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Principle Components & Neural NetworksPowerPoint Presentation

Principle Components & Neural Networks

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Principle Components & Neural Networks

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Principle Components & Neural Networks

How I finished second inMapping Dark Matter Challenge

Sergey Yurgenson, Harvard University

Pasadena, 2011

Scientific view.

Kitching, 2011

Data mining view

Training set

40,000 training examples

Test set

60,000 examples

P

e

P

e

e1= ?

e2= ?

e1=-0.13889

e2=0.090147

g: P -> e

•Regression function g does not need to be justified in any scientific way!

•Supervised learning is used to find g

Neural Network

e1=-0.13889

e2=0.090147

=>

=>

Matlab

RMSE=0.01779

Too many inputs parameters.

Many parameters are nothing more than noise.

Slow training

Result is not very good

Reduce number of parameters

Make parameters “more meaningful”

Principle components to reduce number of input parameters

Neural Network with PC as inputs : RMSE~0.0155

Calculate center of mass with threshold.

Center pictures using spline interpolation.

Recalculate principle components

Fine dune center position using amplitude of antisymmetrical components

Centered

Original

Implicit use of additional information about data set:

2D matrixes are images of objects

Objects have meaningful center.

Principle Components after center recalculation

Principle components - stars

Components # 2 and # 3

Color – 2theta

Color – (a-b)/(a+b)

e1=[(a-b)/(a+b)]cos(2theta) e2=[(a-b)/(a+b)]sin(2theta)

Linear regression using only components 2,3 => RMSE~0.02

•Neural Network:

38 (galaxies PC) + 8 (stars PC) inputs

2 Hidden Layers -12 neurons (linear transfer function) and 8 neurons(sigmoid transfer function)

2 outputs – e1 and e2 as targets

80% random training subset, 20% validation subset

•Multiple trainings with numerous networks achieving training RMSE<0.015

•Typical test RMSE =0.01517 – 0.0152

•Small score improvement by combining prediction of many networks (simple mean):

Combination of multiple networks, training RMSE ~0.0149

public RMSE ~0.01505-0.01509

private RMSE ~0.01512-0.01516

Benefit of network combination is ~0.00007-0.0001

•Best submission – mean of 35 NN predictions

Training set

Test set

std=0.01499

std=0.01518

Questions:

Method is strongly data depended. How method will perform for more diverse data set and real data ?

Is there a place for this kind of methods in cosmology?