Modified Least Squares Methods

At the beginning of Section 11.3, the calculation of constant distributions from experimental data is an ill-posed problem that, if approached by classical solution schemes, leads to serious numerical problems and instabilities. To circumvent these difficulties, least squares minimization procedures are modified by the introduction of constrains, regularization, or both of them, to provide a substantial stabilization of the resulting distributions these approaches have been widely applied (Provencher 1982a Cernik, Borkovec, and Westall 1995 Borkovec et al. 1996 Rusch et al. 1997 Bersillon et al. 2001). In general terms, all modifications are applied over the common least squares problem, where the function to be minimized is the variance  

58 is minimized with the condition that/(log K) 0 for all K. This helps greatly in eliminating many oscillating solutions. However, in many cases, there exists still variability in the solution, so regularization has to be applied. In a regularization procedure, the objective function to be minimized. Equation 11.58, is modified by a regularizor term, which penalizes the unwanted solutions  

FIGURE 11.6 Sorption isotherms of dodecylpiridinium on a soil material (EPA-12). (a) Points experimental data line fitted isotherms (almost indistinguishable) with discrete and continuous affinity spectra (b) discrete affinity spectrum, found using regularization for a small number of sites (c) continuous affinity spectrum, after regularizing for smoothness note that the smaller peaks are magnified by a factor of 30. (Reprinted with permission from Cernik, M. et al.. Environ. Sci. TechnoL, 29,2,413-425. Copyright 1995 American Chemical Society.) 

FIGURE 11.7 (a) Affinity distributions obtained at different ionic strengths from experimental data for purified peat humic acid (PPHA) (b) deconvolution of the distribution obtained for PPHA with Gaussian functions. (Data from Milne. C.J. et al.. Environ. Sci. TechnoL, 37, 958-971, 2003 Reprinted from J. Colloid Interface Sci., 336, Orsetti, S. et al., Application of a constrained regularization method to extraction of affinity distributions Proton and metal binding to humic substances, 377-387. Copyright 2009, with permission from Elsevier.) 

The MaxEnt formalism seeks to find the most probable distribution compatible with all known constraints of a given problan it was introdnced by Jaynes (1957, 1982) and since then applied to many different statistical problems (Christakos 1998 Cheeseman and Stntz 2005 Komnitsas and Modis 2006 Vakarin and Badiali 2006 Orton and Lark 2007). Garces, Mas, and Puy (1998) proposed its application to the extraction of affinity spectra as an alternative to regularization methods as follows. The MaxEnt formalism attempts to find the most probable distribution /J of sites, compatible with the set of the M experimental data points obtained, where 

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Select the method of calculation for tray efficiency. Two methods are presented the O Connell method and the two-film method. In the programs accompanying this book, you may select the O Connell method by entering either an F for fractionator or an A for absorbers. In 1946, O Connell [4] published curves on log-log plots showing both absorber and fractionator efficiencies vs. equilibrium-viscosity-density factored equations. Separate curves for absorbers and fractionators were given. Such data have been curve-fit using a modified least-squares method in conjunction with a log scale setup. The fit is found to be reasonably close to the O Connell published curves. 

Errors in calibration doses. A modified least squares method, taking into account that calibration doses also contain uncertainties, can be applied [37]. 

Dunitz and Seiler (1973) have used the equivalence to modify least-squares weighting, such as to emphasize the fit near the density peak positions, in order to obtain parameters less biased by bonding effects. The resulting weights emphasize high-order reflections, similar to the higher-order refinement method, but with a smoothly varying cut-off rather than a sharp sin 6//. limit. 

Magnetic susceptibilities of 10a and 10b were measured on a SQUID suscep-tometer in microcrystalline form. %T-T plots are shown in Fig. 9.5. The data were analyzed in terms of a modified singlet-triplet two-spin model (the Blea-ney-Bowers-type), in which two spins (S = V2) couple antiferromagnetically within a biradical molecule by exchange interaction J. The best-fit parameters obtained by means of a least-squares method were 2J/kB = -2.2 + 0.04 K for 10a and -11.6 + 0.4 K for 10b. Although the interaction (2J/kB = -2.2 K) between the two spins in the open-ring isomer 10a was weak, the spins of 10b showed a remarkable antiferromagnetic interaction (2J/kB = -11.6 K). 

Kim, K.H. and Martin, Y.C. (1991c). Evaluation of Electrostatic and Steric Descriptors for 3D-QSAR the H and CH3 Probes Using Comparative Molecular Field Analysis (CoMFA) and the Modified Partial Least Squares Method. In QSAR Rational Approaches to the Design of Bioactive Compounds (Silipo, C. and Vittoria, A., eds.), Elsevier, Amsterdam (The Netherlands), pp. 151-154. 

All reactions were carried out in a thermostated, magnetically stirred 50 ml autoclave (baffles, two thermocouples, X-shaped stirrer, 1=25 mm, max. 900 rpm). A reservoir and a pressure regulator allowed experiments to be carried out under isobaric conditions, p and T in the autoclave and in the reservoir were recorded every 10 s. Usually, the reactions were stopped after 10 min corresponding to conversions of 10 - 50%. Conversion and e.e. were determined by GLC-analysis of the crude reaction mixture (30 m capillary 3-Dex 100, 75°C). In all experiments with HCd modifier, the R-enantiomer was the major enantiomer. The initial rates of hydrogenation were calculated with a linear least square method from the pressure drop in the reservoir, accounting for the compressibility of hydrogen. In all series, one experiment was carried out without modifier to determine k (see below). 

Figure 3.18. Plot of GC content of genes against the GC levels of DNA components in which they arc located. The numbers indicate genes (see the original article). The line was drawn using the least-square method. The unit slope line corresponds to the coincidence in GC contents of genes and major components in which genes are located. (Modified from Bernardi et ah, 1985b).

Ridge regression is also used extensively to remedy multicollinearity between the X, predictor variables. It does this by modifying the least-squares method of computing the coefficients with the addition of a biasing component. 

Regularization methods, to invert Equation 11.31 numerically by a modified least-squares scheme... 

Quantifieation of the amount of residual VO(pie)2 (shown in Fig. 10) was eon-dueted through subtraction of normalized time-domain ESEEM speetra of VO(pie)2 in liver (Fig. 10a) from VOSO4 in liver (Fig. 10b), yielding the time-domain speetram of the minor species (Fig. 10c), as described by Eq. (1), where Modnorm eoiTesponds to the normalized time-domain intensity (baekground deeay eurve subtracted) extrapolated to t = 0 and corrected by the preexponential faetor obtained from a modified Block-type relaxation equation, for eaeh speeies. The coefficients a and P were estimated by a linear least-squares method [71] ... 

Aviv a least squares method provided by Aviv, actually a modified program of Chang et al. [78C1]... 

We have seen that PLS regression (covariance criterion) forms a compromise between ordinary least squares regression (OLS, correlation criterion) and principal components regression (variance criterion). This has inspired Stone and Brooks [15] to devise a method in such a way that a continuum of models can be generated embracing OLS, PLS and PCR. To this end the PLS covariance criterion, cov(t,y) = s, s. r, is modified into a criterion T = r. (For... 

Hartley, H.O., "The Modified Gauss-Newton Method for the Fitting of Non-Linear Regression Functions by Least Squares", Technometrics, 3(2), 269-280(1961). 

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