation history (SFH)
is one of the key factors ofgalaxy evolution.
SFH is directly reflected to thespectral luminosity of galaxies.

Galaxy evolution related to theSFH will be well represented in
the multiwavelength (band)
luminosity space.
Wavelength [Å]Bruzual & Charlot (1993)

2.2 Traditional methods and its limitation

Color (u - g)

If we plot galaxy luminosity(absolute magnitude) vs.
color, a clear dichotomy isfound: the color bimodality.

Color (u - g)Color-magnitude relation

Redder galaxies:
red sequence
Bluer galaxies:
blue cloudBoundary: green valley

Absolute magnitude Mg
Blanton (2006)

(log) SFR [M yr-1]

Star formation main sequence of galaxies (Local)(log) Stellar mass [M]
Seymour et al. (2008)

Potential problem in the traditional color-based methods
Colors are basically ratios of two luminosities.
⇒ Selection effect is always too entangled and messy.

Potential problem in the traditional color-based methods
Colors are basically ratios of two luminosities.
⇒ Selection effect is always too entangled and messy.
⇒ Completeness test is almost impossible in a simple way.

Potential problem in the traditional color-based methods
Colors are basically ratios of two luminosities.
⇒ Selection effect is always too entangled and messy.
⇒ Completeness test is almost impossible in a simple way.
What can we do, then?

Potential problem in the traditional color-based methods
Colors are basically ratios of two luminosities.
⇒ Selection effect is always too entangled and messy.
⇒ Completeness test is almost impossible in a simple way.

Suggestion: forget about colors!

Potential problem in the traditional color-based methods
Colors are basically ratios of two luminosities.
⇒ Selection effect is always too entangled and messy.
⇒ Completeness test is almost impossible in a simple way.

Suggestion: forget about colors!
Instead, we can simply use the distribution of galaxies ina multidimensional luminosity (absolute magnitude)
space.

Potential problem in the traditional color-based methods
Instead, we can simply use the distribution of galaxies ina multidimensional luminosity (absolute magnitude)
space.

Color

Luminosity

Since we have a bimodality in color-color space, wemust have an equivalent peaks in the multidimensional
luminosity space. Color-color plots only show reduced
information.

Color

LuminosityPotential problem in the traditional color-based methods
Instead, we can simply use the distribution of galaxies ina multidimensional luminosity (absolute magnitude)
space.
Since we have a bimodality in color-color space, wemust have an equivalent peaks in the multidimensional
luminosity space. Color-color plots only show reducedinformation.
The boundary can be automatically defined by themachine-learning type method.

Potential problem in the traditional color-based methods
Instead, we can simply use the distribution of galaxies ina multidimensional luminosity (absolute magnitude)
space.
Since we have a bimodality in color-color space, wemust have an equivalent peaks in the multidimensional
luminosity space. Color-color plots only show reduced
information.
The boundary can be automatically defined by themachine-learning type method.
Advantage of this idea is that we should simply deal withthe selection at each band, not as an entangled multipleselection.

3. Data: VIPERSVIMOS Public Extragalactic Redshift Survey (VIPERS)• Want a volume and density comparable to a survey like
2dFGRS, but at z = [0.5-1]: cosmology driven, but withbroader legacy return.
• Means Volume ~ 5×107 h-3 Mpc3, ~ 90,000 redshifts, close
to full sampling.
• IAB< 22.5

http://vipers.inaf.it/

VIPERS products• Redshifts• Spectra• Photometry from CFHTLS ugriz (optical), GALEX FUVand NUV (ultraviolet), and additional ZYJHK and Ks (nearinfrared) bands (and other properties derived from SEDfitting).
• Stellar mass from Hyperzmass code
• Line-estimated metallicity for subsamples with high-S/Nspectra• Density field reconstructed from galaxy distribution• Cluster/group membership• Ancillary data (Herschel, WISE, etc.)

4. Classification in Luminosity Space
4.1 Classification in multiwavelength luminosity space
Multiwavelength luminosity space
Construct a subsample with high-S/N data: 52,114
Redshift range: 0.4 < z < 1.3
Twelve rest-frame magnitudes: FUV, NUV, u, g, r, i, z, B,V, J, H, Ks, normalized around unity.
Perhaps impossible to classify by intuition.

Unsupervised machine learning in luminosity space
Fisher Expectation-Maximization (FEM) algorithm(Bouveryron & Brunet 2012)
1. Assign initial cluster (class) centers by k-means++
2. Execute FEM
i. E-step: calculate the complete log-likelihood
under the current value of the Gaussian
mixture modelii. F-step: boundary is chosen to maximize thedistances between groups, and to minimize theinternal scattersiii. M-step: parameters of Gaussian functions areoptimized by maximizing the conditional
expectations of the complete log-likelihood
iv. Back to 2.i (E-step) until the result converges.

