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Short introduction:
QSVD (quotient
singular value decomposition)
and ICA (independent
component analysis)
are techniques used for input data
preprocessing. They give a possibility of better
representation of data (network may learn more
easily on transformed data). Also
dimmensionality of data may be reduced by
recognizing and eliminating noisy components.
This example shows how to take advantages of
these transformation.
QSVD
ICA:
- part I
-
part II
QSVD:
This is
an ortoghonal transformation from the original
space to the space where axes point the
directions of maximum values of
ssig/sbkg
(where
ssig
and
sbkg
are the standard deviations of classes being
separated, so each event should be accompanied
with information about the class it belongs to).
In other words this is just a rotation that
shows the data from the most interesting angle.
|
Transformation matrix is calculated as: |
A = UT, |
|
where
U
is taken from SVD decomposition: |
Cbkg-1 ∙ Csig
=
U ∙ S ∙ VT, |
|
where
Cbkg
and
Csig
are the covariance matrices of
background and signal data respectively; |
|
transformed event feature vectors are
calculated as: |
y = A ∙ x. |
Rows of the matrix
A are
sorted so less informative rows with
corresponding singular values close to
1.0
are placed at the lower rows. Calculated
singular values (that are also values of
ssig/sbkg)
may be displayed through the menu
Edit/Project Tree
where the
Transform
block node should be expanded. Yellow marked
values placed at the end of the list correspond
to the rows of matrix
A
that are safe to remove (more).
If value is on a red label - you have already
excluded corresponding row from calculations.
Example of basic operations with QSVD
transformation is contained in
svd_ica_basics.NetPrj
project file. Signal and background classes in
this example are 3D, but only 2 dimmensions are
class-specific. Third dimmension is noise
generated in the same way for both classes.
Directions of axes of the original feature space
has been chosen randomly. Scatter plots of each
combination of feature vector components are on
the images below. We will try to discover 2
informative dimmensions with QSVD
transformation.
Push
Go!
button in the
Setup
dialog window of the
QSVD
transform block. This will release calculations
of transformation matrix. Then events from the
3D input space
block are transformed into 2D events and sent to
the
2D
reduced space
data block. Directions of extreme values of
ssig/sbkg
are discovered properly and one redundant
dimmension is removed. Check the QSVD/mode node
in the
Project Tree
- there are two singular values (sValues)
corredsponding to transformed space axes and one
sValue
on a red label (ssig/sbkg=1.0786).
The image below shows events in the transformed
sapace.
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--up--
ICA
In many cases simple rotation is not enough to
find an "interesting" point of view. ICA may be
more helpful in these situations. It is a
linear, but non-ortoghonal transformation, where
directions of most non-gaussian data probability
distribution functions (pdf)
are searched for. It is assumed that the more
gaussian is the
pdf,
the more likeli it is just noise in this
direction. Additionaly,
pdf-s
in each direction are uncorrelated to each other
(more).
Following examples show how to separate mixed
signals (part I) and how to expose structures
that may not be visible in a original space
(part II).
ICA - part I
This is most often
published example of ICA application. Its goal
is to separate four source signals basing on
four different linear combinations of them.
Example is contained in
svd_ica_basics.NetPrj
project file. Mixed signals that are the input
to the transformation are presented in the
images below. Also the histograms of each of
these combinations are shown - both, the signals
and theirs histograms look similar, like a noise
with gaussian distribution. ICA requires that
input data has to be centered and normalized.
This condition is met (not precisely, but enough
for the computations in this case...) - values
of
m
and
s
are shown in the legend in the image with
histograms.
text
--top--
--up--
ICA - part II
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