Independent Component Analysis for Beam Measurement

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

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

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

4. source1 Mixing mixture1 source2 mixture2 Mixing source3 mixture3 Mixing source4 mixture4 Mixing S(t) A(t) X(t) Mixing Measured signals Source signals Demixing ? W(t)

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:

6. Principal Component Analysis x : measured signals (mixtures of sources signals) W: demixing matrix Y: hopefully the source signals Y = Wx s11 sij s22 Cov(Y) = sji s33 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 Cst , t = 0,1,2… should be diagonal.

8. BPM data Turn number x11 x12 x13 x14 X1,997 X1,998 X1,999 X1,1000 BPM number X = x21 x22 x23 x24 X2,997 X2,998 X2,999 X2,1000 x31 x32 x33 x34 X3,997 X3,998 X3,999 X3,1000 xm1 xm2 xm3 xm4 Xm,997 Xm,998 Xm,999 Xm,1000 t=3 T AutoCov(X) =

9. ICA using time structure s11 S1 0 s1 s22 = X S2 0 s33 S3 s2 s3 Cst = E{s(t)s(t-t)T} is diagonal !

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

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 t so that the eigenvalues are distinct, but this is not always possible if the source signals have identical power spectra, identical autocovariances.

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.

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

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

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

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

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

18. Study of nonlinear motion -- 2nx 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 3rd and 4th modes at the tune of 2nx

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

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

21. Study of 2nx mode (cont’) Closed Orbit= x-xb-x2n Simple betatron oscillation !

22. Study of 2nx mode (cont’) AGS lattice with two sextuples located at 185m and 420.37m, with strength K2L = 1m-2 and -1.5m-2. Black lines indicate the locations of two sextupoles.

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

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

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

27. 1 experiment

28. With 12 sextupoles in the lattice

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.

30. Thank you.