Least-squares computations

The computer least squares optimization algorithm used is termed Simplex [4] which was programmed using Microsoft QuickBasic 4.5. A version of the program is provided at the end of this section. 

Tables 2—4 (see Appendix, p. 95) list what the author believes are the most accurate values of the dielectric virial coefiBdents obtained to date. The values of the coefiSdents for a particular gas are all taken from a angle paper, except in the case of a few dipolar gases where the authors published values of Ae and B, in separate papers. Emphasis has been placed on accmate values of Be, and for some gases, particularly non-dipolar ones, more accurate values of Ae can be found elsewhere in the literature (e.g. Table II of ref. 61). The uncertainty limits listed are those given by the authors. In most cases involving expansion techniques these reflect deviations of the experimental points from the computed least-squares curve, and contain no estimate of possible systematic errors. The papers listed under other references contain data which will lead to values of the virial coefiSdents considered less accurate than those given in the tables. In many cases these papers contain no actual values of the coefiSdents, but rather data from which values can be obtained.

Once you have selected your function, you will not only find it described, but you will also get help in placing the arguments. For example, when you look under Math Trig => SUMX2MY2, a function we will often use when computing least-squares fits, you will get two windows in which you can place the addresses of the two columns or rows you want to use for X andY. 

Plot versus x the simulated data points y of B9 B58 together with the computed least-squares line ycalc in F9 F58, see Fig. 2.7-1. 

Computing least-squares solutions to over-determined equations is a useful computation in linear algebra. If one is given R and trying to solve for r from... 

Standard Gibbs energy of formation of NiO was found to be the following function of the temperature AfG°(7) = (-233.651 + 0.085 (77K)) kJ-mol". The accuracy calculated from the maximum deviation from the computed least-squares line is equal to 0.209 kJ mol . This corresponds to the maximum deviation in cell potential (emt) of 1.0 mV. 

As Fig. 1 indicates, the plot of C/T vs. shows a dependence similar to 1/T. The straight line through the data points represents a computer least-squares fit to the function A + + The values of the coefficients so obtained differ from... 

The ability of partial least squares to cope with data sets containing very many x values is considered by its proponents to make it particularly suited to modern-day problems, where it is very easy to compute an extremely large number of descriptors for each compound (as in CoMFA). This contrasts with the traditional situation in QSAR, where it could be time-consuming to measure the required properties or where the analysis was restricted to traditional substituent constants. 

Procedure. Compute the slope of the function by a linear least squares procedure and obtain a value of Boltzmann s constant. How many particles do you expect to find 125 pm above the reference point Take the uncertainty you have calculated for the slope, as the uncertainty in k. Is the modem value of = 1.381 x 10 within these enor limits ... 

In multivariate least squares analysis, the dependent variable is a function of two or more independent variables. Because matrices are so conveniently handled by computer and because the mathematical formalism is simpler, multivariate analysis will be developed as a topic in matrix algebra rather than conventional algebra. 

The principal topics in linear algebra involve systems of linear equations, matrices, vec tor spaces, hnear transformations, eigenvalues and eigenvectors, and least-squares problems. The calculations are routinely done on a computer. 

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. 

Goodman, T.P., A Least-Squares Method for Computing Balance Corrections, ASME Paper No. 63-WA-295. 

The parameters and a, depend on the angular momentum (s-, p-, d- etc.) and are determined by least squares fit. Typically between two and seven Gaussian functions are used in the fit many Gaussians improve the fit (and consequently the resulting orbitals) at the price of increased computational time. 

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