240
9 Least-Squares Method
9.3.2 Practical Solution
When the observations have different uncertainties, their variance may be written
Var(y) = Var(ε) = σ
2 W
−1
(9.19)
where W is the (generally) diagonal weight matrix.
The substitutions y → y
= W
1/2 y and J → J
= W
1/2 J allows us to solve the
system of equations by the ordinary least-squares method (it is enough to left-multiply
(9.4) by W
1/2 )
ˆ
β =
J
T WJ
−1 J
T Wr
(9.20)
The variance of the fit is
s
2
= ˆ
σ
2
=
y − J ˆ
β
T
W
y − J ˆ
β
/(n − p) =
n
i=1 W i r
2
i
n − p
(9.21)
The variance-covariance matrix of the parameters is
( ˆ
β) = ˆ
σ
2
(J
T WJ)
−1
(9.22)
9.4 Diagnostics
The least-squares method is known to give good results when the number n of data
is much larger than the number p of parameters to determine. This is rarely true in
the case of a structure determination where the number of degrees of freedom n – p
is usually small. For this reason, it is important to operate with caution. In particular,
it often happens that some parameters are very sensitive to small perturbations in
the data. Several diagnostics are available to detect this problem and to identify the
parameters affected. A thorough discussion of the conditioning diagnostics may be
found in Belsley (1991). The outlier detection is described in Rousseeuw and Leroy
(1987).
9.4.1 Condition Number
One of the most useful diagnostics is the condition number.
The “scaled condition indexes” of the scaled matrix J are defined by:
9 Least-Squares Method
9.3.2 Practical Solution
When the observations have different uncertainties, their variance may be written
Var(y) = Var(ε) = σ
2 W
−1
(9.19)
where W is the (generally) diagonal weight matrix.
The substitutions y → y
= W
1/2 y and J → J
= W
1/2 J allows us to solve the
system of equations by the ordinary least-squares method (it is enough to left-multiply
(9.4) by W
1/2 )
ˆ
β =
J
T WJ
−1 J
T Wr
(9.20)
The variance of the fit is
s
2
= ˆ
σ
2
=
y − J ˆ
β
T
W
y − J ˆ
β
/(n − p) =
n
i=1 W i r
2
i
n − p
(9.21)
The variance-covariance matrix of the parameters is
( ˆ
β) = ˆ
σ
2
(J
T WJ)
−1
(9.22)
9.4 Diagnostics
The least-squares method is known to give good results when the number n of data
is much larger than the number p of parameters to determine. This is rarely true in
the case of a structure determination where the number of degrees of freedom n – p
is usually small. For this reason, it is important to operate with caution. In particular,
it often happens that some parameters are very sensitive to small perturbations in
the data. Several diagnostics are available to detect this problem and to identify the
parameters affected. A thorough discussion of the conditioning diagnostics may be
found in Belsley (1991). The outlier detection is described in Rousseeuw and Leroy
(1987).
9.4.1 Condition Number
One of the most useful diagnostics is the condition number.
The “scaled condition indexes” of the scaled matrix J are defined by: