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Mathematical Best Fit Algorithm for Energy Distributions in VO Context
Andrea LARUELO

2006 Trainee Report Meeting Andrea LARUELO

European Space Astronomy Centre (ESAC) Villafranca del Castillo, MADRID (SPAIN)


Project Setup

Name
Mathematical fitting algorithms for Energy Distributions in VO context

Aim
To implement an algorithm that looks for the theoretical model that best fits a given Spectral Energy Distribution (SED)

Tutors
Pedro Osuna (ESA) Jesus Salgado (INSA)

2006 Trainee Report Meeting Andrea LARUELO

European Space Astronomy Centre (ESAC) Villafranca del Castillo, MADRID (SPAIN)


Virtual Observatory
Its aim is to offer seamless access to astronomical data worldwide Astronomers Multiple data centers Analysis and Visualization tools

Standard framework

Delivering and publishing services

2006 Trainee Report Meeting Andrea LARUELO

European Space Astronomy Centre (ESAC) Villafranca del Castillo, MADRID (SPAIN)


Virtual Observatory (cont.)
In order to fulfill this aim the International Virtual Observatory Alliance (IVOA) was created in June 2002 · Data -management -analysis -distribution -interoperability

· Data Access Layer (DAL) -working group within IVOA -standards for remote observational data access

Simple Spectral Access Protocol (SSAP)

To define a uniform interface to spectral data

2006 Trainee Report Meeting Andrea LARUELO

European Space Astronomy Centre (ESAC) Villafranca del Castillo, MADRID (SPAIN)


Virtual Observatory (cont.)
· Theory Group -interest group within the IVOA -formed en January 2004 -define requirements needed for a full interoperability between observations and models

Theoretical Spectrum Access Protocol (TSAP)

Incorporate theoretical models to the VO

2006 Trainee Report Meeting Andrea LARUELO

European Space Astronomy Centre (ESAC) Villafranca del Castillo, MADRID (SPAIN)


Project environment Virtual Observatory Alliance Data Access Layer Theory Group

Simple Spectra Access Protocol

Theoretical Spectra Access Protocol VOSpec

2006 Trainee Report Meeting Andrea LARUELO

European Space Astronomy Centre (ESAC) Villafranca del Castillo, MADRID (SPAIN)


VOSpec
Tool for handling · VO compliant spectra through SSAP · Theoretical models through TSAP Handling spectra with VOSpec: · Display and superimpose spectra · Automatic unit conversion through dimensional analysis · Multi-wavelength analysis · Polynomial/Black Body/Gaussian fitting

2006 Trainee Report Meeting Andrea LARUELO

European Space Astronomy Centre (ESAC) Villafranca del Castillo, MADRID (SPAIN)


Why Best Fit?
Currently to find the model that best fits a given SED: - Choose parameters manually - Inspect result visually ("Chiby-eye") - Modify parameters manually - Reinspect visually - Loop..... New approach

Best Fit finds the model

automatically
2006 Trainee Report Meeting Andrea LARUELO European Space Astronomy Centre (ESAC) Villafranca del Castillo, MADRID (SPAIN)


Best Fit brief description
It is going to be integrated as a new fitting utility in VOSpec

Java language

Aim

Find the theoretical model that best fits the SED

2006 Trainee Report Meeting Andrea LARUELO

European Space Astronomy Centre (ESAC) Villafranca del Castillo, MADRID (SPAIN)


Best Fit brief description (cont.)
To fulfill this aim: ·Chi-square function measures the agreement between real and theoretical spectrum ·The best fit spectrum is the one that achieves a minimum in the Chi-square function Different ·Problem in minimization sampling in many dimensions
real theoretical

=
2


i =1

N

(

yi - y ( xi ; p1



,..., p M

)2

i

)

SED CUBIC SPLINE interpolation Levenverg-Marquardt method

2006 Trainee Report Meeting Andrea LARUELO

European Space Astronomy Centre (ESAC) Villafranca del Castillo, MADRID (SPAIN)


Levenberg-Marquardt method
Iterative method Standard technique for nonlinear least-square problems
Steepest descent method

Modified for discrete case

Combination of
Gauss-Newton method

Steepest descent method ·Initial estimate ·Successive estimate ·Iterations stop when

x

0

x
2 (x i )

i +1

= x i - c 2 (x i )

is sufficiently small

·Hard to determine the step size ·Slow, but guaranteed convergence
2006 Trainee Report Meeting Andrea LARUELO European Space Astronomy Centre (ESAC) Villafranca del Castillo, MADRID (SPAIN)


Levenberg-Marquardt method (cont.)
Gauss-Newton method ·Taylor expansion

1 (x + x) = (x) + (x) x + xT H (x) x 2
2 2 2
Iterative equation

·Set the derivative of the previous expression equal to 0 ·Quadratic convergence ·If the initial value is too far from the minimum convergence can fail

x

i +1

= x i - H -1 2 (x i )

2006 Trainee Report Meeting Andrea LARUELO

European Space Astronomy Centre (ESAC) Villafranca del Castillo, MADRID (SPAIN)


Levenberg-Marquardt method (cont.)
Steepest descent and Gauss-Newton methods are complementary in the advantages they provide

Levenberg Marquardt Method

x

i +1

= x i - c 2 (x i )

x

i +1

= x i - H -1 2 (x i )

x

i +1

= x i - ( H + I ) ( x i )
2

-1

2006 Trainee Report Meeting Andrea LARUELO

European Space Astronomy Centre (ESAC) Villafranca del Castillo, MADRID (SPAIN)


Levenberg-Marquardt method (cont.)
· Far from the minimum ­ Linear approximation with steepest descent method to the closest minimum ­ Adds a measure of the step in the steepest descent method

Close to the minimum Gauss-Newton convergence

· L-M method works well in practice

2006 Trainee Report Meeting Andrea LARUELO

European Space Astronomy Centre (ESAC) Villafranca del Castillo, MADRID (SPAIN)


Best Fit diagram
Real spectrum from SSAP server

Given
TSAP server spectra

SSAP VOSpec
Best Fit model model + spectra model

TSAP

Spline Best Fit
Parameter adjustment

L-M
2006 Trainee Report Meeting Andrea LARUELO European Space Astronomy Centre (ESAC) Villafranca del Castillo, MADRID (SPAIN)


Conclusions
IVOA standards are nearly mature enough to do real science BF algorithm implementation using TSAP protocol increases VO functionality Nonlinear least-square methods (LM) are very appropriate for the implementation of the algorithm Very easy integration with ESA VOSpec tool

2006 Trainee Report Meeting Andrea LARUELO

European Space Astronomy Centre (ESAC) Villafranca del Castillo, MADRID (SPAIN)