Unsupervised machine learning in luminosity space
Fisher Expectation-Maximization (FEM) algorithm(Bouveryron & Brunet 2012)
1. Assign initial cluster (class) centers by k-means++
2. Execute FEM
i. E-step: calculate the complete log-likelihood
under the current value of the Gaussian
mixture modelii. F-step: boundary is chosen to maximize thedistances between groups, and to minimize theinternal scattersiii. M-step: parameters of Gaussian functions areoptimized by maximizing the conditional
expectations of the complete log-likelihood
iv. Back to 2.i (E-step) until the result converges.

Unsupervised machine learning in luminosity space
Fisher Expectation-Maximization (FEM) algorithm(Bouveryron & Brunet 2012)
1. Assign initial cluster (class) centers by k-means++
2. Execute FEM
i. E-step: calculate the complete log-likelihood
under the current value of the Gaussian
mixture modelii. F-step: boundary is chosen to maximize thedistances between groups, and to minimize theinternal scattersiii. M-step: parameters of Gaussian functions areoptimized by maximizing the conditional
expectations of the complete log-likelihood
iv. Back to 2.i (E-step) until the result converges.

Unsupervised machine learning in luminosity space
Fisher Expectation-Maximization (FEM) algorithm(Bouveryron & Brunet 2012)
1. Assign initial cluster (class) centers by k-means++
2. Execute FEM
i. E-step: calculate the complete log-likelihood
under the current value of the Gaussian
mixture modelii. F-step: boundary is chosen to maximize thedistances between groups, and to minimize theinternal scattersiii. M-step: parameters of Gaussian functions are
optimized by maximizing the conditional
expectations of the complete log-likelihood
iv. Back to 2.i (E-step) until the result converges.

Unsupervised machine learning in luminosity space
Fisher Expectation-Maximization (FEM) algorithm(Bouveryron & Brunet 2012)
1. Assign initial cluster (class) centers by k-means++
2. Execute FEM
i. E-step: calculate the complete log-likelihood
under the current value of the Gaussian
mixture modelii. F-step: boundary is chosen to maximize thedistances between groups, and to minimize theinternal scattersiii. M-step: parameters of Gaussian functions areoptimized by maximizing the conditional
expectations of the complete log-likelihood
iv. Back to 2.i (E-step) until the result converges.

4.2 Classification result
The FEM classification separates VIPERS galaxies intotwelve classes.

NUV - rN.B. Twelfth class is the AGN and not included in thefurther discussion.

r - KsThe UVJ diagram

U-VWilliams et al. (2009) and subsequent authors proposedthat the UVJ diagram provide a separation between
passive and star-forming galaxies.

V-J

The NUVrK 3D diagram
The red galaxies are divided into three subclasses,
separated by the FUV - NUV color.

The redshift (in)dependence at 0.4 < z < 1.0
The color evolution of galaxies is clearly visible, but theclassification is almost unchanged with redshifts.
0.4 < z < 0.5

0.5 < z < 0.6

0.6 < z < 0.7

0.7 < z < 0.8

0.8 < z < 0.9

0.9 < z < 1.0

4.3 Global properties of classified galaxies
Colors

A turnover exists in some colors.

Spectral and morphological features

D4000: indicator of galaxy age (relatively new SFH)EW[OII]: emission from SF region
Sérsic index: shape parameter indicating early (~ 4) or late(~ 1) type

Stellar mass and specific SFR (sSFR)

sSFR: SFR per stellar mass [yr-1]

Indicator of the SF activity normalized with the galaxy size.

SFR [M yr-1]

Main sequence of star-forming galaxies

Stellar mass M* [M]
The main sequence is naturally reproduced from thismethod. Note that this is free from color selection bias,
which introduced a significant confusion in this diagram.

