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The problem of fitting experimental data to a given model function $f(t; p_1,p_2,dots,p_N)$ is conventionally solved numerically by methods such as that of Levenberg-Marquardt, which are based on approximating the Chi-squared measure of discrepancy by a quadratic function. Such nonlinear iterative methods are usually necessary unless the function $f$ to be fitted is itself a linear function of the parameters $p_n$, in which case an elementary linear Least Squares regression is immediately available. When linearity is present in some, but not all, of the parameters, we show how to streamline the optimization method by reducing the nonlinear activity to the nonlinear parameters only. Numerical examples are given to demonstrate the effectiveness of this approach. The main idea is to replace entries corresponding to the linear terms in the numerical difference quotients with an optimal value easily obtained by linear regression. More generally, the idea applies to minimization problems which are quadratic in some of the parameters. We show that the covariance matrix of $chi^2$ remains the same even though the derivatives are calculated in a different way. For this reason, the standard non-linear optimization methods can be fully applied.
Given a linear regression setting, Iterative Least Trimmed Squares (ILTS) involves alternating between (a) selecting the subset of samples with lowest current loss, and (b) re-fitting the linear model only on that subset. Both steps are very fast and
We consider a resampling scheme for parameters estimates in nonlinear regression models. We provide an estimation procedure which recycles, via random weighting, the relevant parameters estimates to construct consistent estimates of the sampling dist
In a regression setting with response vector $mathbf{y} in mathbb{R}^n$ and given regressor vectors $mathbf{x}_1,ldots,mathbf{x}_p in mathbb{R}^n$, a typical question is to what extent $mathbf{y}$ is related to these regressor vectors, specifically,
We consider best approximation problems in a nonlinear subset $mathcal{M}$ of a Banach space of functions $(mathcal{V},|bullet|)$. The norm is assumed to be a generalization of the $L^2$-norm for which only a weighted Monte Carlo estimate $|bullet|_n
We consider the problem of efficiently solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. Whilst some classical approaches are theoretically well founded, they can face difficult