Least-squares process

The procedure of Lifson and Warshel leads to so-called consistent force fields (OFF) and operates as follows First a set of reliable experimental data, as many as possible (or feasible), is collected from a large set of molecules which belong to a family of molecules of interest. These data comprise, for instance, vibrational properties (Section 3.3.), structural quantities, thermochemical measurements, and crystal properties (heats of sublimation, lattice constants, lattice vibrations). We restrict our discussion to the first three kinds of experimental observation. All data used for the optimisation process are calculated and the differences between observed and calculated quantities evaluated. Subsequently the sum of the squares of these differences is minimised in an iterative process under variation of the potential constants. The ultimately resulting values for the potential constants are the best possible within the data set and analytical form of the chosen force field. Starting values of the potential constants for the least-squares process can be derived from the same sources as mentioned in connection with trial-and-error procedures. 

The lion s share of the computer-time for the least-squares process has to be provided for forming the Z-matrix. The elements of this matrix are evaluated partly numerically and partly analytically in the calculations of Lifson and Warshel (17). In certain cases, strong parameter correlations may occur. Therefore caution is demanded when inverting the matrix C. Also from investigations other than consistent force-field calculations it is known that such correlations frequently occur among the parameters for the nonbonded interactions (34,35). Another example of force field parameter correlations was encountered by Ermer and Lifson (19) in the course of the calculation of olefin properties. When... 

The IR methods have progressed from hand-drawn baselines and peak height or area for quantitation, to spectral subtraction, to leastsquares methods. Least-squares analysis eliminates the reliance on single peaks for quantitation and the subjectivity of spectral subtraction. However, negative concentration coefficients are a problem with least-squares analysis, since they have no physical meaning. Negative components can be omitted according to some criterion and the least-squares process iterated until only... 

Technique 7. In Technique 7, the x-t data are digitized into 70 discrete points. A linear fit is made to three adjacent x-t points, and the slope is taken as the velocity at the midpoint of the line. Then one x-t end point is dropped, a new one is added on the other end, a new linear fit is made, and the velocity is found. This running linear least squares process is repeated until all 70 x-t points have been used. The u-x data are then extrapolated to zero thickness (x = 0) to find the initial shock velocity UsQ. All other analysis is done as in Technique 1. ... 

The most commonly used PCA algorithm involves sequential determination of each principal component (or each matched pair of score and loading vectors) via an iterative least squares process, followed by subtraction of that component s contribution to the data. Each sequential PC is determined such that it explains the most remaining variance in the X-data. This process continues until the number of PCs (A) equals the number of original variables (M), at which time 100% of the variance in the data is explained. However, data compression does not really occur unless the user chooses a number of PCs that is much lower than the number of original variables (A M). This necessarily involves ignoring a small fraction of the variation in the original X-data which is contained in the PCA model residual matrix E. 

This alternating least squares processing is continued until some convergence criteria is met. We continue until the square root of the sum of all the residuals squared in the relation (Residuals = A - CK) changes by <0.0001. 

Often, H coordinates determined from a difference-Fourier map are difficult to refine in the subsequent least-squares process. In our experience, the success or failure in the refinement can sometimes be critically dependent on the particular choice of variables in the cycles immediately following the introduction of the H positions 18L... 

Averaging is a least-squares process that reduces the effects of noise, if the noise is zero-mean and fairly random [10], and the moving-average filter removes high-frequency noise well. It is less successful at removing low-frequency noise, since these nonzero-mean variations are less likely to be affected by the averaging. It also... 

For a least-squares solution of the system of Eqs. 40 for all s = 2, Ns, we have to identify the components of the vector of observations y, the components of the vector of variables p and the elements of the Jacobian matrix X as shown below (Eqs. 46—48). A left arrow has been used instead of a sign of equation to indicate that, in general, the dimensions of p, X, and y are preliminary and must be reduced before least-squares processing can take place some of the P/,ma = A/j 1 may not be independent because symmetrically equivalent atoms have been substituted. Other coordinates may be kept fixed intentionally (e.g., at zero when an atom is known to lie on a principal plane or axis). The respective component(s) must then be eliminated from the vector of variables. Also, one or more of the observations may have to be dropped in order to comply with the recommendations given for the Chutjian-type treatment of substitutions on a principal plane or axis [44],... 

The local axis on each atom is defined by the program s user (see Figure 2) this flexibility is very interesting for big molecules possessing non-crystallographic local symmetry and/or containing chemically equivalent atoms. These symmetry and chemical constraints permit to reduce the number of the k, Pv, and Plm electron density parameters in the least-square process (see applications in ref. 13). For example, all atoms of a benzene ring may be constrained to have the same density parameters and a local symmetry mm2 can be applied to each atom. 

In the converse application of the explicit relation between F and G matrices and thermodynamic quantities, a set of effective harmonic force constants covering a certain range of temperature is derived from experimental data on a strongly temperature-dependent property obtained in the same temperature range. The least-squares process becomes non-linear... 

The treatment of least-squares fitting given here is superficial and nonrigorous. Albritton, et al., (1976) Marquardt (1963), Wentworth (1965) and Press, et al., (1996) discuss the least-squares process more completely. 

As a principle of the least squares process, a is chosen such that X e is driven to 0. This leaves X y = X X a, which is known as the normal equation. To find a with... 

For simple spectra, the various interaction parameters are evaluated by a direct comparison of the theoretical formulae for the energy levels with their experimental values. Since in most cases the number of parameters is much smaUer than the number of levels, the energy level calculation is based on an iterative diagonalization-least-squares process. This process comprises four peu ts ... 

Thus both the slope and the intercept of the least-squares line can be calculated from simple sums using eqns (6.4) and (6.9). In practice, few people ever calculate regression lines in this way as even quite simple scientific calculators have a least-squares fit built in. However, it is hoped that this brief section has illuminated the principles of the least-squares process and has shown some of what goes on in the black box of regression packages. 

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