Least squares method, data reduction

Carry out data reduction numerically using least-squares methods and determine probable errors in quantities obtained by these methods. 

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. 

The XPS spectra were recorded on a Surface Science Laboratories small spot system using a monochromatized A1K X-ray radiation source. The take-off angle used for these measurements was 35°. Full details of the methods used in interpreting the XPS data have been described elsewhere [14], Data reduction was done using Surface Science Laboratories software version 8.0. This software utilizes a least squares curve fitting approach with only chi square statistics for goodness of the calculated fit to the experimental data. 

The values of w/v and in (3) were obtained using a least-squares program. In the case where the velocity remains constant, reduction of the initial data yields the values of w/v and with an accuracy of not less than 5 ppm. It follows from (3) that to determine the Lamb frequency v it is necessary to measure the velocity of the 2s atoms, using an independent method. 

Assumptions may be made or models adopted (often by implication) about a system being measured that are not consistent with reality. The selection of the method of data reduction may be partly on the basis of the model adopted and partly on the basis of features such as computation time and simplicity. Kelly classified data processing methods as direct, graphical, minmax, least squares, maximum likelihood, and bayesian. Each method has rules by which computations are made, and each produces an estimate (or numerical result) of reality. 

In this chapter the experimental ECD and NIMS procedures for studying the reactions of thermal electrons with molecules and negative ions are described. Gas phase electron affinities and rate constants for thermal electron attachment, electron detachment, anion dissociation, and bond dissociation energies are obtained from ECD and NIMS data. Techniques to test the validity of specific equipment and to identify problems are included. Examples of the data reduction procedure and a method to include other estimates of quantities and their uncertainties in a nonlinear least-squares analysis will be given. The nonlinear least-squares procedure for a simple two-parameter two-variable case is presented in the appendix. 

We also discussed graphical and numerical data reduction procedures. The most important numerical data reduction procedure is the least squares, or regression, method, which finds the best member of a family of functions to represent a set of data. We discussed the propagation of errors through this procedure and presented a version of the procedure in which different data points are given different weights, or importances, in the procedure. 

Principal component analysis and partial least squares analysis are chemometric tools for extracting and rationalizing the information from any multivariate description of a biological system. Complexity reduction and data simplification are two of the most important features of such tools. PCA and PLS condense the overall information into two smaller matrices, namely the score plot (which shows the pattern of compounds) and the loading plot (which shows the pattern of descriptors). Because the chemical interpretation of score and loading plots is simple and straightforward, PCA and PLS are usually preferred to other nonlinear methods, especially when the noise is relatively high. ... 

Kemsley, E. K. (1996). Discriminant analysis of high-dimensional data a comparison of principal components analysis and partial least squares data reduction methods, Chemom. Intell. Lab. Syst. 33 47-61. 

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