I am new to Mathematica, and I don´t quite understand, what the difference between the fitting algorithms Fit, FindFit, LinearModelFit, NonLinearModelFit are.
Why is the syntax so different ({1,x,x^2} vs. a + b x + c x^2)?
Which should I prefer for which use-case?
Why can I use a non linear Function like a + b x + c x^2 in an LinearModelFit?
I am very confused, I hope you can help me.
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$\begingroup$ Well the difference between Linear and NonLinear should be clear given the name? :-) $\endgroup$– chrisApr 27, 2015 at 16:55
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3$\begingroup$ Fit and FindFit differ in the way they handle the result. One is meant to cut the chase and give the result, the other provide with the whole statistical analysis. $\endgroup$– chrisApr 27, 2015 at 16:55
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$\begingroup$ some of the differences are historical. $\endgroup$– chrisApr 27, 2015 at 16:56
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1$\begingroup$ FindFit \ Properties and Relations section explains a lot. But not everything. And I think question is on topic, at the beginning I was always confused - "great that I can do this in 10 different ways, but which one is appriopriate?" $\endgroup$– Kuba ♦Apr 27, 2015 at 17:19
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1$\begingroup$ Strongly related $\endgroup$– bobthechemistApr 28, 2015 at 1:49
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1 Answer
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Luca, I think that you may be misunderstanding the meaning of "linear" in the names of those functions, and in general in reference to linear models.
Just to make sure that we agree on the nomenclature here, in a fitting model you have parameters, and predictor variables. For instance, in a simple quadratic model such as y = a x^2 + b x + c we typically understand a, b, c as the parameters, and x as the predictor variable.
Linear models are those models in which the predicted value (e.g. y in our example) depends linearly on the parameters of the model. It doesn't matter whether the relationship between y and x is linear. For instance, the model above is linear. This following model is also linear: y = a Sin[x^2]. Model fitting is a search for the parameters, so linear parameter relationships lead to models that are easier to deal with computationally, and in some cases to closed-form solutions to the regression problem.
Non-linear models are those in which the predicted value depends on the parameters in a non-linear fashion. For instance: y = Sin[a] x is non linear because the predicted value depends non-linearly on the value of the parameter a (even though the predicted value depends linearly on x!).
In addition to the interesting reasons mentioned by the commenters to your question, which I think shed some light on the issue as well, the existence of multiple fitting functions also reflects this fundamental divide.
On the basis of my (limited) experience, here is how I view the most common fitting functions:
Fitis the oldest of the bunch; it carries out linear model fitting by linearly combining a list of base functions you explicitly indicate. I don't tend to use it much, since the majority of models I fit to are non-linear.FindFitis newer; it is capable of both linear and non-linear fitting. You feed it the explicit form of the fitting model directly. I find this very intuitive to use.NonlinearModelFitaccomplishes the same results asFindFitand it shares the same syntax. In addition to the fitted model, it also gives you access to a wealth of fitting descriptors: goodness of fit estimates, prediction intervals, residuals, just to name a few. You could think of it as aFindFit"on steroids". It would be really worth your time to take a look at the many options available for this function ("Details and Options" section of its documentation).FindFitis also faster thanNonlinearModelFit: the former is implemented in fast numerical code within the kernel; the latter is written entirely in top-level code, i.e. using Mathematica expressions, and it internally callsFindFitto do the heavy lifting anyway (@OleksandrR. talked about this point more authoritatively in his answer regarding compiled versions of the fitting routines).
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$\begingroup$ Oh ok, my fault. And can I use Fit and FindFit also for non linear fitting? $\endgroup$ Apr 27, 2015 at 18:02
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$\begingroup$ @LucaThiede Good point. Let me add something to that respect to the answer. $\endgroup$– MarcoBApr 27, 2015 at 18:25
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$\begingroup$ Ok, and I assume that LinearModelFit is the same as NonLinearModelFit, but limited to the Linear Models, whereas NonLinearModelFit is for everything, is that correct? Has LinearModelFit any advantage over the NonLinearModelFit? $\endgroup$ Apr 27, 2015 at 18:40
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1$\begingroup$ @LucaThiede You are correct on the
LinearvsNonlinearfunctions. Linear models can typically be solved deterministically (i.e. there is a closed-form solution to find the best fit parameters), whereas non-linear models require an iterative process, so if your model is linear the functions specifically geared to the linear models are going to be faster and more reliable. Of course, if your model is not linear, then you are stuck with the iterative processes. $\endgroup$– MarcoBApr 27, 2015 at 18:47