230
5 Formulation and Implementation of the Gamow Shell Model
One will now implicitly assume that ¯
C is calculated in p 0 . The covariance
between two observables A and B follows as:
ΔA ΔB
N p
α,β=1
∂A
∂p α
p 0
¯
C αβ
∂B
∂p β
p 0
.
(5.81)
In particular, for A = B, Eq. (5.81) gives the statistical uncertainty of the observable
A:
ΔA =
ΔA 2 .
(5.82)
and the dimensionless correlation coefficient [49] is defined as:
c AB =
ΔA ΔB
ΔA ΔB
.
(5.83)
Applying Eqs. (5.81) and (5.83) to the model parameters, their statistical uncertainties reduce to Δp α =
¯
C αα , and the correlation coefficients between two
parameters p α and p β are related to the covariance matrix elements by:
c αβ =
¯
C αβ
¯
C αα ¯
C ββ
.
(5.84)
A demonstration of the validity of the fundamental equations used in linear
regression is done in Exercise VIII. In principle, Eqs. (5.81) and (5.83) are valid
for the model parameters and the observables to be predicted by the model.
However, these do not hold for the observables defining parameters during the
fitting procedures, as those are already well constrained around the minimum.
The uncertainties on the adjusted observables can only be assessed through a full
statistical analysis, such as the Bayesian inference.
Exercise VIII
One will demonstrate the fundamental equations used in linear regression to
calculate statistical errors.
A. Derive Eq. (5.80) directly from the assumption that the covariance matrix of
differences
¯
O
i
=
¯
O i (p) − ¯
O
exp
i
δ ¯
O i
,