Digital least-square computation

The most common digital filtering polynomial methods, according to Savitzky and Golay [22, 23], involve a shortened least-square computation using a sliding window with variable data points but other algorithms are also sometimes practicable. 

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. 

A program was written for an IBM 650 digital computer to obtain the experimental rate constant, k from a given set of kinetic data. This program was written according to a modification of the interpretive system devised by V. M. Wolontis (14). The program was constructed to produce a least-squares fit of the data to the linear equation,... 

Quantitative measurements are carried out by use of a single frequency in the conventional i.r. spectrum, but, with digitized spectra and a computer, the entire frequency spectrum of each component in a mixture can be fitted by curve-fitting techniques using such methods as least-squares refinements,5 which can yield an indication as to the precision of the fit. 

In most fields of physical chemistry, the use of digital computers is considered indispensable. Many things are done today that would be impossible without modem computers. These include Hartree-Fock ab initio quantum mechanical calculations, least-squares refinement of x-ray crystal stmctures with hundreds of adjustable parameters and mar r thousands of observational equations, and Monte Carlo calculations of statistical mechanics, to mention only a few. Moreover computers are now commonly used to control commercial instalments such as Fourier transform infrared (FTIR) and nuclear magnetic resonance (FT-NMR) spectrometers, mass spectrometers, and x-ray single-crystal diffractometers, as well as to control specialized devices that are part of an independently designed experimental apparatus. In this role a computer may give all necessary instaic-tions to the apparatus and record and process the experimental data produced, with relatively little human intervention. 

Elucidation of the constitution of C2-C3 copolymers through digitization of the spectra and subsequent analysis by an iterative least-squares procedure using a computer to provide band positions, intensities and half-widths for the overlapped bands in the methylene and methyl rocking regions. Contribution from each structural unit calculated using band areas derived from computer-resolved band parameters... 

Eor multivariate calibration in analytical chemistry, the partial least squares (PLS) method [19], is very efficient. Here, the relations between a set of predictors and a set (not just one) of response variables are modeled. In multicomponent calibration the known concentrations of / components in n calibration samples are collected to constitute the response matrix Y (n rows, / columns). Digitization of the spectra of calibration samples using p wavelengths yields the predictor matrix X (n rows, p columns). The relations between X and Y are modeled by latent variables for both data sets. These latent variables (PLS components) are constructed to exhaust maximal variance (information) within both data sets on the one hand and to be maximally correlated for the purpose of good prediction on the other hand. From the computational viewpoint, solutions are obtained by a simple iterative procedure. Having established the model for calibration samples. comp>o-nent concentrations for future mixtures can be predicted from their spectra. A survey of multi-component regression is contained in [20],... 

Several different methods have been used to obtain derivative spectra. For modern computer-controlled digital spectrophotometers, the differentiation can be performed numerically using procedures such as derivative least-squares polynomial smoothing, which is discussed in Section. 5C-2. With older analog instruments, derivatives of spectral data could be obtained electronically with a suitable operational amplifier circuit (see... 

Tables of the thermodynamic properties of neon have been computed using an electronic digital computer. An equation of state with 12 constants was fitted to PVT data for neon. The 102 points measured at Leiden were used because they lie in the temperature region of interest, The form of the equation was chosen to allow evaluation of the constants directly from the data using the method of least squares.

Constants may be determined directly by least squares if they are either independent or linearly dependent. After studying the series forms of the equations just discussed, twelve p, T terms with independent constant coefficients were assumed for this work. Values for these coefficients were determined by a least squares fit of lOZ PVT observations on neon made at Leiden in 1915 and 1919. A Datatron electronic digital computer determined the constants for the working equation. These constants are dimensional and have units corresponding to T in and p in gram-mols per liter. Each constant forms a dimensionless group when taken with its proper combination of powers of p and T. 

The Role of Computers in Activation Analysis.— For many years spectral data accumulated by m.c.a. have been analysed by digital computers. Many complex schemes of spectrum analysis have been developed involving least-squares methods, iterative Gaussian fits to peaks, correlation of the subject spectrum with ideal or measured spectrum shapes, convolution methods, and many combinations and variations of these schemes. Quittner... 

---