Transcript Application of Independent Component Analysis (ICA) to

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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Content
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

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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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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
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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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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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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
E{g ( x) f ( y)}  E{g ( x)}E{ f ( y)}
Uncorrelatedness
E{xy}  E{x}E{ y}
The source signals can be identified from measurements
under some assumptions with Independent Component
Analysis (ICA).
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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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ICA with Second-order Statistics*
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The model
with
Measured signals
Random noises
Source signals
Mixing matrix
Note:*See A. Belouchrani, et al, for Second Order Blind Identification (SOBI)
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ICA with Second-order Statistics
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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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ICA with Second-order Statistics
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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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ICA with Second-order Statistics
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Algorithm
D1,D2 are
diagonal
1. Data whitening
 D1
C x (0)  [U1 ,U 2 ]

1


[U1 ,U 2 ]T

D2 
Set to remove
noise
with
0  max(D2 )    min(D1 )
Benefits of whitening:
1. Reduction of dimension
2. Noise reduction
2. Joint approximate diagonalization
3. Only rotation (unitary W) is
T
Cz ( )  W Cs ( )W for   { i | i  1,2,, k} needed to diagonalize.
z  D1 2U1T x  Vx
E{zz T }  In×n
The mixing matrix A and source signals s
s  WVx
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1
2
1
A  (U D )W T
T
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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

x  Ab1s1  Ab 2 s2
 Ab1 

 Ab 2 
  a( Ab21  Ab22 )   tan 1 
2. Dispersion
x  Al sl
Dx  bAl
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
sl
b
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Betatron motion is decomposed to a
sine-like signal and a cosine-like signal
a, b are constants to be
determined
Orbit shift due to synchrotron oscillation
coupled through dispersion
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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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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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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A case study: PCA vs. ICA
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ICA Mode 1,4
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A case study: PCA vs. ICA
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ICA Mode 2,3
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A case study: PCA vs. ICA
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PCA Mode 1,4
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A case study: PCA vs. ICA
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PCA Mode 2,3
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A case study: PCA vs. ICA
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ICA Mode 8, 14
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A case study: PCA vs. ICA
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PCA Mode 8, 14
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Another Case Study with APS data*
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ICA Mode 1,3
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*Data supplied by Weiming Guo
Another Case Study with APS data*
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PCA Mode 1,3
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*Data supplied by Weiming Guo
Booster Results (, )
(b)
(a)
(1915,1000)*, MODE 1: (a)
Spatial pattern; (b) temporal
pattern; (c) FFT spectrum of (b)
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*(Starting turn index, number of turns)
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(c)
Booster Results (, )
(b)
(a)
(c)
(1915,1000)*, MODE 2: (a) Spatial
pattern; (b) temporal pattern;
(c) FFT spectrum of (b)
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Booster Results (, )
120
60
MAD
Measured
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50
110
40
x
x
105
100
30
95
20
90
10
0
0
85
10
20
30
40
BPM index
(a) σ=7%
50
80
0
MAD
Measured
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10
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Period index
(b)
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σ =3 deg
Comparison of (, ) between MAD model and measurements. (a)
Measured  with error bars. (b) phase advance in a period (S-S).
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Note: Horizontal beam size is about 20-30 mm at large ; Betatron
amplitude was about 0.6mm; BPM resolution 0.08mm.
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Booster Results (Dx)
(b)
(a)
1000 turns from turn index 1.
(a)Temporal pattern. (b) Spatial pattern.
(t=0)= -0.3×10-3
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Booster Results (Dx)
5.5
5
MAD
Measured
4.5
4
D
x
3.5
3
2.5
2
1.5
1
0.5
0
10
20
30
BPM index
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50
(a) σD=0.11 m
Comparison of dispersion between MAD model and measurements.
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Summary
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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:
1. Explore new algorithms.
2. Refine the algorithms to suit BPM data.
3. More rigorous understanding of ICA and PCA.
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