Principle Components & Neural Networks How I finished second in Mapping Dark Matter Challenge Sergey Yurgenson, Harvard University Pasadena, 2011.

Principle Components & Neural Networks

How I finished second in Mapping Dark Matter Challenge

Sergey Yurgenson, Harvard UniversityPasadena, 2011

To measure the ellipticity of 60,000 simulated galaxies

Kitching, 2011

Scientific view.

e1=-0.13889e2=0.090147

Training set 40,000 training examples

Test set 60,000 examples

e1= ?e2= ?

g: P -> e•Regression function g does not need to be justified in any scientific way!

•Supervised learning is used to find g

Data mining view

P Pe e

Neural Network

=> =>e1=-0.13889e2=0.090147

RMSE=0.01779

Too many inputs parameters.Many parameters are nothing more than noise.Slow trainingResult is not very good

Reduce number of parametersMake parameters “more meaningful”

Matlab

Principle components to reduce number of input parameters

Neural Network with PC as inputs : RMSE~0.0155

Implicit use of additional information about data set:2D matrixes are images of objectsObjects have meaningful center.

Calculate center of mass with threshold.Center pictures using spline interpolation.Recalculate principle componentsFine dune center position using amplitude of antisymmetrical components

Original Centered

Principle Components after center recalculation

Principle components - stars

Components # 2 and # 3

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

Color – 2theta

Color – (a-b)/(a+b)

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

•Neural Network:38 (galaxies PC) + 8 (stars PC) inputs2 Hidden Layers -12 neurons (linear transfer function) and 8 neurons(sigmoid transfer function)2 outputs – e1 and e2 as targets80% 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 ~0.01505-0.01509private 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

Training RMSE Test RMSE

Original 0.01499 0.01518

7 bit resolution 0.01503 0.01522

6 bit resolution 0.01513 0.01532

5 bit resolution 0.01551 0.01574

4 bit resolution 0.01696 0.01718

Pix size 2 0.01546 0.01571

+ 0.5 noise 0.01684 0.01706

+1.0 noise 0.02120 0.02152

+1.5 noise 0.02873 0.02916

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?