234
9 Least-Squares Method
the rotational constants. Molecular intensities, sM(s), are measured by gas-phase
electron diffraction, and they are nonlinear functions of the interatomic distances.
Finally, ab initio calculations directly give the values of the internal coordinates.
From the statistical point of view, the application of the least-squares method
requires certain assumptions (Gauss–Markov conditions). In particular, the errors
are assumed to have zero mean and to be randomly distributed. Actually, in the
special case of a structure determination, these assumptions are rarely fulfilled.
The problem is mainly due to the rovibrational correction which is either neglected
(effective structure) or approximately taken into account (mass-dependent structures,
etc.), see Chap. 6. In rotational spectroscopy, even when the rovibrational correction is determined experimentally, systematic errors remain due to the approximate
treatment of anharmonic resonances. This caveat is also valid for semiexperimental
method (see Sect. 6.7) because the computed rovibrational corrections are affected
by non-negligible systematic errors, see Sect. 6.7 and Vogt et al. (2011 and 2014).
Estimation of systematic errors is also important in gas-phase electron diffraction,
see Sect. 7.11. Another difficulty is that reliable results are obtained when the number
of data is much larger than the number of parameters. Practically, this is rarely the
case. Nevertheless, the least-squares method is considered to be the best method to
determine a structure, but the derived standard deviations of the parameters are not
always a reliable indicator of their accuracy.
The second Sect. 9.2 presents the general method of nonlinear least squares. In
Sect. 9.3, the case of data of different precision (weighting) is discussed. Section 9.4
is dedicated to the description of the different diagnostics to assess the quality of
a least-squares fit. Section 9.5 describes the iteratively reweighted least-squares
method, which partly avoids the difficulty to know in advance the uncertainty of
the data. Section 9.6 studies the effect of fixed parameters and Sect. 9.7 proposes a
powerful solution: the mixed regression, which improves the accuracy of the parameters and improves the conditioning. After an introduction to the correlated leastsquares method, Sect. 9.8, an alternative method to the mixed regression is presented
in Sect. 9.9: the merged fit. Section 9.10 is devoted to the important problem of
systematic errors. Finally, the accuracy of the parameters is discussed in Sect. 9.11.
9.2 Nonlinear Least-Squares Method
9.2.1 Principle of the Method
The input data are generally of different accuracy but we will assume provisionally
that all data have the same uncertainty, and in the Sect. 9.3, we will describe the
weighted least-squares method that permits to take into account the different uncertainties. We will assume that the experimental data (moments of inertia, …) are a
non-linear function of parameters (internal coordinates) β.
9 Least-Squares Method
the rotational constants. Molecular intensities, sM(s), are measured by gas-phase
electron diffraction, and they are nonlinear functions of the interatomic distances.
Finally, ab initio calculations directly give the values of the internal coordinates.
From the statistical point of view, the application of the least-squares method
requires certain assumptions (Gauss–Markov conditions). In particular, the errors
are assumed to have zero mean and to be randomly distributed. Actually, in the
special case of a structure determination, these assumptions are rarely fulfilled.
The problem is mainly due to the rovibrational correction which is either neglected
(effective structure) or approximately taken into account (mass-dependent structures,
etc.), see Chap. 6. In rotational spectroscopy, even when the rovibrational correction is determined experimentally, systematic errors remain due to the approximate
treatment of anharmonic resonances. This caveat is also valid for semiexperimental
method (see Sect. 6.7) because the computed rovibrational corrections are affected
by non-negligible systematic errors, see Sect. 6.7 and Vogt et al. (2011 and 2014).
Estimation of systematic errors is also important in gas-phase electron diffraction,
see Sect. 7.11. Another difficulty is that reliable results are obtained when the number
of data is much larger than the number of parameters. Practically, this is rarely the
case. Nevertheless, the least-squares method is considered to be the best method to
determine a structure, but the derived standard deviations of the parameters are not
always a reliable indicator of their accuracy.
The second Sect. 9.2 presents the general method of nonlinear least squares. In
Sect. 9.3, the case of data of different precision (weighting) is discussed. Section 9.4
is dedicated to the description of the different diagnostics to assess the quality of
a least-squares fit. Section 9.5 describes the iteratively reweighted least-squares
method, which partly avoids the difficulty to know in advance the uncertainty of
the data. Section 9.6 studies the effect of fixed parameters and Sect. 9.7 proposes a
powerful solution: the mixed regression, which improves the accuracy of the parameters and improves the conditioning. After an introduction to the correlated leastsquares method, Sect. 9.8, an alternative method to the mixed regression is presented
in Sect. 9.9: the merged fit. Section 9.10 is devoted to the important problem of
systematic errors. Finally, the accuracy of the parameters is discussed in Sect. 9.11.
9.2 Nonlinear Least-Squares Method
9.2.1 Principle of the Method
The input data are generally of different accuracy but we will assume provisionally
that all data have the same uncertainty, and in the Sect. 9.3, we will describe the
weighted least-squares method that permits to take into account the different uncertainties. We will assume that the experimental data (moments of inertia, …) are a
non-linear function of parameters (internal coordinates) β.