1. Application of Independent Component Analysis (ICA) to Beam Diagnosis
5th MAP Meeting
Application of Independent Component Analysis (ICA) to Beam Diagnosis
Xiaobiao Huang
Indiana University / Fermilab
5th MAP meeting at IU, Bloomington
3/18/2004
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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.
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3. Review of MIA 1. Organize BPM turn-by-turn data 2. Perform SVD
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
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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
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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
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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).
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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).
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8. ICA with Second-order Statistics*
The model
with
Measured signals
Source signals
Random noises
Mixing matrix
Note:*See A. Belouchrani, et al, for Second Order Blind Identification (SOBI)
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9. 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
(2)
Noises are temporally white and spatially decorrelated.
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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.
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11. ICA with Second-order Statistics
Algorithm
2. Joint approximate diagonalization
1. Data whitening
with
Set to remove noise
D1,D2 are diagonal
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
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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
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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).
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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.)
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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.
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16. A case study: PCA vs. ICA
ICA Mode 1,4
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17. A case study: PCA vs. ICA
ICA Mode 2,3
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18. A case study: PCA vs. ICA
PCA Mode 1,4
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19. A case study: PCA vs. ICA
PCA Mode 2,3
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20. A case study: PCA vs. ICA
ICA Mode 8, 14
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21. A case study: PCA vs. ICA
PCA Mode 8, 14
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22. Another Case Study with APS data*
ICA Mode 1,3
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*Data supplied by Weiming Guo

23. Another Case Study with APS data*
PCA Mode 1,3
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*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)
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25. Booster Results (, ) (b) (a) (c)
(1915,1000)*, MODE 2: (a) Spatial pattern; (b) temporal pattern; (c) FFT spectrum of (b)
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26. Booster Results (, ) (a) σ=7% (b) σ =3 deg
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 mm at large ; Betatron amplitude was about 0.6mm; BPM resolution 0.08mm.
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27. Booster Results (Dx) (b) (a) 1000 turns from turn index 1.
Temporal pattern. (b) Spatial pattern.
(t=0)= -0.3×10-3
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28. Booster Results (Dx) (a) σD=0.11 m
Comparison of dispersion between MAD model and measurements.
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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.
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