Linearity parameter

Marquadt, D. W., An algorithm for least-squares estimation of non-linear parameters, J. Soc. Indust. Appl. Math., 11, 431, 1963. 

In the second group of models, the pc surface consists only of very small crystallites with a linear parameter y, whose sizes are comparable with the electrical double-layer parameters, i.e., with the effective Debye screening length in the bulk of the diffuse layer near the face j.262,263 In the case of such electrodes, inner layers at different monocrystalline areas are considered to be independent, but the diffuse layer is common for the entire surface of a pc electrode and depends on the average charge density <7pc = R ZjOjOj [Fig. 10(b)]. The capacitance Cj al is obtained by the equation... 

Sn + Pb is a two-phase eutectic system in which fine crystals of Pb with a linear parameter of 0.01 to 0.02 fim are localized along the grain boundaries of large Sn crystals (3 to4//m). A comparison of experimental... 

Brubaker, T. A., Tracy, R., and Pomemacki, C. L., Linear Parameter Estimation, Anal. Chem. 50, 1978, 1017A-1024A. 

Instead of using repeated solution of a suitable eigenvalue equation to optimize the orbitals, as in conventional forms of SCF theory, we have found it more convenient to optimize by a gradient method based on direct evaluation of the ener functional (4), ortho normalization being restored after every parameter variation. Although many iterations are required, the energy evaluation is extremely rapid, the process is very stable, and any constraints on the parameters (e.g. due to spatial symmetry or choice of some type of localization) are very easily imposed. It is also a simple matter to optimize with respect to non-linear parameters such as orbital exponents. 

Basically two search procedures for non-linear parameter estimation applications apply. (Nash and Walker-Smith, 1987). The first of these is derived from Newton s gradient method and numerous improvements on this method have been developed. The second method uses direct search techniques, one of which, the Nelder-Mead search algorithm, is derived from a simplex-like approach. Many of these methods are part of important mathematical computer-based program packages (e.g., IMSL, BMDP, MATLAB) or are available through other important mathematical program packages (e.g., IMSL). 

The application of optimisation techniques for parameter estimation requires a useful statistical criterion (e.g., least-squares). A very important criterion in non-linear parameter estimation is the likelihood or probability density function. This can be combined with an error model which allows the errors to be a function of the measured value. A simple but flexible and useful error model is used in SIMUSOLV (Steiner et al., 1986 Burt, 1989). 

Non-linear parameter estimation is far from a trivial task, even though it is greatly simplified by the availability of user-friendly program packages such as a) SIMUSOLV (Steiner et al., 1986), b) ESL, c) a set of BASIC programs (supplied with the book of Nash and Walker-Smith, 1987) or d) by mathematical software (MATLAB). ISIM itself does not supply these advanced features, but ISIM programs can easily be translated into other more powerful languages. 

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. 

In engineering we often encounter conditionally linear systems. These were defined in Chapter 2 and it was indicated that special algorithms can be used which exploit their conditional linearity (see Bates and Watts, 1988). In general, we need to provide initial guesses only for the nonlinear parameters since the conditionally linear parameters can be obtained through linear least squares estimation. 

Another important linear parameter is the excitation anisotropy function, which is used to determine the spectral positions of the optical transitions and the relative orientation of the transition dipole moments. These measurements can be provided in most commercially available spectrofluorometers and require the use of viscous solvents and low concentrations (cM 1 pM) to avoid depolarization of the fluorescence due to molecular reorientations and reabsorption. The anisotropy value for a given excitation wavelength 1 can be calculated as... 

In the example of charging a neutral particle, A, = qt/q is the linear parameter. Choosing intermediate states separated by a constant AX is, however, not a good choice for this problem because, as has been seen in Sect. 2.5, AA is a quadratic function of q. A better choice would be to decrease AX quadratically. Alternatively, one could define M A) as a quadratic function of A. Then, using a constant AX would be appropriate. 

The virial ratio is, as we noted above, 1.3366 for the separate-atom AO basis MO calculation, i.e. not 1.0. Now within the confines of the linear variation method (the usual LCAO approach) there is no remaining degree of freedom to use in order to constrain the virial ratio to its formally correct value (or indeed to impose any other constraint). Thus imposing the correct virial ratio on the linear variation method is, in this case, not possible without simultaneously destroying the symmetry of the wave function. Only by optimising the non-linear parameters can we improve the virial ratio as the above results show. Even at this most elementary level, the imposition of various formally correct constraints on the wave function is seem to generate contradictions. 

