Least squares, method arithmetic

The values of the silanol number, aon, of 100 silica samples, with a completely hydroxylated surface, were established [3-5]. The average silanol number (arithmetical mean) was found to be aoH,av = 4.9 OH/nm. Calculations by the least-squares method yielded aon,av = 4.6 OH/nm. These values are in agreement with those reported by De Boer and Vleeskens [11] as well as with results reported by other researchers. To sum up, the magnitude of the silanol number, which is independent of the origin and structural characteristics of amorphous silicas is considered to be a physicochemical constant. The results fully confirmed the idea predicted earlier by Kiselev and co-workers [13,14] on the constancy of the silanol number for a completely hydroxylated silica surface. This constant now has a numerical value cioH,av = 4.6 0.5 OH/nm [3-5] and is known in literature as the Kiselev-Zhuravlev constant. 

However, multicomponent quantitative analysis is the area we are concerned with here. Regression on principle components, by PCR or PLS, normally gives better results than the classical least squares method in equation (10.8), where collinearity in the data can cause problems in the matrix arithmetic. Furthermore, PLS or PCR enable a significant part of the noise to be filtered out of the data, by relegating it to minor components which play no further role in the analysis. Additionally, interactions between components can be modelled if the composition of the calibration samples has been well thought out these interactions will be included in the significant components. 

This method, because it involves minimizing the sum of squares of the deviations xi — p, is called the method of least squares. We have encountered the principle before in our discussion of the most probable velocity of an individual particle (atom or molecule), given a Gaussian distr ibution of particle velocities. It is ver y powerful, and we shall use it in a number of different settings to obtain the best approximation to a data set of scalars (arithmetic mean), the best approximation to a straight line, and the best approximation to parabolic and higher-order data sets of two or more dimensions. 

Both the numerical and the analytical methods discussed in this chapter can be tedious to carry out, especially with large collections of precise data. Fortunately, the modem digital computer is ideally suited to carry out the repetitive arithmetic operations that are involved. Once a program has been written for a particular computation, whether it be numerical integration or the least-squares fitting of experimental data, it is only necessary to provide a new set of data each time the computation is to be calculated. 

The calibration was represented in the computer program by a fifth-degree polynomial. The conventional method of least-squares was followed to determine the coefficients of the polynomial. The sensitivity of the normal equations made round-off error a significant factor in the calculations. The effect of round-off error was greatly reduced when the calculations were performed with double-precision arithmetic. The molecular weights corresponding to selected count numbers were calculated from the coefficients. The coefficients were input information for the data-reduction program. 

If the functional relationship between one variable and another is linear, a straight-line plot would be obtained on arithmetic-coordinate graph paper. If the relationship approaches a linear one, the best method of fitting the data to a linear model would be through the method of least squares. The resulting linear equation (or line) would have the properties of lying as close as possible to the data. For statistical purposes, close and/or best fit is defined as that linear equation or line for which the sum of the squared vertical distances between the data (values of Y or independent variable) and line is minimized. These distances are called residuals. This approach is employed in the solution below. 

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