Windows, Mac, Linux Description:
Non-linear Iterative Curve Fitting a.
We saw that in some cases a non-linear situation can be converted into a linear one by a coordinate transformation, but this is possible only in some special cases, it may restrict the range of allowable dependent data values, and, in any case, the resulting coordinate transformation of the noise in the data can result in imprecision in the parameters measured in this way.
The most general way of fitting any model to a set of data is the iterative methoda kind kosmische octave custom writing "trial and error" procedure in which the parameters of the model are adjusted in a systematic fashion until the equation fits the data as close as required. This sounds like a brute-force approach, and it is.
In fact, in the days before computers, this method was only grudgingly applied. But its great generality, coupled with advances in computer speed and algorithm efficiency, means that iterative methods are more widely used now than ever before.
Iterative methods proceed in the following general way: This continues until the fitting error is less than the specified error.
One popular technique for doing this is called the Nelder-Mead Modified Simplex. This is essentially a way of organizing and optimizing the changes in parameters step 4, above to shorten the time required to fit the function to the required degree of accuracy.
With contemporary personal computers, the entire process typically takes only a fraction of a second to a few seconds, depending on the complexity of the model and the number of independently adjustable parameters in the model.
The animation on the right shows the working of the iterative process for a 2-peak unconstrained Gaussian fit to a small set of x,y data. This model has four nonlinear variables the positions and width of the two Gaussians, which are determined by iteration and two linear variables the peak heights of the two Gaussians, which are determined directly by regression for each trial iteration.
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In order to allow the process to be observed in action, this animation is slowed down artificially by plotting step-by-step, making a bad initial guess and adding a "pause " statement.
The entire process normally takes only about 0. The main difficulty of the iterative methods is that they sometime fail to converge at an optimum solution in difficult cases. The standard approach to handle this is to restart the algorithm with another set of first guesses; repeat that several times and take the one with the lowest fitting error that process is automated in the peak fitting functions described below.
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Iterative curve fitting also takes longer than linear regression - with typical modern personal computers, an iterative fit might take fractions of a second where a regression would take fractions of a millisecond.
Still, this is fast enough for many purposes. It is not practical to predict the standard deviations of the measured model parameters using the algebraic approach, but both the Monte Carlo simulation and bootstrap methods are applicable.
In Fourier deconvolution, the underlying peak shape is unknown but the broadening function is assumed to be known; whereas in iterative least-squares curve fitting it's just the reverse: Thus the term "spectral deconvolution" is ambiguous: These are different processes; don't get them confused.
Fourier deconvolution vs curve fitting.They are pre owned, and are in very good condition, no dents, cracks, or physical things, but there is a little bit of the upper black.
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