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Independent Component Analysis For Track Classification. Seeding for Kalman Filter High Level Trigger Tracklets After Hough Transformation. Outline of the presentation. What is ICA Results (TPC as a test case) Why ICA has worked ? a. Unsupervised Linear Learning - PowerPoint PPT Presentation
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A K Mohanty 1
Independent Component Analysis For Track Classification
• Seeding for Kalman Filter
•High Level Trigger
•Tracklets After Hough Transformation
A K Mohanty 2
Outline of the presentation
• What is ICA
• Results (TPC as a test case)
• Why ICA has worked ?
a. Unsupervised Linear Learning
b. Similarity with Neural net
(both supervised and unsupervised)
A K Mohanty 3
Let me define the problem
m321 x........x, x,x
N
m
• m---Measurements• N----No. of tracksWe have to decide N good track out of Nm combinations
m321 ,........ss,s,s
S=WX
Find W which is a matrix of m rows and m columns
If si are independent, true tracks have certain characteristic which is not found for ghost tracks
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Definition of Independence
Consider any two random variables y1 and y2. If independent p(y1,y2)=p1(y1)p2(y2) This is true for any n number of variables. This would imply that the independent variables should satisfy
E{f1(y1)f2(y2)…}=E{f1(y1)}E{f2(y2)}
Weaker definition of independence is uncorrelated ness. Two variables are uncorrelated if their covariance zero
E{y1y2}-E{y1}E{y2}=0
A fundamental restriction is independent component must be non Gaussian for ICA to be possible
A K Mohanty 5
How do we achieve Independence ?
H(y)-)H(y)y.....y,I(y im21 m
Define Mutual Information I which is related to the differential Entropy H
Entropy is the basic concept of Information theory. Gaussian variables has the largest entropy among all random variables of equal variance. Look for a transformation which deviates from Gaussianity .
K=E{y4}-3(E{y2})2
. Hyvarinen A and E. Oja, Neural Networks, 13, 411, 2000
A K Mohanty 6
Steps Involved:
1. Centering (Subs tract the mean so as to make X as zero mean variable)
2. Whitening (Transform the observed vector X to Y=AX where Y is white. Its
component are uncorrelated with unity variance.) The above two steps corresponds to the Principal Component
Transformation where A is the matrix that diagonalises the covariance matrix of X.
3. Choose an initial random weight vector W.
4. Let W+=E{Y g(WTY)}-E{g’(WTY)}W5. Let W=W+/||W+||6. If not converged go back to 4
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Projection of fast points on X-Y planeOnly high PT tracks are being considered to start with. Only 9 rows of outer sectors are taken.
X-Y Distribution
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Conformal MappingCircle Straight line
To reduce the number of combinatorics
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Tracklet I
Tracket II
Tracklet III
Global
Generalized Distance after PCA transformation
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Global Tracking after PCA
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In parameter space
At this stage variables are only uncorrelated, not independent. They can be made independent by maximizing the entropy
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Independent
Uncorrelated
A=wT W W is a matrix and w is a vector
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A K Mohanty 14
PCA TransformationICA transformation
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True Tracks
False Tracks
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Input Layer
jx iji
i i d
Hidden layer
Output Layer
• Principal Component Transformation (variables become un-correlated)• Entropy Maximization (variables become independent)
Linear Neural NetUnsupervised Learning
Why ICA has worked ?
A K Mohanty 17
Hidden Layer
Output Layer; 1 if true 0 if false
Non Linear Neural Network (Supervised learning)
Input Layer
•At each node, use a non linear sigmoid function•Adjust the weight matrix so that the cost function is minimized
Nxj /}t-){g(O 2iij
A K Mohanty 18
Independent Inputs
Neural net learns faster when the inputs are mutually independent. This is a basic and important requirement for any multilayer neural net.
Original Inputs
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Out put of neural net during training
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False True
Classification using supervised neural net
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Conclusions:
a. ICA has better discriminatory features which can extract good tracks either eliminating or minimizing the false combinatorics depending on the multiplicity of the events.
b. ICA which learns in a unsupervised way can also be used as a preprocessor for more advanced non-linear neural nets to improve the performance.
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