Can you use a linear spline for extrapolation? Say you have data (6,1.04),(7,1.71),(9,3.28),(10,4.08); and you wanted to evaluate at x=17. I thought that extrapolating using linear splines was a bad idea--is that true?
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I assume that you mean linear extrapolation if you leave your interpolation interval. What makes you think that this is a bad idea? Use whatever fits best your actual problem. It is only a bad idea if the data from the last interplation interval is probably bad (e.g., noisy). In that case you should use approximation anyway and extrapolate the approximation. – Tobias Nov 18 '16 at 07:31
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Okay, so in this case it should be f(x)=3.28+0.8(x-9), in which case f(17)=9.68 ? I thought it may be a bad idea since the interpolation polynomial wasn't for that data point, but since the last interval doesn't have noisy data it should work? – PJS Nov 18 '16 at 07:48
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When you have to extrapolate it means that you need an approximation at a point where you do not have data. Use as much problem relevant information as you get to obtain a good approximation there! Just an example: If you know that the signal you are approximating is limited it is better to use constant extrapolation. If you do not know anything maybe you are good with linear extrapolation. – Tobias Nov 18 '16 at 07:52
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I understand the concept of extrapolation, I guess what I am getting confused on is how else I should think about solving this problem, other than using the interpolation interval for x between 9 and 10, since I don't know anything else other than those 4 data points. Say, if I calculate the other interpolation intervals, how can I use that information to obtain a better approximation than just using the last interpolation interval? Thanks for the help, by the way. – PJS Nov 18 '16 at 08:02