238
9 Least-Squares Method
is popular but controversial. In practice, the mixed estimation (Sect. 9.7) is a more
natural and flexible way for correcting for the ill-conditioning.
9.3 Weighted Least Squares Method
9.3.1 Choice of the Weights
When the data are of equal precision, no weighting is required. Furthermore, the
values of the estimated parameters do not depend on the weighting provided that the
number of data is much larger than the number of parameters. Unfortunately, this
last condition is rarely fulfilled in a structure determination and there are many cases
where a weighting is required, in particular when the mixed regression (Sect. 9.7) is
used. Another case is when the data are of different precision. A typical example is
the C 3v molecules: the axial rotational constant A is generally at least one order of
magnitude less accurate than the B rotational constant, see Sect. 4.5.3. If a correct
weighting is not used, statistical diagnostics will indicate that the data are not compatible. As a consequence the results, and particularly their standard deviations, will
not be reliable. The weighting also allows us to take into account the imperfection
of the model or to reduce the influence of some data (Sect. 9.4.2).
Usually, the weight of an observation is taken as the reciprocal square of its
experimental uncertainty. As the ground-state rotational constants are in most cases
extremely accurate, the main contribution to the error of equilibrium rotational
constants comes from the rovibrational correction B vib . The uncertainty may be
defined as
σ
2
= σ
2
rand + σ
2
model
(9.17)
where σ rand is essentially the random error of the rovibrational corrections, and σ model
is the systematic error induced by the assumptions of the model, i.e., mainly the error
due to the approximate knowledge of the rovibrational correction. Usually, σ model is
larger than σ rand by several orders of magnitude. It is known that the rovibrational
correction is affected by a systematic error of 2–5% or more (Vogt et al. 2011) but what
is needed for the weighting is the random part of error. One way to get a first rough
estimate of the weights is to perform a preliminary non-weighted fit. Sometimes, the
iteratively reweighted least-squares (see Sect. 9.5) may be used. Fortunately, when
the molecule has some symmetry, it is possible to use the invariance of the planar
moments of inertia upon substitution.
Example 1 Phenylacetylene (Rudolph et al. 2013). It is a planar molecule of C 2v
symmetry. Rotational constants are available for 39 isotopologues. For the 15 isotopologues, which keep the C 2v symmetry, the equilibrium A rotational constants are not
affected by a substitution on the a-symmetry axis. With the rovibrational correction
9 Least-Squares Method
is popular but controversial. In practice, the mixed estimation (Sect. 9.7) is a more
natural and flexible way for correcting for the ill-conditioning.
9.3 Weighted Least Squares Method
9.3.1 Choice of the Weights
When the data are of equal precision, no weighting is required. Furthermore, the
values of the estimated parameters do not depend on the weighting provided that the
number of data is much larger than the number of parameters. Unfortunately, this
last condition is rarely fulfilled in a structure determination and there are many cases
where a weighting is required, in particular when the mixed regression (Sect. 9.7) is
used. Another case is when the data are of different precision. A typical example is
the C 3v molecules: the axial rotational constant A is generally at least one order of
magnitude less accurate than the B rotational constant, see Sect. 4.5.3. If a correct
weighting is not used, statistical diagnostics will indicate that the data are not compatible. As a consequence the results, and particularly their standard deviations, will
not be reliable. The weighting also allows us to take into account the imperfection
of the model or to reduce the influence of some data (Sect. 9.4.2).
Usually, the weight of an observation is taken as the reciprocal square of its
experimental uncertainty. As the ground-state rotational constants are in most cases
extremely accurate, the main contribution to the error of equilibrium rotational
constants comes from the rovibrational correction B vib . The uncertainty may be
defined as
σ
2
= σ
2
rand + σ
2
model
(9.17)
where σ rand is essentially the random error of the rovibrational corrections, and σ model
is the systematic error induced by the assumptions of the model, i.e., mainly the error
due to the approximate knowledge of the rovibrational correction. Usually, σ model is
larger than σ rand by several orders of magnitude. It is known that the rovibrational
correction is affected by a systematic error of 2–5% or more (Vogt et al. 2011) but what
is needed for the weighting is the random part of error. One way to get a first rough
estimate of the weights is to perform a preliminary non-weighted fit. Sometimes, the
iteratively reweighted least-squares (see Sect. 9.5) may be used. Fortunately, when
the molecule has some symmetry, it is possible to use the invariance of the planar
moments of inertia upon substitution.
Example 1 Phenylacetylene (Rudolph et al. 2013). It is a planar molecule of C 2v
symmetry. Rotational constants are available for 39 isotopologues. For the 15 isotopologues, which keep the C 2v symmetry, the equilibrium A rotational constants are not
affected by a substitution on the a-symmetry axis. With the rovibrational correction