Least-Squares Deconvolution Methods

The kernel in Eq. (8.31) corresponds to one element of a Voigt circuit, and the time constant distribution function, Eq. (8.32), is continuous. However, the integral in Eq. (8.4) can be substituted by a sum of discrete functions [378, 379]  

A modification of this procedure was proposed in the literature [389] and applied to determine the time constant distribution function [379]. This method is based on the predistribution of time constants uniformly on the logarithmic scale, and to improve the quality of the analysis, a Mmite Carlo technique was used to increase the number of analyzed time constants. Approximation was carried out using a constrained least-squares method and led to a continuous distribution function. This procedure converted the nonlinear problem to a linear one from which versus r , were obtained and produced positive values of the distribution function. The procedure was also applied to the distribution of the dielectric constants [379,389]. 

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Bagli, M. et al.. Mean input times of three oral chlorprothixene formulations assessed by an enhanced least-squares deconvolution method, J. Pharm. Set, 85 434-439, 1996. 

Veng PP. Novel Approach to Bioavailability Testing Statistical Method for Comparing Drug Input Calculated by a Least-square Deconvolution Technique. J Pharm Sci 1980a 69 318-324. 

Considerable effort has gone into solving the difficult problem of deconvolution and curve fitting to a theoretical decay that is often a sum of exponentials. Many methods have been examined (O Connor et al., 1979) methods of least squares, moments, Fourier transforms, Laplace transforms, phase-plane plot, modulating functions, and more recently maximum entropy. The most widely used method is based on nonlinear least squares. The basic principle of this method is to minimize a quantity that expresses the mismatch between data and fitted function. This quantity /2 is defined as the weighted sum of the squares of the deviations of the experimental response R(ti) from the calculated ones Rc(ti) ... 

The least-squares method is also widely applied to curve fitting in phase-modulation fluorometry the main difference with data analysis in pulse fluorometry is that no deconvolution is required curve fitting is indeed performed in the frequency domain, i.e. directly using the variations of the phase shift and the modulation ratio M as functions of the modulation frequency. Phase data and modulation data can be analyzed separately or simultaneously. In the latter case the reduced chi squared is given by... 

Many types of data treatment software allow the composition of a mixture to be obtained from its spectra. Kalman s least squares filter is one of the most widely known of these methods. Using successive approximations, it automatically finds the spectra of the sample solution by addition of the spectra of each compound contained in the spectral library (i.e. by additivity of the absorbances) and use of weight coefficients. These are called deconvolution methods (Fig. 11.26). 

Hence, we adopted their analytical way and extended their method to analyze the v p bands. It is necessary to resolve the overlap of those transitions by least-squares fitting in order to obtain the widths of the component bands. For the least-squares procedure, we have to determine analytical functions for fundamental transitions, v, a, v, p, and for hot band transitions the v,p band is reproduced as a sum of three Lorentzian curves of the v, p, v,Mp, and v,h2p. We also took into account the presence of the v2hla and v2h a band for the v,a band. The observed spectra in the 2320-2220 cm range were deconvoluted using... 

It is important to realise that least squares and maximum entropy solutions often provide different best answers and move the solution in opposite directions, hence a balance is required. Maximum entropy algorithms are often regarded as a form of nonlinear deconvolution. For linear methods the new (improved) data set can be expressed as linear functions of the original data as discussed in Section 3.3, whereas nonlinear solutions cannot. Chemical knowledge often favours nonlinear answers for example, we know that most underlying spectra are all positive, yet solutions involving sums of coefficients may often produce negative answers. 

Veldhuis JD, Evans WS, Johnson ML. Complicating effects of highly correlated model variables on nonlinear least-squares estimates of unique parameter values and their statistical confidence intervals Estimating basal secretion and neurohormone half-life by deconvolution analysis. Methods Neurosci 1995 28 130-8. 

Although deconvolution is a wdl defined mathematical procedure, its a lication to fluorescence decay curves is attended with numerous difficulties owing to the counting enors and instmmental distortions that accompany sin e photon countii data. It is now generally accepted that least squares iterative reconvdution is the most satisfactory method of analymg nano cond decay data In its amplest... 

NIR spectroscopy became much more useful when the principle of multiple-wavelength spectroscopy was combined with the deconvolution methods of factor and principal component analysis. In typical applications, partial least squares regression is used to model the relation between composition and the NIR spectra of an appropriately chosen series of calibration samples, and an optimal model is ultimately chosen by a procedure of cross-testing. The performance of the optimal model is then evaluated using the normal analytical performance parameters of accuracy, precision, and linearity. Since its inception, NIR spectroscopy has been viewed primarily as a technique of quantitative analysis and has found major use in the determination of water in many pharmaceutical materials. 

The final step before structure determination by traditional methods is extraction of integrated intensities from the powder diffraction pattern (see below). Intensities can nearly always be determined only after deconvolution of partially overlapping Bragg reflections. This can be done for individual peaks or for groups of peaks, but usually full decomposition methods are used. There are basically two techniques the iterative Le Bail method [6] and the constrained linear least-squares approach developed by Pawley [7]. 

Quantification requires knowledge of the beam path through the sample. Hence, the sample often needs to be modified to allow for a known geometry. Spectral subtraction, least square regression analysis, PLS and spectral deconvolution are some of the spectroscopic techniques widely used to quantify constituents in a multicomponent sample. For almost any type of spectroscopic analysis, this is usually the first step. It is especially important for polymers given the variations in spectra for the same polymer due to molecular weight, conformation, crystallinity, sample preparation, age and sampling method. 

Figure 7.9 Isotope deconvolution of a deuterated peptide (residues 101-138 from the light chain of an IgCI antibody with 33 backbone amide hydrogens) using least-squares and MEM methods. Spectra on the left are experimental spectra of a peptide with and without deuterium labeling the distributions on the right are calculated deuterium distributions by least-squares and MEM methods separately, as described by Zhang et al. [40]...

Maximum entropy deconvolution, however, is normally iteratively combined with other methods. For example, least squares curve fitting attempts to create structure in a spectrum. Maximum entropy picks solutions with least structure the reason is that in the absence of external evidence, there is no reason to assume structure in the reconstructed spectrum the flat map is the most likely answer. Most algorithms try to decrease mean square errors iteratively. A target residual is selected, according to the desired speed of computational convergence, and the solution with maximum entropy is chosen. Then, the residual is decreased and, again, a maximum entropy solution is found, until convergence. 

To date, the most successful method of combined background correction and peak deconvolution has been the method of digital filtering and least squares fitting of reference peaks to the unknown spectrum [45]. This method is robust, simple to automate, and is applicable to any sample type. 

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