Nonlinear Gauss-Newton method

Kinetic curves were analyzed and the further correlations were determined with a nonlinear least-square-method PC program, working with the Gauss-Newton method. 

As seen in Chapter 2 a suitable measure of the discrepancy between a model and a set of data is the objective function, S(k), and hence, the parameter values are obtained by minimizing this function. Therefore, the estimation of the parameters can be viewed as an optimization problem whereby any of the available general purpose optimization methods can be utilized. In particular, it was found that the Gauss-Newton method is the most efficient method for estimating parameters in nonlinear models (Bard. 1970). As we strongly believe that this is indeed the best method to use for nonlinear regression problems, the Gauss-Newton method is presented in detail in this chapter. It is assumed that the parameters are free to take any values. 

THE GAUSS-NEWTON METHOD - NONLINEAR OUTPUT RELATIONSHIP... 

The above equation represents a set of p nonlinear equations which can be solved to obtain koutput vector around the trajectory xw(t). Kalogerakis and Luus (1983b) showed that when linearization of the output vector is used, the quasilinearization computational algorithm and the Gauss-Newton method yield the same results. 

Least-squares methods are usually used for fitting a model to experimental data. They may be used for functions consisting of square sums of nonlinear functions. The well-known Gauss-Newton method often leads to instabilities in the minimization process since the steps are too large. The Marquardt algorithm [9 1 is better in this respect but it is computationally expensive. 

The Gauss-Newton Method for solving the nonlinear set of equations (19.18) can be expressed as... 

The method of parameter estimation depends on the models involved. All models listed in Table CS3.2 are nonlinear and algebraic. One might employ the Gauss-Newton method of nonlinear least... 

Hartley, H.O. The modified Gauss-Newton method for the fitting of nonlinear regression functions. Technometrics 1961 3 269-280. 

The steepest descent method is very effective far from the minimum of , but is always much less efficient than the Gauss-Newton method near the minimum of . Marquardt (1963) has proposed a hybrid method that combines the advantages of both Gauss-Newton and steepest descent methods. Mar-quardt s method, combined with the Hellmann-Feynman pseudolinearization of the Hamiltonian energy level model, is the method of choice for most nonlinear molecular spectroscopic problems. 

Implementation Guidelines for ODE Models The Gauss-Newton Method — Nonlinear Output Relationship The Gauss-Newton Method - Systems with Unknown Initial Conditions Examples... 

Iterative fitting by the Gauss-Newton method may be applied to equations that are nonlinear in two or more constants, simply by expanding Eq. (18.37). 

The function minimum coincides with the prediction of the Gauss-Newton method applied to the nonlinear system (7.1). Moreover, the gradient of the function (7.64) in d = 0 is equal to the gradient of the merit function (7.12), which is obtained through the relation (7.31). 

When solving nonlinear regression using the Gauss-Newton method, the final parameter estimates are sensitive to the initial guesses. 

This method, known as the Gauss-Newton method, converts the nonlinear problem into a linear one by approximating the function F by a Taylor series expansion around an estimated value of the parameter vector b ... 

The calculation procedure for Newton s method is almost the same as that of Gauss-Newton method with the exception that the vector of corrections to the parameters is calculated from Eq. (7.173). If O is quadratic with respect to b (that is, linear regression), Newton s method converges in only one step. Like all other methods applying Newton s technique for the solution of the set of nonlinear equations, a relaxation factor may be used along with Eq. (7.173) when correcting the parameters. 

Linear and nonlinear regression analyses, including least squares, estimated vector of parameters, method of steepest descent, Gauss-Newton method, Marquardt Method, Newton Method,... 

Least squares multiple nonlinear regression using the Marquardt and Gauss-Newton methods. The program can fit simultaneous ordinary differential equations and/or algebraic equations to multiresponse data. 

The value of the parameters in (12) was obtained by a nonlinear regression method using the Gauss-Newton Algorithm. Those values are ... 

In general, the error e tic-q-i+j, 0) is a non-linear function of the parameter vector 0. Therefore, the above problem is a well-known nonlinear least squares problem (NLSP) that may be solved by various optimisation algorithms such as the Levenberg-Marquardt algorithm [2], the quasi-Newton method or the Gauss-Newton (GN) algorithm [3]. 

All nonlinear regression approaches use numerical methods, such as the Gauss-Newton or Levenberg-Marquardt algorithm optimisaticai algorithms, to search for the optimal point. 

Performs nonlinear regression using the Gauss-Newton estimation method. The jc-data is given as x, while the y-data is given as y. The function, FUN, that is to be fitted must be written as an m-file. It will take three arguments the coefficient values, x, and y (in this order). The function should be written to allow for matrix evaluatitni. The initial guess is specified in bataO. The vector beta contains the estimated values of the coefficients, the vector r contains the residuals, and covb is the estimated covariance matrix for the problem. J is the Jacobian matrix evaluated with the best estimate for the parameters. 

The overall system consists of Eqs. (2.30)- (2.36). This is a set of nonlinear simultaneous algebraic equations that can be solved simultaneously using a combination of Newton s method (Sec. 1.9) and the Gauss-Jordan method to be developed in this chapter (Sec. 2.6). 

Implicit equations cannot be solved individually but must be set up as sets of simultaneous algebraic equations. When these sets are linear, the problem can be solved by the application of the Gauss eiimination methods developed in Chap. 2. If the set consists of nonlinear equations, the problem is much more difficult and must be solved u.sing Newton s method for simultaneous nonlinear algebraic equations developed in Chap. I,... 

Method of Solution The Marquardt method using the Gauss-Newton technique, described in Sec. 7.4.4, and the concept of multiple nonlinear regression, covered in Sec. 7.4.5, have been combined together to solve this example. Numerical differentiation by forward finite differences is used to evaluate the Jacobian matrix defined by Eq. (7.164). 

---