Gauss-Newton algorithm with

Gauss Newton algorithm. For E-1000 columns, the calibration curve is a cubic polynomial with straight lines at the ends, whereas E-linear calibration curves are cubic polynomials. 

The Gauss-Newton algorithm is very well suited to deal with the least squares minimization problem. The numerical solution is found by applying the following iterative process ... 

The best-fitting set of parameters can be found by minimization of the objective function (Section 13.2.8.2). This can be performed only by iterative procedures. For this purpose several minimization algorithms can be applied, for example, Simplex, Gauss-Newton, and the Marquardt methods. It is not the aim of this chapter to deal with non-linear curve-fitting extensively. For further reference, excellent papers and books are available [18]. 

In this section we deal with estimating the parameters p in the dynamical model of the form (5.37). As we noticed, methods of Chapter 3 directly apply to this problem only if the solution of the differential equation is available in analytical form. Otherwise one can follow the same algorithms, but solving differential equations numerically whenever the computed responses are needed. The partial derivations required by the Gauss - Newton type algorithms can be obtained by solving the sensitivity equations. While this indirect method is... 

Many basic algorithms, each with a number of refinements, are useful in the search for a global minimum. Some of these methods are described briefly. These are the grid search, steepest descent, Gauss-Newton, Marquardt, and simplex methods. 

If ARRs can be established in symbolic form, they may be differentiated analytically with respect to the parameters to be estimated. That is, the Jacobian of the cost function to be minimised can be provided and a gradient based method such as the Gauss-Newton, or the Levenberg-Marquardt algorithm can be used. As a result, the number of iteration is much lower in comparison to a gradient free algorithm such as the one of Nelder-Mead which is important in real-time FDI. 

There are a multitude of methods for this task. Those that are conceptually simple usually are computationally intensive and slow, while the fast algorithms have a more complex mathematical background. We start this chapter with the Newton-Gauss-Levenberg/Marquardt algorithm, not because it is the simplest but because it is the most powerful and fastest method. We can t think of many instances where it is advantageous to use an alternative algorithm. 

Instead of developing a program that performs the task as just explained, we move to the 2-parameter case. Subsequently, we generalise to the np-parameter case and then we analyse the relationship with the Newton-Gauss algorithm for least-squares fitting. 

An additional observation for photon counting data there are no fractions of photons and thus the count can only include integer numbers. Thus the measurements in column B are rounded down to the nearest integer. It seems to be reasonable to do the same with the calculated values in column C. However, a test in Excel reveals that such an attempt does not work. The reason is, that the solver s Newton-Gauss algorithm requires the computation of the derivatives of the objective (x2 or ssq) with respect to the parameters. A rounding would destroy the continuity of the function and effectively wipe out the derivatives. 

Chapter 4 is an introduction to linear and non-linear least-squares fitting. The theory is developed and exemplified in several stages, each demonstrated with typical applications. The chapter culminates with the development of a very general Newton-Gauss-Levenberg/Marquardt algorithm. 

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