Question

Here are at least two reasons to use constraints.

* Avoid having the fit result in parameters that apply at a local minimum rather than a global minimum. See this discussion for exactly a case in point. http://www.igorexchange.com/node/6192

* Apply a physically realistic boundary condition to an otherwise unknown parameter in a larger group of parameters. An example is in spectroscopy, where the peak width or position may be rather well-defined based on the chemistry and only the amplitude (amount) of a component is truly unknown.

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J. J. Weimer
Chemistry / Chemical & Materials Engineering, UAHuntsville

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Thank you very much for your reply.......If i understand properly, regarding your first point we are talking about getting rid of ''bad'' or ''not wanted'' data and perform a fit in a more meaningful way to a certain portion of the data.......I guess that was some type of fluctuation and the person wanted to get rid of this discrepancy......Regarding your second point, you are saying that based on the chemistry of a sample, you may know the position and width of a peak but not the height.....However, the position may slightly vary (due to precision error in defining the exact wavenumber, instability of laser etc) which results in a small deviation from what you would expect from literature values for example.....Literature sometimes gives a range of wavenumbers for a specific peak rather than a particular wavenumber value......As a result between various experiments you may see these small variations........I would expect that fitting all peak parameters (position, width, height) in an unconstraned manner would make more sense to me......On the other hand, that varies between materials and maybe sometimes you always expect the same position of a certain peak for example......i work with collagen and my experience indicates these small variations in peak position (for proline peak for example). This is why i prefer unconstrained fit. I may miss something regarding your points though so please feel free to tell me

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In the first example, constraints are needed to keep the peak width from falling to a value that is so low where only one side of the broad Gaussian is fit rather than both sides. The fit by a small peak width is a local minimum. Another would appear when the peak would fit only the right side. The confusion is compounded by the fact that some data points are absent or removed from the set. This latter issue has nothing to do with setting constraints during the fit ... it is a fortuitous lead-in to the reason.

In the second case where multiple peaks are being fit under an envelope (some call the "peak deconvolution" rather than the truer peak fitting), I would state that physical insight should trump unconstrained peak fitting. One can very easily fit a broad Gaussian with two Gaussian peaks of various widths, positions, and heights. The best case is to start with a set of constraints based on physical insight. In some cases, a good approach is to hold one of the three parameters constant (generally width or position) for one or more sub-peaks, iterate to a solution, then allow the constrained parameter to vary within a reasonable bounds. IOW, force the system to a state close to where it should be for a "perfect" physical system, then "tweak" the fit around that point to allow for finer variations in the state of the physical system.

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J. J. Weimer
Chemistry / Chemical & Materials Engineering, UAHuntsville

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Thanks for your feedback.....Sometimes its a bit difficult to know the system just because it may be a new sample that you look and you just need to get a quick idea of your raman spectra......at this point constraints are a bit difficult to be set. Nevetheless, i have seen people performing either unconstrained or constrained fitting......My understanding is that both ways can give you representative data of the material that you are probing.....To make long story short, it does not seem to me that one thing is right and the other is wrong.......i have seen both cases stated in papers with or without constraints

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I would also say that neither is right or wrong. Since you are talking about a chemical spectroscopy, the danger of unconstrained fitting is, you can end with answers that do not represent the real chemistry of the sample because they are one of a set of local minimum. By comparison, the danger of constrained fitting is, one can assume too much about the chemistry of the sample to start and ultimately force an answer that is also unrepresentative of the real chemistry.

So, for chemical spectroscopy in particular, the best peak fitting is done either with some reasonable _a prior_ information about the expected sample chemistry (i.e. constrained fitting) or with some logical _post-fitting_ reasoning about the meaning of the results vis-a-vis the predicted sample chemistry (i.e. unconstrained fitting). The absolute best is to know approximately where the sample chemistry is expected to be to start and also to confirm the fitting results at the end, especially with other analysis methods.

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J. J. Weimer
Chemistry / Chemical & Materials Engineering, UAHuntsville

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