106 Beam-based Correction and Optimization for Accelerators
In the ideal scenario when the model is accurate and all deviations are
random, the uncertainties of the fitted parameters can be derived from measurement errors using error propagation from Eq. (4.30), which relates errors
in the fitting parameters, ∆p, to measurement errors in r 0 . It is straightforward to see that the covariance matrix of the fitting parameters is given
by
Σ p ≡ ≡∆p∆p
T = (J
T J)
−1 J
T ∆r 0 ∆r
T
0 J(J
T J)
−1 = (J
T J)
−1 ,
(4.32)
where ·· denotes ensemble averaging over many possible data sets. It is assumed that ∆r 0 ∆r
T
0 is the identity matrix because ∆r 0 is normalized by
the noise sigma in the measurement and the data points in r 0 are assumed to
be uncorrelated. The covariance matrix Σ p contains information of the uncertainties of the fitting parameters. For example, its diagonal elements give
the error bars of the fitting parameters,
σ
2
pi = [Σ p ] ii , i = 1, 2, · · · , P.
(4.33)
As discussed earlier, the Jacobian matrix J may be degenerate or near
degenerate and hence the inversion of J
T J can be done only after some small
singular values are eliminated. The error sigmas of the fitting parameters
calculated this way under-estimate the uncertainties as errors in the directions
(in the parameter space) corresponding to the eliminated singular values are
not included. In fact, we could calculate the error propagation with Eq. (4.31),
which gives
Σ p =
N th
i=1
1
s 2
i
v i v
T
i .
(4.34)
Therefore, the error bars of the fitting parameters can be given by
σ
2
pi =
N th
j=1
1
s 2
j
v
2
j (i),
(4.35)
where v j (i) is the i’th element of vector v j .
4.2.6 Optics correction and an example
After the least-square fitting, we obtain calibrated values for the fitting parameters, including BPM and corrector gains, BPM and corrector rolls, and
quadrupole and skew quadrupole strengths. The BPM and corrector parameters can be used to update the calibration of the corresponding variables in the
control system. The lattice model can be updated with the fitted quadrupole
and skew quadrupole gradients. The actual lattice parameters that correspond
to the conditions of the machine at the time of data taking can be calculated
with the calibrated lattice model. The updated lattice model can also be used
In the ideal scenario when the model is accurate and all deviations are
random, the uncertainties of the fitted parameters can be derived from measurement errors using error propagation from Eq. (4.30), which relates errors
in the fitting parameters, ∆p, to measurement errors in r 0 . It is straightforward to see that the covariance matrix of the fitting parameters is given
by
Σ p ≡ ≡∆p∆p
T = (J
T J)
−1 J
T ∆r 0 ∆r
T
0 J(J
T J)
−1 = (J
T J)
−1 ,
(4.32)
where ·· denotes ensemble averaging over many possible data sets. It is assumed that ∆r 0 ∆r
T
0 is the identity matrix because ∆r 0 is normalized by
the noise sigma in the measurement and the data points in r 0 are assumed to
be uncorrelated. The covariance matrix Σ p contains information of the uncertainties of the fitting parameters. For example, its diagonal elements give
the error bars of the fitting parameters,
σ
2
pi = [Σ p ] ii , i = 1, 2, · · · , P.
(4.33)
As discussed earlier, the Jacobian matrix J may be degenerate or near
degenerate and hence the inversion of J
T J can be done only after some small
singular values are eliminated. The error sigmas of the fitting parameters
calculated this way under-estimate the uncertainties as errors in the directions
(in the parameter space) corresponding to the eliminated singular values are
not included. In fact, we could calculate the error propagation with Eq. (4.31),
which gives
Σ p =
N th
i=1
1
s 2
i
v i v
T
i .
(4.34)
Therefore, the error bars of the fitting parameters can be given by
σ
2
pi =
N th
j=1
1
s 2
j
v
2
j (i),
(4.35)
where v j (i) is the i’th element of vector v j .
4.2.6 Optics correction and an example
After the least-square fitting, we obtain calibrated values for the fitting parameters, including BPM and corrector gains, BPM and corrector rolls, and
quadrupole and skew quadrupole strengths. The BPM and corrector parameters can be used to update the calibration of the corresponding variables in the
control system. The lattice model can be updated with the fitted quadrupole
and skew quadrupole gradients. The actual lattice parameters that correspond
to the conditions of the machine at the time of data taking can be calculated
with the calibrated lattice model. The updated lattice model can also be used