1.2 Galaxy formation from the cosmic initial condition

Statistical
tical Challenges to Dimensionality
Dimensionin Astronomical Big Data

Galaxies are supposed to have formed from a tiny (order of~ 10-5) fluctuation of matter (mainly dark matter: DM) inthe early Universe.
The initial condition is imprinted on the Cosmic Microwave
Background (CMB) observed at radio wavelengths.

Tsutomu T. TAKEUCHI
1. Division of Particle and Astrophysical Science, Nagoya
University, Japan
2. The Research Center for Statistical Machine Learning,the Institute of Statistical Mathematics
International Symposium on Recent Advances in Theories andMethodologies for Large Complex Data, Tsukuba, 7-9 Dec., 2023
http://www.rssd.esa.int/index.php?project=Planck

Collaborators
Suchetha COORAY, Kai T. KONO (ἙἙ㔝㻌ᾏ)Division of Particle and Astrophysical Science, Nagoya University, Japan
Kazuyoshi YATA (▮⏣ ࿴ၿ), Makoto AOSHIMA(㟷ᔱ㻌ㄔ)Institute of Mathematics, University of Tsukuba, Japan
Kento EGASHIRA (Ụ㢌㻌೺ᩯ), Aki ISHII (▼஭㻌ᬗ)Department of Information Sciences, Tokyo University of Science, Japan

1.4 Internal galaxy evolution
Star formation in galaxies
Galaxies have formed at various epochs in the Universe,
merged, and grown. In parallel, gas has transformed into stars.
Stars die and return back their gas into the ISM, and next
generation of star formation proceeds.

Kohji YOSHIKAWA (ྜྷᕝ㻌⪔ྖ)Center for Computational Sciences, University of Tsukuba, Japan
Kouichiro NAKANISHI (୰す㻌ᗣ୍㑻)
ALMA Project, National Astronomical Observatory of Japan
Kotaro KOHNO (Ἑ㔝㻌Ꮥኴ㑻)Institute of Astronomy, The University of Tokyo, Japan

1. Introduction

Chemical evolution of galaxies: metal and dust

1.1 Structure formation in the Universe

Dust production

Star formation
Heavy element production
Mass ejection
ISM

Dust production
Stellar evolution

Stellar winds
PNe
Heavy element production
Supply of heavy
elements
Mass ejection
SNRs

All the structures in the Universe have emerged from a tiny
fluctuation at very early epoch (380,000 yr).

Protoplanetary
disk

Stellar death
Heavy element production

Dust
Cool star atmosphere
PNe

Dust production
SNRs

Blast wave by SNe
Dust destruction

2. Interstellar Medium (ISM) and Spectroscopy

2.3 Spectroscopy of the ISM

2.1 Matter between stars in galaxies

Electromagnetic spectrum and scales

https://www.americanpharmaceuticalreview.com/Featured-Articles/331616-Optical-Spectroscopy-Where-is-it-Going/

2.2 ISM phases and star formation

Quantum transition to spectral lines
Astronomical spectroscopy brings physical information of theobjects in the remote Universe.

ISM has various phases
1. Plasma (ionized diffuse phase)
2. Neutral gas (mainly neutral hydrogen HI)
3. Molecular gas (mainly molecular hydrogen H2)

Since gas must become dense enough to form stars,
star formation occurs in molecular clouds. Namely,
Atomic gas э Molecular gas э Stars
https://www.yokogawa.com/about/research-d
development/inv_center/spectroscopy/

Kennicutt-Schmidt (K-S) law

2.4 Spectral mapping in astronomy

Stars form in molecular cores.
э It is natural to suppose a relation between the star formation
rate (SFR) and gas density. Schmidt (1959) proposed a relation
SFR ‫ ࢔࣋ ן‬.
i. n = 1 Density controls star formation.
ii. n = 2 Collision-like process plays a role for star formation
э The power-law index contains substantial information onwhat triggers the star formation.
It is crucial to reveal spatially resolved SF law in galaxies!
http://ifs.wikidot.com/what-is-ifs

