Least squares iterative

The calculation of the matrices A and B, the inversion of B, and the matrix multiplications, are the major steps in a least-squares iteration. 

Calibration and mixture analysis addresses the methods for performing standard experiments with known samples and then using that information optimally to measure unknowns later. Classical least squares, iterative least squares, principal components analysis, and partial least squares have been compared for these tasks, and the trade-offs have been discussed (Haaland,... 

Using a least squares iterative method, each spectrum is fitted with two Gaussian lines. The smooth curve seen on Fig. 2 corresponds to such a Gaussian fit. This procedure provides a rather accurate value of Fc, the frequency of the center of the two Zeeman lines. It provides also the Fc uncertainty interval. This interval is related to the difference between the best fit and the experimental spectrum, which is here mainly due to noise. 

Least-Squares Iterations Nonlinear Evaluation of Cyclodextrin Multiple Complex Formation with Static and Ionizable Solutes... 

At last, the finite width of the excitation pulse is accounted by a non linear least square iterative reconvolution procedure. Thus, a good precision on r(t) can be achieved if ... 

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... 

The decay law G(t) was extracted from the experimental decay curve using nonlinear least-squares iterative reconvolution. The parameters were varied until the xl values were minimised and initially the trial function for G(t) was chosen to be a sum of 10 exponential terms viz... 

The common procedure used to calculate the SLD profile from the reflectivity curve is to assume a model profile, calculate the theoretical reflectivity curve using the optical matrix or recursion method, and compare calculated and experimental curves. A least-squares iterative procedure is then used to vary the parameters of the SLD profile until a good fit between the calculated curve and the experimental data is achieved. Although the inversion of the reflectivity data is not unique and... 

A number of approaches to fitting the electrostatic potential (EP) to point charges have been proposed. Early attempts involved a simple least-squares iterative fit to a select set of points about the molecule of interest. Cox and Williams used a cube of points separated by 1.0—1.2 A, excluding the volume within the van der Waals radii and more than 1.0 A beyond the van der Waals radii. Singh and Kollman " chose to use a Connolly surface, a distance of 1.2-... 

The fitted curve Is presented with the observed data, where and F(max) were calculated using a least squares Iteration. Reproducibility between two trials using the same receptor preparation Is 11 lustrated In Table I. Observation of the spectra before and after enhancement showed that the Intensity Increase (F(max)) was greatest on the shorter wavelength portion of the spectrum, corresponding to emission from components In a relatively non-polar environment. The existence of those fluorescent components which were not affected by the binding event provided a convenient method for internal compensation for other variables such as Intensity of the source, sensitivity of... 

FIGURE 1 displays the kinetics of the uptake and clearance of a phenol in fish taking PCPasan example. Upon estimation of the rate constants of the uptake (ki) and the clearance phase (k2) using a non-linear least-squares iterative method (Butte and Blum 1984) the BCFs can easily be calculated. 

One widely used algorithm for performing a PCA is the NIPALS (Nonlineai Iterative Partial Least Squares) algorithm, which is described in Ref [5],... 

A useful method of weighting is through the use of an iterative reweighted least squares algorithm. The first step in this process is to fit the data to an unweighted model. Table 11.7 shows a set of responses to a range of concentrations of an agonist in a functional assay. The data is fit to a three-parameter model of the form... 

FIGURE 11.9 Outliers, (a) Dose-response curve fit to all of the data points. The potential outlier value raises the fit maximal asymptote, (b) Iterative least squares algorithm weighting of the data points (Equation 11.25) rejects the outlier and a refit without this point shows a lower-fit maximal asymptote. 

If the graph y vs. x suggests a certain functional relation, there are often several alternative mathematical formulations that might apply, e.g., y - /x, y = a - - exp(b (x + c))), and y = a-(l- l/(x + b)) choosing one over the others on sparse data may mean faulty interpretation of results later on. An interesting example is presented in Ref. 115 (cf. Section 2,3.1). An important aspect is whether a function lends itself to linearization (see Section 2.3.1), to direct least-squares estimation of the coefficients, or whether iterative techniques need to be used. 

Because of peak overlappings in the first- and second-derivative spectra, conventional spectrophotometry cannot be applied satisfactorily for quantitative analysis, and the interpretation cannot be resolved by the zero-crossing technique. A chemometric approach improves precision and predictability, e.g., by the application of classical least sqnares (CLS), principal component regression (PCR), partial least squares (PLS), and iterative target transformation factor analysis (ITTFA), appropriate interpretations were found from the direct and first- and second-derivative absorption spectra. When five colorant combinations of sixteen mixtures of colorants from commercial food products were evaluated, the results were compared by the application of different chemometric approaches. The ITTFA analysis offered better precision than CLS, PCR, and PLS, and calibrations based on first-derivative data provided some advantages for all four methods. ... 

H. Wold, Soft modelling by latent variables the non-linear iterative partial least squares (NIPALS) algorithm. In Perspectives in Probability and Statistics, J. Gani (Ed.). Academic Press, London, 1975, pp. 117-142. 

There have also been attempts to describe the temporal aspects of perception from first principles, the model including the effects of adaptation and integration of perceived stimuli. The parameters in the specific analytical model derived were estimated using non-linear regression [14]. Another recent development is to describe each individual TI-curve,/j(r), i = 1, 2,..., n, as derived from a prototype curve, S t). Each individual Tl-curve can be obtained from the prototype curve by shrinking or stretching the (horizontal) time axis and the (vertical) intensity axis, i.e. fff) = a, 5(b, t). The least squares fit is found in an iterative procedure, alternately adapting the parameter sets (a, Zi, for 1=1,2,..., n and the shape of the prototype curve [15],... 

The structure of such models can be exploited in reducing the dimensionality of the nonlinear parameter estimation problem since, the conditionally linear parameters, kl5 can be obtained by linear least squares in one step and without the need for initial estimates. Further details are provided in Chapter 8 where we exploit the structure of the model either to reduce the dimensionality of the nonlinear regression problem or to arrive at consistent initial guesses for any iterative parameter search algorithm. 

When the Gauss-Newton method is used to estimate the unknown parameters, we linearize the model equations and at each iteration we solve the corresponding linear least squares problem. As a result, the estimated parameter values have linear least squares properties. Namely, the parameter estimates are normally distributed, unbiased (i.e., (k )=k) and their covariance matrix is given by... 

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