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Sorry if this is a silly question, i'm just getting back into math after a long time away. My question is regarding approximation and interpolation. In which cases is it appropriate for one technique versus the other. If I have a list of data points and i would like to build some sort of model based on this data which is the most appropriate? And then of the choice selected, how do i select among the most many different methods of either approximation or interpolation. I realize this question maybe pretty broad but any help provided is appreciated, i'm just trying to get pointed in the right direction.

Thanks, Mike

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There is not one way that is better than all others. Mainly it will depend on your data whether it seems to be easily approximated by a certain function. – Listing May 24 '11 at 4:58
Thanks everyone for the feedback. To follow up on Listing's comment, how do define "better"? Would I try a few different interpolation or approximation techniques or functions and then calculate the residual error to figure out which was best? – miggety May 24 '11 at 13:42
I addressed this topic in this answer. – J. M. Jul 31 '11 at 16:30
up vote 4 down vote accepted

The quality of your data. If your data points are highly accurate, then it makes sense to respect them as much as possible, and some form of interpolation would be appropriate. On the other hand, if you expect some uniform random noise to be associated with the data, then it makes sense to pay more attention to local averages, to "smooth" it out. In this case an approximation scheme is more appropriate.

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I think the fundamental criterion is whether or not the data points are locally linear. If the data points are the masses and volumes of almost-perfectly-spherical widgets with almost-uniform density, then we can predict the volume of a never-before-seen mass with some confidence by interpolation. But if the data points are the ages and weights of the researchers conducting the prior study, then interpolation is inappropriate: Maybe Bob is 35 years old and weighs 75 kilograms, and Darcy is 37 years old and weighs 125 kilograms, but that is not a good reason to conclude that 36-year-old Clive weighs 100 kilograms.

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