Independent Component Analysis For Track Classification

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Independent Component Analysis For Track Classification

• Seeding for Kalman Filter

•High Level Trigger

•Tracklets After Hough Transformation

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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)

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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

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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

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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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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 ?

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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

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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.