Principal Component Analysis: Why do we use fourier transformation - - PowerPoint PPT Presentation

Principal Component Analysis: Why do we use fourier transformation to analyze flow?

Ziming Liu

Peking University

Collaborators: Huichao Song, Wenbin Zhao

December 16, 2018

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 1 / 30

Overview

1

Motivation of the Question

2

Introduction to PCA

3

PCA in Sciences

4

Model

5

Results(Paper in Preparation)

6

Conclusions

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 2 / 30

Simple Review for Flow

Integrated flow is decomposed under Fourier bases: dN dϕ = 1 2π

∞

• −∞
• Vne−inϕ = 1

2π(1 + 2

∞

• n=1

vn e−in(ϕ−Ψn)) (1)

• Vn = vneinΨn : n-th order flow-vector

vn = cos n(ϕ − Ψn) : n-th flow harmonics Ψn : corresponding event plane angle

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 3 / 30

What makes a good flow

• bservable?

Fourier Transformation?

You are right. But, approximately.

Phys.Rev. C64 (2001) 054901

Z.Phys. C70 (1996) 665-672

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 4 / 30

What makes a good flow

• bservable?

Fourier Transformation?

You are right. But, approximately.

Phys.Rev. C64 (2001) 054901

Z.Phys. C70 (1996) 665-672

Q: How to find good bases to decompose particle distribution?

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 4 / 30

Overview

1

Motivation of the Question

2

Introduction to PCA

3

PCA in Sciences

4

Model

5

Results(Paper in Preparation)

6

Conclusions

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 5 / 30

PCA belongs to Machine Learning

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 6 / 30

One minute for PCA

PCA transform a set of correlated variables to uncorrelated ones via an

• rthogonal transformation:

X = UΣZ U, Z: orthogonal matrices; Σ: Diagonal matrix. X: Original variables; Z: transformed variables.

z1 z2

σ1 σ2

Eigenvectors z: correlations between features Singular values σ: importance of eigenvectors

x1 x2

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 7 / 30

Motivation : Face detection with PCA

Figure: Dataset:different faces Figure: Eigenfaces

Eigenfaces show interesting correlations: More beard/mustache→ man→ tanned face Round face→ baby→ less wrinkle

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 8 / 30

Each face is decomposed into superposition of eigenfaces. Each face can be expressed by number of faces far less than pixels of the original image. Correlations play a huge role!

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 9 / 30

Overview

1

Motivation of the Question

2

Introduction to PCA

3

PCA in Sciences

4

Model

5

Results(Paper in Preparation)

6

Conclusions

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 10 / 30

Classical Mechanics and Atmospheric Sciences

eigenfrequencies in particle motion

• H. Y. Chen, Raphal Ligeois, John R. de Bruyn, and Andrea

Soddu Phys. Rev. E 91, 042308 Published 15 April 2015

Multi-resolution PCA to discover El Nino.

https://arxiv.org/pdf/1506.00564.pdf Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 11 / 30

Condensed matter physics

Machine learning helps discover Correlations between spin configurations Phase transition

C Wang, H Zhai - Physical Review B,96(2017),14,144432 Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 12 / 30

Flow in Heavy Ion Collisions

subleading modes of factorization breaking

Aleksas Mazeliauskas, Derek Teaney Phys.Rev.C93 (2016) no.2, 024913

Nonlinear response coefficients

Piotr Bozek, Phys.Rev. C97 (2018) no.3, 034905

Best linear descriptor ζ(a)

n,pred = εn,n + c1εn,n+2

Rajeev S. Bhalerao, Jean-Yves Ollitrault, Subrata Pal, Derek Teaney Phys.Rev.Lett. 114 (2015) no.15, 152301

Experimental data

CMS collaboration, Phys.Rev. C96 (2017) no.6, 064902 Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 13 / 30

Overview

1

Motivation of the Question

2

Introduction to PCA

3

PCA in Sciences

4

Model

5

Results(Paper in Preparation)

6

Conclusions

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 14 / 30

Previous work and our approach

Previous work1 utilizes Fourier Transformation in the φ direction: dN dp =

+∞

• n=−∞

Vn(p)einφ p = (pt, η) PCA decomposes Vn(p) into eigenmodes: Vn(p) =

k

• α=1

ξ(α)V (α)

n

(p) However, we apply PCA directly to dN/dφ data without FT: dN dφ =

k

• α=1

ξ(α)(dN dφ )(α)

1Rajeev S. Bhalerao, Jean-Yves Ollitrault, Subrata Pal, Derek Teaney Phys.Rev.Lett. 114 (2015) no.15, 152301 Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 15 / 30

Simulations

Pb+Pb collisions at 2.76 A TeV

Trento initial model Vishnew Hydrodynamics iss particle sampling

No hadron rescattering or resonance decays to simplify problem settings.

