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 Motivation of the Question 1 Introduction to PCA 2 PCA in Sciences 3 Model 4 Results(Paper in Preparation) 5 Conclusions 6 Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 2 / 30

Simple Review for Flow Integrated flow is decomposed under Fourier bases: ∞ ∞ d ϕ = 1 V n e − in ϕ = 1 d N � � � v n e − in ( ϕ − Ψ n ) ) 2 π (1 + 2 (1) 2 π n =1 −∞ V n = v n e in Ψ n : n -th order flow-vector � v n = � cos n ( ϕ − Ψ n ) � : n-th flow harmonics Ψ n : corresponding event plane angle Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 3 / 30

Phys.Rev. C64 (2001) 054901 Z.Phys. C70 (1996) 665-672 Fourier Transformation? What makes a good flow observable? You are right. But, approximately. Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 4 / 30

Phys.Rev. C64 (2001) 054901 Z.Phys. C70 (1996) 665-672 Fourier Transformation? What makes a good flow observable? You are right. But, approximately. Q: How to find good bases to decompose particle distribution? Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 4 / 30

Overview Motivation of the Question 1 Introduction to PCA 2 PCA in Sciences 3 Model 4 Results(Paper in Preparation) 5 Conclusions 6 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 orthogonal transformation: X = U Σ Z U , Z : orthogonal matrices; Σ: Diagonal matrix. X : Original variables; Z : transformed variables. x 2 z 1 σ 2 z 2 σ 1 Eigenvectors z : correlations between features Singular values σ : importance of eigenvectors x 1 Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 7 / 30

Motivation : Face detection with PCA Figure: Eigenfaces Figure: Dataset:different faces 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 Motivation of the Question 1 Introduction to PCA 2 PCA in Sciences 3 Model 4 Results(Paper in Preparation) 5 Conclusions 6 Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 10 / 30

Classical Mechanics and Atmospheric Sciences eigenfrequencies in particle Multi-resolution PCA to motion discover El Nino. https://arxiv.org/pdf/1506.00564.pdf H. Y. Chen, Raphal Ligeois, John R. de Bruyn, and Andrea Soddu Phys. Rev. E 91, 042308 Published 15 April 2015 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 Best linear descriptor factorization breaking ζ ( a ) n , pred = ε n , n + c 1 ε n , n +2 Rajeev S. Bhalerao, Jean-Yves Ollitrault, Subrata Pal, Derek Aleksas Mazeliauskas, Derek Teaney Phys.Rev. C93 (2016) Teaney Phys.Rev.Lett. 114 (2015) no.15, 152301 no.2, 024913 Experimental data Nonlinear response coefficients Piotr Bozek, Phys.Rev. C97 (2018) no.3, 034905 CMS collaboration, Phys.Rev. C96 (2017) no.6, 064902 Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 13 / 30

Overview Motivation of the Question 1 Introduction to PCA 2 PCA in Sciences 3 Model 4 Results(Paper in Preparation) 5 Conclusions 6 Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 14 / 30

Previous work and our approach Previous work 1 utilizes Fourier Transformation in the φ direction: + ∞ dN � V n ( p ) e in φ dp = p = ( p t , η ) n = −∞ PCA decomposes V n ( p ) into eigenmodes: k ξ ( α ) V ( α ) � V n ( p ) = ( p ) n α =1 However, we apply PCA directly to dN / d φ data without FT: k dN ξ ( α ) ( dN � d φ ) ( α ) d φ = α =1 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 iss Trento Vishnew particle initial Hydrodynamics sampling model 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 dN Data sets: top eigenvectors:σ 1 ,σ 2 ,σ 3 …… dφ mean μ PCA With PCA, each flow distribution is decomposed into superposition of eigenmodes. = + dN = μ + x z +x z +x z +…… dφ 1 1 2 2 3 3 Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 17 / 30

Overview Motivation of the Question 1 Introduction to PCA 2 PCA in Sciences 3 Model 4 Results(Paper in Preparation) 5 Conclusions 6 Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 18 / 30

Singular values σ Singular values σ pairwise matched 0.2 v ′ 2 v ′ 3 n 0.1 v ′ 4 v ′ 1 v ′ 5 v ′ 6 0.0 1 5 10 15 20 n Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 19 / 30

Eigenvectors z Eigenvectors look similar to sin ( n φ ) and cos ( n φ ). z 1 /z 2 z 3 /z 4 0.2 0.2 dN/d 0.0 0.0 0.2 0.2 2 0 2 2 0 2 z 5 /z 6 z 7 /z 8 0.2 0.2 dN/d 0.0 0.0 0.2 0.2 2 0 2 2 0 2 z 9 /z 10 z 11 /z 12 0.2 0.2 dN/d 0.0 0.0 0.2 0.2 2 0 2 2 0 2 Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 20 / 30

