Linear optics measurement and correction - I 105
Because of the degeneracy in the Jacobian matrix, it is necessary to compute the pseudo-inverse of the square matrix J
T J. This can be done in the
same fashion as was done in Chapter 3 for the orbit correction case, which
results in the solution of ∆p in the form of
∆p = −
N th
i=1
1
s i
v i (u
T
i r 0 ),
(4.31)
where u i and v i are column vectors in matrices U and V for the i’th singular
value, respectively, and N th stands for the number of singular values to be
kept in the calculation of the pseudo-inverse matrix. In orbit response matrix
fitting, the number of rows of the Jacobian matrix is often significantly larger
than the number of columns (i.e., the number of fitting parameters). For
example, for a ring with 180 BPMs, 180 horizontal correctors, and 180 vertical
correctors, the dimension of the full orbit response matrix, and hence the
number of rows, is 360 × 360. The U-matrix in the SVD of the Jacobian
matrix would have 360
4 = 1.68 × 10
10 elements. The space needed to store
such a matrix exceeds the memory size of an ordinary computer. Fortunately,
in Eq. (4.31) we only need the columns of the U-matrix that correspond to
the non-zero singular values. This can be obtained without computing the full
U-matrix.
Starting from an initial solution p 0 and applying Eq. (4.31) to move the
solution in the parameter space iteratively, the objective function may be
brought down to an acceptable level. The final solution p contains the fitted
values of the parameters such as quadrupole gradients and BPM and corrector
gains.
4.2.5 Error analysis
It is important to check the validity of the results when using the least-square
method to fit data to a model. A common measure of the goodness-of-fit is
the value of the final χ
2 . If the fitted model is an exact representation of
the experimental system, any deviation between a measured data point and
the model prediction can only be due to random measurement errors. The
random measurement errors can be assumed to obey Gaussian distributions.
If the standard deviations of the error distributions are used to normalize the
error terms in the χ
2 definition, the final value of χ
2 /(N − P ) after fitting
should approach unity, where N is the number of data points and P is the
number of fitting parameters.
Substantial deviation of the final χ
2 /(N − P ) from unity indicates that
the fitted model is a not a true representation of the system. This is the case
when the model has systematic errors. Systematic errors can be caused by the
omission of some relevant physical processes in the model, or inconsistencies
in the data due to drifting experimental conditions during the time span of
data taking. It could also mean the fitting algorithm has not converged to the
true, global minimum.
Because of the degeneracy in the Jacobian matrix, it is necessary to compute the pseudo-inverse of the square matrix J
T J. This can be done in the
same fashion as was done in Chapter 3 for the orbit correction case, which
results in the solution of ∆p in the form of
∆p = −
N th
i=1
1
s i
v i (u
T
i r 0 ),
(4.31)
where u i and v i are column vectors in matrices U and V for the i’th singular
value, respectively, and N th stands for the number of singular values to be
kept in the calculation of the pseudo-inverse matrix. In orbit response matrix
fitting, the number of rows of the Jacobian matrix is often significantly larger
than the number of columns (i.e., the number of fitting parameters). For
example, for a ring with 180 BPMs, 180 horizontal correctors, and 180 vertical
correctors, the dimension of the full orbit response matrix, and hence the
number of rows, is 360 × 360. The U-matrix in the SVD of the Jacobian
matrix would have 360
4 = 1.68 × 10
10 elements. The space needed to store
such a matrix exceeds the memory size of an ordinary computer. Fortunately,
in Eq. (4.31) we only need the columns of the U-matrix that correspond to
the non-zero singular values. This can be obtained without computing the full
U-matrix.
Starting from an initial solution p 0 and applying Eq. (4.31) to move the
solution in the parameter space iteratively, the objective function may be
brought down to an acceptable level. The final solution p contains the fitted
values of the parameters such as quadrupole gradients and BPM and corrector
gains.
4.2.5 Error analysis
It is important to check the validity of the results when using the least-square
method to fit data to a model. A common measure of the goodness-of-fit is
the value of the final χ
2 . If the fitted model is an exact representation of
the experimental system, any deviation between a measured data point and
the model prediction can only be due to random measurement errors. The
random measurement errors can be assumed to obey Gaussian distributions.
If the standard deviations of the error distributions are used to normalize the
error terms in the χ
2 definition, the final value of χ
2 /(N − P ) after fitting
should approach unity, where N is the number of data points and P is the
number of fitting parameters.
Substantial deviation of the final χ
2 /(N − P ) from unity indicates that
the fitted model is a not a true representation of the system. This is the case
when the model has systematic errors. Systematic errors can be caused by the
omission of some relevant physical processes in the model, or inconsistencies
in the data due to drifting experimental conditions during the time span of
data taking. It could also mean the fitting algorithm has not converged to the
true, global minimum.