Iteratively weighted least squares

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... 

A useful method of weighting is through the use of an iterative reweighted least squares algorithm. The first step... 

One approach is to use weighted least squares in which the weights for each iteration are a function of the sizes of the residuals ... 

The calculation of M estimates is similar in form to that of weighted least squares, where the weights are chosen iteratively on the basis of the current fit. The M estimates satisfy... 

Unlike linear models where normal equations can be solved explicitly in terms of the model parameters, Eqs. (3.14) and (3.15) are nonlinear in the parameter estimates and must be solved iteratively, usually using the method of nonlinear least squares or some modification thereof. The focus of this chapter will be on nonlinear least squares while the problem of weighted least squares, data transformations, and variance models will be dealt with in another chapter. 

The new estimate of 0 is a more efficient estimator since it makes use of the variability in the data. A modification of this process is to combine Steps 2 and 3 into a single step and iterate until the GLS parameter estimates stabilize. This modification is referred to as iteratively reweighted least-squares (IRWLS) and is an option available in both WinNonlin and SAS. Early versions of WinNonlin were limited in that g(.) was limited to the form in Eq. (4.6) where is specified by the user. For example, specifying 4> = 0.5, forces weights... 

We recognize the equation in step 2 as the weighted least squares estimate on the adjusted observations. Iterating through these two steps until convergence finds the iteratively reweighted least squares estimates. 

The PLS approach was developed around 1975 by Herman Wold and co-workers for the modeling of complicated data sets in terms of chains of matrices (blocks), so-called path models . Herman Wold developed a simple but efficient way to estimate the parameters in these models called NIPALS (nonlinear iterative partial least squares). This led, in turn, to the acronym PLS for these models, where PLS stood for partial least squares . This term describes the central part of the estimation, namely that each model parameter is iteratively estimated as the slope of a simple bivariate regression (least squares) between a matrix column or row as the y variable, and another parameter vector as the x variable. So, for instance, in each iteration the PLS weights w are re-estimated as u X/(u u). Here denotes u transpose, i.e., the transpose of the current u vector. The partial in PLS indicates that this is a partial regression, since the second parameter vector (u in the... 

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. 

Although satisfactory criteria for deciding whether data are better analyzed by distributions or multiexponential sums have yet to established, several methods for determining distributions have been developed. For pulse fluorometry, James and Ware(n) have introduced an exponential series method. Here, data are first analyzed as a sum of up to four exponential terms with variable lifetimes and preexponential weights. This analysis serves to establish estimates for the range of the preexponential and lifetime parameters used in the next step. Next, a probe function is developed with fixed lifetime values and equal preexponential factors. An iterative Marquardt(18) least-squares analysis is undertaken with the lifetimes remaining fixed and the preexponential constrained to remain positive. When the preexponential... 

Using least squares statistical techniques, the relationship between the second and third virial coefficients with molecular weight was obtained. This was necessary to make kinetic light-scattering measurements at a constant concentration. An iteration formula was given to calculate Mw from the lima o Kc/Re value at one concentration. 

This chapter uses Gauss 1809 treatment of nonlinear least squares (submitted in 1806, but delayed by the publisher s demand that it be translated into Latin). Gauss weighted the observations according to their precision, as we do in Sections 6.1 and 6.2. He provided normal equations for parameter estimation, as we do in Section 6.3, with iteration for models nonlinear in the parameters. He gave efficient algorithms for the parameter... 

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