GammaVision
®
v7 (A66-BW) 783620H / 1013
The
σ
i
factor is the fractional uncertainty at the i
th
calibration point (as defined for Eq. 104).
σ
i
is not the absolute uncertainty because the actual curve being fitted is not the efficiencies themselves but the natural log of the curve.
Matrix Inversion
From Eq. 7.20 in Bevington, the n
th
fitting parameter in our Eq. 109 or Eq. 106 can be calculated as:
(114)
where
δ is the inverse matrix of α:
(115)
The
δ matrix is called the error matrix because its diagonal elements are the variances of the fitting parameters, and the off-diagonal elements are the covariances of the fitting parameters (see
Eq. 7.25 in Bevington):
(116)
If i = j, is the variance for the fitting parameter a
i
. If i
… j, the fitting parameter a
i
and a
j
.
is the covariance between
Uncertainty of the Fit
From Eq. 3.13 in Bevington, the polynomial fit uncertainty is given by:
(117)
where y is the polynomial given in Eq. 109 and a
i
is the i
th
fitting parameter. The error matrix
δ
i j
has been used instead of for clarity. Equation 117 does not use the factor 2 before the covariance terms, as is done in Bevington Eq. 3.13. This is due to the double-summation notation used in Eq. 117 and because the error matrix is symmetric (
δ
12
=
δ
21
, ...). The diagonal elements are not double-counted in the above equation. (In Bevington Eq. 3.13, is denoted as .)
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783620H / 1013
From Eq. 109, since , we have:
6. ANALYSIS METHODS
(118)
where for a linear/quadratic fit:
j i
1
1
= i
! 1
= j
! 1 and for a polynomial fit:
j i
1
1
= 2
! i
= 2
! j
Finally, the calibration uncertainty is:
(119)
where
ε is the fitted efficiency at energy E (from Eq. 106), and σ
y
σ
c
is zero, then the “sigma above” (
σ
a
) or “sigma below” (
σ
b
is calculated from Eq. 118. If
) value described in the following paragraphs is used.
If the fit type is neither polynomial nor TCC-polynomial, the calibration uncertainty is calculated as:
(120)
(121)
where
σ
a
and
σ
b
are defined as follows.
If the energy E > E
knee
,
(122)
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