Application of Independent Component Analysis (ICA) to Beam Diagnosis

1. Application of Independent Component Analysis (ICA) to Beam Diagnosis 5th MAP meeting at IU, Bloomington 3/18/2004 Xiaobiao Huang Indiana University / Fermilab

2. Content • Review of MIA* • Principles of ICA • Comparisons (ICA vs. PCA**) • Brief Summary of Booster Results *Model Independent Analysis (MIA), See J. Irvin, Chun-xi Wang, et al **MIA is a Principal Component Analysis (PCA) method.

3. Review of MIA Each raw is made zero mean 1. Organize BPM turn-by-turn data 2. Perform SVD 3. Identify modes spatial pattern, m×1 vector temporal pattern, 1×T vector

4. Review of MIA • Features 1. The two leading modes are betatron modes 2. Noise reduction 3. Degree of freedom analysis to locate locale modes (e.g. bad BPM) 4. And more … Comments: MIA is a Principal Component Analysis (PCA) method

5. A Model of Turn-by-turn Data • BPM turn-by-turn data is considered as a linear* mixture of source signals** • (1) Global sources • Betatron motion, synchrotron motion, higher order resonance, coupling, etc. • (2) Local sources • Malfunctioning BPM. Note: *Assume linear transfer function of BPM system. ** This is also the underlying model of MIA

6. A Model of Turn-by-turn Data • Source signals are assumed to be independent, meaning where p{} is joint probability density function (pdf) and pi {si} represents marginal pdf of si. This property is called statistical independence. Independence is a stronger condition than uncorrelatedness. Independence Uncorrelatedness The source signals can be identified from measurements under some assumptions with Independent Component Analysis (ICA).

7. An Introduction to ICA* • Three routes toward source signal separation, each makes a certain assumption of source signals. 1. Non-gaussian: source signals are assumed to have non-gaussian distribution. Gaussian pdf 2. Non-stationary: source signals have slowly changing power spectra 3. Time correlated: source signals have distinct power spectra. This is the one we are going to explore * Often also referred as Blind Source Separation (BSS).

8. with Measured signals Source signals Random noises Mixing matrix ICA with Second-order Statistics* • The model Note:*See A. Belouchrani, et al, for Second Order Blind Identification (SOBI)

9. (2) Noises are temporally white and spatially decorrelated. ICA with Second-order Statistics • Assumptions (1) • Source signals are temporally correlated. • No overlapping between power spectra of source signals. As a convention, source signals are normalized, so

10. ICA with Second-order Statistics • Covariance matrix So the mixing matrix A is the diagonalizer of the sample covariance matrix Cx. Although theoretically mixing matrix A can be found as an approximate joint diagonalizer of Cx() with a selected set of , to facilitate the joint diagonalization algorithm and for noise reduction, a two-phase approach is taken.

11. D1,D2 are diagonal Set to remove noise 1.Data whitening with 2. Joint approximate diagonalization ICA with Second-order Statistics • Algorithm • Benefits of whitening: • Reduction of dimension • Noise reduction • Only rotation (unitary W) is needed to diagonalize. n×n for The mixing matrix A and source signals s

12. Linear Optics Functions Measurements • The spatial and temporal pattern can be used to measure beta function (), phase advance () and dispersion (Dx) 1.Betatron function and phase advance Betatron motion is decomposed to a sine-like signal and a cosine-like signal a, b are constants to be determined 2.Dispersion Orbit shift due to synchrotron oscillation coupled through dispersion

13. Comparison between PCA and ICA • Both take a global view of the BPM data and aim at re-interpreting the data with a linear transform. • Both assume no knowledge of the transform matrix in advance. • Both find un-correlated components. 1. However, the two methods have different criterion in defining the goal of the linear transform. For PCA: to express most variance of data in least possible orthogonal components. (de-correlation + ordering) For ICA: to find components with least mutual information. (Independence) 2. ICA makes use of more information of data than just the covariance matrix (here it uses the time-lagged covariance matrix).

14. Comparison between PCA and ICA • So, ICA modes are more likely of single physical origin, while PCA modes (especially higher modes) could be mixtures. ICA has extra benefits (potentially) while retaining that of PCA method : 1. More robust betatron motion measurements. (Less sensitive to disturbing signals) 2. Facilitate study of other modes (synchrotron mode, higher order resonance, etc.)

15. A case study: PCA vs. ICA • Data taken with Fermilab Booster DC mode, starting turn index 4235, length 1000 turns. Horizontal and vertical data were put in the same data matrix (x, z)^T. Similar results if only x or z are considered. Only temporal pattern and its FFT spectrum are shown. Only first 4 modes are compared due to limit of space. The example supports the statement made in the previous slide.

16. A case study: PCA vs. ICA ICA Mode 1,4

17. A case study: PCA vs. ICA ICA Mode 2,3

18. A case study: PCA vs. ICA PCA Mode 1,4

19. A case study: PCA vs. ICA PCA Mode 2,3

20. A case study: PCA vs. ICA ICA Mode 8, 14

21. A case study: PCA vs. ICA PCA Mode 8, 14

22. Another Case Study with APS data* ICA Mode 1,3 *Data supplied by Weiming Guo

23. Another Case Study with APS data* PCA Mode 1,3 *Data supplied by Weiming Guo

24. Booster Results (, ) (b) (a) (c) (1915,1000)*, MODE 1: (a) Spatial pattern; (b) temporal pattern; (c) FFT spectrum of (b) *(Starting turn index, number of turns)

25. Booster Results (, ) (b) (a) (c) (1915,1000)*,MODE 2: (a) Spatial pattern; (b) temporal pattern; (c) FFT spectrum of (b)

26. Booster Results (, ) (b) σ =3 deg σ=7% (a) Comparison of (, ) between MAD model and measurements.(a) Measured  with error bars. (b) phase advance in a period (S-S). Note: Horizontal beam size is about 20-30 mm at large ; Betatron amplitude was about 0.6mm; BPM resolution 0.08mm.

27. Booster Results (Dx) (b) (a) • 1000turns from turn index1. • Temporal pattern. (b) Spatial pattern. • (t=0)= -0.3×10-3

28. Booster Results (Dx) (a) σD=0.11 m Comparison of dispersionbetween MAD model and measurements.

29. Summary • ICA provides a new perspective and technique for BPM turn-by-turn data analysis. • ICA could be more useful to study coupling and higher order modes than PCA method. • More work is needed to: • Explore new algorithms. • Refine the algorithms to suit BPM data. • More rigorous understanding of ICA and PCA.