# Interpolation at “extreme values.”

I am working with a meteorologist on a project. We are pulling data from METAR Observation stations on several variables (such as temperature, dew point, wind speed, etc.) throughout time. Unfortunately there are occasional missing values for some of these variables at different times, but our model requires there to be no missing data at all.

I am from a mathematics background, so I was investigating some of the "pure math" ways to come up with a replacement value where data is missing. I have looked at Linear Interpolation, Cubic Spline Interpolation, Akima Spline Interpolation, and Polynomial Interpolation.

Other than the polynomial interpolation (because of the oscillatory problems) I have found in my simulations that all of the three methods predict the missing values very well, especially when data points are close together in time. However they all fall short at extreme values. I really thought that since I have past and future values to work with (not just past values), I should be able to find a method of replacement that can give better results at the maximum and minimum values. I have tried different averaging techniques that use values on both sides, and more.

Do you know of a method of interpolation (or maybe regression) that performs well at extreme values?

EDIT: I think that I did not explain my problem well. I know about Runge's phenomenon and that is why I tried three different spline methods, but I am looking for interpolation methods to help approximate the extreme values of the function $f$, not the extreme left and right values of $x$.

• If I recall, scientists working on global warming have the same problem and used some kind of interpolation to extract some additional data too. Looking at some of their papers could probably help. – Hippalectryon Jul 1 '15 at 11:53
• It's called Runge's phenomenon, there are several ways to solve that problem and some of them are already discussed on wiki. – JukesOnYou Jul 1 '15 at 11:54
• A simple approach would be to use weighted least squares, where the weight is a function of the distance between that point and the average. This will reward a function that is most accuruate at the endpoints. – user237392 Jul 1 '15 at 11:57
• I am not familiar with your specific data but you may try combinations of sin/cos and/or wavelets depending on the scope. – AnilB Jul 1 '15 at 16:05
• So, you want to approximate $\max f$ and $\min f$, regardless of where it occurs on the domain? Correct? – user237392 Jul 2 '15 at 1:07