Linear least squares analysis

Lagtime, 75 Laplace transform, 82 Larmor precessional frequency, 155, 165 Laser pulse absorption, 144 Lattice energy, 403 Law of mass action, 60, 125 Least-squares analysis linear, 41 nonlinear, 49 univariate, 44 unweighted, 44, 51 weighted, 46, 51, 247 Leaving group, 9, 340, 349, 357 Lennard-Jones potential, 393 Lewis acid-base adduct, 425 Lewis acid catalysis, 265 Lewis acidity, 426... 

FIGURE 3 (left). Relationship between the % of thylakoid-bound FBPase (ELISA) and its light activation. Data were fit using linear regression least-squares analysis. Linear correlation coefficient was r=0.84 with high significance (p<0.001). 

Lieb, S. G. Simplex Method of Nonlinear Least-Squares—A Logical Complementary Method to Linear Least-Squares Analysis ofData, /. Chem. Educ. 1997, 74, 1008-1011. 

Show that if the relative error ajk is constant, then an unweighted linear Arrhenius least-squares analysis is correct. 

While principal components models are used mostly in an unsupervised or exploratory mode, models based on canonical variates are often applied in a supervisory way for the prediction of biological activities from chemical, physicochemical or other biological parameters. In this section we discuss briefly the methods of linear discriminant analysis (LDA) and canonical correlation analysis (CCA). Although there has been an early awareness of these methods in QSAR [7,50], they have not been widely accepted. More recently they have been superseded by the successful introduction of partial least squares analysis (PLS) in QSAR. Nevertheless, the early pattern recognition techniques have prepared the minds for the introduction of modem chemometric approaches. 

A drawback of the method is that highly correlating canonical variables may contribute little to the variance in the data. A similar remark has been made with respect to linear discriminant analysis. Furthermore, CCA does not possess a direction of prediction as it is symmetrical with respect to X and Y. For these reasons it is now replaced by two-block or multi-block partial least squares analysis (PLS), which bears some similarity with CCA without having its shortcomings. 

These problems are provided to afford an opportunity for the reader to analyze binding data of different sorts. The problems do not require nonlinear least squares analysis, but this would be recommended to those with access to appropriate facilities. It must be emphasized that, while linearizing transformations allow binding data to be clearly visualized, parameter estimation should... 

Number-average molecular weights are Mn = 660 and 18,500 g/ mol, respectively (15,). Measurements were carried out on the unswollen networks, in elongation at 25°C. Data plotted as suggested by Mooney-Rivlin representation of reduced stress or modulus (Eq. 2). Short extensions of the linear portions of the isotherms locate the values of a at which upturn in [/ ] first becomes discernible. Linear portions of the isotherms were located by least-squares analysis. Each curve is labelled with mol percent of short chains in network structure. Vertical dotted lines indicate rupture points. Key O, results obtained using a series of increasing values of elongation 0, results obtained out of sequence to test for reversibility. 

Linear least squares analysis of points 3-8 yields the following slope = 0.0073 intercept = 7.36... 

Linearity is evaluated by appropriate statistical methods such as the calculation of a regression line by the method of least squares. The linearity results should include the correlation coefficient, y-intercept, slope of the regression line, and residual sum of squares as well as a plot of the data. Also, it is helpful to include an analysis of the deviation of the actual data points for the regression line to evaluate the degree of linearity. 

The collection of kinetic modelling programs will be adapted in the subsequent chapter for the non-linear least-squares analysis of kinetic data and the determination of rate constants. 

For the linear least-squares analysis of a monovariate data set equation (4.97) reduces to... 

These plots can also provide information about the assumption of constant error variance (Section III) made in the unweighted linear or nonlinear least-squares analyses. If the residuals continually increase or continually decrease in such plots, a nonconstant error variance would be evident. Here, either a weighted least-squares analysis should be conducted (Section III,A,2) or a transformation should be found to stabilize the error variance (Section VI). 

In the above analysis, y was considered to be a reaction rate. Clearly, any dependent variable can be used. Note, however, that if the dependent variable, y, is distributed with constant error variance, then the function z will also have constant error variance and the unweighted linear least-squares analysis is rigorous. If, in addition, y has error that is normal and independent, the least-squares analysis would provide a maximum likelihood estimate of A. On the other hand, if any transformation of the reaction rate is felt to fulfill more nearly these characteristics, the transformation may be made on y, ru r2 and the same analysis may be applied. One common transformation will be logarithmic. 

