Normal equations nonlinear

Table 6.2. Suitable nonlinear calibration functions, their normal equations and sensitivity functions... 

The normal equations will be written for the least squares regression of the nonlinear equation for N sets of data, (xj.y, ... 

The procedure is repeated until the procedure converges, that is until the correction vector b approaches zero. Methods of modifying the size of the correction vector have been developed to improve convergence (B5) and this method, in theory, should always converge (HI). In practice, nonlinearities in the model and poor parameter estimates can prevent convergence. A modification of this method linearizes the normal equations higher derivatives have also been used. 

Of course, the roots of Eq. (46a) may easily be found. The unknown quantities fi and 2m may be found by substituting hm from Eq. (46a) into the imaging equations (18) and normalization equation (3). Because of the nonlinear nature of these equations, a Newton-Raphson (or other iterative) method of solution would be necessary. 

Carry out the least-squares minimization of the quantity in Eq. (7) according to an appropriate algorithm (presumably normal equations if the observational equations are linear in the parameters to be determined otherwise some other such as Marquardf s ). The linear regression and Solver operations in spreadsheets are especially useful (see Chapter HI). Convergence should not be assumed in the nonlinear case until successive cycles produce no significant change in any of the parameters. 

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... 

Interval estimates for nonlinear models are usually approximate, since exact calculations are very difficult for more than a few parameters. But, as our colleague George Box once said, One needn t be excessively precise about uncertainty. In this connection. Donaldson and Schnabel (1987) found the Gauss-Newton normal equations to be more reliable than the full Newton equations for computations of confidence regions and intervals. 

Approximate regularized solution of the nonlinear inverse problem We can find an approximate solution of the regularized normal equation (5.86) for the optimum step, using the same idea which we applied for the approximate. solution of the linear inverse problem in Chapter 3. Let us assume that the regularization parameter tv is big enough to neglect the term with respect to the term... 

The derivatives of the calculated data with respect to the force constants are then formed, and these are used to construct the normal equations from which corrections to the force constants are calculated in such a way as to minimize the sum of weighted squares of residuals. Because the relations between the data and the force constants are often very nonlinear, it is necessary to cycle this calculation until the changes in the force constants drop to zero when the calculation will have converged and the sum of weighted squares of errors will be minimized. The usual statistical formulas are then used to obtain the variance/covariance matrix in the derived best estimates of the force constants, and the estimated standard errors in the force constants are usually quoted along with their values. The whole procedure is referred to as a force constant refinement calculation. ... 

Another difficulty arises from the nonlinear nature of the calculation, which may often cause the surface of sum of weighted squares of residuals to have multiple minima as a function of the force constants, so that it may be possible to converge onto several different minima by starting from different trial force fields in the refinement. This can be particularly troublesome when the data are only just sufficient to determine the force field, so that the normal equations are somewhat ill-conditioned. The nature of the calculation is reminiscent of S. D. 

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. 

To find the value of 0 that maximizes Eq. (5.24), LL(0) is differentiated with respect to 0 and the resulting expressions set equal to zero. These are, again, the set of normal equations to solve. Like the general nonlinear regression case, the normal equations for Eq. (5.24) cannot be solved directly, but must be solved iteratively using least squares or some variant thereof. 

By applying the variationed principle on this functional we can derive the time-independent nonlinear Schrodinger equation specific the system under scrutiny. If we impose that first-order variations of G with respect to arbitrary variations of the solute wavefunction are zero, and that the latter is normalized, the nonlinear Schrodinger equation becomes ... 

Consider the reflection of a normally incident time-harmonic electromagnetic wave from an inhomogeneous layered medium of unknown refractive index n(x). The complex reflection coefficient r(k,x) satisfies the Riccati nonlinear differential equation [2] ... 

The regression constants A, B, and D are determined from the nonlinear regression of available data, while C is usually taken as the critical temperature. The hquid density decreases approximately linearly from the triple point to the normal boiling point and then nonhnearly to the critical density (the reciprocal of the critical volume). A few compounds such as water cannot be fit with this equation over the entire range of temperature. Liquid density data to be regressed should be at atmospheric pressure up to the normal boihng point, above which saturated liquid data should be used. Constants for 1500 compounds are given in the DIPPR compilation. 

When experimental data is to be fit with a mathematical model, it is necessary to allow for the facd that the data has errors. The engineer is interested in finding the parameters in the model as well as the uncertainty in their determination. In the simplest case, the model is a hn-ear equation with only two parameters, and they are found by a least-squares minimization of the errors in fitting the data. Multiple regression is just hnear least squares applied with more terms. Nonlinear regression allows the parameters of the model to enter in a nonlinear fashion. The following description of maximum likehhood apphes to both linear and nonlinear least squares (Ref. 231). If each measurement point Uj has a measurement error Ayi that is independently random and distributed with a normal distribution about the true model y x) with standard deviation <7, then the probability of a data set is... 

For a polyatomic molecule, the complex vibrational motion of the atoms can be resolved into a set of fundamental vibrations. Each fundamental vibration, called a normal mode, describes how the atoms move relative to each other. Every normal mode has its own set of energy levels that can be represented by equation (10.11). A linear molecule has (hr) - 5) such fundamental vibrations, where r) is the number of atoms in the molecule. For a nonlinear molecule, the number of fundamental vibrations is (3-q — 6). 

Even if we make the stringent assumption that errors in the measurement of each variable ( >,. , M.2,...,N, j=l,2,...,R) are independently and identically distributed (i.i.d.) normally with zero mean and constant variance, it is rather difficult to establish the exact distribution of the error term e, in Equation 2.35. This is particularly true when the expression is highly nonlinear. For example, this situation arises in the estimation of parameters for nonlinear thermodynamic models and in the treatment of potentiometric titration data (Sutton and MacGregor. 1977 Sachs. 1976 Englezos et al., 1990a, 1990b). 

Because the term r(CA T) is exponentially dependent on T and can be nonlinear as well, a numerical solution or piecewise linearization must be used. To simplify the numerical manipulations, equations in Table IX are normalized by = z/L, r = ut/L, and jc = 1 - C,/(C,)0, where i is normally S02. y also is a normalized quantity. The Peclet numbers for mass and heat are written PeM = 2Rpu/D) and PeH = 2Rpcpul t for a spherical particle. They are also written in terms of bed length as Bodenstein numbers. It is... 

Whereas the profile in linear wave equations is usually arbitrary it is important to note that a nonlinear equation will normally describe a restricted class of profiles which ensure persistence of solitons as t — oo. Any theory of ordered structures starts from the assumption that there exist localized states of nonlinear fields and that these states are stable and robust. A one-dimensional soliton is an example of such a stable structure. Rather than identify elementary particles with simple wave packets, a much better assumption is therefore to regard them as solitons. Although no general formulations of stable two or higher dimensional soliton solutions in non-linear field models are known at present, the conceptual construct is sufficiently well founded to anticipate the future development of standing-wave soliton models of elementary particles. 

As introduced in sections 3.1.3 and 4.2.3, the Arrhenius equation is the normal means of representing the effect of T on rate of reaction, through the dependence of the rate constant k on T. This equation contains two parameters, A and EA, which are usually stipulated to be independent of T. Values of A and EA can be established from a minimum of two measurements of A at two temperatures. However, more than two results are required to establish the validity of the equation, and the values of A and EA are then obtained by parameter estimation from several results. The linear form of equation 3.1-7 may be used for this purpose, either graphically or (better) by linear regression. Alternatively, the exponential form of equation 3.1-8 may be used in conjunction with nonlinear regression (Section 3.5). Seme values are given in Table 4.2. 

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