Xiaoying PangIndiana UniversityMarch. 17th , 2010
Independent Component Analysis for Beam
Measurement
Outline
Introduction to ICA Application to linear betatron motionStudy of nonlinear motion – 2x modesBeam-based measurement of sextupole
strength
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Black Source Separation
Without knowing the positions of microphones or what any person is saying, can you isolate each of the voices?
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source1
source2
source3
source4
mixture1
mixture2
mixture3
mixture4
Mixing
Source signals
Mixing Measured signals
Demixing ?
Mixing
Mixing
Mixing
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BPM dataTurn-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:
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Principal Component Analysis
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x : measured signals (mixtures of sources signals)W: demixing matrixY: hopefully the source signals
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ij
ji
Y = Wx
Cov(Y) =
Larger variance modes contain more information.Source signals should have zero correlation.
TTTTTTTTTT DDWUWUWUVDWUDVWWxxYY ,
IYYUW TT 2/1
Singular Value Decomposition (SVD)
Independent Component AnalysisRequirement: ICs have different
autocovariance. Autocovariance: the covarience between the
values of the signals at different time points.For one signalFor two different signalsAll 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 Cs
, t = 0,1,2… should be diagonal.
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BPM data
X =
x11 x12 x13 X1,997 X1,998 X1,999 X1,1000x14
xm1 xm2 xm3 Xm,997 Xm,998 Xm,999 Xm,1000xm4
x21 x22 x23 X2,997 X2,998 X2,999 X2,1000x24
x31 x32 x33 X3,997 X3,998 X3,999 X3,1000x34
AutoCov(X) = T
Turn number
BPMnumber
ICA using time structure
s1
X =
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0
0S1
S2
S3
s2 s3
CsE{s(t)s(t-)T} is diagonal !
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ICAOne 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
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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.
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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.
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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 proper
ty
Temporal
property
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PCA Linear Betatron motion(cont’)
Now consider the betatron motion:
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ICA vs. PCA on Linear Betatron motion(cont’)
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ICA vs. PCA on Linear Betatron motion(cont’)
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ICA vs. PCA on Linear Betatron motion – effect of BPM noise
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Study of nonlinear motion -- 2x modeAGS 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 2x
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Study of 2x mode (cont’)Equation of motion of 2x mode
Hill’s eqn:For a short sextupole, use the localized kickFloquet transformation:
whereSolution: Get the particular solution
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Study of 2x mode (cont’)
Closed Orbit
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Study of 2x mode (cont’)
Closed Orbit= x-x-x2
Simple betatron oscillation !21
Study of 2x mode (cont’)
AGS lattice with two sextupleslocated at 185m and 420.37m, with strength K2L = 1m-2 and -1.5m-2.
Black lines indicate the locations of two sextupoles.
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ICA vs. PCA on 2x 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.
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Beam-based measurement of sextupole strength
BPM1
BPM2
BPM3
SXT
SXT
3x̂
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22
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3
sin
2s
ss
LxKx
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
2sx
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1 experiment
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With 12 sextupoles in the lattice
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ConclusionBasic idea of ICAWe have developed ICA for both linear and
nonlinear betatron motion, particularly beam-based measurement of nonlinear sextupole strength.
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Thank you.
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