The conclusion above that optimisation of the non-linear parameters in the AO basis leads to a basis with correct spatial symmetry properties cannot be true for all intemuclear separations. At R = 0 the orbital basis must pass over into the double-zeta basis for helium i.e. two different 1 s orbital exponents. It would be astonishing if this transition were discontinuous at R = 0. While considering the variation of basis with intemuclear distance it is worth remembering that the closed-shell spin-eigenfunction MO method does not describe the molecule at all well for large values of R the spin-eigenfunction constraint of two electrons per spatial orbital is completely unrealistic at large intemuclear separation. With these facts in mind we have therefore computed the optimum orbital exponents as a function of R for three wave functions ... 

The most difficult problem we face in deciding to use a basis of hybrids which reflects the molecular symmetry is how do we choose such a basis in view of the enormous numerical difficulties involved in optimising the non-linear parameters in molecular calculations The real question is are there any rules for this choice, can the optimisation be done (at least approximately) once and for all The chemical evidence is for us — it is the most basic concept of the theory of valence that particular electronic sub-structures tend to be largely environment-independent. How can we select our basis to reflect this chemical fact ... 

In any book, there are relevant issues that are not covered. The most obvious in this book is probably a lack of in-depth statistical analysis of the results of model-based and model-free analyses. Data fitting does produce standard deviations for the fitted parameters, but translation into confidence limits is much more difficult for reasonably complex models. Also, the effects of the separation of linear and non-linear parameters are, to our knowledge, not well investigated. Very little is known about errors and confidence limits in the area of model-free analysis. 

The minimum of ssq is near the true values slope= 6 and intercepts20 that were used to generate the data (see Data mxb. m). ssq is continuously increasing for parameters moving away from their optimal values. Analysing that behaviour more closely, we can observe that the valley is parabolic in all directions. In other words, any vertical plane cutting through the surface results in a parabola. In particular, this is also the case for vertical planes parallel to the axes, i.e. ssq versus only one parameter is also a parabola. This is a property of so-called linear parameters. 

More careful examination of this shape reveals two important facts, (a) Plots of ssq as a function of k at fixed Io are not parabolas, while plots of ssq vs. Io at fixed k are parabolas. This indicates that Io is a linear parameter and k is not. (b) Close to the minimum, the landscape becomes almost parabolic, see Figure 4-6. We will see later in Chapter 4.3, Non-Linear Regression, that the fitting of non-linear parameters involves linearisation. The almost parabolic landscape close to the minimum indicates that the linearisation is a good approximation. 

Earlier, we promised an explicit solution for the determination of linear parameters. We first change the original notation introduced in equation (4.3) ... 

The prototype application is the fitting of the np linear parameters, a, ...,a p defining a higher order polynomial of degree np-1. The generalisation of equation (4.5) reads as ... 

The example below shows a short Matlab program that fits the function y =tan(x) with a polynomial of degree 3 defined by 4 linear parameters, i.e. the elements of a. 

In this chapter we expand the linear regression calculation into higher dimensions, i.e. instead of a vector y of measurements and a vector a of fitted linear parameters, we deal with matrices Y of data and A of parameters. 

It is probably more realistic to assume that we know neither the rate constants nor the absorption spectra for the above example. All we have is the measurement Y and the task is to determine the best set of parameters which include the rate constants ki and /cj and the molar absorptivities, the whole matrix A. This looks like a formidable task as there are many parameters to be fitted, the two rate constants as well as all elements of A. In Multivariate Data, Separation of the Linear and Non-Linear Parameters (p.162), we start tackling this problem. 

Multivariate Data, Separation of the Linear and Non-Linear Parameters... 

The secret is to realise that there are linear and non-linear parameters and that they can be separated, essentially reducing the number of parameters to be fitted iteratively, to the number of non-linear ones. The vector p defines the matrix C of concentrations and C, in turn, allows the computation of the best matrix A as a linear least-squares fit, A=C+Y, recall equation (4.61). Thus R can be computed as... 

A straightforward way to organise J is as a 3-dimensional array The derivative of R with respect to one particular parameter pi is a matrix of the same dimensions as R itself. The collection of all these nsx nl derivatives with respect to all the np non-linear parameters (e.g. rate constants) can be arranged in a 3-D array of dimensions nsxnlxnp, all individual matrices dR/dpi written slice-wise behind each other. This is illustrated in Figure 4-41. 

Remember, p only comprises the non-linear parameters, i.e. in case of our consecutive reaction the rate constants ki and fe. The linear parameters, the elements of the matrix A containing the molar absorptivities, have effectively been eliminated. 

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