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An automated cross calibration fitting tool- 2
The Basics of Spectral Fitting
Elena Gonzсlez Alonso (Complutense University) Tutors: M.Stuhlinger, M.Kirsch


The project
Develop a tool which enables to fit the data coming from the EPIC cameras and spectrometers, on board XMM-Newton, in an automated way.

Let's pay attention to the fit concept...


...we will discover: Why
the fit concept is needed we

Which mathematical tool
use to do the fitting to know whether we have performed a good fitting or not.

How


The CCDs in the EPIC cameras obtain photon counts (C) within the specific instruments channels (I). The observed spectrum, C(I), is related to the actual one, f(E), as follows:


C (I ) =


0

f ( E ) R ( I , E ) dE

R(I,E) = instrumental response (instruments on board satellites use to suffer from some technical behaviour!)


* The goal: f(E) = actual spectrum * The data: C(I) = observed spectrum, R(I,E) = instrumental response * The relation:
C(I ) =



0

f ( E ) R( I , E )dE

The problem!

It is impossible to invert this equation as these invertions tend to be non-unique and unstable to small changes in C(I) the fit concept is needed.


The solution
1. Choose a model spectrum, described in terms of n parameters, and calculate the predicted count spectrum
f (E, p , p2,..., pn) 1 C (I) = f (E, p , p2,..., pn)R(I,E)dE m 1
0


З

PKS0558-504 observed spectrum C(I) from EPIC cameras, RGS1 and RGS2.

З

PKS0558-504 model spectrum Cm(I): phabs * (power + power)

P1 = nH

P1 = phoIndex P2 = norm


2. Compare the predicted count spectrum to the observed one, to decide if the model spectrum fits the observed data The model parameters are varied to find the best fitting.

(CI ) - Cm (I )) ( = 2 (I) N
2

2


PKS0558-504 simultaneous fitting


The Chi-Squared is the most common statistic in use to fit a function f to data {x, y} :

=
2


N

( yi - f ( xi ))

2



2 i

N = number of data points

i2 = variance (related to the measurement error for yi )
y = independet variable x = dependet variable f = assumed relationship between x and y yi = observed mean f(x i ) = predicted mean


Statistical concepts relevant to data fitting
З The parent distribution has a mean, variance and form that are unknown. By choosing a model we are defining these values: parent distribution = model distribution З The fitting function f describes the assumed functional relationship between x and y. So, f predicts the mean for each data point.


З The difference between the predicted mean and the observed mean is called deviation:

deviation = f ( xi ) - yi = yi (positive value)


The Chi-Squared has been defined in general, so, if we suppose:
N = number of channels ( I ) = error for channel I C(I) = observed spectrum (observed counts) Cm ( I ) = model spectrum (model counts)


The Chi-Squared turns to:

=
2


N

(C ( I ) - C m ( I ) ) 2 (I )

2


The Chi-Squared characterizes the dispersion of the observed frequencies from the expected frequencies, as its components are:
C ( I ) - Cm ( I ) = deviation = spread of the observations

(I) = good measure of the expected spread.

So, for a GOOD MODEL: C ( I ) - Cm ( I ) ( I ) (C ( I ) - Cm ( I )) 2 1 2 (I )

2 =


N

(C ( I ) - Cm ( I )) 2 N 2 (I )


In fact, the Chi-Squared expected value is:

= = N -n = degrees of freedom
2

N = number of channels n = number of model parameters


The general rule to be sure of the "Goodness-of-fit" of the model is:

=1
2
It is called the Reduced ChiSquared


Possible situations for the Reduced Chi-Squared:

З Case 1: >1 2 З Case 2: <1
2


Case 1: >1 2 > = N - n
2

C(I)-Cm ( I (I ) C(I)-Cm (I )

) > 1 channel (I ) > 1 channel

2

There are two possibilities again:


(a) If (I) is "too small" C(I)-Cm ( I ) is "too large" (I ) If the model is "perfect" C(I)-Cm ( I ) is "too small" 2 C(I)-Cm ( I ) 2 = 1 channel (I ) 2 ЕЕcontradiction!! 1 ( I ) UNDERESTIMATED!!!


(b) If C(I)-Cm ( I ) is "too large" C(I)-Cm ( I ) is "too large" (I ) If 2 C = 2 ( I ) is (I)-Cm (I ) 1 also "too large" 2 (I ) 1 channel ЕЕcontradiction!!)

Cm ( I ) INCORRECT!!!


Case 2: <1 2 < = N - n
2

C(I)-Cm ( I ) < 1 channel (I ) C(I)-Cm ( I ) < ( I ) channel

( I ) OVERESTIMATED!!!


Summary: Testing the "Goodness-of-fit"
2 * 1 GOOD FITTING!! 2 * > 1 { (I) UNDERESTIMATED or INCORRECT MODEL
BAD FITTING!!

}

2 * < 1

{

(I) OVERESTIMATED} BAD FITTING!!


PKS0558-504 simultaneous fitting


Testing the "Goodness-of-fit" for PKS0558-304
Best model: phabs * (power + power)

2 = 1.02668 1
Best parameters: * Phabs nH: 4.382314 atoms/cm2 * Powerlaw 1 phoIndex: 2.79705 ( dimensionless ) norm: 6.897345 10-3 photons / keV / cm2 / s at keV * Powerlaw 2 phoIndex: 2.79705 ( dimensionless ) norm: 6.627209 10-4 photons / keV / cm2 / s at keV