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I am not sure I understand the difference between functional data analysis (FDA) and GAM. Or in short, I was reading about GAM and I found the following model which seems like an FDA model (there is a base function and its coefficients).

$$f(x) = \sum_{j=1}^q b_j(x) \gamma_j$$

Could you please help me understand what are the differences between the two methods?

UPDATE

From the discussions here and here it seems that someone can run all flavours of functional data analysis using GAM (or in brms). Also, this and this one give a short description of FDA vs traditional methods.

But still, I don't understand how the FDA is different from GAM.

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    $\begingroup$ I see it is a philosophy. FDA sees a scatterplot as a data point. An additive model sees a scatterplot and fits a curve to it (ditto for a neural network). $\endgroup$
    – Dave
    Dec 22, 2020 at 15:18
  • $\begingroup$ Ok, nice way to think of. Could you please elaborate a bit more? $\endgroup$
    – Lefty
    Dec 22, 2020 at 15:30
  • $\begingroup$ In terms of regression I think that the Wikipedia article on functional regression is actually quite decent. Müller's work is very much focused on this matter. $\endgroup$
    – usεr11852
    Mar 27, 2022 at 22:21

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By trying both methods for some time now I see that a main difference is that in FDA you actually get functional coefficients, meaning that your coefficients are depended on a particular moment of your continuum. So your covariates may have linear effect but you still can get a wiggly function. In contrast, with GAM you may fit a wiggly spline but you miss from interpretability.

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  • $\begingroup$ You describe what is called a functional concurrent model. $\endgroup$
    – usεr11852
    Mar 27, 2022 at 22:16

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