4.4 Spectral properties of classified galaxies

Normalized flux

Stacked spectra of class 1-6

Rest-frame wavelength [Å]Normalized flux

Stacked spectra of class 7-11

Rest-frame wavelength [Å]Summary of spectral features in classified galaxies
Class 1-3 (red)Strong absorption features, D4000 break, redcontinuumClass 4-6 (green)
Balmer absorption, gradually stronger emission
lines
Class 7-11 (blue)
Strong emission lines, blue continuum
Redshift dependence
In each class, higher-z galaxies have gradually bluercontinuum, significant at longer wavelengths

5. Conclusions
With VIPERS multicolor galaxy photometric data with
redshifts, we explored the performance of a machinelearning method to study galaxy evolution.
1. To avoid complicated selection effects and subjective
choice of features, we abandoned the color-based method.
Instead, we performed an unsupervised machinelearning classification directly to the twelve-dimensional
luminosity-redshift space of VIPERS galaxies.
2. Our classification yielded twelve galaxy classes (+ oneAGN class). Class 1-3 corresponds to red passive galaxies,
4-6 to “green” galaxies, and 7-11 to blue star-forming
galaxies, respectively.

5. Conclusions
3. In the redshift range of 0.4 < z < 1.0, the classes remainunchanged with a gradual shift to bluer colors.
4. Classes 1-3 show stronger absorption lines and D4000
and Sérsic index n ~ 4 (spheroid-like), indicating oldstellar population. Classes 7-11 show strong [OII]
emission and n ~ 1 (disk-like), consistent with active SF.
5. Galaxies in each class showed clearly distinguishable
physical properties, as a sequence of features. Stellar
mass decreases from class 1 to 11. In contrast, specific
SFR increases with the class labels.

5. Conclusions
6. The star-forming galaxy main sequence is clearly
reproduced by classes 7-11, with class 1-3 as quiescentnon-star forming galaxies. Note that, in contrast to
previous studies, our result is not affected by thecomplicated color selection and robust.
7. Classified galaxies demonstrated a clear representative
spectrum for each class. The spectra are redshiftdependent: higher-z galaxies have gradually bluer
continuum, corresponding to the evolution.
Take-home message
Machine learning will open a new window to redefine galaxy
evolution, and provide us with a substantially new discovery.

Evolution of the color-magnitude relation

Arnouts et al. (2007)

SFR [M yr-1]

Star formation main sequence of galaxies (high-z)

Stellar mass [M]

Daddi et al. (2007)

VIPERS Public Data Release

http://vipers.inaf.it/3D distribution of VIPERS galaxies

VIPERS spectra: examples

Guzzo et al. (2013); Garilli et al. (2014)

Number

VIPERS redshift distributionRedshiftk-means++ algorithm
1. Choose one center uniformly at random from sample
data points.
2. For each data point x, compute D(x), the distance
between x and the nearest center that has already beenchosen.
3. Choose one new data point at random as a new center,using a weighted probability distribution where a point x
is chosen with probability proportional to D(x)2.
4. Repeat Steps 2 and 3 until k centers have been chosen.
5. Now that the initial centers have been chosen, proceedusing standard k-means clustering.

k-means++ algorithmk-meansk-means++

https://qiita.com/NoriakiOshita/items/cbd46d907d196efe64a3

Gaussian mixture model
For m variables and n Gaussians,x : [ x1, x2, x3, … xm ]μk : mean vector (1 × m) of k-th GaussianΣk : covariance matrix (m × m) of k-th Gaussianπk : mixing coefficient50Gaussian mixture model: example

Scatter plot

PDF

Estimation of the parameters for Gaussian mixture model
Given a dataset x : [ x1, x2, x3, … xm ], we estimate
μk : mean vector,
σk : covariance matrix, andπk : mixing coefficient
via maximum likelihood method.

Estimation of the parameters for Gaussian mixture model
When N data are divided into K clusters, a log-likelihood
function isWe calculate a contribution function 𝜸𝜸(𝒛𝒛𝒏𝒏𝒋𝒋 )(𝒛𝒛 : latent variable, 𝟎𝟎 ≤ 𝒛𝒛 ≤ 𝟏𝟏)8

Selection of best model and optimal number of classes
The number of classes is knot known a priori: a majordifficulty in unsupervised algorithm.
In this work, the best model and the range of possibleclasses are chosen based on three statistical modelselection criteria: AIC (Akaike 1974), BIC (Schwarz1978), and ICL (Baudry 2012).

Siudek et al. (2018)

Selection of best model and optimal number of classesThe result can be intuitively verified on a NUVrKcolor-color diagram.

9 classes

10 classes

11 classes

12 classes
Siudek et al. (2018)

Selection of best model and optimal number of classes
The result can be intuitively verified on a NUVrKcolor-color diagram.
Dusty galaxy

9 classes

10 classes

11 classes

12 classes

High-z AGNs

Siudek et al. (2018)

The NUVrK diagram

NUV - r

The NUVrK diagram (Arnouts et al. 2013) is similar tothe UVJ plane but allows for a better separation between
passive and star-forming galaxies.

r-K