The parameter estimates obtained by a linear least-squares analysis of the original coded data (Cl) are shown in Table XV. After a canonical analysis, this equation becomes... 

If this procedure is followed, then a reaction order will be obtained which is not masked by the effects of the error distribution of the dependent variables If the transformation achieves the four qualities (a-d) listed at the first of this section, an unweighted linear least-squares analysis may be used rigorously. The reaction order, a = X + 1, and the transformed forward rate constant, B, possess all of the desirable properties of maximum likelihood estimates. Finally, the equivalent of the likelihood function can be represented b the plot of the transformed sum of squares versus the reaction order. This provides not only a reliable confidence interval on the reaction order, but also the entire sum-of-squares curve as a function of the reaction order. Then, for example, one could readily determine whether any previously postulated reaction order can be reconciled with the available data. 

With this method, the best straight line is fitted to a set of points that are linearly related as y = mx + b , where/ is the ordinate and x is the abscissa datum point, respectively. The slope (m) and the intercept (b) can be calculated by least squares analysis using Eqs. (50) and (51), respectively [23] ... 

Figure 4 shows plots of pressure reading vs mass of H2P for the pressure readings at temperature and when cold and for both cases a linear relation was observed. Least squares analysis of the data provided the following equations ... 

Tavare and Garside ( ) developed a method to employ the time evolution of the CSD in a seeded isothermal batch crystallizer to estimate both growth and nucleation kinetics. In this method, a distinction is made between the seed (S) crystals and those which have nucleated (N crystals). The moment transformation of the population balance model is used to represent the N crystals. A supersaturation balance is written in terms of both the N and S crystals. Experimental size distribution data is used along with a parameter estimation technique to obtain the kinetic constants. The parameter estimation involves a Laplace transform of the experimentally determined size distribution data followed a linear least square analysis. Depending on the form of the nucleation equation employed four, six or eight parameters will be estimated. A nonlinear method of parameter estimation employing desupersaturation curve data has been developed by Witkowki et al (S5). 

The error term e in the above equation is the deviation of the value of Y, from the true value Y . A least square analysis was carried out for each dependent variable Y with the objective of finding the best linear equation that fits the data with respect to the criteria minimizing the sum of the error square (i.e., minimize... 

The derived enthalpies for isomerization of n-BuOH to f-BuOH are —31.9 kJmol (Iq) and —36.7 kJmol (g). From the enthalpies of formation of t-BuOOH, the calculated enthalpies of formation of n-BuOOH are —261.7 kJmol (Iq) and —209.3 kJmol (g). Using the more recently reported enthalpy of formation of teri-BuOOH in the gas phase, the derived enthalpy of formation of n-BuOOH is —198.2 kJmoU. The linear least-squares analysis of the derived liquid enthalpies of formation of n-PrOOH and n-BuOOH and the experimental enthalpy of formation of n-HexOOH gives an intercept of — 161.3 kJ mol and a slope of —23.4 kJmoU, the latter value very close to that for the C3-C6 n-alkanols of —25 kJmoU. From the regression constants, the liquid enthalpy of formation of n-HeptOOH is —325 kJmoU, a value that is less negative, and thus more plausible when compared to its secondary isomers. 

Figure 4-13 Spreadsheet for linear least-squares analysis.

Figure 2. Gel permeation data for polypeptide linear random coils plotted according to the method of Porath (8) M0,555 is plotted vs. Kd1/3. Lines drawn through the data from each column are lines of best fit determined by linear least-squares analysis. Numerical designation for each curve represents the agarose...

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.

The absorbance at the band maximum for solutions of lithium, sodium, and barium in ND8 is a linear function of the concentration, and extrapolation of the linear function to zero concentration predicts zero absorbance (Figure 5 and Table II). The molar extinction coefficients as calculated by the least squares method for solutions of lithium, sodium, and barium in ND8 at —70° C. are given in Table II. If it is assumed that barium loses two electrons upon solvation, the molar extinction coefficients of lithium, sodium, and barium solutions are the same to within the estimated error of measurement (4%). The residual absorbance, as calculated from the least squares analysis, in each case is... 

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