Eigenvectors z Eigenvectors look similar to sin ( n φ ) and cos ( n φ ). z 1 /z 2 z 3 /z 4 0.2 0.2 dN/d 0.0 0.0 0.2 0.2 2 0 2 2 0 2 z 5 /z 6 z 7 /z 8 0.2 0.2 dN/d 0.0 0.0 0.2 0.2 2 0 2 2 0 2 z 9 /z 10 z 11 /z 12 0.2 0.2 dN/d 0.0 0.0 0.2 0.2 2 0 2 2 0 2 Machines automatically discover fourier transformation for flow! Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 20 / 30

Defining new flow observables v ′ n z k : k -th (normalized) eigenvector x k : amplitude of z k . k dN � d φ = µ + x k z k i =1 Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 21 / 30

′ Compare v n and v n ′ ′ 2 fits really well with v 2 , and v 3 fits really well with v 3 . v ′ 4 is deviated from v 4 . v 4 3 or v ′ 2 v ′ v ′ 0.0 0.1 0.00 0.05 0.00 0.03 v 2 v 3 v 4 Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 22 / 30

FC of eigenvectors z 1 / z 2 contain sin(4 φ ) and cos(4 φ ) bases as well. 1 2 3 4 5 6 7 8 9 10 11 12 v 1 v 2 = ⇒ v 3 Fc v 4 ⇒ = v 5 v 6 Eigenmodes z i Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 23 / 30

SC ( v m , v n ) SC ( v m , v n ) = � v 2 m v 2 n � − � v 2 n �� v 2 m � SC(v 2 , v 3 ) × 10 7 SC(v 2 , v 4 ) × 10 7 SC(v 2 , v 5 ) × 10 7 12.5 0 1.25 10.0 5 1.00 7.5 0.75 10 5.0 0.50 2.5 15 Fourier 0.25 0.0 PCA 20 0.00 SC(v 3 , v 4 ) × 10 7 SC(v 3 , v 5 ) × 10 7 SC(v 4 , v 5 ) × 10 7 0.0 0.04 0.3 0.1 0.2 0.02 0.2 0.1 0.00 0.3 0.0 0.4 0.02 10 20 30 40 50 60 70 10 20 30 40 50 60 70 10 20 30 40 50 60 70 Centrality% Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 24 / 30

Pearson Coefficient: r ( v m , ε n ) r (v 2 , 2 ) r (v 2 , 3 ) r (v 2 , 4 ) r (v 2 , 5 ) 1.0 0.15 0.8 0.10 0.3 0.6 0.10 0.05 0.2 Fourier 0.4 0.05 PCA 0.00 0.1 0.2 0.0 0.05 0.0 0.00 r (v 3 , 2 ) r (v 3 , 3 ) r (v 3 , 4 ) r (v 3 , 5 ) 0.0 0.8 0.3 0.6 0.2 0.1 0.0 0.4 0.1 0.2 0.1 0.2 0.0 0.3 0.0 0.2 0.1 r (v 4 , 2 ) r (v 4 , 3 ) r (v 4 , 4 ) r (v 4 , 5 ) 0.4 0.6 0.10 0.10 0.3 0.4 0.05 0.2 0.05 0.2 0.00 0.1 0.00 0.05 0.0 0.0 r (v 5 , 2 ) r (v 5 , 3 ) r (v 5 , 4 ) r (v 5 , 5 ) 0.25 0.20 0.3 0.20 0.15 0.2 0.15 0.2 0.10 0.10 0.1 0.1 0.05 0.05 0.00 0.0 0.00 0.0 10 20 30 40 50 60 70 10 20 30 40 50 60 70 10 20 30 40 50 60 70 10 20 30 40 50 60 70 Centrality% 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: PCA: 1.0 1.0 ′ ′ ′ ′ ′ 2 3 4 5 6 2 3 4 5 6 0.9 0.9 v 2 v ′ 2 0.8 0.8 v 3 0.7 0.7 v ′ 3 0.6 0.6 v 4 v ′ 4 0.5 0.5 v 5 v ′ 0.4 0.4 5 0.3 0.3 v 6 v ′ 6 0.2 0.2 0.1 0.1 Ziming Liu (PKU) PCA and Hydrodynamics December 16, 2018 26 / 30

Overview Motivation of the Question 1 Introduction to PCA 2 PCA in Sciences 3 Model 4 Results(Paper in Preparation) 5 Conclusions 6 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