1. Xiaoying Pang Indiana University March. 17 th, 2010 Independent Component Analysis for Beam Measurement

2. Outline Introduction to ICA Application to linear betatron motion Study of nonlinear motion – 2 x modes Beam-based measurement of sextupole strength 2

3. Black Source Separation Without knowing the positions of microphones or what any person is saying, can you isolate each of the voices? 3

4. source1 source2 source3 source4 mixture1 mixture2 mixture3 mixture4 Mixing Source signals Mixing Measured signals Demixing ? Mixing 4

5. BPM data Turn-by-turn BPM signals, x(t) is usually a mixuture of betatron motion, synchrotron motion, nonlinear motion and noise. For a turn-by-turn measurements of M-BPMs and N-turns, we construct BPM data matrix: 5

6. Principal Component Analysis 6 x : measured signals (mixtures of sources signals) W: demixing matrix Y: hopefully the source signals  11  22  33  ij  ji Y = Wx Cov(Y) = Larger variance modes contain more information. Source signals should have zero correlation. Singular Value Decomposition (SVD)

7. Independent Component Analysis Requirement: ICs have different autocovariance. Autocovariance: the covarience between the values of the signals at different time points. For one signal For two different signals All these autocovariances for a particular time lag can be grouped into an autocovariance matrix Due to the independence of the source signals, the source signal autocovariance matrices C s , t = 0,1,2… should be diagonal. 7

8. 8 BPM data X = x 11 x 12 x 13 X 1,997 X 1,998 X 1,999 X 1,1000 x 14 x m1 x m2 x m3 X m,997 X m,998 X m,999 X m,1000 x m4 x 21 x 22 x 23 X 2,997 X 2,998 X 2,999 X 2,1000 x 24 x 31 x 32 x 33 X 3,997 X 3,998 X 3,999 X 3,1000 x 34  AutoCov(X) = T Turn number BPM number

9. ICA using time structure s1s1 X =  11     0 0 S1S1 S2S2 S3S3 s2s2 s3s3 C s   E{s(t)s(t-  ) T } is diagonal ! 9

10. ICA One time lag and the AMUSE (Algorithm for Multiple Unknown Signals Extraction) algorithm Consider the whitened data z(t), with the separating matrix W, the source signals s(t) can be found as: Slightly modified time-lagged covariance matrix: The new time-lagged covariance matrix is symmetric. So the eigenvalue decomposition is well defined and easy to compute. W can be obtained by SVD of 10

11. ICA (cont’) Drawbacks of the AMUSE algorithm Requirement: the eigenvalues of matrix have to be uniquely defined. The eigenvalues are given by, thus the source signals must have different autocovariances. Otherwise, ICs can not be estimated. We can search for a suitable time lag  so that the eigenvalues are distinct, but this is not always possible if the source signals have identical power spectra, identical autocovariances. 11

12. ICA (cont’) Using several time-lags An extension of the AMUSE algorithm that improves its performance is to consider several time lags instead of a single one. Then the choice of the time lag is a less serious problem. Using several time lags, we want to simultaneously diagonalize all the lagged covariance matrices. This joint-diagonalization cannot be perfect, but we can define a quantity to express the degree of diagonalization and try to find its minimum/maximum. Minimizing off(M) is equivalent to diagonalizing M. 12

13. PCA to Linear Betatron motion Consider a simple sinusoidal model With M BPMs in the accelerator and N turns, the (i,j) element of the turn-by-turn BPM data matrix X is SVD of x is: Spatial property Temporal property 13

14. PCA Linear Betatron motion(cont’) Now consider the betatron motion: 14

15. ICA vs. PCA on Linear Betatron motion(cont’) 15

16. ICA vs. PCA on Linear Betatron motion(cont’) 16

17. ICA vs. PCA on Linear Betatron motion – effect of BPM noise 17

18. Study of nonlinear motion -- 2 x mode AGS lattice with 12 superperiod of FODO cells Add sextupoles in the lattice. Particle tracking was carried out and the data were analyzed by PCA and ICA. We found totally 6 important modes. We only consider the 3 rd and 4 th modes at the tune of 2 x 18

19. Study of 2 x mode (cont’) Equation of motion of 2 x mode Hill’s eqn: For a short sextupole, use the localized kick Floquet transformation: where Solution: Get the particular solution 19

20. Study of 2 x mode (cont’) Closed Orbit 20

21. Study of 2 x mode (cont’) Closed Orbit= x-x  -x 2 Simple betatron oscillation ! 21

22. Study of 2 x mode (cont’) AGS lattice with two sextuples located at 185m and 420.37m, with strength K 2 L = 1m -2 and -1.5m -2. Black lines indicate the locations of two sextupoles. 22

23. ICA vs. PCA on 2 x mode Compare the spatial function obtained by ICA and PCA After ICA processing, the normalized spatial wave functions of the 3 rd and 4 th modes have simple linear betatron motion outside the sextupole. The spatial function of the 4 th mode obtained by PCA preprocessing is messy, but still important in a proper ICA analysis. 23

24. Beam-based measurement of sextupole strength BPM1 BPM2 BPM3 SXT 24

25.  With single sextupole in the lattice, very point corresponds to one turn of tracking, totally 1000 turns.  The slope indicates strength of the sextupole.  The slope can be accurately determined by the centroid of each bin of  The band width is proportional to noise level.  This method can also be used for other higher order non-linear elements 25

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27. 1 experiment 27

28. With 12 sextupoles in the lattice 28

29. Conclusion Basic idea of ICA We have developed ICA for both linear and nonlinear betatron motion, particularly beam-based measurement of nonlinear sextupole strength. 29

30. Thank you. 30