Linear optics measurement and correction - II 137
5.2.3 Independent component analysis (ICA)
By diagonalizing the covariance matrix of the BPM data, PCA finds an orthogonal basis of the multi-variate data space. The new variables defining the
orthogonal basis are uncorrelated and would be the underlying source signals
if the covariance matrix is not degenerate and there are no noise or bad BPMs.
However, in reality there is always noise and it is not uncommon to have bad
BPMs. The variances of the source signals can also be nearly equal. Under
these circumstances, the uncorrelated variables found by PCA may not necessarily be the source signals. More advanced methods are needed to identify
the source signals.
What PCA achieves is the uncorrelatedness in the new variables after a
linear transformation. Uncorrelatedness is a requirement for the independent
source signals. However, in general, uncorrelatedness does not guarantee independence between the new variables. Two random variables, x 1 and x 2 ,
are uncorrelated if the covariance x 1 x 2 = x 1 x 1 . By definition, the two
variables are statistically independent when their joint probability distribution is the product of their respective probability distribution functions, i.e.,
p(x 1 , x 2 ) = p(x 1 )p(x 2 ), from which one can show that for any two functions
h 1 (·) and h 2 (·), h 1 (x 1 )h 2 (x 2 ) = h 1 (x 1 )h 2 (x 2 ) is satisfied. Therefore, independence is a much stronger condition than uncorrelatedness. Uncorrelatedness implies independence if and only if the probability distribution functions
are Gaussian [64].
The source signals can be identified from the original data by utilizing the
additional requirements for the independent variables. This process is referred
to as independent component analysis (ICA) [70, 24, 64, 63]. A major category
of ICA methods relies on the assumption that the probability distributions
of the independent variables are non-Gaussian. These methods try to maximize the non-Gaussianity of the new variables after a linear transformation,
using appropriate parametric measures of the non-Gaussianity of the random
variables [62].
Other ICA methods exploit the time dependence in the source signals. The
time dependence can be fast oscillations or long-term, slow drifts. Separation
of source signals from mixed signals recorded on multiple sensors using only
the recorded signals without the use of any a priori information is called blind
source separation (BSS) [20, 13]. The fast oscillations of the source signals
can be characterized by their frequency spectra. ICA methods based on the
spectral features are very suitable for the application to turn-by-turn BPM
data because the source signals of beam motion in circular accelerators are
typically oscillations of certain frequencies. The power spectra of the source
signals can be assumed to be narrow-band, for example, a single frequency
or with a narrow frequency spread. The mutual independence of two source
signals requires their frequency contents to have no overlap – otherwise the
two are correlated in the temporal pattern.
5.2.3 Independent component analysis (ICA)
By diagonalizing the covariance matrix of the BPM data, PCA finds an orthogonal basis of the multi-variate data space. The new variables defining the
orthogonal basis are uncorrelated and would be the underlying source signals
if the covariance matrix is not degenerate and there are no noise or bad BPMs.
However, in reality there is always noise and it is not uncommon to have bad
BPMs. The variances of the source signals can also be nearly equal. Under
these circumstances, the uncorrelated variables found by PCA may not necessarily be the source signals. More advanced methods are needed to identify
the source signals.
What PCA achieves is the uncorrelatedness in the new variables after a
linear transformation. Uncorrelatedness is a requirement for the independent
source signals. However, in general, uncorrelatedness does not guarantee independence between the new variables. Two random variables, x 1 and x 2 ,
are uncorrelated if the covariance x 1 x 2 = x 1 x 1 . By definition, the two
variables are statistically independent when their joint probability distribution is the product of their respective probability distribution functions, i.e.,
p(x 1 , x 2 ) = p(x 1 )p(x 2 ), from which one can show that for any two functions
h 1 (·) and h 2 (·), h 1 (x 1 )h 2 (x 2 ) = h 1 (x 1 )h 2 (x 2 ) is satisfied. Therefore, independence is a much stronger condition than uncorrelatedness. Uncorrelatedness implies independence if and only if the probability distribution functions
are Gaussian [64].
The source signals can be identified from the original data by utilizing the
additional requirements for the independent variables. This process is referred
to as independent component analysis (ICA) [70, 24, 64, 63]. A major category
of ICA methods relies on the assumption that the probability distributions
of the independent variables are non-Gaussian. These methods try to maximize the non-Gaussianity of the new variables after a linear transformation,
using appropriate parametric measures of the non-Gaussianity of the random
variables [62].
Other ICA methods exploit the time dependence in the source signals. The
time dependence can be fast oscillations or long-term, slow drifts. Separation
of source signals from mixed signals recorded on multiple sensors using only
the recorded signals without the use of any a priori information is called blind
source separation (BSS) [20, 13]. The fast oscillations of the source signals
can be characterized by their frequency spectra. ICA methods based on the
spectral features are very suitable for the application to turn-by-turn BPM
data because the source signals of beam motion in circular accelerators are
typically oscillations of certain frequencies. The power spectra of the source
signals can be assumed to be narrow-band, for example, a single frequency
or with a narrow frequency spread. The mutual independence of two source
signals requires their frequency contents to have no overlap – otherwise the
two are correlated in the temporal pattern.