3. Astronomical Spectral Map as HDLSS Data

Unusual behavior of high-dimensional data

3.1 General situation in astrophysics

d = 4000Classical statistical analysis
Sample size: nData dimension: dd=4

The following condition is implicitly assumedn >> dBut this is not the case for many cases in scientificresearches. Astronomers and astrophysicists haveever simply given up when they face such type ofproblem.
https://www.math.tsukuba.ac.jp/~aoshima-lab/research.html

3. Astronomical Spectral Map as HDLSS Data
3.1 General situation in astrophysics
High-dimensional low-sample size (HDLSS) data analysis

3.3 High-dimensional PCA
A specially designed PCA, the high-dimensional PCA, cansweep out the noise sphere and extract features of the data.

Sample size: nData dimension: d
For the HDLSS data, the condition is
Feature spacen << dNoise space

This condition is often found in e.g., genomic analysis,
medical analysis, etc.
In astrophysics, for example,
spectroscopy has this property.

integralfield
Extracted feature (PCs)
Feature space embedded in a noise space

3.2 Unusual behavior of high-dimensional data
For high-dimensional data, classical limit theorems do notwork. If we wrongly assume them, we would be lead to awrong conclusion.
Simplest example: for the sample mean

1. as d /nĺ 0

Aoshima (2012)

3.4 Actual data: ALMA data cube of NGC253
NGC 253: prototypal starburst

Disk (out to > 20 kpc)
Starburst nucleus(~ 1kpc)

2. as d /nĺ This striking property is referred to as the strong
inconsistency.
2MASS (JHK) Jarrett et al. (2003)

4. Analysis of Starburst Region in NGC253

Close up of the starburst

4.1 Analysis of Raw Data
Eigenvalues of the PCA (contribution)
Eigenvalue (normalized)

0.6
0.5

0.3
0.2
0.1
0

Ando et al. (2017)19

The huge amount of information on the
ALMA spectra are basically determinedby two largest eigenvalues.

0.4

0

10

15

20

25

Index of eigenvalues

Result: automatic sparse PCA (A-SPCA) PC2

Rich in molecular lines
ALMA resolved diverse star-forming activities at ~ 10 pc
scale.

PC1

ALMA Band7 spectra
Ando et al. (2017)

3.5 Structure of the Data

PC1 and 2 consist of ~ 20 elements (spectral features on
the resolution units). The key features may be reduced
only to a few to several lines!

Responsible spectral features for PC1, PC2 and PC3

Data: Ando et al. (2017)~ spatial dimension 231 㽢 spectral dimension 2248
э A case with n = 231 and d = 2248 (n << d)Problems from astrophysical side
• Too much information on spectra.
• Too large variety of spectral lines compared to n.
We apply the high-dimensional statistical analysis to the
ALMA spectral mapping data of NGC253.

Takeuchi et al. (2023)
Now PC1 more clearly represents the total intensity, andPC2 and 3 represent smaller-scale velocity structures.

Systemic rotation and Doppler shift

Redshift

PC1

Dec (y coordinate䠅䠅

Spatial map of PC1

RestframeO

BlueshiftOO

If the system is rotating as a whole, the observed wavelengthis affected by the Doppler shift. PC2 beautifully describes the
Doppler shift!

R.A. (x coordinate)

4.3 Main analysis

Spatial map of PC1 and PC2

PC2

Dec (y coordinate䠅䠅Doppler shift correction

Takeuchi et al. (2021)

R.A. (x coordinate)

Eigenvalues of the NGC253 after Doppler correction

Dec (y coordinate䠅䠅Eigenvalue (normalized)

Velocity field of the systemic rotation

We estimated the peculiar velocity field (mainly due to thesystemic rotation of the central region of NGC253) byaveraging the results from HCN(4-3), HNC(4-3) and CS(7-6)
lines, and corrected the Doppler shift.
Due to this correction, the final data dimension is d = 1971.

HCN(4-3)

0.5
0.4
0.3
0.2
0.1
0

0

10

15

20

25

Index of eigenvalues
R.A. (x coordinate)э Doppler shift correction to remove the systemic rotation.

