Linear optics measurement and correction - II 131
where vectors x(t) and ξ have M elements, one for each BPM, s has P
elements, one for each source signal, and the matrix A contains all the a ij
coefficients. Matrix A is called the mixing matrix and is M × P in dimension.
Turn-by-turn BPM data for T turns on all BPMs can be put in a matrix
X =
x 1 (1) x 1 (2) · · · x 1 (T )
x 1 (1) x 1 (2) · · · x 1 (T )
. . .
. . .
. . .
. . .
x M (1) x M (2) · · · x M (T )
,
(5.21)
where x i (n) indicates the orbit reading on BPM i for the n’th turn.
Pulse-by-pulse orbit data collected on one-pass systems can be similarly
cast into the model of Eq. (5.20). The difference is that the time variation of
the source signals does not represent beam motion as each data point is for
a different beam pulse. The drifting of the machine conditions with time and
the shot to shot jitter will show up on the time variation.
With all turn-by-turn BPM data put in one matrix, the challenge is to
separate out the source signals without any additional input.
5.2.2 Principal component analysis (PCA)
Principal component analysis (PCA) is a statistical method for multi-variate
data reduction. Suppose the data are samples of M potentially correlated variables, the goal of PCA is to find an orthogonal transformation of the variables
to a new set of M variables that are mutually uncorrelated. Furthermore, the
new variables are sorted in the descending order in their variances, with each
new variable having the maximum possible variance in the remaining subspace after the preceding variables are excluded. In other words, the first new
variable is the linear combination of the original variables with the highest
variance; the second new variable is the linear combination with the highest
variance in the subspace that is orthogonal to the first new variable; and so
on.
Mathematically, with the data arranged in a form like Eq. (5.21), PCA is
achieved by performing SVD on the data matrix [97],
X = UΛV
T ,
(5.22)
with orthogonal matrix U and V, and diagonal matrix Λ = diag(λ 1 , λ 2 , · · · ,
λ M ) (padded with zeros to make an M × T matrix). The column vectors in
U represent the distribution of the SVD modes over the BPMs and are called
the spatial patterns. The column vectors in V represent the turn-by-turn
variation of the modes and are called the temporal patterns. The orthogonal
transformation from the original variables to the new variables is given by
Z = U
T X,
(5.23)
where Z is the data matrix in the new variables.
where vectors x(t) and ξ have M elements, one for each BPM, s has P
elements, one for each source signal, and the matrix A contains all the a ij
coefficients. Matrix A is called the mixing matrix and is M × P in dimension.
Turn-by-turn BPM data for T turns on all BPMs can be put in a matrix
X =
x 1 (1) x 1 (2) · · · x 1 (T )
x 1 (1) x 1 (2) · · · x 1 (T )
. . .
. . .
. . .
. . .
x M (1) x M (2) · · · x M (T )
,
(5.21)
where x i (n) indicates the orbit reading on BPM i for the n’th turn.
Pulse-by-pulse orbit data collected on one-pass systems can be similarly
cast into the model of Eq. (5.20). The difference is that the time variation of
the source signals does not represent beam motion as each data point is for
a different beam pulse. The drifting of the machine conditions with time and
the shot to shot jitter will show up on the time variation.
With all turn-by-turn BPM data put in one matrix, the challenge is to
separate out the source signals without any additional input.
5.2.2 Principal component analysis (PCA)
Principal component analysis (PCA) is a statistical method for multi-variate
data reduction. Suppose the data are samples of M potentially correlated variables, the goal of PCA is to find an orthogonal transformation of the variables
to a new set of M variables that are mutually uncorrelated. Furthermore, the
new variables are sorted in the descending order in their variances, with each
new variable having the maximum possible variance in the remaining subspace after the preceding variables are excluded. In other words, the first new
variable is the linear combination of the original variables with the highest
variance; the second new variable is the linear combination with the highest
variance in the subspace that is orthogonal to the first new variable; and so
on.
Mathematically, with the data arranged in a form like Eq. (5.21), PCA is
achieved by performing SVD on the data matrix [97],
X = UΛV
T ,
(5.22)
with orthogonal matrix U and V, and diagonal matrix Λ = diag(λ 1 , λ 2 , · · · ,
λ M ) (padded with zeros to make an M × T matrix). The column vectors in
U represent the distribution of the SVD modes over the BPMs and are called
the spatial patterns. The column vectors in V represent the turn-by-turn
variation of the modes and are called the temporal patterns. The orthogonal
transformation from the original variables to the new variables is given by
Z = U
T X,
(5.23)
where Z is the data matrix in the new variables.