Linear least-squares inversion

Figure B2.4.8. Relaxation of two of tlie exchanging methyl groups in the TEMPO derivative in figure B2.4.7. The dotted lines show the relaxation of the two methyl signals after a non-selective inversion pulse (a typical experunent). The heavy solid line shows the recovery after the selective inversion of one of the methyl signals. The inverted signal (circles) recovers more quickly, under the combined influence of relaxation and exchange with the non-inverted peak. The signal that was not inverted (squares) shows a characteristic transient. The lines represent a non-linear least-squares fit to the data.

To extract the agglomeration kernels from PSD data, the inverse problem mentioned above has to be solved. The population balance is therefore solved for different values of the agglomeration kernel, the results are compared with the experimental distributions and the sums of non-linear least squares are calculated. The calculated distribution with the minimum sum of least squares fits the experimental distribution best. 

Figure 3. Gel permeation data for linear randomly coiled polypeptides on various agarose resins, plotted according to the method of Ackers (9). M0 555 is plotted vs. the inverse error function complement of Kd (erfc 1 Kd). Lines drawn through the data points represent best fits obtained from linear least-squares analysis of the data. Numerical designation of each curve represents the percent agarose composition for the resin used. Filled triangles on the curve for the 6% resin, and the filled squares on the curve for the 10% resin are points determined using fluorescent proteins. Data for the labeled polypeptides were not included in the least-squares analysis.

These maxima in the Fourier transform data, which correspond to the different chromium coordination shells, were isolated using a filter window function. The inverse transform of each peak was generated and fitted using a non-linear least squares program. The amplitude and phase functions were obtained from the theoretical curves reported by Teo and Lee (2 ). The parameters which were refined included a scale factor, the Debye-Waller factor, the interatomic distance, and the threshold energy difference. This process led to refined distances of 1.97(2) and 2.73(2) A which were attributed to Cr-0 and Cr-Cr distances, respectively. Our inability to resolve second nearest neighbor Cr-Cr distances may be a consequence of the limited domain size of the pillars. 

Yano, Y. Yamaoka, K. Tanaka, H. A non-linear least squares program, MULTI(FILT), based on fast inverse laplace transform for microcomputers. Chem. Pharm. Bull. 1989, 37, 1035-1038. 

Fig. 2. Plots of the logarithm of resistivity versus inverse temperature for a film with doping in the range 1 < x s 3. These data correspond to the arrows in Fig. 1. The lines are linear least-squares fits to the data and are extrapolated to T " =0, to show the activated behavior of the resistivity. The doping prt esses from top to bottom with the lowest curve corresponding to x 3.

Finally, for this section we note that the valence interactions in Eq. [1] are either linear with respect to the force constants or can be made linear. For example, the harmonic approximation for the bond stretch, 0.5 (b - boV, is linear with respect to the force constant If a Morse function is chosen, then it is possible to linearize it by a Taylor expansion, etc. Even the dependence on the reference value bg can be transformed such that the force field has a linear term k, b - bo), where bo is predetermined and fixed, and is the parameter to be determined. The dependence of the energy function on the latter is linear. [After ko has been determined the bilinear form in b - bo) can be rearranged such that bo is modified and the term linear in b - bo) disappears.] Consequently, the fit of the force constants to the ab initio data can be transformed into a linear least-squares problem with respect to these parameters, and such a problem can be solved with one matrix inversion. This is to be distinguished from parameter optimizations with respect to experimental data such as frequencies that are, of course, complicated functions of the whole set of force constants and the molecular geometry. The linearity of the least-squares problem with respect to the ab initio data is a reflection of the point discussed in the previous section, which noted that the ab initio data are related to the functional form of empirical force fields more directly than the experimental data. A related advantage in this respect is that, when fitting the ab initio Hessian matrix and determining in this way the molecular normal modes and frequencies, one does not compare anharmonic and harmonic frequencies, as is usually done with respect to experimental results. 

For the iterative linearized localization method that was presented above localization errors are described by the symmetric covariance matrix C. For the least squares inversion method, C ean be easily calculated from matrix G as (Flirm 1965)... 

The procedure described for calibration of K and a is laborious because of the required fractionation process. The two constants are derived as described from the intercept and slope of a linear least squares fit to [t ]-A/ values for a series of fractionated polymers. Experimentally, K and a are found to be inversely correlated. If different laboratories determine these MHS constants for the same polymer-solvent combination, the data set which yields the higher K value will produce the lower a. Thus, My from Eq. (3-44) is often essentially the same for different K and a values provided the molecular weight ranges of the samples used in the two calibration processes overlap. 

For temperature fit, a linear least squares fit of logarithmic radiant exposures aud inverse temperatures was used, implying an Anhenius approach. Separate fits for the respective N 30 content resulted in overlapping activation energies, within their uncertainty ranges. Therefore, a common activation energy was calculated, by... 

S Wold, A Ruhe, H Wold, WJ Dunn III. The collmearity problem m linear regression. The partial least squares (PLS) approach to generalized inverses. SIAM I Sci Stat Comput 5 735-743, 1984. 

Inverse least-squares (ILS), sometimes known as P-matrix calibration, is so called because, originally, it involved the application of multiple linear regression (MLR) to the inverse expression of the Beer-Lam be rt Law of spectroscopy ... 

Multiple Linear Regression (MLR), Classical Least-Squares (CLS, K-matrix), Inverse Least-Squares (ILS, P-matrix)... 

To summarize, Wiener inverse-filter is the linear filter which insures that the result is as close as possible, on average and in the least squares sense, to the true object brightness distribution. 

Wold, S Ruhe, A., Wold, H Dunn, W. J. I. SIAM J. Sci. Stat. Comput. 5, 1984, 735-743. The collinearity problem in linear regression. The partial least squares approach to generalized inverses. 

Spin-lattice relaxation times were measured by the fast inversion-recovery method (24) with subsequent data analysis by a non-linear three parameter least squares fitting routine. (25) Nuclear Overhauser enhancement factors were measured using a gated decoupling technique with the period between the end of the data acquisition and the next 90° pulse equal to eibout four times the value. Most of the data used a delay of eibout ten times the Ti value. (26)... 

---