Transformed least squares criterion

Fourier domain fitting. The Fourier transform of the experimental elution curve is calculated. The parameters a and 3 are then determined using a fitting procedure in the Fourier domain that is equivalent to a least-squares criterion in the time domain. With Fourier domain estimation, model parameters are chosen to minimize the difference between the Fourier transforms of experimental and theoretical elution curves. The Fourier transform of a bounded, time varying response curve, f(t), is defined as... 

The best criterion for minimizing the difference between the theoretical and experimental transform is not immediately obvious. In the time domain, a least-squares criterion is usually preferred (221 that is, parameters are selected that minimize the least-squares objective function... 

With logarithmic, power, or exponential models, one may modify the data by considering the logarithm of one or both of the variables instead of the original data, in order to reduce a nonlinear model to a linear one. This method is sometimes called the transformed least squares criterion. Another criterion, namely the Chebyshev... 

If enough data is acquired at large q values, and 4 can be determined using nonlinear least-squares fitting and, therefore, the ideal intensity. However, the problem arises in the selection of q limits, since the choice of both the lower and upper limits significantly affects the validity of the Porod s law. In this case, the upper limit was set to ( upp = 2, and the lower limit q was varied until the minimum area of the interference function G iq) was obtained (Eq. 19.53). Once this criterion is achieved, the interface distribution function gi(r) (Eq. 19.54) is calculated from the Eourier transform of G iq). 

The minimization process use ICP algorithm (Iterative Closest Point) described by Greespan [8] and Besl et al. [9]. Defining D the set of data points of the surface Sj and M the set of points of the model or surface S2, this method establish a matching of D and Mpoints. Thus for each point of D there is a point (the nearest) of the model M. By the correspondence established above, the transformation that minimizes the distance criterion is calculated and applied to the points of the set D and the overall error is calculated using least squares method. 

The hyperbohc saturation function of the form ax i [x + b) often arises in biophysical and biochemical appHcations. It is more obvious that this represents a hyperbola, if it is written in a double-reciprocal form, as in a Lineweaver-Burk transformation of the MichaeHs-Menten enzyme kinetics that employs this type of function. Other contexts where this function appears are monomolecular photochemical kinetics and visual physiology. In the present context of action spectroscopy, x would stand for the fluence or, in some cases, the fluence rate. When x = b, the function is at the half-maximum level that is often chosen to be the criterion response. So, when one performs least-squares fits using such a function, the parameter b is the estimate of the fluence needed for the criterion response, and the effectiveness (action spectrum ordinate) is just h -... 

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