Takeuchi et al. (2023)

Spatial map of PC3 after Doppler correction

PC1

Takeuchi et al. (2023)

PC3

PC2

Dec (y coordinate䠅䠅

PC1 and PC2 from sparse PCA

R.A. (x coordinate)

Butterfly-like pattern completely disappeared.

Anomaly regions in the velocity field

Dec (y coordinate䠅䠅Responsible spectral features for PC1, PC2 and PC3

Takeuchi et al. (2023)
Now PC1 more clearly represents the total intensity, andPC2 and 3 represent smaller-scale velocity structures.

Spatial map of PC2 after Doppler correction

R.A. (x coordinate)

What do we see from the Doppler-corrected map?

PC2

Dec (y coordinate䠅䠅NGC253• Pure starburst: SFR in the central molecular zone is 2 Mۨyr-1 (Rieke et al. 1980; Keto et al. 1999)• Intense outflow (Matsubayashi et al. 2009; Bolatto et al.
2013)
Indeed the outflow phenomenon is mainly delineated by PC3.

R.A. (x coordinate)5. Galaxy Evolution in Multiwavelengths
5.1 Galaxy evolution from a modern point of view
Galaxies evolve in various aspects:

6.2 Classification in multiwavelength luminosity space
Generate a subsample with all 11 rest-frame magnitudes(FUV, NUV, u, g, r, i, z, Y, J, H, K) ~ 800,000 galaxies
э Construct a volume limited sample that is representativeof the whole galaxy sample of ~ 30,000 galaxies.
Perhaps impossible to classify by human intuition.

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.

5.2 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 the
SFH will be well represented in
the multiwavelength (band)
luminosity space.

Galaxy manifold
Historically, in 80’s, astronomers introduced a method ofclassical multivariate analysis such as PCA to find and unify
various scaling relations (e.g., Djorgovski 1992).

However, since classical PCAtype analysis could only findlinear structure in the feature
space, the idea worked only to alimited problems, and havebeen once forgotten.

Djorgovski (1992)

Wavelength [Å]6. Galaxy Evolution in Multiwavelengths

Galaxy manifold

6.1 Data: RCSED

Some preceding studies have suggested the existence of asmooth relation of galaxies in the 3D color–color–magnitude
space smoothly continuing from the blue cloud to the red
sequence (e.g. Chilingarian et al. 2012).

• Reference Catalog of galaxy Spectral Energy
Distributions (RCSED) (Chilingarian et al. 2016)• Catalog of galaxies produced as join between GALEX,
SDSS, and UKIDSS catalogs, and processed with state-ofthe-art spectral analysis methods• Covers approximately 25% of the sky and contains kcorrected ultraviolet-to-near-infrared photometry (11bands of FUV, NUV, u, g, r, i, z, Y, J, H, K) of some 1
million galaxies, as well as some of their physicalproperties

Chilingarian et al. (2012)

http://rcsed.sai.msu.ru

э general idea of a low dimensional submanifold existingin a higher dimensional feature space: revival of the galaxy
manifold!

6.3 Galaxy manifold in the space of three bands

7.3 Algorithms
We applied two algorithms of manifold learning:
IsomapUMAP
Both methods keep connectivity of a point data cloud.

Takeuchi et al. (2023c)

N.B. These methods do not determine the dimension of the
manifold, because they reduce the dimension of the datacloud to two or three for any dataset.

A similar structure was found in the luminosity space of our data.
We want to quantify and examine the manifold.

7. Application of Manifold Learning to Galaxies
7.1 Features of the galaxy manifold
1. Not uniformly distributed along the manifold

7.4 Result: galaxy manifold from Isomap
Isomap preserves the density of the data point cloud (i.e.,
distribution of galaxies) on the manifold (i.e. dense regions
remain dense).

Dense regions on the manifold imply that galaxies spendlong time there in their evolution, and less dense regions
suggest a rapid evolution there, and galaxies move on toa much more stable state.
2. Noisy in feature space
Input features depend on the spectral energy
distribution (SED) modeling methods (dust needs to beconsidered carefully).

7.2 Manifold learning
Manifold Learning (or non-linear dimensionality reduction)embeds data that originally lies in a high dimensional space ina lower dimensional space, while preserving characteristic
properties.
A manifold is a topological space that locally resembles
Euclidean space near each point.

