Regular Least Square

Cemik, M., Borkovec, M. and Westall, J. C. (1995). Regularized least-squares methods for the calculation of discrete and continuous affinity distributions for heterogeneous sorbents, Environ. Sci. Technol., 29, 413-425. 

The steepest descent method for nonlinear regularized least-squares inversion To solve the problem of minimization of the parametric functional using the steepest descent method, let us calculate the first variation of P (m,d), assuming that the operator A(m) is differentiable, so that... 

Thus, the Newton algorithm for the nonlinear regularized least-squares inversion can be expressed by the formula... 

Discrete least square Ordinary least square Regular least square Weighted least square... 

The inversion in all cases including that of Eqs. 27 and 28 can be implemented by the regularized least-squares solution already described. In general, because the anti-Stokes radiation generated by pulse 1 and pulse 2 overlap in frequency, a solution like that implemented for Eq. 17 will not suffice for this case. 

The standard way to answer the above question would be to compute the probability distribution of the parameter and, from it, to compute, for example, the 95% confidence region on the parameter estimate obtained. We would, in other words, find a set of values h such that the probability that we are correct in asserting that the true value 0 of the parameter lies in 7e is 95%. If we assumed that the parameter estimates are at least approximately normally distributed around the true parameter value (which is asymptotically true in the case of least squares under some mild regularity assumptions), then it would be sufficient to know the parameter dispersion (variance-covariance matrix) in order to be able to compute approximate ellipsoidal confidence regions. 

Firstly, it has been found that the estimation of all of the amplitudes of the LI spectrum cannot be made with a standard least-squares based fitting scheme for this ill-conditioned problem. One of the solutions to this problem is a numerical procedure called regularization [55]. In this method, the optimization criterion includes the misfit plus an extra term. Specifically in our implementation, the quantity to be minimized can be expressed as follows [53] ... 

These factors are used in the equations given in Table I. The computation requires only that the variance ratios be accurately known. The absolute precision of the method may change from day to day without affecting the validity of either the least-squares curve-of-best fit procedure or the confidence band calculations. (It is not practical to regularly monitor local variances, and errors may develop in variance ratios. Eowever, the error due to incorrect ratios will almost always be much less than the error due to assuming constant variance. Even guessed values of, say, S a concentration are likely to yield more precise data.)... 

We can distinguish easily among the three types of solutions discussed for Gji with a plot of In (71/72) against Xi. For an ideal solution, the result is a value of 0 for aU points. For a regular solution, the plot is a straight line with slope equal to 2B, which passes through 0 at Xi = 0.5. For correlations that require 2 or more parameters, the coefficients can be obtained from a least-squares analysis of the data for as a function of Xi (see Section (A.l)). 

By far the best way to refine structures using electron diffraction data is to use multislice calculations. These will be discussed in the next chapter. However, some useful information can be obtained by regular crystallographic least squares with the assumption of kinematic data. 

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