04/19/23 1

Application of Independent Component Analysis (ICA) to Beam Diagnosis

5th MAP meeting at IU, Bloomington

3/18/2004

Xiaobiao Huang

Indiana University / Fermilab

04/19/23 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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Review of MIA

1. Organize BPM turn-by-turn data

2. Perform SVD

3. Identify modes

spatial pattern, m×1 vector

temporal pattern, 1×T vector

Each raw is made zero mean

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Review of MIA

Features1. The two leading modes are betatron modes2. Noise reduction3. 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**

Note: *Assume linear transfer function of BPM system.** This is also the underlying model of MIA

(1) Global sources Betatron motion, synchrotron motion, higher order resonance, coupling, etc.(2) Local sources

Malfunctioning BPM.

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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.

The source signals can be identified from measurements under some assumptions with Independent Component Analysis (ICA).

Independence is a stronger condition than uncorrelatedness.)}({)}({)}()({ yfExgEyfxgE

}{}{}{ yExExyE

Independence

Uncorrelatedness

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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.

2. Non-stationary: source signals have slowly changing power spectra

3. Time correlated: source signals have distinct power spectra.

* Often also referred as Blind Source Separation (BSS).

Gaussian pdf

This is the one we are going to explore

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ICA with Second-order Statistics*

The model

Note:*See A. Belouchrani, et al, for Second Order Blind Identification (SOBI)

with

Measured signals Source signals

Random noises Mixing matrix

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ICA with Second-order Statistics

Assumptions

(1)

• Source signals are temporally correlated.• No overlapping between power spectra of source signals.

(2)

Noises are temporally white and spatially decorrelated.

As a convention, source signals are normalized, so

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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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ICA with Second-order Statistics

Algorithm

The mixing matrix A and source signals s

WCWC sT

z )()(

TT WDUA )( 2

1

11WVxs

IzzE T }{

2. Joint approximate diagonalization

Tx UU

D

DUUC ],[],[)0( 21

2

121

VxxUDz T

12

1

1

1. Data whitening

)min()max(0 12 DD with

Set to remove noise

D1,D2 are diagonal

},,2,1|{ kii for

n×nBenefits of whitening:1. Reduction of dimension2. Noise reduction3. Only rotation (unitary W) is

needed to diagonalize.

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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)

2211 sAsAx bb

)( 22

21 bb AAa

2

11tanb

b

A

A

llsAx

lx bAD b

sl

2. Dispersiona, b are constants to be determined

Betatron motion is decomposed to a sine-like signal and a cosine-like signal

Orbit shift due to synchrotron oscillation coupled through dispersion

1. Betatron function and phase advance

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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 BoosterDC 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

ICA Mode 1,4

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A case study: PCA vs. ICA

ICA Mode 2,3

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A case study: PCA vs. ICA

PCA Mode 1,4

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A case study: PCA vs. ICA

PCA Mode 2,3

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A case study: PCA vs. ICA

ICA Mode 8, 14

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A case study: PCA vs. ICA

PCA Mode 8, 14

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Another Case Study with APS data*

*Data supplied by Weiming GuoICA Mode 1,3

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Another Case Study with APS data*

*Data supplied by Weiming GuoPCA Mode 1,3

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Booster Results (, )

(1915,1000)*, MODE 1: (a) Spatial pattern; (b) temporal pattern; (c) FFT spectrum of (b)

(a)

(b)

(c)

*(Starting turn index, number of turns)

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Booster Results (, )

(1915,1000)*, MODE 2: (a) Spatial pattern; (b) temporal pattern; (c) FFT spectrum of (b)

(a)

(b)

(c)

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Booster Results (, )

Comparison of (, ) between MAD model and measurements. (a) Measured with error bars. (b) phase advance in a period (S-S).

(a) (b)σ=7% σ =3 deg

Note: Horizontal beam size is about 20-30 mm at large ; Betatron amplitude was about 0.6mm; BPM resolution 0.08mm.

0 10 20 30 40 500

10

20

30

40

50

60

BPM index

x

MADMeasured

0 5 10 15 20 2580

85

90

95

100

105

110

115

120

Period index

x

MADMeasured

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Booster Results (Dx)

(a) (b)

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)

0 10 20 30 40 500.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

BPM index

DxMADMeasured

(a) σD=0.11 m

Comparison of dispersion between MAD model and measurements.

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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:1. Explore new algorithms.2. Refine the algorithms to suit BPM data.3. More rigorous understanding of ICA and PCA.