The arrows show the continuous evolution of less-massive,
actively star-forming galaxies to massive, quiescent galaxies.

Isomap manifold axes and physical properties

Manifold axis 1 represents
the stellar mass well.

Isomap manifold axes and physical properties

UMAP manifold axes and physical properties

Manifold axis 2 represents
the star formation rate to acertain extent.

7.5 Result: galaxy manifold from UMAP
UMAP expands dense regions and contracts sparse
regions, i.e., uniform representation of the observational
galaxy parameters.

The arrows show the continuous evolution of less-massive,
actively star-forming galaxies to massive, quiescent galaxies.

UMAP manifold axes and physical properties

Manifold axis 2 represents
the star formation rate
better than Isomap.

8. Interpretation and modeling
8.1 Physical interpretation of the manifold

Takeuchi et al. (2023d)
The galaxy manifold obtained with UMAP. The vector field
represents the direction of galaxy evolution by star formation.

8.2 Interpretation by a physical model
Assuming that galaxies move on the manifold with time, we

Manifold axis 1 represents
the stellar mass better than
Isomap.

model galaxies as a walk process.
The next location is decided from the following equations:'t = 10 [Myr], r (return fraction) = 0.35, Ș (mass loading
factor) ~ 3. A manifold location that satisfies the above ischosen as the next position.
Star
Formation
Rate

Stellar
Mass

Total Gas
Mass

Physical interpretation of the star formation history

SFH from the Main Sequence

9. Summary
8. The high-dimensional sparse PCA successfully chose two
PCs that reproduce the general properties of the ALMA
spectroscopic map of NGC253.
9. The controlling feature was HCN(4-3) rotational lines.
PC1 describes the total intensity of the lines, and PC2
represents the Doppler shift caused by the systemicrotation.

1 Gyr

SFMS (z=0)(Kashino+21)

10. After correcting the Doppler shift due to the systemicrotation, we could obtain information on the smaller-scale
velocity field described by PC2 (new) and PC3. These may
be caused by outflow phenomena of starburst regions.
Star formation rate

Stellar mass

cf. Cooray et al. (2023)

9. Summary
1. Galaxies ubiquitously exist in the present-day Universe,
but they have been formed from a tiny fluctuation ofmatter in the early Universe.
2. Evolution of galaxies is mainly driven by the starformation, a transition from ISM to stars.
3. Various phases of the ISM are related, and the evolutionof the ISM is a key to complete the understanding of thegalaxy evolution.
multiwavelength
photometric
4. Spectroscopic
andobservations are of vital importance to extract andinterpret the information of matter in galaxies.

9. Summary

9. Summary
Part IIIWe demonstrated that manifold learning is a powerful
method to extract physical laws embedded in a highdimensional feature space of galaxies.
11. Main difficulty of the analysis of multiwavelength data
was that the dimension of the feature space is high andclassical methods do not give a good intuition.
12. The manifold learning is useful to find a hidden relation
in a high-dimensional feature space.
13. Algorithms that keep the connectivity are desirable forthe purpose of the analysis of galaxy evolution. Isomap
and UMAP are promising for this purpose.

9. Summary

Part II
Spectroscopic mapping and similar methods arefundamentally important to reveal the ISM physics, but thedata are high-dimensional low sample size.

14. Isomap and UMAP give a consistent result, and we
found that the galaxy evolution is well described only by
two physical quantities, the stellar mass and the starformation rate (SFR).

6. We applied the high-dimensional PCA on the NGC253
spectral map. ALMA mapping data are typically HDLSS
in general, and in this case n = 231 and d = 2227.

15. It is crucial to describe the discovered nonlinear relation
by interpretable equations. This is not, however, an easytask, and at this moment a comparison with classicaltheory works fairly well.

7. Very large variety in the molecular line spectra of
NGC253 map can be described only by two PCs! Each PC
consists of ~ 20 elements, much fewer than d. Because
these elements may be a part of same features, the key
features may be reduced to several.

Dimensionality reduction in astrophysics plays an importantrole for new discoveries. Further investigation is needed.

Stay tuned!