Least mean squares deviations

One of the simplest approaches is to scale by the same factor all the calculated frequencies in order to obtain the least mean-squared deviation between the experimental and the theoretical frequencies [14], This procedure is equivalent to homogeneous scaling of all the elements in Fl m. The advantage of this method is that it uses only one adjustable parameter. However, this is payed out by the necessity of very extensive quantum-mechanical calculations large atomic basis sets, and appropriate account for the electron correlation effects. If simpler theoretical schemes are used the homogeneous scaling may result in improper assignment of the experimental frequencies. 

Convergence of the iterative process to a true minimal value of the least mean square deviation. 

It is a straightforward matter to fit various model profiles to realistic, exact computed profiles, selecting a greater or lesser portion near the line center of the exact profile for a least mean squares fit. In this way, the parameters and the root mean square errors of the fit may be obtained as functions of the peak-to-wing intensity ratio, x = G(0)/G(comax)- As an example, Fig. 5.8 presents the root mean square deviations thus obtained, in units of relative difference in percent, for two standard models, the desymmetrized Lorentzian and the BC shape, Eqs. 3.15 and 5.105, respectively. 

In order to determine the ion pair dissociation constant Kd, of a salt it is necessary therefore to measure X as a function of C and obtain a roughly extrapolated value for X0. Calculation of the variables F(z)/X and f 2 F(z)CX is usually accomplished with a small computer program, and hence a more accurate value for X0 and a first value for Kd obtained from a straight line plot of these functions. It is, however, more convenient to carry out the whole process by computer with iteration accompanied by a least mean square calculation to obtained the most accurate value for X0 and Kd. For solvents of low dielectric constant, and if sufficiently dilute solutions are not examined, Fuoss plots deviate downward at higher concentrations, because of triple ion formation. This can lead to an excessively low estimate for X0 and too high a value for Kd. 

The crystal structures of the E. coli DHFR-methotrexate binary complex (Bolin et al., 1982), of the Lactobacillus casei (DHFR-NADPH-methotrexate ternary complex (Filman et al., 1982), of the human DHFR-folate binary complex (Oefner et al., 1988), and of the mouse (DHFR-NADPH-trimethoprim tertiary complex (Stammers et al., 1987) have been resolved at a resolution of 2 A or better. The crystal structures of the mouse DHFR-NADPH-methotrexate (Stammers et al., 1987) and the avian DHFR—phenyltriazine (Volz et al., 1982) complexes were determined at resolutions of 2.5 and 2.9 A, respectively. Recently, the crystal structure of the E. coli DHFR—NADP + binary and DHFR-NADP+-folate tertiary complexes were resolved at resolutions of 2.4 and 2.5 A, respectively (Bystroff et al., 1990). DHFR is therefore the first dehydrogenase system for which so many structures of different complexes have been resolved. Despite less than 30% homology between the amino acid sequences of the E. coli and the L. casei enzymes, the two backbone structures are similar. When the coordinates of 142 a-carbon atoms (out of 159) of E. coli DHFR are matched to equivalent carbons of the L. casei enzyme, the root-mean-square deviation is only 1.07 A (Bolin et al., 1982). Not only are the three-dimensional structures of DHFRs from different sources similar, but, as we shall see later, the overall kinetic schemes for E. coli (Fierke et al., 1987), L. casei (Andrews et al., 1989), and mouse (Thillet et al., 1990) DHFRs have been determined and are also similar. That the structural properties of DHFRs from different sources are very similar, in spite of the considerable differences in their sequences, suggests that in the absence, so far, of structural information for ADHFR it is possible to assume, at least as a first approximation, that the a-carbon chain of the halophilic enzyme will not deviate considerably from those of the nonhalophilic ones. 

Table 1. Structural parameters for solvation shell of lithium ion in DMF as determined by XD measurements and least-squares fitting procedure average distances r, , root mean square deviations /y, and coordination numbers ny. Solution of LiQ (left) and of LiNCS (right).

Hindman et al. (28, 29, 30) have investigated the effect of D2O on the Np(IV)-Np(VI) reaction and redetermined the acid dependence. The previously determined dependence in H2O (30) could be interpreted in terms of either consecutive or parallel rate-determining reactions (51). We have now used the newer data and a least-squares procedure to compare the two mechanisms. In H2O, consecutive reactions fit the data better than parallel reactions the root-mean-square deviations are 3.63 and 3.81% respectively. In D2O the corresponding deviations are 7.79% and 4.18%. It is concluded that, unlike the analogous U(IV)-Pu( VI) reaction (51), there is no strong evidence for consecutive reactions and a binuclear intermediate. This reaction has been reinvestigated by Rykov and Yakovlev (76), who report higher rate constants under comparable conditions. 

Average of three measurements. Error quoted for this value is the root mean square deviation of the three measurements. Other errors are based on the results of the least-squares fits. 

The results of the study are summarized in Table 7.5, along with a brief account of the features of each pj-parameter model. Each model was fitted by least squares to 283 observations of the functions InA iu, where Niuz is the measured axial flux of species i in the wth event, in g-moles per second per cm of particle cross section. This corresponds to using the same variance for each response function lnA j 2. Lacking replicates, we compare the models according to Eq. (7.5-16) with a variance estimate = 0.128/(283 — 6), the residual mean-square deviation of the observations... 

In applications to held data, the systems of equations (13.11) and (13.12) are usually overdetermiiied because there are more chemical components measured than sources. That is, there are more equations than unknowns. At least two approaches can be used to solve for the source contributions, m/. One can choose a set of chemical components equal in number to the number of unknown sources. The chemical components chosen for this purpose are the ones with minimum uncertainty in the experimentally measured values and which are present in high concentrations in the aerosol. The quality of the fit with the rest of the data is measured by the mean square deviation between the measured and calculated chemical component concentrations. 

Structure comparison methods are a way to compare three-dimensional structures. They are important for at least two reasons. First, they allow for inferring a similarity or distance measure to be used for the construction of structural classifications of proteins. Second, they can be used to assess the success of prediction procedures by measuring the deviation from a given standard-of-truth, usually given via the experimentally determined native protein structure. Formally, the problem of structure superposition is given as two sets of points in 3D space each connected as a linear chain. The objective is to provide a maximum number of point pairs, one from each of the two sets such that an optimal translation and rotation of one of the point sets (structural superposition) minimizes the rms (root mean square deviation) between the matched points. Obviously, there are two contrary criteria to be optimized the rms to be minimized and the number of matched residues to be maximized. Clearly, a smaller number of residue pairs can be superposed with a smaller rms and, clearly, a larger number of equivalent residues with a certain rms is more indicative of significant overall structural similarity. 

Root-mean-square deviation of 0-4 atoms from their least-squares plane. 

The results obtained were analyzed by the least-squares method. We obtained equations of the E — A + BT type. These equations, the mean-square deviations of the experimental values from the linear dependences, and the equations for the calculation of AZ are all listed in Table 2. 

The errors in the estimates of the values obtained were assumed to be twice as large as the mean-square deviations calculated by the least-squares method. These estimates corresponded to the 95% confidence interval for the values within 2A. 

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