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 16 / 30

Our approach

PCA for flow analysis

Data sets: PCA

mean μ

top eigenvectors:σ1,σ2,σ3…… With PCA, each flow distribution is decomposed into superposition of eigenmodes. = = + + μ x z +x z +x z +……

1 1 2 2 3 3

dN

dφ

dN

dφ

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 17 / 30

Overview

1

Motivation of the Question

2

Introduction to PCA

3

PCA in Sciences

4

Model

5

Results(Paper in Preparation)

6

Conclusions

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 18 / 30

Singular values σ

Singular values σ pairwise matched

1 5 10 15 20

n

0.0 0.1 0.2

n

v′

2

v′

3

v′

4

v′

1 v′ 5 v′ 6

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 19 / 30

Eigenvectors z

Eigenvectors look similar to sin(nφ) and cos(nφ).

2 2 0.2 0.0 0.2

dN/d z1/z2

2 2 0.2 0.0 0.2

z3/z4

2 2 0.2 0.0 0.2

dN/d z5/z6

2 2 0.2 0.0 0.2

z7/z8

2 2 0.2 0.0 0.2

dN/d z9/z10

2 2 0.2 0.0 0.2

z11/z12

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 20 / 30

Eigenvectors z

Eigenvectors look similar to sin(nφ) and cos(nφ).

2 2 0.2 0.0 0.2

dN/d z1/z2

2 2 0.2 0.0 0.2

z3/z4

2 2 0.2 0.0 0.2

dN/d z5/z6

2 2 0.2 0.0 0.2

z7/z8

2 2 0.2 0.0 0.2

dN/d z9/z10

2 2 0.2 0.0 0.2

z11/z12

Machines automatically discover fourier transformation for flow!

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 20 / 30

Defining new flow observables v ′

n

zk: k-th (normalized) eigenvector xk: amplitude of zk. dN dφ = µ +

k

• i=1

xkzk

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 21 / 30

Compare vn and v

′

n

v

′

2 fits really well with v2, and v

′

3 fits really well with v3.

v

′

4 is deviated from v4.

0.0 0.1

v2 v′

2 v′ 3 or v′ 4

0.00 0.05

v3

0.00 0.03

v4

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 22 / 30

FC of eigenvectors

z1/z2 contain sin(4φ) and cos(4φ) bases as well.

1 2 3 4 5 6 7 8 9 10 11 12 Eigenmodes zi v1 v2 v3 v4 v5 v6 Fc

= ⇒ = ⇒

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 23 / 30

SC(vm, vn)

SC(vm, vn) = v2

mv2 n − v2 nv2 m

20 15 10 5

SC(v2, v3) × 107

Fourier PCA 0.0 2.5 5.0 7.5 10.0 12.5

SC(v2, v4) × 107

0.00 0.25 0.50 0.75 1.00 1.25

SC(v2, v5) × 107 10 20 30 40 50 60 70

0.4 0.3 0.2 0.1 0.0

SC(v3, v4) × 107 10 20 30 40 50 60 70

0.0 0.1 0.2 0.3

SC(v3, v5) × 107 10 20 30 40 50 60 70 Centrality%

0.02 0.00 0.02 0.04

SC(v4, v5) × 107 Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 24 / 30

Pearson Coefficient: r(vm, εn)

0.0 0.2 0.4 0.6 0.8 1.0

r (v2,

2)

Fourier PCA

0.05 0.00 0.05 0.10

r (v2,

3)

0.0 0.1 0.2 0.3

r (v2,

4)

0.00 0.05 0.10 0.15

r (v2,

5)

0.3 0.2 0.1 0.0

r (v3,

2)

0.0 0.2 0.4 0.6 0.8

r (v3,

3)

0.2 0.1 0.0

r (v3,

4)

0.1 0.0 0.1 0.2 0.3

r (v3,

5)

0.0 0.1 0.2 0.3 0.4

r (v4,

2)

0.05 0.00 0.05 0.10

r (v4,

3)

0.0 0.2 0.4 0.6

r (v4,

4)

0.00 0.05 0.10

r (v4,

5) 10 20 30 40 50 60 70

0.00 0.05 0.10 0.15 0.20 0.25

r (v5,

2) 10 20 30 40 50 60 70

0.0 0.1 0.2 0.3

r (v5,

3) 10 20 30 40 50 60 70

0.00 0.05 0.10 0.15 0.20

r (v5,

4) 10 20 30 40 50 60 70 Centrality%

0.0 0.1 0.2

r (v5,

5)

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 25 / 30

Closer look : centrality 10% − 20% data

PCA correlators has a more diagonal pattern. Fourier:

2 3 4 5 6

v2 v3 v4 v5 v6

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

PCA:

′

2

′

3

′

4

′

5

′

6

v

′

2

v

′

3

v

′

4

v

′

5

v

′

6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 26 / 30

Overview

1

Motivation of the Question

2

Introduction to PCA

3

PCA in Sciences

4

Model

5

Results(Paper in Preparation)

6

Conclusions

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 27 / 30

Conclusions

PCA helps visualize data. PCA automatically discovers flow observables. PCA provides a new perspective that relates better to initial profile.

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 28 / 30

Prospectives

PCA helps reveal structure of data with its strong power of visualization. PCA aids in designing observables in complicated systems. Carefully applying PCA to real experimental data in relativistic heavy-ion experiments.

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 29 / 30

Thanks!

Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 30 / 30